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Kinetic Modeling of Reactions in Foods

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FOOD SCIENCE AND TECHNOLOGY Editorial Advisory Board Gustavo V. Barbosa-Cánovas Washington State University–Pullman P. Michael Davidson University of Tennessee–Knoxville Mark Dreher McNeil Nutritionals, New Brunswick, New Jersey Richard W. Hartel University of Wisconsin–Madison Lekh R. Juneja Taiyo Kagaku Company, Japan Marcus Karel Massachusetts Institute of Technology Ronald G. Labbe University of Massachusetts–Amherst Daryl B. Lund University of Wisconsin–Madison David B. Min The Ohio State University Leo M. L. Nollet Hogeschool Gent, Belgium Seppo Salminen University of Turku, Finland John H. Thorngate III Allied Domecq Technical Services, Napa, California Pieter Walstra Wageningen University, The Netherlands John R. Whitaker University of California–Davis Rickey Y. Yada University of Guelph, Canada

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Kinetic Modeling of Reactions in Foods Martinus A. J. S. van Boekel

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-57444-614-2 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Van Boekel, Martinus A. J. S. Kinetic modeling of reactions in foods / Martinus A. J. S. van Boekel. p. cm. -- (Food science and technology) “A CRC title.” Includes bibliographical references and index. ISBN 978-1-57444-614-2 (alk. paper) 1. Food--Analysis. 2. Chemical kinetics--Mathematical models. 3. Food adulteration and inspection. I. Title. II. Series. TX545.V36 2008 664.001’54--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

2008044086

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Dedication To my wife Corrie For her patience and understanding

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Contents Preface ................................................................................................................................................. xiii Author ................................................................................................................................................ xvii

1

Kinetic View on Food Quality ............................................................................................ 1-1 1.1 Introduction .......................................................................................................................................... 1-1 1.2 Food Quality ......................................................................................................................................... 1-1 1.3 Foods as Complex Reaction Media ................................................................................................. 1-6 1.4 Outline of the Book ............................................................................................................................ 1-7 Bibliography and Suggested Further Reading ......................................................................................... 1-8

SECTION I The Basics

2

Models and Modeling ........................................................................................................... 2-1

3

Chemical Thermodynamics in a Nutshell ....................................................................... 3-1

2.1 Introduction .......................................................................................................................................... 2-1 2.2 Models and Modeling ......................................................................................................................... 2-1 2.3 Concluding Remarks ....................................................................................................................... 2-12 Bibliography and Suggested Further Reading ...................................................................................... 2-12

3.1 Introduction .......................................................................................................................................... 3-1 3.2 Quantiﬁcation of Reactants and Products ..................................................................................... 3-1 3.3 Thermodynamics of Reactions ......................................................................................................... 3-6 3.3.1 Heat and Work ..................................................................................................................... 3-6 3.3.2 Energy ..................................................................................................................................... 3-8 3.3.3 Enthalpy ................................................................................................................................. 3-9 3.3.4 Entropy ................................................................................................................................. 3-11 3.3.5 Free Energy .......................................................................................................................... 3-15 3.3.6 Chemical Potential ............................................................................................................. 3-18 3.3.7 Ideal Solutions .................................................................................................................... 3-20 3.3.8 Ideal Dilute Solutions ........................................................................................................ 3-21 3.3.9 Real, Nonideal Solutions: Activity Concept ................................................................. 3-22 3.3.10 Standard States ................................................................................................................... 3-27 3.3.11 Solvent Activity and Water Activity .............................................................................. 3-29 3.3.12 Chemical Potential and Equilibrium ............................................................................. 3-33 vii

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3.3.13 Equilibrium Constants .................................................................................................... 3-36 3.3.14 Thermodynamic Potentials and Conjugate Variables ............................................. 3-42 3.3.15 Nonequilibrium or Irreversible Thermodynamics ................................................... 3-48 3.4 Concluding Remarks ..................................................................................................................... 3-52 Appendix 3.1 Datasets Used for Examples in This Chapter ......................................................... 3-53 Bibliography and Suggested Further Reading ...................................................................................... 3-57

4

Chemical Reaction Kinetics ................................................................................................ 4-1

Introduction ....................................................................................................................................... 4-1 Foods as Chemical Reactors? ......................................................................................................... 4-2 Rate and Extent of Reactions in Closed Systems ....................................................................... 4-4 4.3.1 Kinetics of Elementary Reactions .................................................................................. 4-9 4.3.2 Kinetics of Experimentally Observed Reactions ....................................................... 4-16 4.3.3 Steady-State Approximation and Rate-Controlling Steps ...................................... 4-28 4.4 Catalysis ............................................................................................................................................ 4-33 4.4.1 General Catalysis ............................................................................................................. 4-33 4.4.2 Acid–Base Catalysis ......................................................................................................... 4-34 4.5 Kinetics of Radical Reactions ...................................................................................................... 4-37 4.6 Kinetics of Photochemical Reactions ......................................................................................... 4-41 4.7 Diffusion-Limited Reactions in Aqueous Solutions ............................................................... 4-42 4.8 Kinetics in Open Systems ............................................................................................................. 4-46 4.9 Concluding Remarks ..................................................................................................................... 4-53 Appendix 4.1 Datasets Used for Examples in This Chapter ......................................................... 4-54 Bibliography and Suggested Further Reading ...................................................................................... 4-61 4.1 4.2 4.3

5

Temperature and Pressure Effects ..................................................................................... 5-1

6

Charge Effects ......................................................................................................................... 6-1

5.1 Introduction ....................................................................................................................................... 5-1 5.2 van’t Hoff Equation .......................................................................................................................... 5-1 5.3 Transition State Theory ................................................................................................................... 5-3 5.4 Arrhenius’ Law .................................................................................................................................. 5-8 5.5 Empirical Relations to Describe Temperature Dependence ................................................ 5-15 5.6 Activation Energy and Catalysis ................................................................................................. 5-16 5.7 Parameters Used in Food Science .............................................................................................. 5-18 5.8 Enthalpy=Entropy Compensation .............................................................................................. 5-21 5.9 Variable Temperature Kinetics ................................................................................................... 5-23 5.10 Effect of Pressure ............................................................................................................................ 5-32 5.11 Concluding Remarks ..................................................................................................................... 5-36 Appendix 5.1 Datasets Used for Examples in This Chapter ......................................................... 5-36 Bibliography and Suggested Further Reading ...................................................................................... 5-41

6.1 6.2

Introduction ....................................................................................................................................... 6-1 Models for Ion Activities ................................................................................................................. 6-1 6.2.1 Debye–Hückel Type Models ........................................................................................... 6-4 6.2.2 Mean Spherical Approximation Theory ....................................................................... 6-8 6.2.3 Pitzer Equations ............................................................................................................... 6-11

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Ion Pairing Models ........................................................................................................................ 6-12 6.3.1 Mass Action Law ............................................................................................................. 6-15 6.3.2 Pytkowicz Model .............................................................................................................. 6-18 6.3.3 Binding MSA Model ....................................................................................................... 6-23 6.4 Kinetics of Reactions between Ions ........................................................................................... 6-25 6.4.1 Primary Salt Effect ........................................................................................................... 6-25 6.4.2 Secondary Salt Effect ....................................................................................................... 6-29 6.4.3 Examples Showing the Primary Salt Effect on Kinetics ......................................... 6-31 6.5 Concluding Remarks ..................................................................................................................... 6-40 Appendix 6.1 Datasets Used for Examples in This Chapter ......................................................... 6-40 Bibliography and Suggested Further Reading ...................................................................................... 6-46 6.3

7

Kinetics and Statistics ........................................................................................................... 7-1

Introduction ....................................................................................................................................... 7-1 Some Background on Statistical Approaches ............................................................................. 7-2 7.2.1 Classical Sampling Theory ............................................................................................... 7-3 7.2.2 Maximum Likelihood ....................................................................................................... 7-3 7.2.3 Bayesian Statistics .............................................................................................................. 7-4 7.2.4 Resampling Methods ........................................................................................................ 7-7 7.3 Experimental Design: Statement of the Problem ....................................................................... 7-9 7.4 On Errors and Residuals .............................................................................................................. 7-13 7.4.1 Deterministic and Stochastic Models .......................................................................... 7-13 7.4.2 Least Squares Regression ............................................................................................... 7-14 7.4.3 Sums of Squares and ANOVA ..................................................................................... 7-15 7.4.4 Error Structure of Data: A Variance Model .............................................................. 7-16 7.5 Linear and Nonlinear Models ..................................................................................................... 7-20 7.6 A Closer Look at Assumptions for Parameter Estimation ................................................... 7-21 7.7 Normal Probability Plots and Lag Plots ................................................................................... 7-25 7.8 Goodness of Fit and Model Discrimination ............................................................................ 7-29 7.9 Precision of Regression Lines and Parameter Estimates ....................................................... 7-40 7.9.1 Jackknife Method ............................................................................................................. 7-50 7.9.2 Bootstrap Method ............................................................................................................ 7-50 7.9.3 Grid Search Method ........................................................................................................ 7-53 7.9.4 Monte Carlo Method ...................................................................................................... 7-57 7.9.5 Bayesian Analysis Using Markov Chain Monte Carlo Methods .................................................................................................................. 7-57 7.10 Variability and Uncertainty ......................................................................................................... 7-64 7.11 Transformation of Parameters: Reparameterization .............................................................. 7-68 7.12 Propagation of Errors .................................................................................................................... 7-72 7.13 Sensitivity Analysis ........................................................................................................................ 7-74 7.14 Experimental Design ..................................................................................................................... 7-76 7.14.1 Systematic and Random Errors: Accuracy and Precision .................................................................................................................... 7-77 7.14.2 Experimental Design for Kinetic Models ................................................................... 7-78 7.15 Concluding Remarks ..................................................................................................................... 7-87 Appendix 7.1 Datasets Used for Examples in This Chapter ......................................................... 7-87 Bibliography and Suggested Further Reading ...................................................................................... 7-94 7.1 7.2

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SECTION II

8

Application of the Basics to Chemical, Biochemical, Physical, and Microbial Changes in the Food Matrix

Multiresponse Kinetic Modeling of Chemical Reactions ............................................. 8-1

Introduction ....................................................................................................................................... 8-1 What Is Multiresponse Modeling? ................................................................................................ 8-1 Determinant Criterion ..................................................................................................................... 8-3 Model Discrimination and Goodness of Fit for Multiresponse Models .............................. 8-5 Examples of Multiresponse Modeling of Reactions in Foods ................................................. 8-7 8.5.1 Heat-Induced Acid Hydrolysis of Sucrose ................................................................... 8-7 8.5.2 Degradation of Chlorophyll ............................................................................................ 8-8 8.5.3 Aspartame Degradation ................................................................................................. 8-16 8.5.4 Maillard Reaction ............................................................................................................ 8-19 8.6 Concluding Remarks ..................................................................................................................... 8-26 Appendix 8.1 Datasets Used for Examples in This Chapter ......................................................... 8-27 Bibliography and Suggested Further Reading ...................................................................................... 8-29 8.1 8.2 8.3 8.4 8.5

9

Enzyme Kinetics .................................................................................................................... 9-1

Introduction ....................................................................................................................................... 9-1 Michaelis–Menten Kinetics ............................................................................................................. 9-4 9.2.1 Linearized Plots ................................................................................................................ 9-13 9.3 Enzyme Inhibition ......................................................................................................................... 9-16 9.4 Progress Curves .............................................................................................................................. 9-20 9.5 Kinetics of Two-Substrate Reactions ......................................................................................... 9-29 9.6 Other Types of Enzyme Kinetics ................................................................................................ 9-32 9.7 Temperature Effects ....................................................................................................................... 9-36 9.8 pH Effects ........................................................................................................................................ 9-39 9.9 Experimental Design for Enzyme Kinetics .............................................................................. 9-42 9.10 Enzyme Kinetics in Foods ............................................................................................................ 9-43 9.11 Concluding Remarks ..................................................................................................................... 9-47 Appendix 9.1 Datasets Used for Examples in This Chapter ......................................................... 9-48 Bibliography and Suggested Further Reading ...................................................................................... 9-58 9.1 9.2

10

Kinetics of Protein and Enzyme Denaturation ............................................................ 10-1

11

Kinetics of Physical Changes ............................................................................................ 11-1

10.1 Introduction ..................................................................................................................................... 10-1 10.2 Protein Stability .............................................................................................................................. 10-1 10.3 General Kinetic Schemes Describing Enzyme Inactivation ................................................ 10-13 10.4 Food Matrix Effects ..................................................................................................................... 10-26 10.5 Concluding Remarks ................................................................................................................... 10-29 Appendix 10.1 Datasets Used for Examples in This Chapter .................................................... 10-29 Bibliography and Suggested Further Reading .................................................................................... 10-36

11.1 Introduction ..................................................................................................................................... 11-1 11.2 Kinetics of Diffusion ...................................................................................................................... 11-2 11.2.1 Fick’s Laws ....................................................................................................................... 11-2 11.2.2 Maxwell–Stefan Approach ............................................................................................ 11-8

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11.3 Kinetics of Changes in Dispersity ............................................................................................ 11-14 11.3.1 Kinetics of Aggregation of Colloids ......................................................................... 11-14 11.3.2 Kinetics of Creaming or Settling ............................................................................... 11-19 11.3.3 Kinetics of Coalescence ............................................................................................... 11-21 11.3.4 Kinetics of Ostwald Ripening .................................................................................... 11-23 11.3.5 Kinetics of Gelation of Particles ................................................................................ 11-24 11.3.6 Kinetics of Crystallization ........................................................................................... 11-28 11.4 Kinetics of Texture Changes ...................................................................................................... 11-29 11.5 Partitioning Phenomena ............................................................................................................. 11-32 11.5.1 Partition Coefﬁcients ................................................................................................... 11-33 11.5.2 Partitioning of Volatiles .............................................................................................. 11-33 11.5.3 Partitioning of Weak Acids ........................................................................................ 11-46 11.6 Concluding Remarks ................................................................................................................... 11-49 Appendix 11.1 Datasets Used for Examples in This Chapter .................................................... 11-49 Bibliography and Suggested Further Reading .................................................................................... 11-59

12

Kinetics of Microbial Growth ........................................................................................... 12-1

13

Kinetics of Inactivation of Microorganisms .................................................................. 13-1

14

Modeling the Food Matrix ................................................................................................ 14-1

12.1 Introduction ..................................................................................................................................... 12-1 12.2 Primary Growth Models ............................................................................................................... 12-2 12.2.1 Differential Equations .................................................................................................... 12-3 12.2.2 Algebraic Equations ....................................................................................................... 12-6 12.3 Secondary Models ........................................................................................................................ 12-11 12.4 Nonisothermal Growth Modeling ............................................................................................ 12-20 12.5 Bayesian Modeling ....................................................................................................................... 12-22 12.6 Experimental Design ................................................................................................................... 12-28 12.7 Effects of the Food Matrix ......................................................................................................... 12-28 12.8 Concluding Remarks ................................................................................................................... 12-29 Appendix 12.1 Datasets Used for Examples in This Chapter .................................................... 12-30 Bibliography and Suggested Further Reading .................................................................................... 12-40

13.1 Introduction ..................................................................................................................................... 13-1 13.2 Kinetics of Inactivation of Vegetative Cells ............................................................................. 13-1 13.3 Kinetics of Inactivation of Spores ............................................................................................ 13-10 13.4 Temperature Dependence of Microbial Inactivation ........................................................... 13-16 13.5 Food Matrix Effects ..................................................................................................................... 13-24 13.6 Concluding Remarks ................................................................................................................... 13-26 Appendix 13.1 Datasets Used for Examples in This Chapter .................................................... 13-26 Bibliography and Suggested Further Reading .................................................................................... 13-42

14.1 Introduction ..................................................................................................................................... 14-1 14.2 Speciﬁc Effects in Aqueous Solutions ........................................................................................ 14-3 14.2.1 Water Activity and the Effect of Cosolutes .............................................................. 14-4 14.2.2 Water Activity and Food Stability ............................................................................ 14-11 14.2.3 Ionic and Nonionic Solute Interactions .................................................................. 14-14 14.2.4 Signiﬁcance of pH in Food ......................................................................................... 14-18 14.3 Transport Phenomena and Molecular Mobility in the Food Matrix ............................... 14-23 14.4 Micellar Effects ............................................................................................................................. 14-32

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14.5 Effect of Molecular Crowding in the Food Matrix .............................................................. 14-34 14.6 Concluding Remarks ................................................................................................................... 14-36 Appendix 14.1 Datasets Used for Examples in This Chapter .................................................... 14-37 Bibliography and Suggested Further Reading .................................................................................... 14-53

15

Retrospective and Outlook ................................................................................................ 15-1

15.1 Introduction ..................................................................................................................................... 15-1 15.2 Shelf Life Modeling as an Integrative Approach .................................................................... 15-1 15.2.1 Shelf Life from the Product Point of View ............................................................... 15-2 15.2.2 Shelf Life from the Consumer Point of View .......................................................... 15-4 15.3 Some Developments .................................................................................................................... 15-12 15.4 Concluding Remarks ................................................................................................................... 15-14 Appendix 15.1 Datasets Used for Examples in This Chapter .................................................... 15-15 Bibliography and Suggested Further Reading .................................................................................... 15-19

Appendix A

Some Calculus Rules .......................................................................................... A-1

Appendix B

Ways to Express Amounts of Reactants and Products .............................. B-1

Appendix C

Interconversion of Activity Coefﬁcients Based on Mole Fractions, Molalities, and Molarities ............................................. C-1

Appendix D

Differential and Integrated Rate Equations for Kinetic Models of Complex Reactions ........................................................................................ D-1

Appendix E

McMillan–Mayer and Lewis–Randall Framework and Equations for the Mean Spherical Approximation Theory ........................................... E-1

Appendix F

Probability Laws and Probability Models ...................................................... F-1

Appendix G Use of Matrix Notation in Model Representation and Regression Analysis .................................................................................... G-1 Appendix H Some Thermodynamic Activity Coefﬁcient Models ................................... H-1 Appendix I

Reliability Engineering and the Weibull Model ............................................ I-1

List of Symbols and Units ........................................................................................................... 16-1 Index ................................................................................................................................................... I-1

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Preface The topic of food quality is receiving ever-increasing attention. Consumers are concerned about the quality of their food and have high demands. At the same time, consumer demands are rapidly changing, and the food supply chain needs to match these changing demands in order to be able to deliver food of a desired quality at the end of the chain. However, the quality of a food changes continuously along its way through the food chain. It is therefore important to have tools to control and predict food quality (including food safety) and to be able to quickly change food design according to changing consumer expectations. This is useful for consumers because it helps to ensure that their needs are fulﬁlled and that they obtain safe food. Obviously, it is helpful for the food industry because it provides a suitable tool to connect physical product properties with consumer wishes. I am convinced that the use of mathematical models for modeling of quality attributes of foods is going to be of great help in these matters. This book is about how to model changes taking place in foods, for which the scientiﬁc term is kinetics. The aim of this book is to introduce appropriate kinetic models and modeling techniques that can be applied in food science and technology. It is fair to say that mathematical modeling is already used to some extent in the food science and technology world, but in the author’s opinion there are many more opportunities than those currently applied. This book aims to indicate directions for the use of modeling techniques in food science. It will be argued that modeling of food quality changes is in fact kinetic modeling. However, this is not just another book on kinetics. Rather, it integrates food science knowledge, kinetics, and statistics, so as to open the possibility to predict and control food quality attributes using computer models. Moreover, much more information can be extracted from experiments when quantitative models are used. I hope to show with this book that the quality of modeling can be improved considerably with proper mathematical and, especially, statistical techniques. The choice of topics reﬂects my research interests. Obviously, this choice is subjective and reﬂects my ideas about how modeling of food quality should be done. Quality changes in foods are related to the chemical, biochemical, physical, and microbiological changes taking place in the food, in relation to processing conditions. I have attempted to apply kinetic models using general chemical, physical, and biochemical principles, but allowing for typical food-related problems. The general principles mentioned are usually derived for only very simple, dilute, and ideal systems. Foods are all but simple, ideal, and dilute. Another important point in my view is that allowance should be made for variability and uncertainty, and therefore I consider the use of statistics as indispensable. A substantial part of this book is devoted to the use of statistical techniques in kinetic modeling, which is another reason it is not a typical kinetics book. I introduce the concept of Bayesian statistics, which is hardly known in the food science world. I feel it has great potential, and I intend to show that in this book. The book is ﬁrst of all meant for food scientists who want to learn more about modeling. It was written with two objectives in mind. The ﬁrst was to introduce the topic of kinetics and its application to foods to students and graduates in food science and technology. I teach kinetics to food science students in an advanced MSc course called ‘‘Predicting Food Quality’’ and in an advanced PhD course called ‘‘Reaction Kinetics in Food Science’’ at Wageningen University. The response of the students is encouraging. xiii

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Preface

The book could therefore be helpful as a textbook in advanced MSc and PhD courses at other universities. The second objective was to write a reference book to be used by professional researchers active in food-related work. It should be useful, therefore, for graduates working in the food industry who have a keen interest in modeling, and who are willing to apply modeling concepts in food product design. It could even be useful for nonfood disciplines such as biotechnology, pharmacy, nutrition, and general biology and chemistry. It is on an advanced level in the sense that it builds upon basic food science and technology knowledge, as well as basic mathematical knowledge of calculus and matrix algebra. Also, basic statistical knowledge is assumed, although some introduction is given to Bayesian statistics because this will be new to most food scientists. As a reminder for the reader, appendices consisting of the basic background on all these matters are provided. The ultimate aim is to guide students, graduates, and postgraduates in such a way that they can understand and critically read articles in the literature concerning this topic, and can apply the principles in their own research, be it fundamental or applied. It is, of course, unavoidable that there are many equations in this book since it deals with mathematical models. Fortunately, mathematical complexity can be kept to a minimum using appropriate software such as Mathematica, MathCad, Maple, and even well-known spreadsheets such as Microsoft Excel. I used MathCad and Excel quite extensively for this book, as well as some specialized software where indicated. The reader should try to look beyond the equations and math involved and it will be very helpful to work out the examples given. Wherever possible, I will express in words also what is expressed in an equation. Nevertheless, I do realize that the many mathematical and statistical equations are not easy to digest. Therefore, I have strived to illustrate the concepts introduced with many real-life examples rather than using hypothetical data, or examples that are less relevant for food science problems. The data for the examples were either read directly from tables published in papers, or digitally scanned by computer from graphs. Occasionally, authors supplied me with data, for which I am very grateful, and I also used my own data. All datasets used are supplied in appendices to the chapters, including their sources, so that the interested reader can work with these examples by himself or herself. I would like to stress that the examples chosen are not meant to criticize results; they are chosen because they illustrate the points I want to make. I am actually quite grateful that authors made it possible to extract data from the publications; this is actually as it should be. I have used many references from literature in compiling my own text, by going well beyond the food science and technology literature. However, I decided not to indicate literary references in the text itself to improve readability. Rather, whenever substantial use was made of a particular reference that reference was mentioned at the end of the chapter. I do acknowledge all the excellent articles that are available and which substantially helped me to formulate my own text. Finally, I would like to acknowledge several persons who have been instrumental in helping me realize this book. First of all, I would like to acknowledge Professor Dr. Bronek Wedzicha from the Procter Department of Food Science, University of Leeds, United Kingdom. Thanks to his hospitality, I have been able to spend two sabbatical periods of three months at the University of Leeds in the summers of 1999 and 2004 and during these periods we had very intensive discussions over the topics covered in this book. Moreover, he and his wife Glenis have been very generous to me on a personal level by inviting me to many lovely dinners at their house, and for entertaining walks in beautiful Yorkshire. I do regret not having Professor Wedzicha as a coauthor; the book would have been much better had this been the case. However, his critical spirit has been essential for my writing and many of his thoughts are reﬂected in this book. This is especially true for Chapter 14, which has been inspired strongly by his ideas and lectures on this topic. Furthermore, I would like to thank Professor Pieter Walstra from Wageningen University for stimulating me to take this path in my academic career, and for critically reading several drafts of the chapters. I would also like to thank Professor Willem Norde from Wageningen University for very useful comments on the chapter on thermodynamics. Having acknowledged Bronek Wedzicha, Pieter Walstra, and Willem Norde for their invaluable contributions, I am of course fully responsible for the text,

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including all errors and mistakes. I would very much appreciate remarks, criticism, and corrections from readers. Last but not least, I would like to take this opportunity to thank my wife Corrie for being very patient with me, for not complaining about my physical absence of two periods of three months abroad, not to mention the countless evening and weekend hours, just so that I could do my writing. It is well appreciated and I dedicate this book to her.

M.A.J.S. (Tiny) van Boekel Wageningen

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Author Martinus van Boekel received his BSc, MSc, and PhD in food science and technology from Wageningen University, Wageningen, the Netherlands. Immediately after, from 1980 to 1982, he worked at the Food Inspection Service at Rotterdam, the Netherlands, as a food chemist. He then returned to Wageningen University to work as an assistant professor from 1982 to 1994, as an associate professor from 1994 to 2001, and as a full professor from 2001 onward in the ﬁeld of food science and technology. In 2006, he became the scientiﬁc director of the graduate school VLAG (Food, Nutrition, Agrotechnology, and Health) for 4 years. His research and teaching encompass modeling of food quality attributes in an integrative way, that is, integrating the various food science disciplines but also nutrition, marketing, economics, and quality management. He has been a visiting professor at the University of Madison, Wisconsin, and the Procter Department of Food Science, University of Leeds, United Kingdom. He is the author and coauthor of about 160 refereed scientiﬁc papers and author= editor of six books.

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1 Kinetic View on Food Quality

1.1

Introduction

The aim of this book is to discuss kinetics of reactions in foods in relation to food quality. By reactions we mean all type of change taking place in the food whether they be chemical, enzymatic, physical, or microbial. Kinetics is about change. For the moment it sufﬁces to describe kinetics as the translation of knowledge (theoretical as well observational) on a time-dependent chemical, physical, microbial, reaction into an equation describing such changes in mathematical language. The mathematical relations result in models that we can use to design, optimize, and predict the quality of foods. It should also be helpful in choosing the technology to produce them. We thus need chemical, physical, microbial knowledge to build mathematical models as well as knowledge on composition and structure of foods, i.e., food science; it is assumed that the reader is familiar with basic principles of food science and technology. The major part of the book is concerned with modeling the kinetics of relevant reactions in foods and deals with questions such as: what is kinetics, what are models, how do we apply kinetics to practical problems in foods, what are pitfalls and opportunities, how to deal with uncertainty, and how to interpret results. A key question to be answered is why the kinetics of reactions in foods is often different from, say, that of chemical engineering processes. In this chapter, we discuss some important determinants of food quality. While the subject of quality deserves a book in its own right, the purpose here is to put the relationship between kinetic modeling and food quality in perspective, to be developed in subsequent chapters.

1.2

Food Quality

What then is food quality? There are many deﬁnitions and descriptions of quality. One useful but very general description is ‘‘to satisfy the expectations of the consumer.’’ Although the idea of quality seems to be somewhat elusive, it is important to understand the concept because, as food technologists, we need to be able to control and predict food quality attributes. Food quality attributes are all those product attributes that are relevant in determining quality. The ultimate test for quality is acceptance or rejection by the consumer. When a consumer evaluates a product, a ﬁrst impression arises from so-called quality cues: attributes that can be perceived prior to consumption and that are believed to be indicative of quality. Examples are red color of meat, or information concerning the origin of the product. This leads to certain quality expectations. When the consumer starts eating, he is confronted with the physical 1-1

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Quality attributes (intrinsic and extrinsic)

Quality cues

Physical product properties

Quality expectations

Quality experience

Quality assignment If higher than quality limit

If less than quality limit

Acceptance

FIGURE 1.1

Rejection

Schematic picture of aspects involved in quality evaluation by a consumer.

product properties (e.g., texture, taste, ﬂavor) and this leads to a quality experience. If the quality expectation and the quality experience, integrated with each other, exceed a certain quality level, the consumer will accept the product, if not he will reject it. Figure 1.1 shows this process schematically, but the reader is advised that this scheme is an oversimpliﬁcation. Quality is multidimensional, it contains both subjective and objective elements, it is situation speciﬁc and dynamic in time. A consumer however does not analyze all elements of food quality consciously but gives an integrated response based on complex judgments made in the mind. In order to make quality more tangible for the food scientist, it is suggested to make a division into intrinsic quality attributes, i.e., inherent to the product itself, and extrinsic attributes, linked to the product but not a property of the food itself. Extrinsic factors are, for instance, whether or not a food is acceptable for cultural=religious or emotional reasons, or whether the way it is produced is acceptable (with or without fertilizer, pesticides, growth hormones, genetically modiﬁed, etc.) and its price. Extrinsic factors are therefore not part of the food itself but are deﬁnitely related to it (as experienced by the consumer). On the other hand, the chemical composition of the food, its physical structure, the biochemical changes it undergoes, the microbial and chemical condition (hazards from pathogens, microbial spoilage, presence of mycotoxins, heavy metals, pesticides, etc.), its nutritional value and shelf life, the way packaging interacts with the food, are intrinsic factors. We can propose a hypothetical quality function Q: Q ¼ f (Qint , Qext )

(1:1)

In words, this equation states that quality can be decomposed in intrinsic and extrinsic quality attributes. The nature of this function remains as yet obscure. We do not know, for instance, whether we are allowed to sum intrinsic and extrinsic quality attributes, or that we need to multiply them, or do yet something else. In terms of modeling, quality assignment is usually done either from the consumer perspective or from the product perspective. It would be better if the two approaches were integrated. Techniques like quality function deployment (QFD) try to do this. We will not discuss this further in this book.

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Raw materials and ingredients

Product formulation

Processing

Number, type of bacteria

Bacterial growth

Interactions in the food matrix

Ii =

Taste

Flavor

Shape, color

Nutritional Convenience value

Texture

Shelf life

Food safety

Packaging Intrinsic quality Qint

FIGURE 1.2

Schematic presentation of intrinsic quality attributes Ii.

It helps however to disentangle intrinsic and extrinsic quality attributes to make clear which factors are controllable by a technologist. Figure 1.2 shows a further decomposition of Qint into intrinsic quality attributes Ii: Qint ¼ f (I1 , I2 , . . . , In )

(1:2)

Figure 1.3 does the same for extrinsic quality attributes Ei: Qext ¼ f (E1 , E2 , . . . , En )

Ei = Values, norms

Regulations

Availability

Brand name

Extrinsic quality Qext

FIGURE 1.3

Schematic depiction of extrinsic quality attributes Ei.

(1:3)

Price

After sales service

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As with the overall quality function Q, we do not know the nature of the functions Qint and Qext. In other words we do not know how the quality attributes interact and are integrated by the consumer into one ﬁnal quality judgment; moreover it will differ from consumer to consumer. Much more can be said about quality, but that is beyond the scope of this book. We focus now on intrinsic quality attributes. To be sure, we will not attempt to ﬁnd a relation for Qint in this book; rather we focus on how to characterize the listed quality attributes from a technological point of view. Even though the ﬁnal quality judgment is not based on intrinsic factors alone, measurable objective quality attributes such as food safety, nutritional value, and color are of utmost importance. In food science literature, intrinsic factors such as those mentioned in Figure 1.2 are usually called quality attributes, though this is not strictly correct as shown in Figure 1.1. To satisfy the (dynamic) expectation of consumers, with diversity in needs and markets, a producer must be prepared to be very ﬂexible with respect to intrinsic quality attributes. Insight in these quality attributes is thus a prerequisite to survive in a competitive market. We propose that with the kinetic tools presented in this book these intrinsic quality attributes can be controlled and predicted. Intrinsic food quality attributes can be studied at several levels as shown in Figure 1.4. With reference to Figure 1.4, this book will deal mainly with modeling activities at levels 1 and 2, with some attention to level 3 concerning the design of experiments for food product design. Kinetic modeling of food quality attributes can be a powerful tool as part of the steps to be taken in food product development. Also, it can be the basis for the development of expert systems and management systems, especially with reference to risk analysis and food safety issues. Certain chemical reactions may serve as indicators for speciﬁc quality attributes. For instance in milk, the concentration of lactulose (an isomerization product of lactose) is an indicator of the heat treatment to which the milk has been exposed but it is not a quality attribute by itself. Clearly, the kinetics of the chemical reaction that serves as a quality indicator need to be closely related to the kinetics of the chemical or physical changes that determine the relevant quality attributes it represents. The way quality is monitored and safeguarded

Product market Level 4

Increasing complexity

Food design

Level 3

Functional properties, model system, processing

Properties of real foods, product development

Component interactions, kinetics

Level 2

Chemical, physical, microbial behavior of compounds Level 1

FIGURE 1.4

Consumer + sensory research

Several levels at which food quality can be studied.

Basic (bio)chemical, physical research

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is a particular aspect of food quality. This involves quality management, the introduction of systems such as hazard analysis and critical control points (HACCP), ISO systems, and good manufacturing practices (GMP). The statement of quality is made with reference to speciﬁc technical speciﬁcations, in other words such an approach requires integration of technological and management knowledge (technomanagerial approach). Basically, this comes down to realizing the fact that food quality is not only determined by the product itself or the technology applied but also by the people that handle the product. We will not discuss these aspects here; some references are given at the end of this chapter. The basic message is thus that quality is not a property of the food but is determined by the consumer who translates his perception into quality attributes. Some of these attributes can be related to measurable properties of the food though this is not always possible for user-related factors. The crispness of potato crisps, for instance, relates to mechanical properties of the (fried) potato cell wall; the sweetness of pastry is related to its sugar content as well as the sweetening intensity of the sugar used; the color of a food is for the most part attributable to components that absorb light at a particular wavelength and=or scatter light. There are also intrinsic factors that cannot be perceived directly by the consumer, such as the presence of toxic components or pathogenic bacteria. Such ‘‘hidden’’ quality attributes can, however, in most cases be measured. This book is concerned only with intrinsic factors, and particularly how we can ‘‘capture’’ these quality attributes within mathematical models. The advantage of using such models is that they can be linked to other models describing for instance stimulus–response relationships and consumer preference. The following intrinsic quality attributes are the most important ones for the food scientist: . . . . . . . .

Safety (microbial, toxic, mutagenic) Wholesomeness, nutritional value Usage (handling) properties Storage stability=shelf life Texture Color Appearance Flavor, taste compounds

Some of these attributes are the result of the interaction of stimuli picked up by the senses and are called sensory properties. Sensory properties can be estimated using sensory panels (though this is a different type of measurement process than using laboratory instruments). It is, however, important to make a distinction between product properties and the perception of these properties. Sensory measurements are, therefore, the result of product properties (causing stimuli) and the processing of these stimuli by the consumer. Some quality attributes are the resultant of several phenomena. For instance, the color of a food may be the result of the presence of several components absorbing or reﬂecting light of a certain wavelength. Even though color can be measured instrumentally, it is not immediately obvious which compound is responsible for the color observed. Another example is the quality attribute nutritional value, which is determined not only by vitamin content but also by the type and amounts of amino acids, type and amounts of fatty acids, etc. That is why we propose to decompose quality attributes further into quality performance indicators. In the above examples, a quality performance indicator for color may be the concentration of a carotenoid, and the content of the amino acid lysine may be one of the quality indicators for nutritional value. Many quality indicators can be measured directly using physical or chemical measurements. Examples include the presence=absence of pathogenic microorganisms, the protein content and the biological value of the protein, vitamin content, bioavailability, etc. These indicators clearly cannot be determined via sensory panels; they are hidden to the consumer, although they may have a subliminal effect on food choice. Kinetic approach. When we speak of food quality in this book we address these physical, chemical, biochemical, and microbial quality indicators. We accept that this is only a part of the quality perceived

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by the consumer. However, we limit ourselves deliberately to the indicators mentioned because we consider them the principal domain of the food technologist. An important consideration is that these indicators tend to change with time, and therefore they have to be characterized by a kinetic approach, the subject of this book. Food technology is, in short, concerned with the transformation of raw materials into foods and their stabilization (preservation), taking into account all boundary conditions of food safety and quality mentioned above. Raw materials and foods are subject to change because of their thermodynamic instability: reactions take place driving the system toward thermodynamic equilibrium (as will be discussed in Chapter 3). Foods may deteriorate soon after harvesting (sometimes even during harvesting), and deterioration should be read as loss of quality. Prevention and control of this thermodynamic instability is the main task of food technologists. It is the characterization of the changes taking place that is important because this provides us with possibilities to control quality. This is then the domain of kinetics. Kinetics plays thus an important part in the modeling of food quality. The purpose of this book is to explain how kinetics and kinetic models can be used in a meaningful way, thus to supply valuable tools to describe changes in quality performance indicators and attributes, and most importantly to supply tools to control and predict these quality indicators and attributes. Still, foods are so incredibly complex from a chemical and physical point of view that we need to resort frequently to systems mimicking foods. Otherwise, there will be so many interfering factors that the predictive capabilities of mathematical models will be very limited. Model systems mimicking foods are by their very nature simpliﬁcations but, on the other hand, they need to approach real foods in some sense. Ignoring speciﬁc properties of foods when designing model systems may lead to serious mistakes when one extrapolates from the model systems to real foods. Since this is not straightforward, a special chapter (Chapter 14) discusses this in detail for some relevant food aspects. Overall, the philosophy presented in this book is that it is essential to understand what is happening at the molecular level (occasionally the colloidal level) and for this reason the material presented is at the fundamental level of thermodynamics and chemical kinetics. It is the author’s view that such understanding is needed in order to come to models that will be able to control and predict food quality. In addition, kinetic modeling as such is a tool in understanding what is going on because proposed mechanisms need to be confronted with experiments, and if the two do not match something was apparently wrong with the proposed mechanism. Having said that, it is also appreciated that we sometimes have to resort to empirical models due to the complexity of foods. This statement may seem contradictory to the philosophy that fundamental insight is needed but it is not. It is merely a recognition of the fact that our understanding of what is going on in foods is far from complete, and it would be foolish to stick to models that are derived from situations in very simple and ideal systems while they are not capable to grasp the real situation. Especially if we want to be able to predict real-life situations in a realistic way, empirical models may actually perform better than mechanistic models in some situations. That is why the reader will also be introduced to empirical models. Admittedly, empirical models will not directly provide molecular insight. It is therefore important to have attention for both approaches.

1.3

Foods as Complex Reaction Media

When considering reactions in foods, the medium in which these reactions take place is obviously of importance. We may have solid, liquid, and vapor phases in and around foods. Most of the relevant reactions in foods will take place in the liquid phase. In many cases this will be an aqueous phase but also lipid phases are possible, or ethanol may be present which gives different properties to the reaction medium. There may be partitioning between phases. Solid phases may become of importance because they may result from exceeding solubility products; an important solid phase is, of course, ice, but also salts and sugars may be present as crystalline material, or sometimes as amorphous materials. Moreover, solid phases may induce adsorption of reactants and products and catalyze or inhibit reactions. Then we have the presence of amorphous phases, like in glasses. The vapor phase is of importance when a

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Food Food composition composition and and structure structure

Chemicalpotential, potential, Chemical Activities activities

Ionic Ionic

Compartments within a continuous phase

Nonionic Nonionic

Foambubbles bubbles Foam

FIGURE 1.5

Cells Cells

Molecular mobility mobility Molecular

Viscosity Glassy Glassystate state Viscosity

Emulsiondroplets droplets Emulsion

Networks Networks

Crystals Crystals

Overview of the complexity of foods as reaction media.

headspace is present, or in the case of foams, and the partitioning of volatiles is a very relevant phenomenon in relation to sensorial aspects. When we want to study kinetics in foods, we have to take all these various aspects into account, but there are few, if any, theoretical frameworks available with which to do this. Most theories have been developed for ideal, dilute, and homogeneous systems. Foods are multicomponent, concentrated systems with various phases present at the same time, and consequently foods behave all but ideal. Complications with foods arise because of deviations from simple diffusion laws, complications with molecular mobility, partitioning phenomena, and volume exclusion effects. These questions will be addressed in the book. The food matrix is usually very complex, consisting of water-insoluble material (e.g., cell membranes), a complicated aqueous solution of ionic and nonionic compounds of high and low molar mass, amorphous materials, various phases (fat globules, foam bubbles, crystals), and to complicate matters further, foods can also be in a glassy state. All this has a large impact on kinetics. Figure 1.5 summarizes the aspects involved, and serves as a guideline for the topics to be discussed in this book.

1.4

Outline of the Book

After this introduction, the book is divided in two parts. The ﬁrst part is called ‘‘The basics’’ and attempts to describe the ﬁrst principles of modeling (Chapter 2), thermodynamics (Chapter 3), chemical kinetics (Chapter 4), temperature and pressure effects (Chapter 5), and charge effects (Chapter 6). Chapter 7 introduces the use of statistics in kinetics. In the author’s view this is a crucial topic that deserves a great deal of attention and the topic is therefore treated at some length. Though the treatment in Part I is basic and general, food examples are used wherever possible. Part II is called ‘‘Application of the basics to chemical, biochemical, physical, and microbial changes in the food matrix,’’ in which we direct our attention subsequently to chemical, physical, biochemical, and microbiological aspects relevant for foods. Thus, Chapter 8 discusses the possibilities and advantages of multiresponse modeling, a topic that lends itself very well for food science problems, especially when they are of a chemical nature. Surprisingly, the concept is hardly used in food science literature, and accordingly we describe the principles, applications, and potential problems in detail and apply it to some chemical changes. As indicated above, there is more to food quality than chemical changes. The chapters to follow are devoted to enzyme kinetics and kinetics of protein and enzyme inactivation (Chapters 9 and 10), kinetics of physical processes (Chapter 11), kinetics of microbial growth as well as inactivation (Chapters 12 and 13, respectively). Chapter 14 attempts to address speciﬁc problems arising in the food matrix when dealing with kinetics. This concerns discussions as to why kinetics in foods can be quite different from reactions in simple model

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systems in test tubes, how we can identify such problems and take them into account when using model systems (to get around the problem of variability and complexity). Finally, we give a retrospective and an outlook by discussing some trends and developments in modeling in general and with some attention for shelf life modeling in particular because that requires integration of several aspects (Chapter 15).

Bibliography and Suggested Further Reading About Quality Aguilera J.M. and Lilford P. Food Materials Science. Principles and Practice. New York: Springer, 2008. Bruin S. and Jongen T.R.G. Food process engineering: The last 25 years and challenges ahead. Compr Rev Food Sci Food Saf 2:42–81, 2003. Damodaran S., Parkin K.L., and Fennema O.R. Fennema’s Food Chemistry. 4th ed. Food Science and Technology, p. 1144. Boca Raton: CRC Taylor & Francis, 2008. Jongen W.M.F. and Meulenberg M.T.G. Innovation in Agri-Food Systems. Product Quality and Consumer Acceptance, p. 399. Wageningen: Wageningen Academic Publishers, 2005. Fito P., LeMaguer M., Betoret N., and Fito P.J. Advanced food process engineering to model real foods and processes: The ‘‘SAFES’’ methodology. J Food Eng 83:173–185, 2007. Gaonkar A.G. and McPherson A. Ingredient Interactions. Effects on Food Quality. 2nd ed., p. 554. Boca Raton: CRC Taylor & Francis, 2006. Kind M. Product engineering. Chem Eng Process 38:405–410, 1999. Labuza T.P. and Riboh D. Theory and application of Arrhenius kinetics to the prediction of nutrient losses in foods. Food Technol 36:66–74, 1982. Labuza T.P. Application of chemical kinetics to deterioration of foods. J Chem Educ 61:348–358, 1984. Linnemann A.R. and Van Boekel M.A.J.S. Food Product Design. An Integrated Approach, p. 236. Wageningen: Wageningen Academic Publishers, 2007. Lund D. Predicting the impact of food processing on food constituents. J Food Eng 56:113–117, 2003. Luning P.E., Marcelis W., and Jongen W.M.F. Food Quality Management. Wageningen, the Netherlands: Wageningen Pers, 2002. Luning P., Devlieghere F., and Verhé R. Safety in the Agri-Food Chain, p. 688. Wageningen: Wageningen Academic Publishers, 2006. Martens H. and Martens M. Multivariate Analysis of Quality. Chichester: John Wiley & Sons, 2001. Molnar P.J. A model for overall description of food quality. Food Qual Pref 6:185–190, 1995. Niranjan K. Chemical engineering principles and food processing. Trends Food Sci Technol 5:20–23, 1994. Norton I., Fryer P., and Moore S. Product=process integration in food manufacture: Engineering sustained health. AIChe J 52:1632–1640, 2006. Saguy I. and Karel M. Modeling of quality deterioration during food processing and storage. Food Technol 34:78–85, 1980. Sloof M., Tijskens L.M.M., and Wilkinson E.C. Concepts for modelling the quality of perishable products. Trends Food Sci Technol 7:165–171, 1996. Tijskens, L.M.M. Discovering the Future: Modelling Quality Matters. PhD thesis, Wageningen University, the Netherlands, 2004. Van Boekel M.A.J.S. Kinetic modeling of food quality. Compr Rev Food Sci Food Saf 7:144–157, 2008. Van Trijp H.C.M. and Steenkamp J.B.E.M. In: Innovations in Food Production Systems. Jongen W.M.F. and Meulenberg M.T.G. (Eds.). Wageningen: Wageningen Press, 1998.

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2 Models and Modeling

2.1

Introduction

Since this book is about kinetic modeling, it is appropriate to explain the philosophy about models. Models are certainly not a panacea for all problems. They offer opportunities but also have limitations. It is essential that the reader be aware of this and it is the intention of this chapter to provide this basic awareness.

2.2

Models and Modeling

So, what are models and what is modeling? The answer is not straightforward because it depends on the goals of modeling and the type of model used. Generally speaking, models attempt to formulate the behavior of systems from knowledge of the properties of their component parts. Invariably, models are simpliﬁcations of the real world, designed to facilitate predictions and calculations. They are a tool to help us handle complex situations. Thus, the modeler should be under no illusion with regard to the physical reality of models. Models exist in the mind of the scientist, not in nature. Modeling is an attempt to approximate the real world (the truth), but the truth (whatever that is) will never be reached (if we would know the truth it would not be necessary to use models). This does not detract at all from the usefulness of models but an awareness of the nature of models will help us to see the opportunities as well as the limitations. Thus, models can be seen as a way of communicating a view of the world and they are open to scientiﬁc debate. This applies, of course, equally well to kinetic modeling of reactions in foods. Let us try to picture the various ways in which we can use models to describe a system. Suppose that an input is given to a system that will respond with an output: see Figure 2.1. If we know the input I and we can measure the response R, we can use a model to learn about the system S. For instance, if we heat a food (heat is the input) and we measure the effect on protein denaturation (the response) we could learn something about the behavior of proteins in that particular food matrix (the system). If we know the input I as well as the system S, we can use a model to predict the response. For instance, if we know how much heat we put into a system and we know how the proteins in the system respond to this, we can predict the level of denaturation. If we know the system S as well as the response R it produces upon a certain input we can use a model to control, or to design, which input we need to produce a desired output. For instance, if we want to achieve a certain level of protein denaturation in a food, then we can calculate how much heat is needed to achieve this. These simple examples show that models can be used for various goals. In relation to food quality, all three goals are important. Our system is the food, inputs can be processing conditions, and responses can be changes in food quality attributes. We can use models to learn about the ‘‘physical’’ processes taking place in the food that govern food quality attributes, to 2-1

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Input I

FIGURE 2.1

System S

Response R

System S responding to an input I by a response R.

predict food quality attributes as a function of inputs for a given food, and to control food quality for a given food by managing the input. It is perhaps meaningful to spend a few more words on the concept of prediction because there seems to be some confusion on this in literature. A distinction should be made between ‘‘model ﬁt’’ and ‘‘prediction.’’ A model ﬁt is obtained when comparing the performance of a model with the experimental data. Prediction means that it is possible, via models, to predict events or situations that were not in any way used in setting up the model. This can be future events, or events that were obtained independently in other studies. To be clear, we use the words model ﬁt and prediction in this sense throughout the book. Schemes such as in Figure 2.1 are sometimes referred to as conceptual models, i.e., a hypothesis about how a system works and responds to changes in inputs. In other words, it is a set of qualitative assumptions. If we are able to turn somehow these qualitative assumptions into quantitative ones, and if we can describe this with mathematical equations, a conceptual model changes into a mathematical one. Throughout this book, we will conﬁne ourselves to models that describe ‘‘physical’’ phenomena in a mathematical way, i.e., chemical, physical, or microbial events are translated into mathematical equations. Examples include the nonenzymatic browning of foods, the growth of bacteria in a food, and the sedimentation of cocoa particles in a chocolate drink. Mathematical models relate responses to variables via parameters in one or more equations. Say that we are interested in the fate of a vitamin in a food during processing and storage, as a measure for a change in nutritional quality. What is useful to know then is the change in concentration of such a vitamin over time at a certain temperature, or possibly at ﬂuctuating temperature. As we will see in later chapters, a possible mathematical relation that describes the vitamin concentration (denoted as [vitamin]) as a response to the variables time t and temperature T could be: Ea t [vitamin] ¼ [vitamin]0 A exp RT

(2:1)

[vitamin]0 represents the initial concentration. A and Ea represent parameters characteristic for the degradation of the vitamin in the food under study (these are the pre-exponential factor A and the activation energy Ea in the Arrhenius equation, Chapter 5). Such parameters can be estimated in controlled experiments. The parameter R is a fundamental physical constant (the gas constant), which is known and need not be estimated. Preferably, the parameters should be physically interpretable. Time and temperature are controllable variables that can be manipulated by the experimenter or operator. If an equation such as Equation 2.1 is established and the parameters estimated from experimental results, one could then, in principle at least, control and predict the change in vitamin concentration at any relevant time and temperature because we can control time and temperature, and thereby control this food quality indicator. Quality change modeling. We can generalize this further by following the food along its way in the food production chain. Generally, there will be changes in food quality when the food moves from the producer to the consumer; quality is not static. There may be losses, for instance of vitamins during storage of fruits and vegetables. On the other hand, the nutritional value of a processed food may increase as compared to the raw material, for instance, bioavailability or digestibility may be enhanced. To illustrate this change in quality, suppose that a quality performance indicator (for instance the concentration of a vitamin) is built up during the growth of a vegetable or a fruit. Loss of quality usually starts immediately after harvesting, so postharvest storage may already result in some losses. Processing may perhaps result in

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Processing

Concentration

Postharvest Primary storage production

Storage Distribution

Time

FIGURE 2.2 Schematic depiction of change in a quality performance indicator (e.g., vitamin concentration) along the food processing chain.

much higher losses, while storage and distribution may give a further gradual decrease of quality. Figure 2.2 gives a schematic example of such a quality loss. For modeling purposes, it is convenient to identify and quantify the various factors leading to quality changes in each chain element. The output of one chain element is the input for the next (Figure 2.3); a production chain can be seen in this way as a cascade of unit operations. We propose to describe this as quality change modeling. In doing so, various models need to be connected to each other with proper use of mass and energy balances. However, if we want to maintain a high quality at the end of the chain, i.e., when the food arrives at the consumer, the trick is to optimize quality all over the chain, rather than locally in one of the chain elements. By analyzing quality in this way, it becomes possible to optimize quality from an analysis of what happens in the various elements in the food chain. Analogous to the term HACCP, we propose to describe this as Quality Analysis Critical Control Points (QACCP). In the case of a situation as in Chain element 1

Quality attribute in

Chain element 2

Chain element 3

Processes affecting quality indicators comprising the quality attribute

Chain element 4

Chain element 5

Quality attribute out

Chain element 6

FIGURE 2.3

Schematic presentation of quality models in elements of the food chain.

Factors affecting quality indicators

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Figure 2.2, it is clear that much could be improved in the processing step, so that attempts to increase quality should focus on processing conditions. In other cases, losses during storage or distribution may actually be larger than in processing, and then storage conditions may have to be optimized (for instance, by changing temperature, or relative humidity). In order to be able to derive graphs such as Figure 2.2, one must understand what is happening to that particular quality attribute. The main theme of this book is to apply methods correctly to describe such changes quantitatively in every element of the food chain. Mass and energy balances may be helpful in this respect because terms in such balance equations involve kinetics. To be sure, changes in quality attributes as depicted in Figure 2.3 apply, of course, also to microbial growth as a (negative) quality factor, which is the subject of much research nowadays, sometimes referred to as predictive microbiology, or quantitative microbiology. Of course, much work has already been done and published in the past. Unfortunately, many of these studies published cannot be used to develop predictive models because external conditions as well as essential information on the food were not reported. It is also essential that quantitative data are reported in full rather than as averages for developing and validating models. Deterministic empirical and mechanistic models. If we now generalize equations such as Equation 2.1 in a more abstract way, a description of a mathematical model can be given as: h ¼ f (u, jv )

(2:2)

where h represents measurable response(s) (such as vitamin concentration in Equation 2.1) u symbolizes the parameters of the model (A and Ea in Equation 2.1) jv represents the controllable variables (t and T in Equation 2.1) The notation f(..) should be read as: ‘‘is a function of.’’* In kinetic models, h would thus represent concentrations or rates; u rate constants, activation energies, diffusion constants; and jv reaction time, temperature, pressure, or initial concentrations. The main purpose of kinetic modeling is to cast the relevant quality attribute in some mathematical equation and to ﬁnd the actual form of Equation 2.2, followed by estimation of the characteristic parameters. There may be two different objectives for setting up a mathematical model in the form of Equation 2.2: 1. To obtain an estimate of responses over a range of variables that are of interest, either by interpolation between experimental measurements or in a predictive way. 2. To determine the underlying physical mechanism of the process under study, i.e., to ﬁnd the nature and signiﬁcance of the function h ¼ f(u,jv). For objective 1, a theoretical model is not really needed, although it could be useful if one is to stray outside the boundaries of experimental measurements. All one needs is a suitable mathematical function (such as a polynomial function) that accurately describes the experimental results. This is often referred to as empirical modeling or response surface methodology (RSM). It can be very useful for situations where an underlying mechanism is not readily available. The approach obviously has its limitations. It cannot be used to build a mechanistic model because the parameters have no physical signiﬁcance. It is also very dangerous to extrapolate outside the region of variables for which the function was derived (and sometimes even interpolation is tricky). The situation is different for objective 2. Here a scientiﬁc theory is required on which to base a mathematical function. While a model can never represent the complete real world, an adequate mechanistic model should be based nevertheless on a scientiﬁc theory and the model should be able to predict experimental results or commonly observed phenomena accurately. The parameters in the model * The notation in Equation 2.2 using Greek symbols is commonly used in the statistical literature. We adopt this here because we will apply statistics frequently in this book.

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should have physical signiﬁcance and in the ﬁeld of kinetics they include, for instance, rate constants and diffusion coefﬁcients. It is also of importance to state the conditions clearly under which the parameters have been deﬁned. It should be less dangerous in this case to extrapolate outside the experimentally tested regions. The two types of models, empirical and mechanistic, represent extremes; in reality the situation is somewhere in between, certainly for foods with all their complexity. Thus, even with empirical models, one may have some idea of the underlying mechanism. For instance, some microbiological growth models are, strictly speaking, empirical because the manner in which water activity or pH affect microbial growth is not (fully) understood. On the other hand, the functional behavior of the response, e.g., whether linear or logarithmic with respect to pH, may provide clues as to the underlying mechanism. Conversely, a model that is claimed to be mechanistic may still contain unexplained aspects; a rate constant, for instance, can be apparent, i.e., reﬂect more than one reaction step, as discussed in Chapter 4. Stochastic models. At this point it is essential to introduce yet another element in the discussion about models. Mathematical models as such are deterministic, i.e., they produce a certain outcome, usually expressed in a number (e.g., a vitamin concentration). The model displayed in Equation 2.1 produces a so-called point estimate (when the parameters are known and the controllable variables time and temperature are set). However, we do not live in a deterministic but rather in a stochastic world (from the Greek word ‘‘stoxastikos,’’ meaning guessing, surmising) and a number as such can be misleading because it suggests certainty. In other words, deterministic models provide an answer that is in a sense not realistic because it ignores (random) variability. When we use models to predict something, we have to accept that there will be an element of uncertainty in our prediction. Suppose, for instance, that we are able to predict the content of a vitamin as predicted by Equation 2.1 as a function of time and temperature. We want to use this to predict the shelf life of a product; when the concentration falls below a certain level the product is not deemed acceptable anymore. This could result in a graph as depicted in Figure 2.4A: A critical time tc can be estimated from this. At a time longer than tc the product is not acceptable anymore. However, because there is uncertainty in the value of the parameters A and Ea, there will be uncertainty in the outcome as well and this results in variation in the prediction and consequently the estimation of critical times tc is also variable (Figure 2.4B). If we are somehow able to estimate this variation, it will be possible to predict the uncertainty, and this will usually be in the form of a probability distribution, in this case of critical times tc (Figure 2.4C). Incidentally, this probability distribution need not be a normal distribution. Variability and uncertainty. Uncertainty, in other words, can and should be modeled! In this respect, it is useful to subdivide the total uncertainty in its two constituents variability and uncertainty. Variability comprises the natural variation in the real world. For foods, this comes down to the biological variation in the composition of raw materials and in the behavior of living materials, especially microorganisms. It can also relate to such things as a slightly varying temperature in a supply chain: even though the temperature may be ﬁxed at a certain value, it will show some stochastic variation that will have an effect on the outcome of our prediction. This variation is inherent in the nature of our physical world. We can measure this variation via statistical methods, but we cannot reduce it (at least not without changing the system). Incidentally, this is often the very purpose of using controlled model systems that simulate behavior of foods, for instance by using a solution of an amino acid and a reducing sugar to simulate the Maillard reaction occurring in foods. In this way we can control or even eliminate biological variation and direct our attention to the reaction of interest, which is very useful to understand the mechanism at hand, but when we translate the results back to real foods we should not forget the biological variation. In fact, variability does give important and essential information about the system under study and should be studied accordingly. The other element is uncertainty. This reﬂects the state of our knowledge (or ignorance) about the system. For instance, a parameter (such as an activation energy) in a mathematical model can be estimated from data but there will be an error involved in this estimate because the data are obtained by using an error-prone method. By doing more experiments (and perhaps better designed when we get

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80

60 Minimum acceptable level

40

20

0 (A)

100 Concentration (arbitrary units)

Concentration (arbitrary units)

100

0

tc Time (arbitrary units)

80

60

20

0 25 (B)

Minimum acceptable level

40

0

tc tc tc Time (arbitrary units)

tc

25

Frequency

100

0 (C)

10

tc

20

FIGURE 2.4 Hypothetical example showing the prediction of the change in vitamin content as a function of time at constant temperature. tc is the time at which the minimum acceptable level is reached. Deterministic result (A), variable result because of uncertainty in the parameters (B), frequency distribution for the critical times tc (C).

to know the system better) we can reduce this uncertainty. This is the very reason why it is useful to split total uncertainty up into variation and uncertainty. If total uncertainty is determined mainly by variation, it makes no sense to try to reduce the uncertainty by doing more measurements. If it is, however, determined by uncertainty we can reduce total uncertainty by doing more and better measurements. Such considerations are very important for risk–beneﬁt analyses in a broad sense, i.e., not only microbial risk assessment but also optimization of concentration or bioavailability of certain food components (beneﬁt assessment). A very good impression of total uncertainty can nowadays be obtained via Monte Carlo simulation (for which we need probability models, to be discussed later). It is the author’s opinion that this way of thinking will become increasingly important for food design problems. It means that we should be prepared to introduce elements of stochastic modeling into our mathematical models, i.e., to introduce probability distributions rather than point estimates in our model. So, instead of a ﬁxed value for the activation energy in Equation 2.1 we could insert the probability distribution of the activation energy in the equation (reﬂecting our state of ignorance) and simulate stochastic variation in the prediction by drawing random numbers by computer. This is done typically thousands of times

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(i.e., Monte Carlo simulation) and results in a probability distribution of the outcome, i.e., a description of the range of values that the outcome may take together with the probability that the variable will take any speciﬁc value. A probability can be seen as a numerical measurement of the likelihood of an outcome. This stochastic nature of modeling is the reason why we spend considerable attention to statistics in this book. Not every scientist seems convinced of the usefulness of statistics, sometimes expressed in the phrase ‘‘how to lie with statistics.’’ This is unfortunate because statistics should be seen as an important tool in the scientiﬁc learning process, to cope with the phenomena of variability and uncertainty, and to be able to draw general conclusions from a limited amount of data. A very useful branch of statistics is the so-called Bayesian statistics, especially in relation to modeling. Bayesian statistics treats probability as plausibility of a hypothesis in view of data obtained and expresses this as a so-called posterior probability, whereas ‘‘classical’’ statistics interprets probability in terms of frequency (a proportion in a large number of repetitions of the random process), and uses signiﬁcance tests to see if a hypothesis can be conﬁrmed. There are fundamental differences in the two approaches and few food technologists appreciate Bayesian statistics, as they have been trained, most likely, in classical statistics. We consider Bayesian statistics important enough to introduce it and discuss some of its elements in Chapter 7. It is also important for risk–beneﬁt analysis and decision analysis concerning food safety. Model uncertainty. We now come to a very important philosophical point in modeling. What we actually are trying to do is to approximate truth or reality with our models. However, it is important to realize that we will never be able to capture reality fully (if we could we would not need a model). The only thing we can do is to infer something from the data that we have obtained (either by observational studies or by doing planned experiments, but we will not consider observational studies in this book). So, in other words, we try to capture the truth behind the data, i.e., the processes or mechanisms that cause the data to be as they are; we do not model the data themselves. It is the information contained within the data in which we are interested and that is expressed in mathematical models. How do we know that we select models that come as close to the truth as possible? Information theory is quite helpful in this respect, providing tools to aid in model selection. We will discuss this in some detail in Chapter 7. For the moment it is important to realize that more than one model may come close to the truth (even though we will never know what truth is). We stress this point because this is the essence of modeling: we will never reach truth (and we do not need to!) as models are just approximations. The important consequence is however that this aspect adds to uncertainty, namely the uncertainty as to how far the model is away from the truth. Figure 2.5 gives some hypothetical situations. With reference to Figure 2.5 it is obvious that model 1 comes reasonably close to reality, model 2 follows the trend to some extent but with considerable bias, and model 3 is completely off. Obviously, we

Model 2

Model 1

Reality

Model 3

FIGURE 2.5

Hypothetical examples of ‘‘reality’’ and three models that approximate reality.

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would like to reduce this model uncertainty but, because we do not know the truth we cannot measure this in an absolute sense, only in a relative sense. One such measure is the so-called Akaike criterion, discussed in Chapter 7. The Akaike criterion focuses on predictive accuracy and provides a methodology to see which model performs the best in predictive accuracy. The topic of model discrimination, i.e., differentiation between a good model and one that is less good, or a bad model, is thus another essential topic. A good model is able to extract the relevant structural information from the data and separate this essential information from noise. It may be that more than one model applies, and the choice for a particular model introduces again uncertainty in our endeavor to approximate to the truth. In some cases it may be better to do some form of model averaging rather than choosing just one model. Some methods of model discrimination are discussed in Chapter 7. By comparing models we actually evaluate the amount of information in the data relative to the information capacity of the model (the more complex the model, the more information capacity it has). So, it should be clear by now that models are always wrong, but some of them may be useful (to paraphrase the famous statistician George Box). Box and coworkers (see bibliography at the end of the chapter) suggest that one should ‘‘tentatively entertain a model’’ rather than assume it to be correct. This implies that one should always be prepared to put models in jeopardy, and subsequently revise them in the light of new evidence. This is the very basis of the scientiﬁc method, where hypotheses and theories are subject to peer review and amended (or indeed rejected). The process of modeling is, therefore, iterative in nature (Figure 2.6). All the elements in this iterative cycle are essential for modeling. Although a cycle is depicted in Figure 2.6 with no apparent starting point, we suggest that, whenever possible, the cycle is started with the box called conjecture. The reason for this is that this is the point where science comes in (for food science basically (bio)chemistry, physics, and microbiology). In the case of a chemical reaction, for instance the Maillard reaction, it helps enormously if the researcher is aware of the possible basic mechanisms because that will give structure to the planning of subsequent experiments. It also means that the researcher should propose some possible models already at this stage, which will be tested and compared later on. This conjecture can be a very simple idea based on literature, observation of the phenomena, basic chemical knowledge, or even intuition. In any case, it is important to think hard before doing experiments and to ask the right questions and apply the appropriate science. Admittedly, there may be situations in which it is impossible to pose models beforehand, and that one needs to do some starting experiments in order to get a feel for the problem at hand. However, in most cases, there will be some idea of an approximating model. Experiments are designed to test the original idea.

Conjecture, hypothesis

Analysis of experimental results

Design of experiments

Experiments

FIGURE 2.6

Scheme showing the iterative nature of the various stages in modeling.

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Experimental design is an essential part of modeling, and its importance is often overlooked. Statistical methods are available to support this stage of experimental design. The design determines and limits the information that can be obtained from a data set. The goal of experimental design is to optimize the information content of a data set with the least possible effort. Experimental design is likely to depend on the purpose of the investigation, whether it be model discrimination or parameter estimation. Chapter 7 pays attention to this aspect of kinetic modeling. Doing the actual experiment can be relatively straightforward in principle, but may be complicated in practice. For instance, with an experiment designed at one temperature the heating-up time may be considerable and cannot be neglected. For the analysis of the data, the use of statistics is indispensable because experiments always contain noise (i.e., noninformation caused by unexplainable variation) and we need to be able to differentiate between this noise and essential information contained in the data. Once again we stress that we are trying to model the information contained within the data, not the data themselves. With properly analyzed results, the original idea can be tested for its validity (or if more than one model had been proposed, model discrimination is accomplished). This may well lead to adjustment of the original idea (the objective is not to accept or to reject a model but to improve it). Experimental data only become meaningful in the framework of a model, as data in isolation do not provide this type of information. However, a model is never deﬁnitive and we must accept the iterative nature of modeling. The already mentioned Bayesian statistical approach ﬁts very well into this philosophy because it describes this learning process in a mathematical way: prior knowledge and data are combined in posterior knowledge, as discussed in Chapter 7. In any case, the combined use of statistics and mechanistic understanding is needed here because it will be necessary to differentiate between noise in the data and the information contained within the data. Model parameters. Model parameters constitute the core of a model. One should always strive for the lowest number of parameters possible in a given model because, as it happens, any model will ﬁt a data set if the number of parameters is made high enough. The penalty for this so-called overparameterization is that the model will be indiscriminate and often worthless: the variance of the parameters will increase too much for proper use of the model (e.g., making predictions). Fortunately, proper use of statistics could signal this and appropriate measures can be taken, as discussed in Chapter 7. On the other hand, if the number of parameters is lowered, the bias between the model and the data increases. Modeling is thus a delicate balance between over- and underparameterization. Figure 2.7 illustrates this. It is in fact a depiction of a famous quote from Einstein: ‘‘Models should be as simple as possible but no simpler than

Bias between model and data

Uncertainty in parameter estimates

Number of parameters p ( p ≥ 1)

FIGURE 2.7 Schematic picture showing the bias between a model and experimental data and the uncertainty in parameter estimates as a function of the number of parameters in a model.

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that.’’ As simple as possible refers to the idea that models should simplify reality taking into account the details that matter and neglecting the details that are not so important. No simpler than that means it should be possible to do calculations with the model that tell the essential things about reality. An interesting situation may arise when mechanistic insight requires a certain parameter, whereas the statistical analysis tells us that that parameter is redundant. It may be that the data set does not contain enough information to estimate the parameter, or it may be that the mechanistic insight needs revision. In any case, a sensible interplay between statistics and mechanistic knowledge (chemical, physical, microbiological, and biochemical in the case of foods) is required. The following guidelines are of importance, and are discussed in more detail in following chapters. 1. The art of keeping the number of parameters in a model at a minimum is called the principle of parsimony, or Ockham’s Razor (after the fourteenth century English philosopher William of Ockham) stating that ‘‘things should not be multiplied beyond necessity,’’ or ‘‘shave away all things unnecessary.’’* In other words, a simple model is better than a complex one, but as indicated above a right balance needs to be found between over- and underﬁtting. A proper procedure for model discrimination will contain a penalty function for increase in the number of parameters. This is discussed in Chapter 7. Also of importance in this respect is that the greater the number of parameters, the greater the extent of nonlinear behavior (in the case of nonlinear models), also discussed further in Chapter 7. 2. Parameterization. The extent to which parameters in nonlinear models behave nonlinearly varies greatly. It may be that some parameters need to be reparameterized in order to ﬁnd the best estimation properties. This is discussed further in Chapter 7. 3. Range of applicability. The data should cover the full range over which the model is applied. This is further explained in subsequent chapters. 4. Stochastic speciﬁcation. It is very important to model not only the underlying mechanism but also the error terms involved (i.e., uncertainty). This is discussed extensively in Chapter 7. 5. Interpretability. Preferably, the parameters should have a physical meaning and not just be ﬁt parameters. This is in fact one of the main themes of the book. In relation to kinetics it is discussed in depth in the following chapters. A complicating factor may be that a conﬂict arises between interpretability of parameters and their statistical estimation properties. To summarize, the aims of modeling in the food science area are as follows: 1. Models can be very helpful to control and predict food quality and provide a tool to optimize quality and costs. 2. Critical points determining quality along the various elements of the food production chain can be identiﬁed. 3. Research results in different domains of the food chain can be combined. 4. Models provide tools to identify biological variability as well as uncertainty of parameters and thus provide a basis for structured data acquisition. It is very helpful to state a goal as well as a purpose when building a model, e.g., to develop a microbial growth model (the goal) to predict microbial shelf life (the purpose). The goal can be different for different purposes. It is quite common to make several assumptions when applying models, for instance, that a constant pH exists, etc. It is necessary to state these assumptions explicitly and to consider them again after a model is used: were the assumptions reasonable for the problem at hand, or are perhaps one or more of the assumptions violated? This may help greatly in evaluating the usefulness of a model. * The relevant statements that can be found in William of Ockham’s writings are ‘‘Pluralitas non est ponenda sine necessita’’ (one should not pose more things than is necessary) and ‘‘Frustra ﬁt per plura quod potest ﬁeri per pauciora’’ (it is vain to do with more what can be done with less). The statement ‘‘Entia non sunt multiplicanda praeter necessitatem’’ (entities should not be multiplied unnecessarily) was probably made by a later scholar. See: www.weburbia.com=physics=occam.html

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Modeling and mathematical terminology. Mathematics should be seen as a language to express relations in a concise, logical, and straightforward way. It may be helpful for the remainder of the book to explain brieﬂy some commonly used terms. Mathematical models in relation to kinetics can appear in several forms. When systems are not changing in time, the state of that system remains constant (static). Variables describing the state of the system (e.g., temperature, concentration) do not change in time in such cases. Models describing such a condition are named static models or steady-state models. Incidentally, steady state is not synonymous with equilibrium, and equilibrium is a special case of a steady-state situation. Equilibrium is a thermodynamic concept. A system is in equilibrium when its free energy cannot be decreased any further under the conditions applied (Chapter 3). A mass balance is typically a static model. A chemical reaction in equilibrium is described also by a static model. Static models can be described with algebraic equations. When a model describes a system that changes in the course of time, as will be the case with most (if not all) reactions in foods, we speak of a dynamic model. Dynamic models are typically described by ordinary differential equations (ODEs), relating the state of a system (such as a concentration) to the rate of change of that state (change in concentration). Another classiﬁcation is that of spatial models, when things are not only changing in time but also as a function of space. These can be described by partial differential equations (PDEs). It is important to realize that much information can be obtained from systems that are changing. A system in steady state that is disturbed at a certain moment will respond to this disturbance and ﬁnd its TABLE 2.1

Overview of Terminology Used with Mathematical Models

Term

Description

Mechanistic model

Mathematical model based on a mechanism; a translation of a physical, chemical, and biological theory

Empirical model

Mathematical model that is optimized to give the best ﬁt to the observed data without an underlying chemical, physical, or biological theory

Deterministic model

Gives outcomes as exact numerical values (point estimates); it produces always the same output with the same input

Probabilistic model, stochastic model Static=steady state=stationary model

Gives an outcome with associated total uncertainty (as a probability distribution); it does not produce exactly the same output with the same input A solution of the state equations when the time derivatives of the state variables are all set to zero, i.e., described by algebraic equations. No description of future states

Dynamic model

Results depend on time and space, described by differential equations. Representation of future system states or conditions

Linear model

A model that is linear in the parameters (not necessarily a linear relationship between x and y)

Nonlinear model

A model that is nonlinear in the parameters

Spatial models

Objects have a position in space (or a ﬁnite region in space)

State variable

Quantity describing the state of the system (e.g., concentration at a certain time)

Independent variable

Variable that determines the change in the state of the system (usually time and space) and is controlled by the modeler=experimenter Equations that specify the particular solution (initial values, for instance) for state variables as a function of the independent variables

State equations Transient

Temporal proﬁle of the state variables after a perturbation on the boundary conditions

Boundary conditions

Constraints that apply to the solution of state equations (e.g., initial values, mass balances)

Parameters Sensitivity coefﬁcient

Constants in the state equations (constant in a particular case, but may vary in different cases) Partial derivative of a state variable with respect to variations in a parameter

Black box model

A model that is not based on any supposed mechanism

White box model

A model that is based on a supposed mechanism

Gray box model

A model that contains both empirical and mechanistic elements

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way to a new steady state. If we were to look only at the two steady states without looking at how the system changes in between, we would lose much information. Relations that describe these changes are very informative. Physical and biological models often arise as solutions of differential equations. Therefore, some knowledge of calculus is needed (more details can be found in Appendix A). Depending on the complexity of the problem at hand, analytical solutions can or cannot be found. If not, one has to resort to numerical solutions, but this is relatively easily achieved with appropriate software packages. Another important factor is that regression models are frequently nonlinear in the parameters, the implications of which are discussed in Chapter 7. To conclude this chapter, Table 2.1 gives an overview of terminology used with mathematical models.

2.3

Concluding Remarks

This chapter has attempted to put the opportunities and limitations of models in perspective of the scientiﬁc method. The most important ‘‘take home message’’ is that models are tools to get a grip on reality, but they are deﬁnitely not the truth or reality. Obvious as this may seem, one sometimes gets the impression that researchers prefer models over reality, and this is, of course, a capital sin in modeling. In that sense, data should never be ﬁtted to models as is sometimes stated in literature; it should always be the other way around!

Bibliography and Suggested Further Reading About Modeling Box G.E.P. and Hunter W.G. A useful method for model building. Technometrics 4:301–318, 1962. Box G.E.P. and Hunter W.G. The experimental study of physical mechanisms. Technometrics 7:23–42, 1965. Box G.E.P. Science and statistics. J Am Stat Assoc 71:791–799, 1976. Burnham K.P. and Anderson D.R. Model selection and inference. A Practical Information-Theoretic Approach. New York: Springer Verlag, 1998. Chatﬁeld C. Problem Solving. London: Chapman and Hall, 1995. Cullen A.C. and Frey H.C. Probabilistic Techniques in Exposure Assessment: A Handbook for Dealing with Variability and Uncertainty in Models and Inputs. New York: Plenum Press, 1999. Haefner J.W. Modeling biological systems. Principles and Applications. 2nd edn. New York: Springer, 2005. Kohn M.C. Use of sensitivity analysis to assess reliability of metabolic and physiological models. Risk Anal 22:623–631, 2002. Phair R.D. Integrative bioinformatics. Practical kinetic modelling of large scale biological systems. www. bioinformaticsservices.com=bis=resources=cybertext=Ibcont.html Ratkowsky D.A. Handbook of Nonlinear Regression Models. New York: Marcel Dekker, 1990. Tijskens L.M.M., Hertog M.L.A.T.M., and Nicolaï B.M. (Eds.). Food Process Modelling. Cambridge, United Kingdom: Woodhead Publishing Limited, 2001. Vose D. Risk Analysis: A Quantitative Guide. Chichester, United Kingdom: Wiley & Sons, 2000. Zwietering M.H. and Hastings A.P.M. Modelling the hygienic processing of foods—a global process overview. TransIChemE 75(C3):159–167, 1997. About Ockham’s Razor Jefferys W.H. and Berger J.O. Ockham’s razor and Bayesian analysis. Am Scientist 80:64–72, 1992.

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3 Chemical Thermodynamics in a Nutshell

3.1

Introduction

Although the topic of this book is on kinetics, we consider it appropriate to include a short chapter on chemical thermodynamics, which is the branch of thermodynamics that studies how chemical reactions come to equilibrium. The reason to do this is that thermodynamics and kinetics are complementary to the study of chemical reactions. Thermodynamic parameters are also used in the formulation of chemical kinetics, as we shall see later on in this book. Thermodynamics is, in short, the science of conversion of energy and matter. Reversible thermodynamics makes statements about systems in equilibrium, and gives an answer to the question: what drives chemical reactions toward equilibrium or to completion? Reversible thermodynamics is reasonably established by now. In contrast, irreversible thermodynamics, which makes statements on processes and systems that are not in equilibrium, is still debated. Most of this chapter will be on reversible thermodynamics, with some remarks about irreversible thermodynamics at the end of the chapter. Irreversible thermodynamics makes a nice link to kinetics. Kinetics gives an answer to the question, how does it happen and at what rate, and that will be addressed in subsequent chapters. The chapter is organized as follows. Both for thermodynamics and kinetics we need to be able to express quantitatively the progress of a reaction, so we start with a section on how to quantify reactants and products. This is not only needed for this chapter but also for the rest of the book. Then we move to chemical thermodynamics, by ﬁrst highlighting the concepts of energy, enthalpy, entropy, and free energy. Then we discuss the difference between ideal systems and real, nonideal systems and how we cope with nonideal systems via the activity concept. We continue by considering how these concepts can be used to state something about direction of processes and equilibrium positions of reactions. Finally, we conclude the chapter by making the move from reversible to irreversible thermodynamics, and from there we will make a connection to kinetics.

3.2

Quantiﬁcation of Reactants and Products

Ways to express amounts and concentrations. The amount of chemicals present in a solution can be expressed in many ways. The chemical present in the highest amount is called the solvent, and the chemical present in lower amount the solute (obviously there can be more than one solute in a solvent), 3-1

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and the whole of solutes and solvent is called a solution. We will mostly refer to thermodynamics and kinetics in solution, and the most common way to express amounts is then via molarity, i.e., the number of moles of solute per liter of solution. This is indicated by the symbol M (molar). Another way is via molality, which is the number of moles of solute per kilogram of solvent, indicated by the symbol m (molal). Yet another way is molinity, i.e., the number of moles per kilogram solution, but this unit is hardly ever used. In most cases relevant for foods, the solvent is water. Since the density of water is about 1 kg dm3, there will not be too much difference between molarity and molality for diluted aqueous solutions. However, it should be realized that it depends on the density of the solution in how far molality and molarity differ, and for foods this can make quite a difference. Also, molarities do depend on temperature because the volume changes with temperature while molalities do not. Furthermore, for some foods, such as cheese, or tomato paste, it is hard to envisage what a liter of solution actually means. So, it would not be a bad idea to use molalities instead of molarities for foods, but it is not very common. Another possible way of expressing amounts is via the mole fraction, i.e., the number of moles of solute divided by the total number of moles present in the system. Mole fractions are mostly used for more concentrated systems, but this unit only makes sense for well deﬁned, simple systems, not for foods. Molarities=molalities are more used for dilute systems. Then, it is of course also possible to express amounts in mass fractions, indicated by the symbol w=w (e.g., g solute=kg solution), mass–volume fractions w=v (e.g., g solute=L solution), and volume–volume fractions v=v (e.g., mL solute=L solution); these are expressed often as percentages, and sometimes also as ppt (parts per thousand, e.g., g solute=kg solution) or ppm (parts per million, e.g., mg solute=kg solution). In this book, we use mostly concentrations expressed as molarity or molality, but occasionally other measures will also be used. In any case, it will always be indicated how amounts are expressed as this is a prerequisite for thermodynamics and kinetics. Of course, the various ways to express amounts can be converted into each other, but extra information may be needed. For instance, to convert molar concentration into molal concentration, the density of the solution is needed. Appendix B gives an overview of ways to express amounts and some formulas for conversions. To show the intricacies involved for foods, the example of milk is a nice one. A liter of milk is not a pure solution; it also contains fat globules and casein micelles, i.e., these components are not dissolved but dispersed. Lactose is really dissolved and is part of the milk serum (the part of milk that can be considered as aqueous solution). So, the concentration of a solute like lactose can be expressed in several ways, as shown in Table 3.1, and it does make a difference which unit is used to express the amount of lactose present in milk. Sometimes, components are quantiﬁed in solutions that are ﬁrst separated from a food before the analysis. It may be that the ratio of a component to water in such a separated solution is larger than that ratio in the food from which the solution is obtained. This is due to the phenomenon of steric exclusion, which implies that the volume occupied by one particle is not available for another particle. Solute molecules need to keep a distance from the surface of large particles, and this distance depends on the size of the solute. In other words, a part of the volume is not accessible to solute molecules but is accessible to water molecules: this is the nonsolvent water. Obviously, the effect is larger for larger particles. Figure 3.1 illustrates this effect graphically. Corrections should be applied when concentrations

TABLE 3.1

Various Ways to Express the Amount of Lactose in Full-Fat Milk

Expression

Unit

Result

Mass percentage (%)

g=100 g milk (%)

Concentration (molinity)

mol kg1 milk

0.134

Concentration (molarity)

mol L1 milk (M)

0.138

Concentration (molality)

mol kg1 water (m)

0.154

3

Note: 4% fat, 12.8% dry matter, density 1030 kg m .

4.6

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No water

Water molecules Protein

No solute

FIGURE 3.1

Solute molecules

Graphical illustration of the steric exclusion effect resulting in the presence of nonsolvent water.

determined in a solution obtained from a food are recalculated to the corresponding concentration in the food itself. Stoichiometry of reactions. In a chemical reaction, molecules interact (react) with each other and the result is the formation of products in which atoms are arranged differently than in the original reactant molecules. When a chemical reaction occurs, the proportions of the amounts of participating molecules change. Suppose we have a reaction represented symbolically as nA A þ nB B ! nP P þ n Q Q

(3:1)

where nA, nB, nP, nQ represent the number of molecules or moles of reactants A and B, and products P and Q, respectively (it is just a convention to call the components on the left-hand side reactants and the ones on the right-hand side products). The equation is a standard notation for a chemical transformation and is called a stoichiometric equation (from the Greek words stoixeoy [stoicheon], meaning element, and metroy [metron], meaning measure). The importance of a stoichiometric equation is that it deﬁnes the exact change from initial number of molecules to the ﬁnal composition; in other words, such a change is not arbitrary. The bidirectional arrows indicate that reactants can also be formed from the products, and when there is a balance between formation of products and reactants a steady state is reached, which may or may not be an equilibrium; a more formal deﬁnition for equilibrium will be given later on. nA,B,P,Q are called the stoichiometric constants, which can be seen as physical quantities representing the change in the number of molecules of a component per chemical transformation as indicated by the reaction equation. They indicate ﬁxed proportions of the number of moles upon chemical transformation. We can generalize Equation 3.1 for more reactions in the following way. Suppose there are n species Ai (i ¼ 1, . . . , n) and r independent reactions ( j ¼ 1, . . . , r). An example is the decarboxylation of amino acids, an important reaction in ripening cheese with respect to ﬂavor development. The following reactions are possible: Reaction 1:

R-CH-NH2 ! R-CH2 -NH2 þ CO2

Reaction 2:

j COOH 2R-CH-NH2 þ O2 ! 2R-CO-COOH þ 2NH3 j COOH

Reaction 3:

R-CO-COOH ! R-CHO þ CO2

(3:2)

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3-4 TABLE 3.2

Stoichiometric Table for the Reactions in Equation 3.2

Species i Reaction j

R-CH(COOH)NH2 (i ¼ 1)

O2 (i ¼ 2)

R-CH2NH2(i ¼ 3)

CO2 (i ¼ 4)

R-COCOOH (i ¼ 5)

R-CHO (i ¼ 6)

NH3 (i ¼ 7)

j¼1

1

0

þ1

þ1

0

0

0

j¼2

2

1

0

0

þ2

0

þ2

j¼3

0

0

0

þ1

1

þ1

0

So, we have three reactions (r ¼ 3) and seven species (n ¼ 7). We can set up a table for the stoichiometric constants (Table 3.2). The convention is that stoichiometric constants are indicated with a negative sign for reactants (because the number of these molecules decreases) and a positive sign for products (because the number of product molecules increases as the reactions starts with reactants A and B). Such a table can be summarized in the so-called stoichiometric matrix 2

1 4 2 0

0 1 0

1 1 0 0 0 2 0 1 1

3 0 0 0 25 1 0

(3:3)

Matrices are just convenient ways to represent a bunch of numbers. A concise way to represent balanced reaction equations such as Equation 3.2 is n X

nij Ai ¼ 0

(3:4)

i¼1

In the case of just one reaction, r ¼ 1, Equation 3.4 reduces to n X

n j Ai ¼ 0

(3:5)

i¼1

A balanced reaction equation such as in Equation 3.1 does not necessarily represent the actual mechanism of a reaction. What it does represent is the proportion in which changes occur in amounts of reactants and products. It is important to be aware of this distinction. It may well be that at the molecular level (i.e., elementary reactions) the reaction depicted in Equation 3.1 is as follows: nA A ! nA A1 nA A1 þ nB B ! nB B1 nB B1 ! nP P þ nQ Q

(3:6)

A1 and B1 represent transient intermediates that do not appear in the overall stoichiometric equation. An example of such a reactive species is a radical, for instance formed in several oxidation processes in foods. The danger in using an overall reaction equation such as Equations 3.1 and 3.2 is that it represents the ﬁnal outcome of a reaction. One is, however, not allowed to add up reaction equations (such as Equation 3.6) as if they were algebraic equations. This is especially so for a kinetic treatment: sometimes the intermediates are very important from a kinetic point of view. It can cause much confusion if reaction (Equation 3.1) is taken as the actual mechanism when it only reﬂects the balance in amounts of reactants and products. When we want to study how a reaction progresses we are in need of a parameter that describes this progress. Such a parameter is j, the extent of the reaction.

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Extent of reaction. With the number of moles expressed as ni, and ni,0 the initial number of moles for component i, the extent of reaction is deﬁned as j¼

ni ni,0 Dni ¼ ni ni

(3:7)

or expressed in terms of the number of moles ni ¼ ni,0 þ ni j

(3:8)

The stoichiometric constant ni indicates the amount of component i in the reaction. j is a timedependent variable describing the advancement of a reaction and is proportional to the net number of transformations in going from left to right in Equation 3.1. j could be referred to as the amount of chemical transformations (or number of events) expressed in number of moles. j ¼ 0 means that the amount of (at least one) product is zero. A maximum value jmax is reached when at least one reactant is exhausted; this will happen for a reaction going to completion. The exhausted reactant is called the stoichiometrically limiting reagent. The amount of a limiting component consumed in the reaction must be equal to the amount initially present. The limiting reagent is therefore the component that has the ni,0 and jmax is numerically equal to this lowest value (as follows from Equation 3.7). An lowest value of jni j example may clarify this a bit more. Suppose we have a reaction depicted by: 2A þ B þ 3C ! P þ Q and the reaction starts with initial amounts as shown in Table 3.3. The limiting reagent for the initial conditions indicated is thus component C. In the case of a reaction not going to completion, an equilibrium jeq will be reached as discussed below in the section on equilibrium. j speciﬁes the composition of a reaction mixture during the course of the reaction; it is a useful parameter because it describes the extent of reaction regardless of which compound is considered. It should be realized that j can be larger than 1. For more than one reaction ( j ¼ 1, . . . , r) with n components (i ¼ 1, . . . , n), Equation 3.8 becomes ni ¼ ni,0 þ

r X

nij ji

(3:9)

j¼1

It is possible to turn the parameter j into a dimensionless parameter, for instance by using jmax ar ¼

j jmax

(3:10)

ar is called the degree of reaction, with a numerical value between 0 (only reactants present) and 1 (only products present). TABLE 3.3 Component ni,0 (mol) ni jmax (mol)

Example of Calculation of jmax for the Reaction 2A þ B þ 3C ! P þ Q A

B

C

P

Q

1

1

1

0

0

2 1=2

1 1=1

3 1=3

1

1

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Another frequently used measure for the progress of a reaction is the fractional conversion parameter, fc fc ¼

ni,0 ni ni,0

(3:11)

In contrast to j, fc depends on the species i and is related to the extent of reaction as follows (cf. Equations 3.7 and 3.11): j¼

fc ni,0 ni

(3:12)

These parameters are needed to calculate the amounts of reactants and products as a reaction progresses. We will use them in the sections and chapters to follow. Now that we have learned about the parameters needed to express quantities we can turn to thermodynamics and, subsequently, kinetics.

3.3

Thermodynamics of Reactions

Thermodynamics is the science of energy; it includes all aspects of energy and energy transformation and relationships among the properties of matter. For instance, a cup of hot coffee will cool down but it will never reach a higher temperature by itself, and thermodynamics offers an explanation for that. Thermo is the Greek word for heat (uermh), and dynamics for power (dynamis). Chemical thermodynamics is the study of the interrelation of heat with chemical reactions or with a physical change of state. It can provide information about the equilibrium position in a chemical process, i.e., whether or not the reaction can take place when one starts with the reactants A and B, to what extent, and in which direction. The possible exchanges of work, heat, or matter between a system and its surroundings take place across a boundary proceeding from an initial state to a ﬁnal state. Then, one may want to know how this equilibrium can be inﬂuenced by, for instance, temperature and pressure changes. It is for these reasons that we would like to remind the reader of some relevant thermodynamic principles for (chemical) reactions.

3.3.1

Heat and Work

As mentioned, the basic considerations for thermodynamics are heat effects (thermo) and work effects (dynamics). Heat is the reﬂection of random molecular motion, work is organized motion (force 3 distance). Work can appear in various forms: gravitational, expansion, tension, electrical, dissipative force of friction, and also chemical. Heat and work are processes by which energy is transferred between a reactive system and its surroundings. Work is the energy needed to do something. Thermodynamics allows one to account quantitatively for how much energy goes where. In principle, thermodynamics is model independent and does not refer necessarily to molecular events; this model independency is the case of classical thermodynamics. Thermodynamics can be built also upon quantum mechanics and this is called statistical thermodynamics. Statistical thermodynamics determines the distribution of a given amount of energy E of n identical systems. The goal of statistical thermodynamics is to understand and to interpret the measurable macroscopic properties of materials in terms of the properties of their constituent particles and the interactions between them. This is done by connecting thermodynamics functions to quantum mechanic equations. Since it does help in understanding to consider molecules, the discussion given below contains elements of both classical and statistical thermodynamics. It is important to realize that thermodynamics only applies to macroscopic amounts, i.e., large numbers of molecules. Before we discuss some aspects of thermodynamics it may be useful to recall some basic terminology (Table 3.4). The reader is advised that the following thermodynamic considerations are not as rigorous as they perhaps should be. It is not the intention of this book to discuss thermodynamics in great detail; it merely aims to make the link with kinetics. There are many excellent textbooks on the topic and some suggestions for further reading are given at the end of this chapter.

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3-7

Descriptions of Some Terms Used in Thermodynamics

Term System

Meaning Part of the universe that is studied

Surroundings

Part of the universe outside the system

Open system

System can exchange mass and energy with surroundings

Closed system

System can exchange energy with surroundings but cannot exchange mass

Isolated system

System can neither exchange mass nor energy with surroundings

Adiabatic system Isothermal system

System that does not exchange heat with surroundings Uniform temperature, i.e., no temperature gradients in the system

Isobaric system

Uniform pressure, i.e., no pressure gradients in the system

Isochoric system

System at constant volume

Diathermic system

System that permits heat transfer

Intensive variable

Does not depend on the amount of the quantity it refers to

Extensive variable

Does depend on the amount of the quantity it refers to

Equilibrium state

No net production or consumption of components (chemical reactions), no unbalanced forces (mechanical), no temperature gradients (thermal)

Spontaneous process

A process that occurs in a system without any work being done on the system

Reversible process

The system as well as the surroundings return into the same state when a process is reversed by inﬁnitesimal changes of a variable

Irreversible process

Net changes in the state of a system and surroundings have occurred as a result of the process that cannot be reversed

System, surroundings, and universe. First of all, we divide the universe into system and surroundings (Figure 3.2). A system can be any part of the universe that we want to study, so it can be a single reaction, an aggregate, an emulsion droplet, a whole food, a food factory, etc. The system and its surroundings are distinguishable because one can imagine some type of boundary between them, and this boundary does or does not permit exchange of matter and energy (see Table 3.4 for the possibilities and the terminologies). For instance, a can containing food can exchange heat with the surroundings but not matter, and canned foods are therefore closed systems. Hot coffee in a closed Dewar ﬂask forms an isolated system (in principle at least, though in practice not completely because eventually the coffee temperature will assume the temperature of the surroundings). Freshly baked bread left on the table is an open system: there will be exchange of heat (the bread cools down, the surroundings warm up) and of matter (water will System evaporate from the bread). Incidentally, these simple boundary examples demonstrate why packaging of foods is so important, because one can inﬂuence then the interaction of a food with its surroundings through manipulating the boundaries (gas permeation, solubility of System Universe components in the packaging material, etc.), and thereby one can control reactions. The conditions that describe a system are called collectively the state of the system. A change in conditions implies thus a change of state. Conditions that must be speciﬁed to establish the state of a system are Surroundings called state variables (such as pressure, temperature, FIGURE 3.2 Schematic representation of the volume, number of molecules). An equation of state universe and its constituents: a system and its describes mathematically how state variables are related. A change from one state to another is called a surroundings, separated by a boundary.

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process. A reversible process in thermodynamic terms is brought about by inﬁnitesimal changes of a variable and it means that going along the same path in reverse restores the system as well as the surroundings to its original state. With irreversible processes, permanent changes have occurred in the system and=or surroundings. This terminology may cause some confusion because a reversible reaction in chemical terms refers to a reaction in which products are formed from reactants but reactants can also be formed from products; an irreversible reaction means essentially that reactants are completely converted into products. The difference is thus in the words ‘‘process’’ and ‘‘reaction.’’ A reversible process does not really occur in nature, it is rather an idealized process, a thought experiment, which is nevertheless useful in practice because it helps in determining the limits of real processes. Real, natural processes are irreversible, though some real processes may approximate reversible processes, such as the melting of ice around 08C. An important point is, however, that some thermodynamic functions can only be evaluated when considering reversible processes. We describe ﬁrst classical thermodynamics for reversible processes; at the end of this chapter we will spend some words on irreversible thermodynamics because of its practical importance. Another important consideration is that of ideality; ideal gases and solutions are characterized by the absence of interactions between molecules. They represent limiting cases of real behavior. For instance, ideal solutions do not show a heat or volume effect upon mixing, whereas real solutions do. This is particularly of importance when trying to describe properties of foods in thermodynamic terms. Foods behave by no means as ideal solutions and gases; they are usually inhomogeneous, concentrated, and show many interactions between components. Ideal solutions and gases could be the starting point to derive trends but one should expect complications when applying such concepts to foods. In this book we consider reactions in foods mainly as if they take place in solution, at least as the reference point, acknowledging however, that solid (e.g., crystals) and gaseous phases (e.g., ﬂavor volatiles in headspaces, bubbles in foam) are also important. We will come back to some speciﬁc food complications in Chapter 14. A physical change of state (such as melting, or expansion) implies that the atoms or molecules involved do not change. A chemical change of state implies that the amounts and identities of reactants and products change, in addition to a possible physical change of state. A system is characterized by intensive and extensive parameters. An intensive parameter (or quantity, variable) such as temperature is independent of the amount of substance in the sample and an extensive parameter such as mass, or volume, or energy does depend on the amount of substance. However, the ratio of two extensive parameters, such as density calculated from the ratio of mass and volume, yields an intensive parameter. When chemical reactions occur it is useful to distinguish between species and components. A component can be the sum of several species. For instance, a salt (the component) can consist of several ions (the species). An important concept in thermodynamics is about state and path functions. State functions describe variables that only depend on the difference between one state or another but not on how the change occurs. Path functions on the other hand describe variables that depend on how the changes have occurred. Heat and work are path functions. Energy is a state function. This brings us to the topic of energy.

3.3.2

Energy

The basic thermodynamic property is the internal energy E. We cannot measure this internal energy in absolute terms, but we can measure energy changes. This is an important point to note because it explains the need for a reference point, and that is why standard states are introduced in thermodynamics. We will come back to this shortly. Energy comes in several forms: kinetic energy (i.e., translational, vibrational, or rotational energy of motion of molecules), potential energy (gravitational, chemical, electrical), thermal energy (characterized by incoherent motion of molecules), radiant energy, and even mass is related to energy according to the Einstein equation E ¼ mc2. In terms of chemical reactions, energy mainly refers to potential and kinetic energy in molecules, namely the bond energies of molecules, and translational, rotational, and vibrational molecular motions, respectively. The forms of energy mentioned are all

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interchangeable, and the ﬁrst law of thermodynamics states that energy is conserved throughout. This means that energy cannot be lost. However, the quality of energy, when expressed as its ability to do work, can diminish, as we will discuss shortly. Exchange of energy between system and surroundings can happen via heat (q), or via work (w); work ¼ force 3 distance. There are different kinds of work possible: chemical, mechanical, electrical, and magnetic. A ﬂow of thermal energy is known as heat; thermal energy itself is an energy content. It is important to understand that heat q and work w are modes of energy transfer; they depend on the path the process takes in going from the initial to the ﬁnal state. This path dependency means mathematically that the integral of the differential of work or heat depends not only on the initial and ﬁnal states but also on the path connecting them; therefore the differentials are so-called inexact differentials (indicated by the operator D). In contrast, the integral of an exact differential (indicated by the operator d) is independent of the path between initial and ﬁnal states, and therefore depends only on the initial and ﬁnal states. The result is a state function. However, the sum of these two inexact differentials is an exact differential. So we should write dE ¼ Dq þ Dw

(3:13)

But when the ﬁnal state is reached we can write for the change in energy DE ¼ q þ w

(3:14)

Energy of a system depends only on the current state, not on how it reached that state, in other words, the sum of q and w is always independent of the path between a given initial and a given ﬁnal state. Heat and work relate to processes; the energy content of a system cannot be divided into a heat part and a work part after the process. According to the ﬁrst law of thermodynamics, the following relation should hold: DEsystem ¼ qsystem þ wsystem ¼ DEsurroundings

(3:15)

Expressed in words, if the system gains energy the surroundings will lose the same amount of energy and vice versa, and these energy changes are brought about by heat and work. Once again, this refers to changes in energy content (symbolized by the symbol D), not to the energy content itself. Once the transfer of energy is completed, the contributions of heat and work are no longer distinguishable. One cannot say that a system contains so much heat or so much work; it only contains energy. Temperature is not energy but a measure for the average kinetic energy of atoms or molecules. There is a relation between temperature change (DT), the amount of heat transferred (q), the amount of material that is involved (N), and the type of material (expressed as molar heat capacity at constant pressure CP) DT ¼ Tfinal Tinitial ¼

3.3.3

q NCP

(3:16)

Enthalpy

Most chemical reactions occur at constant pressure, which means that volume changes may occur, so that work is done against the external pressure. For convenience, a new thermodynamic property called enthalpy is deﬁned H E þ PV

(3:17)

The equation that accounts for changes in enthalpy at constant pressure is DH ¼ DE þ PDV

(3:18)

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If the only work done is work against pressure (wsystem ¼ PDV, which is negative by convention because the work is done by the system on the surroundings), the combination of Equation 3.18 with Equation 3.15 shows that DH ¼ qsystem þ wsystem þ PDV ¼ qsystem PDV þ PDV ¼ qsystem

(3:19)

In words, the enthalpy change accounts for the heat ﬂow at constant pressure. Enthalpy is an extensive quantity like energy. Incidentally, volume changes for solids and liquids are usually negligible, so that for these systems DE DH. One can calculate enthalpy changes for chemical reactions using tabulated values for known compounds. Since we are interested in differences rather than absolute values, so-called standard states are used as reference points. Standard states refer to the state of an element, of a gas, a solid or a liquid, or a solvent, or a solute, at a pressure of 1 bar.* It is the convention in thermodynamics to assign zero enthalpy to all chemical elements (not components!) in their most stable state at 1 bar pressure. It should be realized that this is entirely arbitrary and it is just a matter of agreement. What may be confusing is that standard states for components can be deﬁned in various ways, as we shall see later when discussing chemical potentials. For the moment, let us denote a standard state by the superscript ‘‘ 8,’’ so that H8 indicates the value of the enthalpy at the standard state, for instance of one mole of gas at P ¼ 1 bar. Later on we will make a distinction between the various standard states and indicate this also by a different superscript. Next to the standard state, temperature has to be speciﬁed, and usually the temperature chosen is 258C (298.15 K), but again, this is arbitrary (a standard state refers to a concentration or pressure, not to a temperature). The standard reaction enthalpy change DrH8 is thus the change in enthalpy when reactants in their standard states change to products in their standard states (the subscript ‘‘r’’ indicates reaction). Besides reaction enthalpies there are also enthalpies of physical change, for instance, enthalpies of fusion or evaporation. A standard reaction enthalpy covers the overall process from pure unmixed reactants in their standard states to pure separated products in their standard states. A thermochemical equation is a combination of a chemical equation and a standard reaction enthalpy change. Referring to Equation 3.1, a general formula to calculate the standard reaction enthalpy is Dr H ¼ nP HP þ nQ HQ nA HA nB HB

(3:20)

where HA , HB , HP , HQ are the standard enthalpies of A, B, P, Q per mole, respectively, with the unit of J mol1. For instance, the balanced chemical equation for the oxidation of glucose is C6 H12 O6 þ 6O2 ! 6CO2 þ 6H2 O

(3:21)

The difference in enthalpies of reactants and products in their standard state is: 1 Dr H ¼ (6 Hcarbon dioxide þ 6 Hwater ) (1 Hglucose þ 6 Hoxygen ) ¼ 2820 kJ mol

The negative sign indicates that the system releases energy when this reaction takes place. This energy was stored as chemical energy in the glucose molecules (it is in fact an exothermic reaction; we will come back to this shortly). Incidentally, this energy was originally captured by plants from sunlight via photosynthesis, and since energy is conserved it can be regained back from glucose. As indicated, the reference state of an element is its most stable state at the speciﬁed temperature and at 1 bar and is deﬁned as zero by convention. The standard enthalpy of formation of a substance is the standard reaction

* Many authors use a pressure of 1 atm. According to International Union of Pure and Applied Chemistry (IUPAC) recommendations it should be 1 bar.

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enthalpy for the formation of the compound from its elements in their reference states, expressed as enthalpies per mole of compound. Enthalpy changes can be measured as heat exchanges at constant pressure via thermochemical experiments. Also, reaction enthalpies can be calculated from enthalpies of formation DfH8 of reactants and products, which are tabulated for many components Df H ¼

X products

nDf H

X

nDf H

(3:22)

reactants

These tabulated values are based on many accurate thermochemical experiments done in the past. Thus, it is now very easy and convenient to calculate enthalpy changes for almost every reaction. The ﬁrst law of thermodynamics allows one to calculate energy changes when reactions take place, but there are limitations on the ability to convert heat energy into work. Also, the ﬁrst law does not indicate the direction of processes: it applies to any process in which energy is conserved, but there is a direction for processes in real life, as is everyone’s experience. For example, sugar will dissolve easily in hot coffee, but the reverse process that sugar crystallizes all of a sudden in a hot cup of coffee will not happen (or put more correctly, the probability that this will happen is so extremely low that we can safely state that it will never happen). This is where the second law of thermodynamics comes in: there is a certain direction in which reactions go. In other words, there is a reason why sugar does not crystallize in hot coffee and this reason is called entropy S.

3.3.4

Entropy

Entropy is a rather abstract concept. It is based on the Greek words ey and troph (meaning: turning into). It is sometimes referred to as disorder, but this has lead to much confusion, and serious misinterpretation. Even though in some limited cases disorder could be a right term, it is preferable and unambiguous to refer to entropy of a system at constant energy as the number of ways energy can be stored in that system; increase of entropy reﬂects dispersal of energy at some temperature T. So, the entropy of a substance at a temperature T is a measure of the total quantity of energy that had to be dispersed within that substance from T ¼ 0 K to T to exist as a solid, or liquid, or gas at that temperature T. With such a notion of entropy, phase changes and colligative properties (such as freezing point depression, boiling point elevation, and osmosis) can be explained. In addition, it also allows us to understand why some reactions occur and others do not, as will be discussed shortly. The consequence of real-world processes, which are irreversible, is that in every energy transformation the entropy of the universe increases and because of that the potential of the energy available to do work diminishes. So, its ‘‘quality’’ is reduced. Hence, there is a good reason to be careful with the use of energy in society, even though energy cannot be lost according to the ﬁrst law. A suitable measure for the quality of energy is called exergy, and exergy can be lost. We will not discuss this any further here, except to state that exergy calculations are a useful tool for the food industry to design processes as efﬁcient as possible with regard to energy conversion. The second law of thermodynamics states that there is only one direction for irreversible processes, namely those resulting in an increase in entropy S in an isolated system and the entropy is maximal at the equilibrium state. If changes occur spontaneously (i.e., without any work being done on the system) the total entropy of the system and its surroundings (i.e., of the universe) must increase. This does not mean that the entropy of a system cannot decrease; this actually happens during crystallization or condensation, or in a refrigerator, but the requirement remains that the total entropy of the universe must increase. However, the entropy of an isolated system cannot decrease, as will be made clear shortly. The change in entropy determines whether or not a reaction will occur; the energy changes in a reaction have an effect in as much as they increase or decrease the entropy of the system and its surroundings. The dispersal of energy is the ‘‘driving force’’ for physical as well as

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chemical reactions but it is of a probabilistic nature. That means that it is possible for energy to be concentrated temporarily in a system, for instance because there is some barrier that prevent things from happening (otherwise life would not be possible!). This does not conﬂict with the second law of thermodynamics. We will come back to this when discussing kinetics. A frequently used term in relation to entropy is spontaneous process (Table 3.4). A spontaneous process is something that occurs without the input of additional energy from outside the system, but there is a direction indicated by entropy, namely the only direction is the one in which the entropy of the universe increases. Although it is not wrong per se to talk about a spontaneous reaction, the adjective spontaneous does not add much. A nonspontaneous reaction does not occur, so the word is actually superﬂuous. Another possible confusing notion of the word spontaneous is that it should occur immediately, and this is deﬁnitely not the case. A spontaneous reaction such as the oxidation of glucose can take a very long time. In equilibrium thermodynamics, time is not a variable. So far, entropy has been discussed as a qualitative measure, and to make it a quantitative measure, two possibilities exist, in principle at least. The ﬁrst is to count the number of possibilities for energy dispersal via statistical thermodynamics. From molecular thermodynamics and statistical mechanics, it follows that energies of particles and the probability of position in space are coupled inseparably. Such combinations of energy and space are called microstates. A macrostate is the result of many different microstates of individual particles and a macroscopically observable quantity is an average of these microstates (which depend on the conditions of the system); variables such as temperature, volume, number of molecules, measure a macroscopic state. Microstates are thus quantum mechanical descriptions of the ways that molecules can differ in their energy distribution and probable location. Energy dispersion relates to the way in which the energy of particles (i.e., atoms, ions, molecules) are distributed over vibrational, rotational, and translational energy levels. When particles have the ability to access a higher number of energy levels, they can spread out their energy and as a result the entropy is increased. Hence, the entropy of a macrostate (such as a solution) is a measure of the number of ways in which a system can be different in the energetic distribution of the constituting molecules. The number of possible microstates, V, that correspond to a given macrostate is linked to entropy quantitatively via Boltzmann’s relationship S ¼ kB ln V

(3:23)

in which kB is Boltzmann’s constant (1.381 3 1023 J K1), and V the number of possible microstates. V represents the different ways of distributing particle energies over levels leading to the same macrostate. To make this more comprehensible, the following explanation may help. Dispersion of total energy stored in particles is highest when the number of occupied energy levels is as large as possible, while at the same time the distance between the energy levels is as low as possible. Entropy is a measure of the number of occupied energy levels, in other words a measure of the dispersion of energy among accessible microstates. V, the number of microstates, is actually the number of possible permutations of the particles while keeping the number of particles the same in the different energy levels P ni ! V ¼ Qi i ni !

(3:24)

Q ni is the number of particles occupying an energy level i. The operator i represents the continued Q product: i ni ¼ n1 n2 n3 n3 . The lowest energy level is most populated, the highest accessible energy level the least. This is quantitatively expressed by the Boltzmann distribution

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ei e 0 ni ¼ n0 exp kB T

(3:25)

where e0 is the lowest energy level n0 the number of particles occupying energy level e0 An important effect is the distance between energy levels. If this distance increases, quantum leaps will be larger, energy levels less accessible, and consequently entropy will decrease. The decreased accessibility of the higher energy levels limits the number of microstates. An increase in temperature makes higher energy levels accessible to more particles, which explains why entropy increases with T. It should be clear that there is only one kind of entropy change in a system, namely the spreading of energy among a changed number of accessible microstates regardless whether this is due to a change in volume, composition, or temperature. When processes such as phase transitions and chemical reactions take place, this may result in a change in the number of particles occupying a certain energy level, in a changed accessibility of energy levels, and in a change in distance between energy levels. Figure 3.3 illustrates the number of microstates for 10 particles distributed over 3 and 4 energy levels, respectively, while the distance between the energy levels is kept the same. (Incidentally, Figure 3.3 is inaccurate by depicting only energy levels; quantum levels involve energy as well as position in space and the latter are not shown in the ﬁgure.) Using Equation 3.24 it follows that for the situation in Figure 3.3A the number of microstates V ¼ 12,600 while for the situation in Figure 3.3B V ¼ 3,150, which implies that the situation depicted in Figure 3.3A is much more probable than the one in Figure 3.3B, in other words the entropy of the system A is much higher than in B. To be sure, the situation depicted in Figure 3.3B is not impossible, but the probability that it occurs is very low. The example of Figure 3.3 is of course purely hypothetical. For real systems, the number of microstates becomes incredibly high and difﬁcult to comprehend, even when expressed logarithmically as in Equation 3.23. This statistical interpretation of the second law leads to the description of an irreversible process as that process that moves the system toward a state of greater probability in the absence of external forces. In reversible processes, the system remains in a state of maximum probability (or very close to it). A change in entropy between an initial and ﬁnal state is expressed as Vfinal DS ¼ kB ln Vinitial

(3:26)

Entropy change is thus expressed as the spreading of energy among a changed number of accessible microstates. This can happen, for instance, due to a change in volume (increased density of microstates),

e3

n3 = 1

e2

n2 = 2

n2 = 2

e1

n1 = 3

n1 = 4

e0

n0 = 4

n0 = 4

(A)

(B)

FIGURE 3.3 Schematic depiction of 10 particles occupying energy levels. Higher energy levels (>e0) are less accessible than lower energy levels. The situation depicted in Figure 3.3A is more likely than the one in Figure 3.3B (see text).

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composition, or temperature (more energy levels accessible at higher temperature). Boltzmann’s formula applies strictly speaking only to isolated systems, and there are some other restrictions as well that we will not discuss here. It sufﬁces here to think of the statistical interpretation of entropy in terms of microstates that belong to the macrostate the system is in. The second way to look at entropy change is a phenomenological view according to classical thermodynamics, namely by linking the change in energy dispersal to the heat ﬂow between a system and its surroundings. Clausius proposed that for a reversible process a change in entropy is proportional to the heat absorbed and the proportionality constant is 1=T. As remarked before, heat ﬂow is path dependent and we have to work with inexact differentials (indicated by the operator D) dS ¼

Dq T

(3:27)

1=T acts as a so-called integrating factor turning the inexact differential Dq into an exact one for dS. Entropy is thus a state function and we can consider the entropy change DS by looking at the difference in initial and ﬁnal state. Hence, for a heat ﬂow (q) at constant temperature, it follows that the entropy change corresponds to the heat transferred divided by absolute temperature DS

q T

(3:28)

This is the so-called Clausius inequality. The equality sign ( ¼ ) refers to reversible reactions, the ‘‘higher than’’ sign (>) to irreversible reactions. q=T is the entropy change induced in the surroundings by transferred heat. Changes in the entropy function can only be measured along reversible paths. Hence, the entropy change of a reversible process at constant T can be determined from measurements of the heat transferred and the temperature at which this occurs. The entropy change of a system is thus related to the heat transfer by q=T DSsys. For a reversible process q=T ¼ DSsys. For an irreversible process q=T < DSsys and the heat transfer is less than TDSsys. So, the phenomenological point of view for entropy (as opposed to the statistical mechanical interpretation) is that entropy is a measure of energy degraded during irreversible processes. For the universe the following relation holds: DSuniv ¼ DSsys þ DSsur 0

(3:29)

If one sums up all of the entropy changes in an isolated system, the entropy change DS 0. It is zero if there is equilibrium between all of the subsystems and it is positive when an irreversible change has taken place. Thermodynamic processes which develop so slowly as to allow each intermediate step to be an equilibrium state are said to be reversible processes. In a reversible process the system changes such that the system and surroundings can be put back in their original states by exactly reversing the process. These changes are inﬁnitesimally small. If an irreversible process has occurred the entropy change of the universe must have increased. Statistical mechanics deals with elementary steps which are perfectly reversible. Elementary reactions are reversible. But for a large system, irreversibility becomes important. This is nicely illustrated by looking at the dissolution of a sugar cube in a hot drink. The sugar molecules can in principle move in and out of the crystal lattice in a reversible manner, but the dissolution of the sugar cube as a whole is an irreversible process, because that is the most probable distribution for a large system. We conclude this section on entropy by summarizing the second law in Table 3.5.

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3-15

Overview of Entropy Changes dS as a Function of Conditions Reversible Process

3.3.5

Irreversible Process

Isolated system

dS ¼ 0

dS > 0

Nonisolated system

dS ¼ Dq=T

dS > Dq=T

Free Energy

Since the second law reveals the direction of the reaction, namely that which increases the entropy of the universe, we need to be able to calculate this. However, this is an impossible task in terms of the universe. One can get around this problem by deﬁning two other derived thermodynamic quantities called free energy. Free energy can be deﬁned in two ways. At constant volume and temperature we have the so-called Helmholtz free energy F F E TS

(3:30)

At constant pressure and temperature, the so-called Gibbs free energy G is deﬁned as G H TS

(3:31)

We will concentrate here on the Gibbs free energy, but what will follow is valid for the Helmholtz free energy as well. Following Equation 3.31 one can consider the change in Gibbs free energy of a system as follows at constant pressure and temperature DGsystem ¼ DHsystem D(TS)system

(3:32)

Let us now consider a process at constant T so that D(TS)system ¼ (TS)final (TS)initial ¼ TDSsystem

(3:33)

So, at constant T it follows that DGsystem ¼ DHsystem TDSsystem

(3:34)

Now, when also the pressure is constant DHsystem ¼ qsystem (cf. Equation 3.19) and qsystem ¼ qsurroundings (Equation 3.15), it follows that DHsystem ¼ qsurroundings

(3:35)

From Equation 3.28 it follows that for a reversible process qsurroundings ¼ TDSsurroundings. Then it also follows that DHsystem ¼ TDSsurroundings

(3:36)

and now Equation 3.34 can be rewritten, at constant T and P, as DGsystem ¼ TDSsurroundings TDSsystem ¼ T(DSsurroundings þ DSsystem ) ¼ TDSuniverse

(3:37)

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So, the result of this algebraic exercise is that the change of the Gibbs energy at constant temperature and pressure is the change in free energy of the system and it equals the entropy change of the universe multiplied by the temperature. T and DSuniverse can only be positive, hence DGsystem always has to be negative for a process to occur (once again: at constant temperature and pressure). The second law of thermodynamics has been rephrased now in the sense that there is only one direction for processes, namely the direction of decrease in free energy of the system. This is the reason why entropy can never decrease in an isolated system: the entropy of a system can only decrease if energy is put into it and this is not possible for an isolated system by deﬁnition (Table 3.4). Upon increase of entropy, energy is still conserved but something is lost (namely part of the capacity to do work; in other words, useful energy is converted into less useful energy) and this is associated with the increase in entropy (namely TDS). Thus, entropy change is a convenient measure of the loss of capacity of the system to do work. This all means that if we can measure this free energy change, we can state something about the direction of a process. Equation 3.34 is a very important equation because we now have a tool to calculate whether or not a reaction will take place, provided of course that we can calculate free energy changes. DG is a state function. We are interested in changes, and standard states are also deﬁned for free energies, so the molar standard free energy change DGo can be calculated from the standard values of enthalpy and entropy for a particular change DGo ¼ DH o TDSo

(3:38)

Reactions in the nonstandard state. Of course, reactions do not always occur under standard conditions, and if a reaction cannot occur under standard conditions, it may take place under nonstandard conditions (e.g., by changing concentrations and=or temperature). Therefore, one must determine how DG depends on temperature and concentration under nonstandard conditions. This can be done as follows. At higher temperature, energy can be dispersed over more levels, so entropy is clearly temperature dependent. However, temperature affects entropies of reactants and products roughly in the same way, so the effect of temperature on the entropy change DrS due to a reaction is usually small. This means that DrS can be considered independent of temperature. For the same reason, DrH can be considered also independent of temperature. Thus, to account for a temperature different from the one speciﬁed for the standard condition while keeping the concentration at standard condition, this can be calculated as o o Dr GoT Dr HT¼298 K TDr ST¼298 K

(3:39)

The next step is to account for a concentration different from standard conditions. As it happens, DHo depends hardly on concentration, but entropy does: a more dilute sample has a larger entropy because of more accessible translational energy levels. This can be expressed as a function of pressure as S ¼ So R ln

P Po

(3:40a)

in which Po ¼ 1 bar (standard pressure). A similar equation holds for the dependence of entropy on concentration S ¼ So R ln

c co

where c is the molar concentration co is the unit standard concentration (chosen as 1.0 whichever unit is chosen) R is the gas constant

(3:40b)

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Since Po and co have by deﬁnition a numerical value of 1, they are usually omitted; however, it should be realized that c must be expressed in the same units as the chosen standard concentration in order to cancel units. So, for the free energy change under nonstandard conditions Equation 3.34 changes to Dr G ¼ DH o TDr S ¼ DH o T(DSo R ln c) ¼ DH o TDSo þ RT ln c ¼ Dr GoT þ RT ln c

(3:41)

To summarize, if concentrations do not refer to the standard state at the speciﬁed temperature, free energy calculations must be done in two stages. First, one has to correct for the temperature being different from the speciﬁed one (usually 298 K), using Equation 3.39 to obtain Dr GoT , and then that result can be used in Equation 3.41. These equations show clearly that DrG depends on concentration as well as on temperature, and this implies that the direction of a reaction can be manipulated by manipulating temperature and concentration. In this respect, it is instructive to consider both enthalpy and entropy changes. If the reaction enthalpy is negative, implying heat transfer from the system to the surroundings, this is called an exothermic reaction. If the reaction enthalpy is positive, this implies a heat ﬂow from the surroundings to the system, and then it is called an endothermic reaction. An endothermic reaction is only possible if the change in reaction entropy is positive. So, for a reaction as depicted in Equation 3.1 the possibilities are as shown in Table 3.6. An irreversible adiabatic process (no heat exchange between system and surroundings, see Table 3.4) necessarily leads to an increase in entropy of the system in which the process takes place. The tendency of increasing entropy in a given system can be counteracted by putting energy into that system. The total energy of the system plus surroundings remains, of course, constant (ﬁrst law) and the total entropy increases (second law), but the entropy of the system receiving the energy may increase, decrease, or remain constant. If the entropy change of the system is negative, the system must lose heat such that the entropy of the surroundings increases by at least the same amount; in that case the process is necessarily exothermic. If the entropy change of the system is positive the system can absorb heat such that the decrease in entropy of the surroundings matches the increase in the system; the chemical process can then be either exothermic or endothermic. It is thus possible that an endothermic reaction proceeds, if the gain in entropy (dispersal of energy) is sufﬁcient. This gain in entropy in the system is able to overcome the loss of entropy in the surroundings brought about by the inﬂux of heat (or other forms of energy, such as electric energy) from the surroundings into the system. For the sake of completeness we also mention the third law of thermodynamics which states that the entropy of a perfect crystal would be zero at 0 K, where all motion would cease. This gives a reference point to calculate absolute entropy values. Both the Gibbs and Helmholtz energy are deﬁned in terms of the entropy function, implying that they must be evaluated along reversible paths via reversible processes. For most reactions in foods, the TABLE 3.6

Direction of a Reaction A þ B > P þ Q as Dictated by Values of DH, DS, and the Resulting DGa

DH

DS

DG

Negative (exothermic)

Positive

Always negative: reaction from left to right at all temperatures

Negative (exothermic)

Negative

Negative at low temperature: reaction from left to right Positive at high temperature: reaction from right to left

Positive (endothermic)

Positive

Positive at low temperature: reaction from right to left

Positive (endothermic)

Negative

Negative at high temperature: reaction from left to right a

Referring to the reaction going from left to right.

Positive at all temperatures: reaction only from right to left

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differences between using the Gibbs or Helmholtz free energy will be small. We will use mainly Gibbs energy in this book. The Gibbs free energy change in a reversible process at constant temperature and pressure equals the work done exclusive of PdV work. It thus represents the maximum quantity of useful energy from a chemical reaction at constant temperature and pressure in a reversible process, and the Helmholtz free energy that at constant volume; useful energy means interconvertible energy (electrical, chemical, mechanical). If there is excess of free energy, or if the free energy cannot be harnessed as useful energy, the free energy represents the maximum extent of energy dispersal to the universe. Gibbs energy and dissolution. Before considering reactions in more detail, it is also of interest to consider what happens when a compound dissolves in a solvent. For foods, the solvent water is the most important one. The dissolution can be in principle endothermic, exothermic, or a-thermic. It all depends on changes in interactions between solute–solute molecules, solute–solvent molecules, and solvent– solvent molecules. For instance, the dissolution of a crystal depends on the decrease in enthalpy when solvation occurs and the increase in enthalpy needed to disrupt the solvent structure and possibly enthalpy effects of dissociation in the case of electrolytes. Table 3.7 gives a few examples. Table 3.7 shows that the dissolution of NaCl in water is endothermic but with a positive entropy change. The dissolution of CaCl2 on the other hand is exothermic but it leads to a negative entropy change. In both cases the Gibbs energy is negative, indicating that dissolution is possible under these conditions. At temperatures higher than 258C, NaCl would have a greater tendency to dissolve and CaCl2 less, whereas at lower temperatures the opposite is the case. To conclude this section, two remarks must be made. 1. Being state functions, changes in DE, or DH, or DS, or DG, or DF, can be added. However, this can only be done for processes that are independent. This is a requirement that may not always be easy to realize in practice. 2. It should be realized that variables such as enthalpy, entropy, Gibbs, and Helmholtz energies describe huge numbers of particles; these variables do not have any meaning at the level of the individual particles.

3.3.6

Chemical Potential

Every substance has the tendency to change, that is to say, it can . . .

React with other substances Transform into another state of aggregation Migrate to another place

This tendency is described by a single physical quantity, the chemical potential m. Its value depends on a speciﬁc substance; it also depends on T, P, and in solution also on concentration and the kind of solvent, and the state of aggregation. So, when we want to evaluate chemical reactions in thermodynamic terms, the chemical potential of components is a very useful concept. In general, a potential indicates the ability

TABLE 3.7 Examples of Standard Molar Enthalpies, Entropies, and Gibbs Energy of Solution in Water at 258C Compound NaCl CaCl2 CH3OH (methanol) C2H5OH (ethanol)

DsolH8 (kJ mol1) 3.89 81.3 7.28 10.6

DsolS8 (J mol1 K1)

DsolG8 (kJ mol1)

43.1

8.9

44.8

67.9

6.3

9.2

12.1

7.0

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to do something, and a chemical potential indicates the ability for a component to react, or to move to another phase, etc. The chemical potential m of a pure substance is deﬁned at constant temperature and pressure as m¼

@G @n T, P

(3:42)

This shows the chemical potential to be the molar Gibbs energy, i.e., it shows how the Gibbs free energy changes as a function of the change in number of moles at constant temperature and pressure. Using Equations 3.40a and 3.37, it can be shown that the following relation holds P G ¼ Go þ RT ln o P

(3:43)

As remarked before, thermodynamic quantities can be evaluated only relative to different states, so we need a reference, which is called the standard state, at which Go in Equation 3.43 is evaluated. For a perfect gas, this is the state at which Po is the standard pressure of 1 bar. In a perfect gas there are no intermolecular interactions; the pressure is purely the result of kinetic energy of the molecules. It follows then that for a perfect gas P m ¼ m þ RT ln o P o

(3:44)

mo is the chemical potential of the component in its standard state. Moving on now to liquids, imagine a pure liquid compound i that is in equilibrium with its vapor having a vapor pressure P*. The chemical potential of i in the vapor phase is then (with (g) indicating the gaseous state) P* o ¼ m (g) þ RT ln m*(g) i i Po

(3:45)

The superscript ‘‘*’’ indicates that we are dealing with the pure compound. At equilibrium the chemical potential of component i in the vapor phase (g) must be equal to that of the component in the liquid ¼ m*(g), hence Equation 3.45 can be written also as phase (l): m*(l) i i ¼ m*(l) i

moi (g)

P* þ RT ln o P

(3:46)

Imagine now that component i is not present as a pure compound but is one of the components in a binary liquid mixture causing a partial vapor pressure Pi. Equation 3.45 can then be written as Pi mi (g) ¼ moi (g) þ RT ln o P

(3:47)

Again, at equilibrium mi(l) ¼ mi(g), hence Pi mi (l) ¼ moi (g) þ RT ln o P

(3:48)

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By eliminating moi (g) from Equations 3.46 and 3.48 it follows that Pi þ RT ln mi (l) ¼ m*(l) i Pi*

(3:49)

We now have an equation that describes the chemical potential of a component in a liquid mixture as a function of vapor pressures. The next step is to connect this to the composition of liquid mixtures. We can do this via the concept of ideal solutions.

3.3.7

Ideal Solutions

As shown, thermodynamic relations are frequently based upon ideal gases. When discussing equilibria in liquid mixtures, it is necessary to know how the chemical potential of a compound in a solution varies with composition. We can make a link between solutions and gases via Raoult’s law, which is formulated as follows for each component i: Pi ¼ Xi Pi*

(3:50)

Pi is the partial vapor pressure of compound i present in the solution at mole fraction Xi and Pi* is the vapor pressure of pure compound i. If the total vapor pressure of a solution is the sum of the partial pressures related to mole fractions in solution as expressed by Raoult’s law, we speak of an ideal solution. Figure 3.4 shows Raoult’s law graphically. Substituting Equation 3.50 in Equation 3.49 results in the following equation: þ RT ln Xi mi (l) ¼ m*(l) i

(3:51)

This equation describes the dependence of the chemical potential of component i in an ideal solution on its composition expressed as mole fraction, which is what we were looking for.

Pvap P ∗2 P ∗1 P1 =

∗

X1 P ∗ 1

0

P

P2 =X2 2

X2

1

FIGURE 3.4 Illustration of the vapor pressure Pvap above an ideal solution consisting of components 1 and 2 as a function of mole fraction X2. The solid line represents the vapor pressure of the vapor in equilibrium with a liquid mixture of components 1 and 2. The broken lines indicate the partial vapor pressures of components 1 and 2 according to Raoult’s law.

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Even though only very few mixtures behave as ideal solutions, they are the starting points to describe the behavior of real mixtures. An ideal solution does not have the same properties as a perfect gas. In a perfect gas, there are no interactions between molecules. In an ideal solution, there is interaction between molecules, but on average the interaction between the solute–solvent, solvent–solvent, and solute–solute molecules are the same as in the pure liquids. In an ideal solution there are no volume and heat effects upon mixing, i.e., DmixV ¼ 0 and DmixH ¼ 0. For the entropy change DmixS of mixing in an ideal solution the following relation holds: Dmix S ¼ R

X

Xi ln Xi

(3:52)

This shows that the mixing entropy is always positive for an ideal solution. If a solute at mole fraction X2 is dissolved in a liquid with mole fraction X1, then in an ideal solution the following relation holds (known as the Hildebrand equation): R ln X2 ¼ Dfus H2

1 1 T Tfus,2

(3:53)

The subscript ‘‘fus’’ indicates fusion. Likewise, for an ideal solution of a gas in a liquid the following holds: R ln X2 ¼ Dvap H2

1 Tvap,2

1 T

(3:54)

The subscript ‘‘vap’’ indicates evaporation. As shown, these relations are independent of the solvent in an ideal solution.

3.3.8

Ideal Dilute Solutions

There are solutions in which at very low concentrations of the solute (i.e., X1 1, X2 1, whereas 0 < ar < 1, which is more convenient for the present discussion), the reaction Gibbs energy under standard conditions can be expressed as ð Þ ¼ G G ¼ 1 a þ G G þ G þ G þ a G þ G G þ G þ a G r r r r A B P Q A B P Q A B

(3:89)

This can be rewritten as G r ¼ GA þ GB þ ar Dr G

(3:90)

So, when G r is plotted as a function of ar, a straight line with slope Dr G is obtained (Figure 3.15). Of course, it is a hypothetical situation for reactants and products to remain both in their standard state during the reaction. Reactants and products mix due to the very reaction: at ﬁrst there are only reactants but products start to be formed as the reaction progresses. Therefore, reactants and products cannot remain in the standard state. So, we have to study what will happen under nonstandard conditions. An extra contribution to the free energy comes from the mixing effect so that Equation 3.90 becomes G ¼ G A þ GB þ ar Dr G þ Dmix G

(3:91)

As discussed above, for a real mixture the relation holds mi ¼ m*i þ RT ln fi Xi

(3:92)

and for the mixing term Dmix G ¼

X

Xi (mi m*) i ¼

X

Xi RT ln Xi þ

X

Xi RT ln fi

(3:93)

GA + GB

G

∆rG

0

GP + GQ

ar

1

FIGURE 3.15 Free energy of a reaction is depicted as a function of the degree of reaction ar with both reactants and products in the standard state. A, B represent reactants, P, Q represent products.

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3-35

As before, the deviation from ideality is accounted for in the activity coefﬁcient. The ﬁrst term in the right-hand side of Equation 3.93 reﬂects ideal mixing and is always negative while the second term reﬂects nonideal mixing. This second term is called the excess Gibbs energy of mixing. Recall that we are looking for an equation that expresses how G changes with ar. For ideal solutions, there is no volume change when the composition changes and DmixH ¼ 0, so in that case the extra contribution in DmixG is due to entropy effects, as was shown already in Equations 3.40. Equation 3.40 shows that DS contains both a concentration-independent term and a concentration-dependent term; see also Equation 3.52. The concentration-dependent term is called the entropy of mixing. It reﬂects the change in number of accessible microstates upon mixing of two or more chemical species. It is the dispersal of energy of each species within a larger volume due to mixing that affects entropy. Hence, the entropy of mixing is the difference between the entropy of the mixed state and the entropy of the corresponding unmixed state. Expressed in mole fractions Xi the expression for a reaction as depicted in Equation 3.1 taking place in an ideal solution would be (cf. Equation 3.52) Dmix S ¼ NR(XA ln XA þ XB ln XB þ XP ln XP þ XQ ln XQ ) ¼ NR

X

Xi ln Xi

(3:94)

i

N is the total number of moles in the system, and R is the gas constant, as usual. When a reaction starts with reactants A and B, the reaction mixture becomes more mixed when products are formed, and consequently, the entropy of mixing increases. When the reaction would go to completion, i.e., no reactants left anymore, some demixing occurs, leading to a decrease in mixing entropy. The change in entropy of mixing with ar can be derived from Equation 3.94 and when activities are used instead of mole fractions it results in nP nQ aP a d(Dmix S) ¼ R ln nA QnB aA aB dar

(3:95)

DmixS ¼ 0 in the unmixed state and 0 when the reaction progresses. It reaches its maximum value when reactants and products become maximally mixed. Using the relation DG ¼ DH TDS, and remembering that DmixH ¼ 0 for an ideal solution, it follows that nP n Q aP a d(Dmix G) d(TDS) ¼ ¼ RT ln nA QnB ¼ RT ln Qr aA aB dar dar

(3:96)

Thus, the reaction quotient Qr is deﬁned as n

Qr ¼

anPP aQQ anAA anBB

(3:97)

The expression of how G changes with ar can now be found by evaluating Equation 3.90 with respect to ar and substituting Equations 3.96 and 3.97 dG d(ar Dr G ) d(Dmix G) ¼ þ ¼ Dr G þ RT ln Qr dar dar dar

(3:98)

This equation shows the contribution of the entropy of mixing to the change in free energy but now expressed as RT ln Qr. One can thus calculate the change in free energy as a function of the extent of reaction

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Gar Gar ¼0 ¼ ar Dr G þ RT ln Qr

(3:99)

If dG=dar < 0 the reaction will proceed from left to right because that is the direction of free energy decrease, and if dG=dar > 0 for such a reaction it will move in the opposite direction from right to left, again because that decreases free energy. Equilibrium is attained when dG=dar ¼ 0 and then the reaction quotient Qr has become equal to the equilibrium constant Keq n

anPP aQQ anAA anBB

(3:100)

n i mi ¼ 0

(3:101)

Keq ¼ Another expression that describes equilibrium is X

3.3.13

Equilibrium Constants

Recalling Equation 3.1 nA A þ nB B ! nP P þ nQ Q Equations 3.97 and 3.100 are an expression of the famous law of mass action; we will come back to this when discussing chemical kinetics in Chapter 4. Equation 3.100 shows how activities are related to each other when equilibrium is attained. Note that Keq is dimensionless because activities are dimensionless. This is called a thermodynamic equilibrium constant and it is this constant that is reported in tables. In practice, however, it may be more convenient to use constants based on pressures, concentrations, or molalities but it should be realized that these are actually not true constants (except at very low concentrations) because they depend on activity coefﬁcients, and these vary with composition. The following analysis shows how practical equilibrium constants are related to the true thermodynamic equilibrium constant. It is easiest to start with considering the reaction in Equation 3.1 in the gas phase, and consider it as an ideal mixture of perfect gases so that we can use pressures instead of fugacities, and Equation 3.100 can be written as nP nQ PP PQ Dn (PP )nP (PQ )nQ 1 nP þ nQ nA nB 1 Po Po ¼ K Keq ¼ nA nB ¼ P nA nB o PA PB (PA ) (PB ) P Po o o P P

(3:102)

KP is the equilibrium constant in terms of pressure. Since Po ¼ 1 bar, this equation shows that Keq ¼ KP, as it should be, because we have assumed an ideal gas. To change from pressures to concentrations, we can use the ideal gas law that reads P¼

N RT V

(3:103)

The concentration is in fact N=V so that Equation 3.100 changes to Keq ¼

Dn [P]nP [Q]nQ RT Dn RT ¼ Kc o o nA nB [A] [B] P P

(3:104)

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Chemical Thermodynamics in a Nutshell

3-37

This shows that the equilibrium constant in terms of concentration (with a hypothetical standard state of 1 mol dm3) is different from Keq unless Dn ¼ 0. The relation between Kc and KP follows from Equations 3.102 and 3.104 Kc ¼ KP (RT)Dn

(3:105)

Similarly, the relation between KX, the equilibrium constant in terms of mole fraction and KP is KX ¼ KP PDn

(3:106)

(P is the total pressure). Unless Dn ¼ 0, these equations show that the mole fractions or the concentrations in an equilibrium mixture will depend on the pressure P, even though KP does not. As mentioned, thermodynamic equilibrium constants are tabulated for activities, not for concentrations. The expression for Keq as in Equation 3.100 based on activities is universally valid and Keq is a true constant (though depending on temperature) but the difﬁculty is of course in relating activities to measurable quantities. To convert activities to concentrations we need activity coefﬁcients, and as we have seen these depend on the choice of standard state. Practical equilibrium constants such as Kc depend on concentration because activity coefﬁcients depend on concentration. This becomes clear from the following analysis: n

Keq ¼

n

n

anPP aQQ [P]nP [Q]nQ gnPP gQQ yPnP yQQ nA nB ¼ n A n B ¼ Kc n A n B nA nB aA aB [A] [B] gA g B yA yB

(3:107)

Activity coefﬁcients depend on concentration which implies that Kc must also depend on concentration to keep Keq constant. The same is of course true for Km based on molalities. So, Km and Kc are not really true equilibrium constants unless the activity coefﬁcients are equal to one; such a situation may occur in (very) diluted solutions. The standard states need not be the same for all components that are involved in the reaction. However, it is essential that the same standard state is used for the standard chemical potential and for the activity coefﬁcient because these two determine the chemical potential (as is shown in several equations above, e.g., in Table 3.8). The reader is reminded that activities in Equation 3.100 are by deﬁnition equal to 1.0 for pure solids and pure liquids while for gases it is equal to pressure (at a standard pressure Po ¼ 1 bar). Since the activities of pure liquids and solids hardly depend on pressure, we do not need to include activities of pure liquids and solids in calculation of equilibria because they are approximately equal to 1.0 at moderate pressures. For instance, the equilibrium constant for dissociation of water þ H2 O(l) ! H (aq) þ OH (aq)

Keq ¼

aHþ aOH ¼ aHþ aOH a H2 O

(3:108)

with aH2O ¼ 1 and this constant is commonly called the water dissociation constant Kw. For the dissolution of a salt in water, assuming a saturated solution 2þ CaCl2 (s) ! Ca (aq) þ 2Cl (aq)

the equilibrium constant is Keq ¼

aCa2þ a2Cl ¼ aCa2þ a2Cl aCaCl2

(3:109)

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The latter equilibrium constant is usually referred to as the solubility product. For the formation of water from oxygen and hydrogen: 2H2 (g) þ O2 (g) ! H2 O(l) The equilibrium constant is Keq ¼

aH2 O 1 ¼ 2 2 P PH P H2 PO2 2 O2

(3:110)

Perhaps it is useful to come back here brieﬂy to the biochemists’ standard state. To recall, the hydrogen ion concentration for the standard state is deﬁned as 107 M, and as a result the change in standard Gibbs free energy will be different for reactions in which Hþ ions are produced or consumed. So, if we have a reaction: A þ B ! P þ nHþ Hþ Equation 3.98 becomes [P]

dG 0 c ¼ Dr G þ RT ln dar

nHþ

[Hþ ] 107 M [A][B]

c c

(3:111)

Comparing this with the ‘‘normal’’ standard state of c ¼ 1 mol dm3, it follows that Dr G ¼ Dr G þ nHþ RT ln

1 107

(3:112)

0

If, for instance, nHþ ¼ 1, the difference between DrG and DrG is 40 kJ mol1. Similarly, if Hþ ions are taken up in the reaction, the difference is 40.0 kJ mol1 for nHþ ¼ 1. For reactions in which no Hþ is involved, there is no difference in standard free energies. Equilibrium constants and free energy changes. When equilibrium is attained so that dG=dar ¼ 0 and Qr ¼ Keq, it follows from Equation 3.98, that the equilibrium constant of a reaction is linked to the standard Gibbs energy change Dr G ¼ RT ln Keq

(3:113)

This is a very important relationship because it shows that equilibrium constants can be predicted from standard Gibbs energy changes, which are tabulated for many reactions, or can be calculated from standard Gibbs energies of formation (by analogy of Equation 3.20). Table 3.10 shows some numerical examples and gives a feel what the values imply for the relative amount of reactants and products. Incidentally, the fact that DrG appears in Equation 3.113 is due to the relation between the chemical potential and molar Gibbs energy (Equation 3.42). In the case of constant temperature and volume, we should use the Helmholtz energy F, but DrG then still appears in the equations. For instance, Equation 3.98 in the case of constant T, V becomes dF ¼ Dr G þ RT ln Qr dar

(3:114)

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Chemical Thermodynamics in a Nutshell TABLE 3.10

3-39

Relationship between DrG and Keq at 208C

DrG (kJ mol1)

Keq

Composition of the Reaction Mixture

75

2.4 3 10

Only products, amount of reactants negligible

15

472

Mainly products

5

7.8

More products than reactants

0

1

Equal amounts of products and reactants

5

0.13

More reactants than products

15 75

2.1 3 103 4.3 3 1014

Mainly reactants Only reactants, amount of products negligible

13

In other words, Equation 3.113 describes the equilibrium condition for both constant T, P and constant T, V. Distinction between DrG , DrG, and dG=dar. It is essential to appreciate the difference between DrG , DrG, and dG=dar. DrG indicates the difference in free energy between reactants and products in their standard state when they are unmixed. If DrG is positive, it does not mean that a reaction cannot take place. A positive DrG implies an equilibrium constant smaller than 1 (cf. Equation 3.113, and Table 3.10), so the concentration of products may be small but products can be formed nevertheless, despite a positive DrG . It all depends on concentrations of reactants and products whether or not a reaction will take place in a speciﬁed direction, in other words on the value of the reaction quotient Qr. There is a difference between Qr and Keq. Qr can have any value for a reaction, whereas Keq has a speciﬁed, constant value. Qr is a useful parameter to predict the direction of change at a certain composition of the reaction mixture, as indicated in Table 3.11, which shows the connection between Qr, Keq, and dG=dar. Table 3.11 shows the meaning of dG=dar in terms of the direction of a reaction. dG=dar indicates how much the Gibbs free energy changes with the advancement of the reaction. DrG indicates the difference in free energy of ﬁnal and initial states at equilibrium at nonstandard state conditions. As for DrG, if it is positive, the reaction cannot take place, in contrast to when DrG is positive. There is much confusion about this in papers and textbooks, especially when it concerns so-called coupled reactions in biochemistry. Frequently, it is claimed that a reaction under conditions for which DrG < 0 can ‘‘drive’’ a coupled reaction for which DrG > 0. This is simply not true. A reaction showing a positive free energy change just does not occur. However, it is possible that a reaction has a positive standard free energy DrG and if the concentrations of reactants and products are such that dG=dar is negative, then the reaction will occur. What can be said about coupling of reactions is that a reaction having DrG < 0 can drive a reaction having DrG > 0. Another way of making the difference clear between DrG and DrG is that DrG is a constant, while DrG is constantly changing depending on the concentrations, until the reaction composition reaches the point that DrG ¼ 0. A case in point is the formation of glucose-6-phosphate in the body from phosphate and glucose. The standard free energy change DrG for this reaction is þ12.5 kJ mol1. The cellular concentration of glucose is about 0.0001 M, that of phosphate 0.01 M and that of glucose-6phosphate 0.0001 M. The DrG can then be calculated as þ24 kJ mol1. This reaction is therefore not possible in the way that it is written. As mentioned, it is frequently suggested that this positive free energy can be overcome by thermodynamic coupling to the hydrolysis of ATP, which has a negative DrG ¼ 50.4 kJ mol1 at cellular conditions. The overall reaction is depicted as:

TABLE 3.11

Relation between Qr, Keq, and dG=dar for a Reaction A þ B > P þ Q

Reaction Goes to the Right When

There Is Equilibrium When

Reaction Goes to the Left When

Qr< Keq

Qr ¼ Keq

Qr > Keq

dG=dar < 0

dG=dar ¼ 0

dG=dar > 0

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3-40 TABLE 3.12

Data for Mutarotation of Glucose in the Standard State at 258C

Reactant

DHu kJ mol1

a-glucose

1.15

b-glucose

1.15

DSu J mol1 K1

DGu kJ mol1

Keq

0.92

1.425

1.77

0.92

1.425

1.77

Dr G ¼ þ24 kJ mol1 Glucose þ phosphate ! glucose-6-phosphate ! Dr G ¼ 50:4 kJ mol1 ATP ADP þ Pi 1 ! Glucose þ ATP glucose-6-phosphate þ ADP Dr G ¼ 26:4 kJ mol Such a presentation may lead to a wrong conclusion because the ﬁrst reaction cannot take place under any condition! The error made here is that the reactions shown are overall reactions, but they do not take place at the molecular level as they are written down, in other words they may not be added up like this. What really happens is that glucose is phosphorylated via an indirect reaction path in several steps, and for each step the free energy change DrG is necessarily negative; incidentally, these steps take place enzymatically in living organisms but that is not essential for this discussion because also enzymes cannot overcome positive free energy changes! A further example may be helpful here. It concerns the mutarotation of glucose at pH 7 and 258C a-glucose ! b-glucose

(3:115)

Table 3.12 gives some data needed for the calculation. With these data we can calculate the change in G as a function of ar. It is instructive to plot the contribution of the mixing and the nonmixing contribution to the free energy plot as a function of the degree of reaction (Figure 3.16).

0 Nonmixing contribution

Gαr −Gαr=0/kJ mol–1

−0.5

−1

∆rG

Mixing contribution −1.5

∆rG −2 Total

−2.5

Equilibrium position −3 0

0.2

0.4

0.6

0.8

1

ar

FIGURE 3.16 Change in free energy for the mutarotation of glucose at 258C as a function of the degree of reaction ar.

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3-41

The equilibrium is at ar ¼ 0.64 in this example, where a minimum in free energy is reached. The minimum in the mixing contribution, at ar ¼ 0.5 in this example, does not coincide with the equilibrium position, which makes sense because the equilibrium not only depends on DmixS but also on enthalpy and temperature. The minimum in the mixing contribution is always reached at Qr ¼ 1, as follows from Equation 3.96. At this position the afﬁnity of the reaction, dG=dar, which is the slope of the graph of Gar Gar ¼ 0 versus ar, equals DrG , see Equation 3.98. Figure 3.16 also shows what would happen if the standard free energy change is positive. This would be the case if the reaction is started with b-glucose as reactant: then DrG ¼ þ1.425 kJ mol1 (cf. Table 3.12). Even though this is a positive standard free energy change, the reaction will nevertheless take place and a-glucose will be formed from b-glucose up to ar ¼ 0.36 because DrG is negative in going from pure b-glucose to this equilibrium. One may now wonder whether reactions can go ever to completion to the point that ar ¼ 1, when it takes place in solution and when reactants or products are not removed. The answer is no, not completely, but the equilibrium position can lie very far to the right or to the left. In that case, the nonmixing contribution to the free energy is largely determining the free energy change. It is perhaps instructive to show this in a graph (Figure 3.17). This is for the same reaction as in Figure 3.16 but now the standard free energy has been changed, for the sake of argument, from 1.425 to 14.25 kJ mol1, leaving DrS ¼ 0.92 J mol1 K1 so that DrH ¼ 14,249.08 J mol1. The equilibrium would then be found at ar ¼ 0.997 mol, hence the reaction would have gone almost to completion but there is still a tiny contribution from the mixing entropy that prevents it from reaching total completion. Calculation of equilibrium concentrations. With knowledge of standard Gibbs energies, or equivalently, equilibrium constants, equilibrium concentrations can be calculated. This is most easily done using the degree of reaction, ar. An example will show this. It concerns the dissociation of benzoic acid, a food preservative, in pure water. The equilibrium constant at 258C is 6.4 3 105. We can set up the following table (Table 3.13). The following relation holds at equilibrium: Kc ¼

ar c0 ar c0 a2r c0 ¼ 6:4 105 ¼ (1 ar )c0 (1 ar )

(3:116)

0 −2

Mixing contribution

Gαr - Gαr= 0/kJ mol–1

−4

Nonmixing contribution

−6

∆rG

−8 ∆rG

−10 Total

−12 −14

Equilibrium position

−16 0

0.2

0.4

ar

0.6

0.8

1

FIGURE 3.17 Change in free energy when the nonmixing contribution would be 14.25 kJ mol1 rather than 1.425 kJ mol1 as in Figure 3.15.

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TABLE 3.13 Calculation of Concentration as a Function of ar for the Dissociation of Benzoic Acid in Water at 258C Degree of Reaction

C6H5COOH

C6H5COO

Hþ

ar ¼ 0

c0

0

0 (10

ar ¼ 1

0

c0

c0

0 < ar < 1

(1 ar)c0

arc0

arc0

7

in water)

This can be rearranged to a2r c0 þ 6:4 105 ar 6:4 105 ¼ 0

(3:117)

TABLE 3.14 Degree of Reaction at Equilibrium for Various Initial Concentrations for the Dissociation of Benzoic Acid c0 (mol L1)

ar

0.01

0.077

0.1

0.025

1

0.008

Such an equation can be solved following the algebraic rules to derive the roots for a square root equation ar ¼

6:4 105

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (6:4 105 )2 4c0 6:4 105 2c0

(3:118)

This shows that the degree of reaction at equilibrium depends on the initial concentration in this case. Table 3.14 shows this dependence of ar for some values of c0. It is interesting to note that the degree of reaction at equilibrium does not depend on the initial concentration if the number of moles of reactants and products is the same (Sni ¼ 0); this would be the case for mutarotation of glucose, for instance. Perhaps, it is helpful to list some more terms that are used in addition to Table 3.4. Especially in biochemistry textbooks, the terms exergonic and endergonic are used. This means a negative and a positive standard free energy change DrG (but not a negative or positive DrG as is frequently suggested). The terms exergonic and endergonic should be distinguished from exothermic and endothermic; the latter two terms pertain to enthalpy changes at constant pressure (and no work other than by volume changes). Table 3.15 lists the possibilities, including the meaning of enthalpy- and entropy-driven reactions. In conclusion, many reactions do not go to completion in the sense that ar ¼ 1 because the entropy of mixtures of reactants and products is greater than that of separate products or reactants. However, the magnitude of this effect strongly depends on how exothermic the reaction is. A highly exothermic reaction causes a much greater entropic effect to the surroundings than the mixing entropy effect. It is stressed once again that the equations used are for ideal solutions. For real solutions, one also has to take enthalpy and volume changes into account. The trends will be the same but the magnitude of the effect may be different.

3.3.14

Thermodynamic Potentials and Conjugate Variables

Here, we brieﬂy recapitulate the meaning of thermodynamic potentials, which are the quantitative measures of the energy contained within a system. This section forms the bridge between sections on

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Chemical Thermodynamics in a Nutshell TABLE 3.15

3-43

List of Terms Used in Thermodynamics of Chemical Reactions

Term

Symbol

Meaning

Exothermic

DH < 0

Release of energy by the system to the surroundings

Endothermic

DH > 0

Uptake of energy by the system from the surroundings

Exergonic

DrG < 0

Negative free energy difference between reactants and products in their standard state; Keq > 1

Endergonic

DrG > 0

Positive free energy difference between reactants and products in their standard state; Keq < 1

Enthalpy-driven reaction

If jDHj >> jTDSj

Enthalpic effect is much stronger than entropic effect in determining the magnitude of DG

Entropy-driven reaction

If jDHj 0 and dS 0 it follows that Af dj 0. This is called the de Donder inequality. The equation for Gibbs free energy for a system with changing composition is (already shown in Equation 3.67) dG ¼ SdT þ VdP þ

X i

mi dni

(3:130)

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For a system in which chemical reactions occur, it can be changed into, analogous to Equation 3.128 dG ¼ SdT þ VdP þ

X

mi dni 00 Af dj

(3:131)

i

At constant temperature and pressure it follows that for a closed system (dG)T, P ¼ Af dj

(3:132)

Or with k multiple reactions (dG)T, P ¼

X

Af ,k djk

(3:133)

k

Similarly for the Helmholtz free energy dF ¼ SdT PdV þ

X

mi dni 00 Af dj

(3:134)

i

(dF)T,V ¼ Af dj

(3:135)

The afﬁnity is a state variable and the advantage in using it is that the conditions under which a reaction is taking place do not need to be speciﬁed like with Gibbs free energy (constant T,P) or Helmholtz free energy (constant T,V). A positive afﬁnity indicates that the reaction takes place in the direction of j (forward reaction), a negative afﬁnity points at the reverse reaction. If Af ¼ 0, the system is at equilibrium. This is shown schematically in Figure 3.18; note that the slope is negative, i.e., dAf=dj is always negative, regardless of which variables are held constant. Some useful relations for the chemical afﬁnity are @G @F ¼ Af ¼ @j T,P,{ni 00 } @j T,V,{ni 00 } @E @H ¼ ¼ @j S,V,{ni 00 } @j S,P,{ni 00 } @S ¼ @j E,n,{ni 00 }

Af +

0

−

xeq

x

FIGURE 3.18 Relation between afﬁnity Af and extent of reaction j. The equilibrium position is indicated at jeq and Af ¼ 0.

(3:136)

Conjugate pairs. Energy is the ability to do work. In mechanics, energy transfer results from a force Fi that causes a displacement ui, with Fi 3 ui being the amount of energy involved. Similarly, changes in thermodynamic systems can be thought of as the result of a generalized force causing a generalized displacement, with the product of the two being the amount of energy transferred. These thermodynamic force–displacement pairs are known as conjugate pairs or variables. The thermodynamic force is always an intensive variable, and the

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3-46 TABLE 3.16 Conjugate Pair

Overview of Some Common Conjugate Pairs Force F (Intensive Variable)

Displacement u (Extensive Variable) Volume V

PdV

Pressure P

TdS

Temperature T

Entropy S

mdn

Chemical potential m

Number of particles n

Afdj

Afﬁnity

Advancement of reaction j

MdH

Magnetization M

Magnetic ﬁeld H

gdA Efde

Surface tension g Electromotive force Ef

Surface area A Amount of charge e

displacement is always an extensive variable, and the energy transfer is always extensive. The above equation for energy (as well as those for enthalpy, Gibbs, and Helmholtz energy) can thus in general be displayed as a product of force and displacement dE ¼

X

Fi dui

(3:137)

i

The product Fidui represents conjugate variables or pairs, e.g., TdS, PdV, midni, etc. The above equations show that, next to TdS and PdV, mdn as well as Afdj can be considered as conjugate pairs. These conjugate pairs all represent the ﬂow of energy across the boundary of the system. Table 3.16 gives an overview of conjugate pairs discussed so far, along with some other ones, less important perhaps for the topic of this book, but shown anyway to illustrate the general application of this concept. The magnitude of an intensive variable determines the magnitude of the resulting energy change. For instance, a change in entropy has a much higher effect on the energy change at a high temperature than at a low temperature. The sum Fidui in the left-hand side of Equation 3.137 indicates that there are several ways in which work can be done on or by the system. Change of any of the conjugate variables Fi or ui may affect equilibrium activities of reactants and products and thereby equilibria will shift. One can choose a forcing parameter Fi for perturbation to which the equilibrium is sensitive. The relaxation time is generally shorter if during equilibration the extensive variables ui instead of the intensive variables are held constant. However, experimentally it is easier to vary the intensive variables Fi. A remark should be made about the meaning of the words ‘‘force and displacement’’ in relation to chemical reactions and diffusion. Of course, molecules do not feel a force because of a difference in chemical potential; they just move about randomly and when there are spatial concentration differences this will eventually disappear because of this random motion, not because of a force. Similarly, a chemical reaction is basically a stochastic process and molecules do not feel a force that pulls them toward a chemical reaction in the mechanical sense. So, it should be understood that thermodynamic forces and displacements are just convenient ways to state quantitatively what will happen; they should not be taken literally in the case of diffusion and chemical reaction. Analogies have their limitations! In the case where gravity is important, particles do ‘‘feel’’ a real force (for instance, cocoa particles in a chocolate drink) and in the case of electric potentials charged particles also experience a force (ions, proteins, and charged colloidal particles such as casein micelles). We consider some of these cases in Chapter 11. Le Châtelier’s principle. The well-known Le Châtelier’s principle can be formulated in various ways. A popular one is if a chemical system in equilibrium is subjected to a perturbation, the system will shift to a new equilibrium such that the perturbation is partially undone. This principle is often used as a rule of thumb to make qualitative predictions. Unfortunately, it is not a very rigorous principle and is in principle only valid for a perfect gas. It is actually much better to work with de Donder’s afﬁnity Af. If a system is in equilibrium, Af ¼ 0. It implies that one of the thermodynamic functions is at an extremum (i.e., at a maximum or a minimum). In the case of an isolated system the entropy is maximal at

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3-47

Af +

0 T2 < T0 T0 T1 > T0 x1

FIGURE 3.19

x0

x2

x

Af versus j for different temperatures in the case of an exothermic reaction. See text for explanation.

equilibrium, and the Gibbs free energy is minimal at constant temperature and pressure, etc. A useful rule that replaces Le Châtelier’s principle is the following. If in a system at equilibrium a particular extensive variable is changed, the equilibrium will shift in the direction that tends to reduce the change in the corresponding intensive variable. Similarly, if in a system at equilibrium a particular intensive variable is changed, the equilibrium will shift in the direction that tends to increase the change in the corresponding extensive variable. It is possible to make quantitative predictions on how an equilibrium will shift upon perturbation by deriving equations of jeq as a function of the perturbed variable and of Af versus j for various values of the perturbing parameter. In that way, it can be derived, for instance, how a reaction at equilibrium will respond to a temperature change: see Figure 3.19 where the situation is depicted for an exothermic reaction. When an exothermic reaction originally at equilibrium at T0 and j0 is perturbed by a sudden temperature increase (T1 > T0), Af becomes negative and the reaction will move in the reverse direction toward a new equilibrium (j1 < j0). The opposite will happen when the temperature is reduced (T2 < T0): j2 > j0. Similarly, it can be derived quantitatively how equilibria will respond to pressure changes and to addition or removal of species. For details on this approach, we refer the interested reader to literature references given at the end of this chapter. Limits of equilibrium thermodynamics. What are the limits to which equilibrium thermodynamics may be used to treat nonequilibrium transformations in real life? In the equilibrium state there are no unbalanced potentials=driving forces. The focus is on changes that begin and end at equilibrium states and the system is considered uniform throughout, deﬁned macroscopically by quantities such as T, P, V. To consider transformations between different equilibrium states, additional constraints are imposed, or removed. We can, for instance, change the constraints temperature and pressure, or add a chemical to a system. There are speciﬁc expressions that relate the effect of temperature and pressure to the equilibrium constant. This is discussed in more detail in Chapter 5. When a constraint is removed from or added to a system that is at equilibrium, the system will then relax to a new equilibrium state. Reversing such a process would require performing work on the system to reimpose the removed constraint. Work done on a system irreversibly leaves the universe in a higher entropy state. The usefulness of reversible thermodynamics is that it gives information for open systems whether or not a process can occur. Diagrams such as Figures 3.15 and 3.16 allow one to predict the value of j or ar at a particular composition for the reaction, and hence to predict the direction of the reaction, but not the rate at which this occurs. In other words, we do not know anything about dj=dt. Time is not a variable in

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thermodynamics of reversible processes as equilibrium is a time-independent state. Equilibrium positions will change as a function of temperature and pressure. Reversible thermodynamics is about equilibria; the equations are invariant with respect to time. However, entropy has a time element in it: for real processes, entropy increases, so it is in fact a function of time. It is impossible to bring a system subject to an irreversible process backwards in time to its original state without changing anything. The time dependence of entropy generation is made explicit in irreversible thermodynamics.

3.3.15

Nonequilibrium or Irreversible Thermodynamics

A stated repeatedly, real processes are irreversible, proceeding toward an equilibrium state, dissipating the driving force. As a result, such systems are nonhomogeneous: at least some of the intensive parameters are functions of time and position. As mentioned in Section 3.3.14, entropy production is an essential feature of irreversible processes, so we need to take a look at this phenomenon in somewhat more detail. Entropy generation. It is, of course, always possible to bring a system back to its initial state after a process, whether it is reversible or irreversible, but in the case of a reversible process no trace is left whereas for irreversible processes the surroundings do usually some work on the system and will not return to their original state. Obviously, reversible processes do not occur in reality, rather they are idealizations of the real processes. Reversible processes can be seen as theoretical limits for the corresponding irreversible processes. If reversible processes would be possible, they would deliver the most work in the case of work-producing devices (think of engines and turbines) and consume the least work for work-consuming devices (such as pumps, fans, compressors). What makes real processes irreversible? This is due to such factors as friction, mixing of ﬂuids, and heat transfer over ﬁnite temperature differences, electrical resistance, inelastic deformation, and also chemical reactions; these factors are sometimes called irreversibilities. The magnitude of generated entropy depends on the process, it is not a property of the system. As a reminder, the ﬁrst law makes no difference between heat transfer and work; they are considered as equals. The second law, however, does make a distinction: an energy interaction accompanied by entropy transfer is heat transfer, and an energy interaction not accompanied by entropy transfer is work. Mass contains entropy as well as energy and so, energy and entropy contents of a system are proportional to mass. These considerations are very general, and they do of course also apply to the speciﬁc branch of chemical thermodynamics that deals with systems whose chemical composition changes during a process, which is why we are interested in it in the framework of this book. If in the course of a reaction free energy is released, this is either lost as entropy or used as work. In living organisms, for instance, this free energy is used to perform work, e.g., by transport of ions, muscle contraction, etc. However, irreversibilities do occur and not all free energy will be harnessed into useful work. The entropy of a system can change because of two reasons: an internal entropy change diS from within the system (e.g., due to a chemical reaction, or a phase change, etc.) and an external entropy change across the boundary from the surroundings to the system (or vice versa) deS due to heat transfer: deS ¼ Dq=T, hence dSsys ¼ di S þ de S

(3:138)

We can also write for the entropy change of the system dSsys ¼ dSuniv dSsur

(3:139)

By combining the previous two equations we ﬁnd di S þ de S ¼ dSuniv dSsur

(3:140)

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The external entropy change for the system must mirror that of the surroundings de S ¼ dSsur

(3:141)

di S dSsur ¼ dSuniv dSsur

(3:142)

di S ¼ dSuniv

(3:143)

Hence, Equation 3.140 can be written as

We arrive at the ﬁnding that

In other words, the internal entropy change in the system equals the entropy change of the universe. For a reversible process, of course, diS ¼ 0 but for an irreversible process diS > 0. In other words, entropy generation is a measure of the magnitudes of the irreversibilities present during that process. Entropy transfer from surroundings to system and vice versa can happen by two mechanisms: heat transfer and mass ﬂow. In contrast, there is no entropy transfer associated with energy transfer due to work. Figure 3.20 shows all this schematically; it is in fact a more detailed picture of Figure 3.2. Entropy production is thus as a measure for dissipation taking place in a process: how much of the energy is used for work, how much is dissipated to the environment. An irreversible process occurs in response to a thermodynamic force (driving force) which results from the system’s nonequilibrium state. Hence, in a nonequilibrium situation something must ﬂow, i.e., change its spatial distribution. Fluxes tell something about the speed of the process. As we have seen in Section 3.3.14, generalized forces could be differences in chemical potential, difference in temperature, difference in electrical potential, etc., and these are all very relevant for irreversible processes. Examples of generalized ﬂuxes are chemical reactions moving toward equilibrium, heat ﬂow, diffusion, electrical current, etc. The rate of entropy production < (J K1 s1) is deﬁned as > k2, Equation 4.94 reduces effectively to a consecutive irreversible ﬁrst- and second-order reaction and it follows from Equation 4.94 that kobs k1

(4:95)

When however k3[C] k2, it follows from Equation 4.93 that d[P] k1 k3 [A] [C] dt k2

(4:96)

k1 k3 ¼ Kc k3 k2

(4:97)

and kobs

with Kc being the equilibrium constant for the reversible reaction A ! B. If, again, [C] is made large, and thereby effectively constant, the reaction becomes ﬁrst order in [A]. The observed rate constant is in this case seen to be determined by all three elementary rate constants, even though the actual ratelimiting step is the second reaction to form P. Only when the ﬁrst of several sequential steps is rate-controlling, is the overall rate determined by the smallest elementary rate constant. Otherwise, rate constants of all steps prior to and including the rate-limiting one are included in the observed rate constant. The dependence of the overall rate on [C] also shows that the rate-controlling step may depend on the concentration of reactants. As stated above, steady-state approximations are not necessary anymore. It is also possible to ﬁnd out whether or not a particular step is rate-controlling via computer simulation using the parametric sensitivity function as a quantitative measure. The parametric sensitivity function is the partial derivative of the overall rate r with respect to a rate constant ð@[email protected] Þkj . The parameter that has the highest sensitivity can be identiﬁed as the rate-controlling step. In a simple, irreversible consecutive reaction, the rate-controlling step is, as intuitively expected, the one with the smallest rate constant: this step will determine the overall rate. So, for instance for the following consecutive reaction, where the overall rate refers to the formation of end product D:

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18

Parametric sensitivity for [D]

Concentration (arbitrary units)

1 [A]

0.8

[B] 0.6

0.4

[D]

0.2 [C] 60

20 40 Time (arbitrary units)

(A)

d[D]/dk2

14 12 10 8 6 4

d[D]/dk3

d[D]/dk1

2 0 0.00

0 0

16

20.00

(B)

40.00

60.00

Time (arbitrary units)

FIGURE 4.20 Graph of the reaction shown in Equation 4.98 for k1 ¼ 0.1, k2 ¼ 0.02, k3 ¼ 0.5. [A]0 ¼ 1, [B]0 ¼ [C]0 ¼ [D]0 ¼ 0 (arbitrary units) (A) and the respective parametric sensitivity functions (B). k1

k3

k2

A ! B ! C ! D

(4:98)

the course of the reaction is simulated in Figure 4.20A for k1 ¼ 0.1, k2 ¼ 0.02, k3 ¼ 0.5, [A]0 ¼ 1, [B]0 ¼ [C]0 ¼ 0 (arbitrary units). The sensitivity functions are shown in Figure 4.20B and indeed k2, having the lowest value, shows the highest parametric sensitivity for the formation of end product D. However, when the rate constants are of comparable magnitude, a single rate-controlling step cannot be identiﬁed as shown in Figure 4.21A and B, for k1 ¼ 0.05, k2 ¼ 0.06, k3 ¼ 0.07. For slightly more complex reactions, the observed rate is not always solely determined by the smallest elementary rate constant. For instance, if a reversible step is introduced in a consecutive reaction: k3

k1

A ! B ! C k

(4:99)

1

6

0.8

5 [A]

Parametric sensitivity

Concentration (arbitrary units)

2

[D] 0.6 0.4

[B] [C]

0.2

d[A]/dk2

4 3

d[A]/d k3

2 1

0

0 0

(A)

d[A]/d k1

20

40

60

Time (arbitrary units)

80

100

0 (B)

20

40 60 Time (arbitrary units)

80

100

FIGURE 4.21 Graph of the reaction shown in Equation 4.98 for k1 ¼ 0.05, k2 ¼ 0.06, k3 ¼ 0.07, [A]0 ¼ 1, [B]0 ¼ [C]0 ¼ [D]0 ¼ 0 (arbitrary units) (A) and the respective parametric sensitivity functions (B).

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1

8

0.8

6

Parametric sensitivity for (C)

Concentration (arbitrary units)

d[C]/d k1

[A]

0.6

[C]

0.4

0.2 [B]

(A)

20

d[C]/dk3

2 0

0

20

40

60

80

100

−2 d[C]/dk2 −4

0 0

4

40

60

80

100

Time (arbitrary units)

(B)

Time (arbitrary units)

FIGURE 4.22 Graph of the reaction shown in Equation 4.88 for k1 ¼ 0.05, k2 ¼ 0.01, k3 ¼ 0.1, [A]0 ¼ 1, [B]0 ¼ [C]0 ¼ 0 (arbitrary units) (A) and the respective parametric sensitivity functions (B).

it now depends on the relative magnitude of each rate constant which step becomes rate determining. For the case of k1 ¼ 0.05, k2 ¼ 0.01, k3 ¼ 0.1 for instance, the parametric sensitivity is highest for parameter k1, even though k2 is smaller, and the formation of C is also sensitive to the other parameters (see Figure 4.22). In other words, there does not appear to be a clear rate-controlling step. Just another example is the case of reversible parallel reactions, discussed in Equation 4.29, and Figure 4.4B. Here it can be seen that the rate-controlling step can change during the course of a reaction (Figure 4.23): at ﬁrst the decrease in [A] depends strongly on k2, later on much more on k4. In conclusion, this analysis shows that the concept of a rate-controlling step is not an easy one, and, most importantly, cannot be predicted beforehand. The availability of modern simulation software and the possibility to do parametric sensitivity analysis seems to be helpful.

5

d[A]/d k4

Parametric sensitivity

4

3

2 d[A]/dk1,3 1 d[A]/dk2 0

0

1000

2000 3000 Time (arbitrary units)

4000

FIGURE 4.23 Parametric sensitivity for the reaction depicted in Equation 4.29 with k1 ¼ 1, k2 ¼ 0.01, k3 ¼ 0.1, and k4 ¼ 0.0005, [A]0 ¼ 1, [B]0 ¼ [C]0 ¼ 0 (arbitrary units).

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Catalysis

The IUPAC description of the process of catalysis is as follows: ‘‘A catalyst is a substance that increases the rate of a reaction without modifying the overall standard Gibbs energy change of the reaction. The process is called catalysis and a reaction in which a catalyst is involved is known as a catalyzed reaction.’’ Hence, the position of the equilibrium is not affected by the catalyst. The mechanism by which a catalyst operates is to effectively reduce the activation free energy for a reaction (Chapter 5) by providing an alternative pathway to the ‘‘normal,’’ uncatalyzed one. Consequently, the reaction proceeds much faster at the same temperature. A catalyst does not undergo a chemical change itself, or if it does, it can reversibly return to its original state, and, as already mentioned, it does not change the position of the equilibrium (DG remains the same). One can distinguish between homogeneous catalysis (in which the reactants and the catalyst are in the same phase, it concerns protons, ions, atoms, molecules) and heterogeneous catalysis (in which the reactants and the catalyst are in different phases, it concerns solid surfaces, interfaces of emulsion droplets, membranes). Both types of catalysis can occur in foods. Very important catalysts are of course enzymes. The behavior of enzymes can be so different from ‘‘normal’’ catalysts that they are discussed in a separate chapter (Chapter 9); their action is very important for food quality.

4.4.1

General Catalysis

Suppose we have a monomolecular uncatalyzed reaction k1

A ! B

(4:100)

which can also be catalyzed by component C to give the same product via an alternative, catalyzed route: kc

A þ C ! B þ C

(4:101)

in which kc is the second-order catalytic constant. The rate of disappearance of A is now: d[A] ¼ k1 [A] kc [A] [C] ¼ (k1 kc [C]) [A] ¼ kobs [A] dt

(4:102)

with kobs the observed rate constant, which is constant if [C] is constant. It thus means that the catalyzed and uncatalyzed reactions take place in the proportion kc[C]=k1 and that the relative contributions of both routes can be changed by changing [C]. A completely different type is heterogeneous catalysis where the transformation takes place at surfaces. This is very common in the chemical industry, but for foods this may be less relevant because catalysts are not normally added (except of course enzymes). One exception is the hardening of fats in which hydrogen is added to unsaturated fatty acids with the aid of a solid catalyst. It can also happen with reactions at, for instance, precipitated milk salts in heat exchangers, in which case, for instance, the isomerization of lactose is promoted. With heterogeneous catalysis several processes take place: transport of reactant molecules to the surface, adsorption of reactant molecules at the surface, reaction at the surface, desorption of products from the surface, and transport of the products away from the surface. Models that describe such catalysis are mostly of an empirical nature, such as the Langmuir–Hinshelwood–Hougen–Watson (LHHW) relation for a reaction in the gas phase A þ B ! C catalyzed at the surface of a catalyst: r¼

kKA PA KB PB 1 þ KA PA þ KB PB þ KC PC

(4:103)

with PA,B,C the partial pressures of components A, B, C, and KA,B,C their adsorption coefﬁcients.

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Acid–Base Catalysis

Many reactions are catalyzed by acids and=or bases. The classical example is the hydrolysis of esters. Acid and base catalysis can take place in two ways, namely speciﬁc acid (base) catalysis and general acid (base) catalysis. The acid (proton donor) and bases (proton acceptor) are characterized as: þ HA þ H2 O ! H3 O þ A

(4:104)

A þ H2 O ! HA þ OH

(4:105)

The practical equilibrium constants for ionization are respectively: KHA ¼

[H3 Oþ ] [A ] [HA] [H2 O]

(4:106)

KA ¼

[HA] [OH ] [A ] [H2 O]

(4:107)

Taking the concentration of water as essentially constant (in fact, we may take the water activity to be unity), we can deﬁne an apparent equilibrium constant as follows: Ka ¼ KHA [H2 O] ¼

[H3 Oþ ] [A ] [HA]

(4:108)

Kb ¼ KA [H2 O] ¼

[HA] [OH ] [A ]

(4:109)

These two equilibrium constants are linked via the ionization constant of water Kw: Kw ¼ Ka Kb ¼ [H3 Oþ ] [OH ]

(4:110)

Now, suppose we have a very simple reaction of a reactant turning into a product (R ! P) and we want to consider what happens if this reaction is catalyzed by acids and bases. The uncatalyzed reaction is k0

R ! P

(4:111)

The acid-catalyzed reaction is kHþ

R þ H3 Oþ ! P þ H3 Oþ

(4:112)

The base-catalyzed reaction is kOH

R þ OH ! P þ OH

(4:113)

If the reaction is also catalyzed by the undissociated acid, we have: kHA

R þ HA ! P þ HA

(4:114)

and likewise for the reaction with the conjugate base: kA

R þ A ! P þ A

(4:115)

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Based on the above equations, the following rate equations can be derived: r0 ¼ k0 [R]

(4:116)

rHþ ¼ kHþ [H3 Oþ ] [R]

(4:117)

rOH ¼ kOH [OH ] [R]

(4:118)

rHA ¼ kHA [HA] [R]

(4:119)

rA ¼ kA [A ] [R]

(4:120)

The overall rate of the reaction R ! P is then the sum of all these contributions because these reactions run in parallel, cf. Equation 4.87: r ¼ r0 þ rHþ þ rOH þ rHA þ rA ¼ (k0 þ kHþ [H3 Oþ ] þ kOH [OH ] þ kHA [HA] þ kA [A ]) [R] ¼ kc [R]

(4:121)

The catalytic rate constant kc is deﬁned in the above equation as the sum of all separate rate constant contributions. Now we can consider several possibilities concerning general acid–base and speciﬁc acid– base catalysis. First of all, we can consider the case of general acid–base catalysis. This means that the rate is not affected by Hþ or OH ions so that kHþ and kOH can be neglected, as well as k0. Hence, the main contribution to the catalytic rate constant is kc ¼ kHA [HA] þ kA [A ] ¼ kHA [HA] þ kA

Ka [HA] ¼ [Hþ ]

kHA þ kA

Ka [HA] [Hþ ]

(4:122)

So, the result is that the rate constant is proportional to the undissociated acid concentration [HA]. Ka is known, and [Hþ] can be set so that kHA and kA can be calculated from various [HA] and pH experiments. For the case of speciﬁc acid catalysis, the rate is proportional to [Hþ], and now kHA, kA, k0, and kOH can be neglected. Consequently, the catalytic rate constant becomes: kc ¼ kHþ [Hþ ]

(4:123)

Similarly for speciﬁc base catalysis: kc ¼ kOH [OH ] ¼ kOH

Kw [Hþ ]

(4:124)

So, a plot of log kc versus pH yields a straight line with slope 1 and intercept log kþ H for speciﬁc acid catalysis and a slope of þ1 and intercept log(Kw k OH ) for speciﬁc base catalysis. Such a plot is called a pH–rate proﬁle. A slope of about 0 shows that the rate is independent of pH. Figure 4.24 gives a schematic example. Many variations are possible, i.e., speciﬁc acid catalysis at low pH, no effect of pH at higher pH and vice versa, and of course, also slopes that are different from 1 are found. In any case, plots such as the ones in Figure 4.24 can be the starting point for further mechanistic research. A food-related example of speciﬁc acid catalysis is the hydrolysis of sucrose, and an example of speciﬁc base catalysis the isomerization of sugars at alkaline pH or near neutral pH as occurs concurrently in the Maillard reaction. Figure 4.25 gives yet another example of speciﬁc acid catalysis for the

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0 −1

log k

−2 −3 −4 −5 −6 −7 0

FIGURE 4.24

2

4

6 pH

8

10

12

Schematic example of a pH–rate proﬁle.

hydrolysis of fructo-oligomers, which is straightforward. Figure 4.26 gives a rather intricate example of a pH–rate plot of the demethylation of aspartame, an artiﬁcial sweetener. Aspartame can lose methanol via hydrolysis to yield the dipeptide L-a-aspartyl-L-phenylalanine, or it can yield 3-carboxymethyl-6benzyl-2,5-diketopiperazine via intramolecular aminolysis. At low pH, the decomposition is typically speciﬁc acid catalysis leading to L-a-aspartyl-L-phenylalanine; the species that reacts is positively charged because of the protonated amino group. At high pH (which is actually unrealistic for foods) it is a typical speciﬁc base-catalyzed reaction, leading again to L-a-aspartyl-L-phenylalanine; in this case the reactive species is negatively charged due to the dissociated carboxyl group. At intermediate pH 6–10, the main reaction product is diketopiperazine and the effect of increasing [OH] is to increase the concentration of the negatively charged species of aspartame. When this has reached its maximum concentration at about pH 8–9, there is no longer an effect of pH on this reaction and the

0

log (rate)

−1

−2

Slope = −1

−3

−4

−5 1

2

3 pH

4

5

FIGURE 4.25 Example of a reaction showing speciﬁc acid catalysis: acid hydrolysis of a fructo-oligomer. Dataset in Appendix 4.1, Table A.4.9.

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0 −1 Specific base-catalysis: slope = 1

−2

log k

−3 −4 −5

Specific acid-catalysis: slope = −1

−6 −7 −8 0

5

10 pH

FIGURE 4.26 pH–rate proﬁle of the demethylation of aspartame as a function of pH. Speciﬁc acid catalysis of demethylation of aspartame leading to L-a-aspartyl-L-phenylalanine (^), speciﬁc base catalysis of formation of L-a-aspartyl-L-phenylalanine (&), formation of 3-carboxymethyl-6-benzyl-2,5-diketopiperazine (~). Dataset in Appendix 4.1, Table A.4.10.

speciﬁc base-catalyzed reaction takes over. The aspartame example shows that one has to be careful with the interpretation of pH–rate proﬁles. It requires mechanistic insight in what is actually happening before one can draw conclusions about catalytic effects.

4.5

Kinetics of Radical Reactions

Radical reactions in foods are of importance, especially in relation to autoxidation of fats. Such reactions lead to quality loss, ﬁrst of all because of the formation of undesired ﬂavor compounds (oxidative rancidity), and second because of possible formation of hazardous compounds. Radical reactions are also of importance in the body because reactive oxygen species that are formed in all kinds of biochemical reactions may cause damage to DNA, proteins, and cell membranes. Antioxidants in foods could be considered as health-promoting compounds, because they may slow down radical reactions in the body if and when they are absorbed in the body. However, there is currently much debate in the literature whether or not antioxidants in foods can be considered as health promoting. Antioxidants are added also to foods to prevent autoxidation. In any case, radical reactions are of importance and so we spend some attention to the kinetics of radical reactions. However, we limit ourselves to the basics because radical reactions are very complex and it would take too much space to discuss all intricacies. Readers who are interested in more details are referred to some selected references suggested at the end of this chapter. A radical is an atom or a group of atoms possessing one or more unpaired electrons, sometimes also called a free radical (the word free seems to be superﬂuous). A radical is usually indicated with a dot, e.g., H or OH. Radicals can be formed from so-called homolytic scissions of covalent bonds: A ! B þC

(4:125)

as opposed to heterolytic scissions that result in ions: A ! Bþ þ C

(4:126)

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Homolytic scissions are more likely to occur in the absence of solvent. Radicals are unstable and they recombine with each other or attack other groups. In general, the kinetics of radical reactions are quite intricate, and it is not well possible to give a general scheme. In most cases, derivations are made based on steady-state assumptions. This could result in a scheme such as ki

A! B þC

Initiation: Propagation:

kp

A þ B ! B þ D k0p

D ! B þE kt

k0t

C þC ! C C Overall reaction:

(4:128) (4:129)

B þB ! B B

Termination:

(4:127)

k

A! B þ E þ B B þ C C

(4:130) (4:131) (4:132)

Suppose we are interested in the kinetics of formation of compound B: d[B] ¼ kp [A] [B] dt

(4:133)

It is necessary to ﬁnd an alternative equation for this because we cannot easily determine the concentrations of radicals. If we apply the steady-state approximation, which implies that we assume that d[radicals]=dt ¼ 0: d[B] ¼ ki [A] kp [A] [B] k0p [D] 2kt [B]2 0 dt

(4:134)

d[D] ¼ kp [B] [A] k0p [D] 0 dt

(4:135)

It then follows that k0p [D] ¼ kp [B] [A] and substituting this in Equation 4.134 results in: ki [A] kp [A] [B] þ kp [B] [A] 2kt [B]2 ¼ ki [A] 2kt [B]2 ¼ 0

(4:136)

Hence: [B] ¼

ki 2kt

1=2 [A]1=2

(4:137)

We now have an expression for [B] which we can substitute in Equation 4.133: 1=2 d[B] ki [A]3=2 ¼ k[A]3=2 ¼ kp 2kt dt

(4:138)

A kinetic equation having a fractional order is a typical result for radical reactions. Instead of using the steady-state approximation, one can also resort to numerical solutions of the differential equations that describe each step in the proposed reaction mechanism.

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Chemical Reaction Kinetics

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Lipid peroxidation. For foods, lipid peroxidation is most important; it is an oxidative degradation of lipids such as unsaturated fatty acids, sterols, carotenoids, phospholipids, etc. It is called autoxidation because the reaction takes place with molecular oxidation via a self-catalytic mechanism. The classical scheme for lipid peroxidation is divided in three phases, namely initiation, propagation, and termination. A special role is played by oxygen. Normal oxygen is in the so-called triplet state (3 O2 ), with two unpaired electrons in the 2pp orbital and this is not a very reactive oxygen species. However, oxygen can also exist in the singlet state (1 O2 ), which has two paired electrons in the 2pp orbital at one atom and none at the other atom, and this is a very reactive, electrophilic reagent. Singlet oxygen can be formed in various reactions, such as electromagnetic radiation, photodecomposition (see Section 4.6), or irradiation with g-rays, enzymatic reactions involving such enzymes as lipoxygenase, peroxidase, xanthine-oxidase, and due to catalytically acting metals (especially iron and copper). If RH represents a lipid, and X a radical, this results in the following scheme: Initiation:

3

O2 ! 1O2

(formation of singlet oxygen)

RH þ 1 O2 ! ROOH (formation of hydroperoxides) ROOH ! ROO þH Propagation:

(formation of peroxyl radicals)

ROO þRH ! ROOH þ R

R þ 3 O2 ! ROO

Termination:

2ROO ! ROOR þ O2

R þROO ! ROOR R þR ! RR

(4:140) (4:141) (4:142) (4:143)

RH þ 3 O2 ! ROOH

Overall reaction:

(4:139)

(formation of peroxides)

(formation of peroxides)

(formation of peroxides)

(4:144) (4:145) (4:146) (4:147)

The products formed in the termination step are nonradicals. Once radicals are formed, also triplet oxygen may react in the propagation step (Equation 4.143). The hydroperoxides formed (ROOH) are very unstable and decompose into aldehydes and ketones, which are the end products and the cause of rancid ﬂavor. Figure 4.27 shows a general, schematic proﬁle for the course of lipid oxidation. There are many empirical kinetic expressions available in literature describing the course of oxidation as a function of oxygen concentration, the type of lipids (the amount of unsaturated bonds and their location are particularly important), presence of metals, temperature, and notably water activity.

Concentration

Oxygen consumption

Peroxides formation End products formation

Time

FIGURE 4.27 Schematic picture showing the course of lipid autoxidation.

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Lipid oxidation is much faster at low water activity. The empirical expressions differ in the assumptions made to come to a kinetic expression. It is not well possible to list all derived expressions here. The interested reader is referred to references cited at the end of this chapter. Antioxidants. Antioxidants are very important in foods as they can protect fat autoxidation to some extent. Antioxidants are present naturally in foods, such as tocopherols, vitamin C and vitamin E, polyphenols, carotenoids; also synthetic antioxidants are sometimes added, such as butylated hydroxytoluene (BHT). Primary antioxidants interfere directly in the radical reactions as radical scavengers, secondary antioxidants sequester trace metals or quench singlet oxygen, so that the initiation phase is inhibited. Antioxidants are obviously reducing agents, i.e., they donate electrons, and they act in combination with an oxidizing agent. Denoting an antioxidant as AH, a simple scheme for a primary antioxidant would be: R þAH ! RH þ A

(4:148)

A þR ! AR

(4:149)

The radical A that is formed out of the antioxidant is considered less unstable because the unpaired electron is stabilized because of electron resonance in the resulting molecule structure. Figure 4.28 shows the effect of an antioxidant (a-tocopherol) on b-carotene oxidation. Once the antioxidant is consumed, autoxidation can still proceed, which is the reason that the stability of carotene shown in Figure 4.28 depends on the concentration of a-tocopherol. To indicate the complexity of the reactions taking place, 17 differential equations formed the basis for a kinetic model that was able to ﬁt the data shown in Figure 4.28 (the model is not shown here). In fact, this type of kinetic analyses is well suited for the multiresponse technique (Chapter 8).

0.8

[β-carotene] (mol L–1)

0.6

0.4

0.2

0 0

20

40

60

80

100

Time (h)

FIGURE 4.28 Oxidation of b-carotene at an oxygen composition of 40 mol% at 608C, without a-tocopherol (~), with 3.8 mmol L1 (&) and 7.5 mmol L1 a-tocopherol (). Dataset in Appendix 4.1, Table A.4.11.

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Besides in lipid oxidation, radical reactions occur also in the Maillard reaction, during high temperature treatments such as roasting, and during photochemical reactions. Also reactions associated with the heme group in the blood protein hemoglobin are radical reactions, of importance for meat products.

4.6

Kinetics of Photochemical Reactions

As such, photochemical reactions are of utmost importance because that is the way sunlight is captured in photosynthesis of plants. In relation to foods, photochemical reactions are less desired because some components in foods are light sensitive. For instance, beer develops a stale ﬂavor when it is exposed to light, which is the reason why beer is bottled in dark brown or green bottles. Riboﬂavin (vitamin B2) in milk is degraded when exposed to light, so it is not wise to put a bottle of milk in the sunlight. The riboﬂavin decomposes into two biologically inactive products (loss of nutritional value) but in addition it leads to ﬂavor defects. These reactions can be classiﬁed as photochemical. Photochemical reactions can also lead to formation of radicals, which can be the initiation of lipid peroxidation. A photochemical reaction starts with the absorption of a photon hn by a reactant molecule; this is then called an excited molecule. The excited molecule can loose its excitation by collision with other molecules or the wall of a system, or it may turn into a product. This is summarized in the scheme: k1

A þ hv ! A* k2

A* ! A k3

A* ! P

(4:150) (4:151) (4:152)

In photochemistry one uses the concept of primary quantum yield fq, which is the number of reactant molecules producing speciﬁed primary products, and the concept of overall quantum yield Fq, which is the number of reactant molecules that react for each photon absorbed. We can write the following rate law for the rate of disappearance of A:

d[A] ¼ Iabs dt

(4:153)

Iabs is the rate at which photons are absorbed divided by the volume in which absorption occurs. A quantitative expression for the amount of moles of photons absorbed is n¼

Pw Dt (hP cl =l)NAV

(4:154)

In this equation, Pw is the power of the light source in watt, Dt the time over which absorption takes place, hP Planck’s constant, cl the speed of light, l the wavelength, and NAV Avogadro’s number. Using Lambert’s Beer law and denoting the molar absorption coefﬁcient as «M, l the length over which absorption occurs, and the incoming light intensity as I0, Equation 4.153 can be rearranged into:

d[A] ¼ I0 [1 exp («M [A]l)] dt

(4:155)

When the light absorption is high, the rate is proportional to I0 and does not depend on [A], hence it is zero order. If the light absorption is low, the rate is proportional to [A] and the equation turns into a ﬁrstorder reaction. If we include the reactions depicted in Equations 4.150 through 4.152 and assume that [A*] is in steady state, it follows that:

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2

2

1.5

1.5

log (c/c0)

log (c/c0)

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1

1

0.5

0.5

0

0 0

(A)

FIGURE 4.29 Table A.4.12.

0.5

1 Time (days)

1.5

2

0 (B)

1

2 Time (days)

3

4

Photodecomposition of riboﬂavin in water (A) and in milk serum (B). Dataset in Appendix 4.1,

d[A] k2 ¼ Iabs ¼ Fq Iabs k2 þ k3 dt

(4:156)

An example of photochemical degradation is that of riboﬂavin in milk. Figure 4.29 shows some results, which conform to a ﬁrst-order reaction. A typical phenomenon is that in milk serum a two-phase decomposition is observed while there is a one-phase decomposition of riboﬂavin in water. This appears to be linked to the generation of reactive oxygen species in the case of milk serum, which initiates another decomposition reaction of riboﬂavin but at another rate. Another example is the photodecomposition of aspartame, as shown in Figure 4.30. In this case, a zero-order reaction is observed. Photosensitization is the phenomenon that a molecule that does not absorb photons itself can be stimulated by another absorbing molecule. The energy is then transferred by the excited molecule. An example of this is the photosensitization of ascorbic acid by riboﬂavin, in which probably also singlet oxygen is involved. Figure 4.31 shows an example of the magnitude of the effect. It is clear that such phenomena are not desirable for food quality and should therefore be minimized.

4.7

Diffusion-Limited Reactions in Aqueous Solutions

As remarked before, up until now we have assumed that solutions are ideally mixed and that the rate with which reacting species meet each other is not limiting. We will now investigate the conditions when this is not the case. If the observed rate depends on the rate at which solutions mix, this is called macroscopic diffusion control. If the rate is inﬂuenced by the rate of molecular diffusion this is called microscopic diffusion control. We consider this latter condition in more detail here: the rate is then limited by the encounter frequency of reactant molecules. If two species are going to react, they will have to come together ﬁrst. The situation may arise that if the reaction itself is very fast that the actual rate is determined by the encounter frequency and not by the chemical or physical reaction itself. We take reactions in solution as the starting point for discussion. Therefore, solvent effects are important to consider. Solvent effects are classiﬁed as ‘‘physical’’ when the solvent molecules cause the elementary reactions between reactants to be different from what they would have been in the gas phase. The most

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1.3

[Aspartame] (1⫻10−3 M)

1.2

1.1

1

0.9

0.8

0.7

0.6 0

2

4 6 Time (days)

8

10

FIGURE 4.30 Photodecomposition of aspartame in a phosphate buffer, pH 7, 258C, as a function of different light intensities. No light (^), 1100 lx (&), 3300 lx (~), 5500 lx (). Dataset in Appendix 4.1, Table A.4.13.

1.2

[Ascorbic acid] (M ⫻ 10−4)

1

0.8

0.6

0.4

0.2

0 0

5

10

15

Time (min)

FIGURE 4.31 Photosensitization of 1.2 3 104 M ascorbic acid as a function of riboﬂavin content: 0 ppm (^), 1.2 ppm (&), 2.4 ppm (~), 3.6 ppm (3), 6.0 ppm (). Dataset in Appendix 4.1, Table A.4.14.

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important physical effect of solvents is that they can induce ionization (if the solvent is polar enough, certainly true for water), and another effect is the rapid energy transfer of the abundant collisions between reactant and solvent molecules. The solvent molecules are of course also subject to diffusion. A third physical effect is due to the dielectric character of the solvent, which has a large inﬂuence on interactions between ions (Chapter 6). Furthermore, if reactants and products are surrounded by large numbers of solvent molecules, they are more or less trapped; this is called the ‘‘cage effect.’’ Solvent effects are classiﬁed as ‘‘chemical’’ if the solvent molecules participate in the mechanism. These can be catalytic effects, or the solvent molecules can act as reactant or product, which means that solvent molecules can be permanently consumed or generated. An example of this is the hydrolysis of a disaccharide or an ester, in which the solvent water acts also as a reagent. Bimolecular reactions require that molecules will have to come very close before they can interact. Encounters may result because of diffusion and ﬂow (though ﬂow becomes only important in this respect when particles are of colloidal size [Chapter 11]). Here we consider mainly molecular mobility. There are three types of molecular motions: vibrational, rotational, and translational. Vibration concerns changes in size and shape of the molecule by stretching, bending, and rotation of bonds; these are intramolecular effects. Rotational motion is about spinning of the molecule about its axes in three-dimensional space. Translational motion is about the change in location in three-dimensional space. We are mainly concerned with translational motion, which is a random walk characterized by a mean square distance x2, where x is the net distance traveled by the molecule in a time span Dt in a given direction. Einstein derived the following relation: x2 ¼ 2Df Dt

(4:157)

Df is the diffusion coefﬁcient, for which the following relation holds: Df ¼

kB T fs

(4:158)

fs is the friction factor. For spheres, fs was derived by Stokes, and combination of this friction factor with Equation 4.158 results in the Stokes–Einstein equation: Df ¼

kB T 6phv RP

(4:159)

RP is the hydrodynamic radius of the molecule, hv the viscosity. Df is called also the self-diffusion coefﬁcient. For the sake of completion, we also mention the expression for rotational diffusion, the Debye–Stokes– Einstein relation: Df ,r ¼

kB T 8phv R3P

(4:160)

Note that Df,r has units of s1, not m2 s1. Translational motion exists in the gas phase and liquid phase but hardly at all in the solid phase. Based on the Smoluchowski theory for Brownian motion in a viscous liquid, the encounter rate per unit volume and time due to diffusion in solution, rdif, is rdif ¼ 4p(DA þ DB )(RA þ RB )NA NB

(4:161)

This is derived for the case that the particles react immediately upon the encounter. In this equation DA, DB represent the diffusion coefﬁcients, RA and RB the particle radii, and NA and NB the number of

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particles=molecules A, B per unit volume, respectively. For two spherical particles with RA ¼ RB ¼ RP, DA ¼ DB ¼ Df, and assuming that the hydrodynamic radius and the collision radius are the same, it follows that the rate constant for bimolecular encounters, kdif is kdif ¼ 4p(DA þ DB )(RA þ RB )

(4:162a)

After some rearrangement to convert the units of kdif to that of the usual dm3 mol1 s1 the equation becomes: kdif ¼ 16pDf RP 103 NAV (dm3 mol1 s1 )

(4:162b)

in which NAV is the Avogadro number. Combination of this equation with the Stokes–Einstein relation yields an expression for kdif: kdif ¼

8 103 RT (dm3 mol1 s1 ) 3hv

(4:163)

hv is the viscosity of the solution as ‘‘sensed’’ by the diffusing particles (i.e., not necessarily equal to the bulk viscosity). The size of the molecules appears to have no effect as shown in Equation 4.163, because larger molecules move more slowly but are a larger target for collision and as it happens, these two effects compensate each other. If particles would react immediately upon an encounter, it means that the rate of reaction is controlled by diffusion; such reactions are called diffusion controlled. If we take Equation 4.163 as the measure for the fastest bimolecular reaction possible, it is found that for hv ¼ 1 mPa s (viscosity of water at 208C) kdif ¼ 6.6 3 109 dm3 mol1 s1 and at 1008C kdif ¼ 3 3 1010 dm3 mol1 s1. These should be roughly the upper limits for bimolecular reaction rate constants in aqueous solutions at the temperature indicated. The effect of temperature on the encounter rate is incorporated via the effect of temperature on the viscosity of the solvent. A drawback of Equation 4.163 is that it is valid for initially well-mixed systems, and it is quite likely that highly reactive systems are not well mixed initially. If, in addition, the actual chemical reaction takes time, a rate constant with a lower value will be found. For monomolecular dissociation in solution, the rate is determined by the rate at which the products can diffuse away. For uncharged reactants it follows that kdif ¼

kB T phv R3P

(4:164)

Therefore, the upper limit for monomolecular reaction rate constants (uncharged species) would be roughly 1012 s1. When a chemical reaction occurs it is obvious that ﬁrst the reactants must meet before they can react. The overall reaction rate is thus the resultant of a reaction in series and we can apply Equation 4.86. It is of interest to compare the chemical reaction rate to the diffusion rate to see when the chemical reaction is rate limiting and when diffusion becomes rate limiting. Following Equation 4.86: 1 1 1 ¼ þ koverall kchem kdif

(4:165)

If we take a value for kdif, e.g., kdif ¼ 7 3 109 dm3 mol1 s1, we can plot koverall as a function of kchem. This plot is shown in Figure 4.32. It shows clearly that when kchem is 1012 the reaction rate is completely dominated by diffusion.

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7.3E + 09

kdif = 7 ⫻ 109

6.4E + 09

koverall

5.5E + 09 4.6E + 09 3.7E + 09 2.8E + 09 1.9E + 09 1.0E + 09 1.0E + 08 8.00

9.00

10.00 log kchem

11.00

12.00

FIGURE 4.32 Overall reaction rate constant as a function of chemical reaction rate constant, assuming a diffusion rate constant kdif ¼ 7 3 109 dm3 mol1 s1.

When 109 < kchem < 1012, the overall rate constant is determined by both processes. This is then called partial microscopic diffusion control. Since in most reactions in solutions kchem < 109 dm3 mol1 s1, it appears that the rate of encounters is then not limiting the overall rate. Notable exceptions are radical reactions and photochemical reactions where diffusion may become rate-limiting indeed. When diffusion is hindered, however, the picture may become different. This can be the case particularly for foods. We will discuss this further in Chapter 14. When molecules encounter each other they need not react immediately. This can be due to the activation energy barrier (Chapter 5), but also to a geometrical constraint. It may be that the molecules did not approach each other in the right way for a reaction to be possible, and they may need to realign. This is where rotational diffusion can be important. In order to compensate for this delay, it is sometimes proposed to introduce a factor that takes this into account. Although this effect will no doubt play a role at the molecular level, it is currently not clear how to apply this concept to reactions in foods where many other complications also lead to a reduction in rate. Therefore, we will not discuss this in further detail. The rates of elementary reactions are seldom measured directly, since the experimental rate measurements, and the effects of concentration changes etc., on it, reﬂect usually the overall reaction. This will certainly be true for foods with its many components, reaction cascades and complicated structure. We will pay more attention to these phenomena in Chapter 14.

4.8

Kinetics in Open Systems

So far, we have assumed that reactions take place in closed systems, that is to say, without mass transfer to and from the surroundings. While this may be valid for many cases, there are also examples where this is not the case. It is especially important for processing when foods are treated in continuous processes such as in heat exchangers. The topic is vast and we will only show the basic principles as to how the kinetics change when mass transfer comes into play. The reader who is interested in more details is referred to some literature references given at the end of this chapter.

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A reactor allowing mass transfer was already introduced in Figure 4.1, in which the average residence time t is t¼

V w

(4:166)

Suppose next that we are able to introduce a certain amount of a component in the inlet of the CSTR as a pulse. Since the system is considered ideally mixed, the concentration is instantaneously equal in the system at t ¼ 0. We can consider then how the concentration changes at the outlet of the system, supposing that the component is not subject to a reaction, so rA ¼ 0 in Equation 4.1. The following mass balance should hold: win cin wout cout V

dcreactor ¼0 dt

(4:167)

Because the reactor is ideally mixed, the exit stream has the same composition as that in the reactor, i.e., creactor ¼ cout, hence: dcout ¼0 dt

(4:168)

dcout w 1 ¼ dt ¼ dt cin cout V t

(4:169)

win cin wout cout V This can be rearranged to:

The boundary conditions are that at t ¼ 0 cout ¼ c0 (because the reactor is considered ideally mixed) and at t > 0: cin ¼ 0. This leads to the following expression: t cout ¼ c0 exp t

(4:170)

Figure 4.33 shows the change in concentration in this situation. It basically shows how a compound is washed out of a reactor due to mass transfer in the case that there is no further chemical change occurring. The next question is what happens if the compound is subject to a chemical reaction. We need to know, of course, the kinetics of the reaction in order to calculate the change quantitatively. If we suppose a ﬁrst-order decay reaction in A, assuming certain values for the parameters of interest, the equation becomes: t 1 ¼ cA,in exp exp (kt) ¼ cA,in exp t þ k t t

cA,out

(4:171)

The change in the amount of component A is shown in Figure 4.34 for the conditions assumed, as well as the concentration of component B formed out of A. Both component A and B are washed out of the reactor. This result should be compared to the rate of change of A in a closed system as depicted in Figure 4.5, in which component B accumulates. Equation 4.171 shows that the chemical reaction will dominate the disappearance of the compound if k >> 1=t, while the physical removal due to mass transport will dominate if k 1=t. That gives the opportunity to direct the desired change via the ﬂow rate w and the volume of the reactor V.

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1 0.9

Introduction of a certain amount of component A at t = 0

0.8

f in

fout

cout (mol dm−3)

0.7 rA = 0

[A]

0.6 0.5 0.4 0.3

Amount of component A at time t

0.2 0.1 0

0

5

10 t (s)

15

20

FIGURE 4.33 Change in concentration at the outlet of a CSTR when a compound A is added as a pulse at t ¼ 0 and when the compound is not subject to reaction. V ¼ 10 dm3, w ¼ 2 dm3 s1, [A]0 ¼ 1 mol dm3.

Now, consider the situation that an amount of A is constantly supplied at the entrance of the CSTR. Then the concentration of A at the outlet will be building up and eventually reaches the concentration at the inlet if no reaction in the reactor takes place: this is the point where a steady state is reached. The equation describing this situation is t cout ¼ cin cin exp t

(4:172)

1 0.9 Introduction of a certain amount of component A at t = 0

0.7 A→B rA = − d[A] = k[A] dt

fout

cout (mol dm−3)

fin

0.8

0.6 0.5

[A]

0.4 0.3 0.2

Amount of component A and B at time t

[B]

0.1 0 0

5

10 t (s)

15

20

FIGURE 4.34 Schematic representation of a CSTR into which an amount of component A is introduced and for which a ﬁrst-order decomposition reaction is assumed. V ¼ 10 dm3, w ¼ 2 dm3 s1, [A]0 ¼ 1 mol dm3, k ¼ 0.1 s1.

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1 0.9 0.8

cout (mol dm−3)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10 t (s)

15

20

FIGURE 4.35 Pre-steady-state concentration change at the outlet of a CSTR reactor into which a continuous input of A is given from t ¼ 0 onwards, without a chemical reaction taking place. V ¼ 10 dm3, w ¼ 2 dm3 s1, [A]0 ¼ 1 mol dm3.

A graphical representation is given in Figure 4.35. When a compound is also subject to a reaction in the reactor, the concentration changes because of mass transfer and the reaction. For a ﬁrst-order decay reaction in a CSTR, the situation is described by the following equation: t k exp cout ¼ cin cin exp t t

(4:173)

Assuming certain values for the parameters, this equation is depicted in Figure 4.36. The result displayed in Figure 4.36 shows that eventually a steady-state situation is reached (which is not the same as equilibrium!). This situation is distinctly different from that in Figure 4.33 in which the concentration of A and B becomes zero eventually. A different type of open system is a plug ﬂow reactor. It is then assumed that material ﬂows through, say, a pipe such that the residence time is the same for each element in the material and it is assumed that there is no mixing, i.e., no axial dispersion (see Figure 4.37). If a reaction takes place inside the reactor, a concentration gradient will develop in the direction of the ﬂow. We can set up a mass balance over an inﬁnite small slice perpendicular to the direction of the ﬂow having a volume dV. At the beginning of the slice the concentration of a component A is cA, at the end it will have changed to cA þ dcA. At steady state, when there is no change in the amount of component A, the following equation should hold (see Equation 4.1): 0 ¼ wcA w(cA þ dcA ) þ rA dV

(4:174)

This equation can be transformed to: dcA dV ¼ rA w

(4:175)

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1 0.9 0.8

cout (mol dm−3)

0.7 0.6

[A]

0.5 0.4 [B]

0.3 0.2 0.1 0 0

5

10 t (s)

15

20

FIGURE 4.36 Pre-steady-state concentration changes at the outlet of a CSTR reactor into which a continuous input of A is given from t ¼ 0 onwards and in which a ﬁrst-order reaction A ! B takes place. V ¼ 10 dm3, w ¼ 2 dm3 s1, [A]0 ¼ 1 mol dm3, k ¼ 0.1 s1.

Integration over the whole reactor leads to: cA,out ð

cA,in

dcA ¼ rA

ðV 0

dV w

(4:176)

dcA rA

(4:177)

Or alternatively: V t¼ ¼ w

cA

cA,out ð

cA,in

cA + dcA

f , cA,in

f , cA,out

x=0 t=0

FIGURE 4.37

Schematic drawing of a plug ﬂow reactor.

x=L t=τ

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The kinetics in a plug ﬂow reactor are actually the same as in a batch reactor, except that the reaction time is ﬁxed at the residence time t. This becomes apparent if we integrate the expression in Equation 4.175 for, e.g., a zero-order reaction, for which rA ¼ k: cA,out ð

t¼ cA,in

dcA 1 ¼ (cA,out cA,in ) k k

(4:178)

! cA,out ¼ cA,in kt The same exercise for a ﬁrst-order reaction (rA ¼ kcA) yields: cA,out ð

t¼ cA,in

dcA 1 ¼ ln (cA,out cA,in ) kcA k

(4:179)

! ln cA,out ¼ ln cA,in kt These equations are thus exactly the same as for a batch reactor, given earlier in this chapter. Figure 4.38 shows the change in a compound subject to a ﬁrst-order reaction in a plug ﬂow reactor. If one knows the kinetics of a reaction and the initial concentration, one can calculate the outcoming concentration straightforwardly using the equations given in this chapter. Of course, one can vary the residence time by changing the volume V and the ﬂow rate w. The residence time for a plug ﬂow reactor is also given by Equation 4.166. Although this is also an idealized situation, it can be approximated quite closely if the ﬂow inside the reactor is turbulent. In real life, however, there is a distribution of residence times that,

cA + dcA

cA

f , cA,out

f, cA,in 1 0.9

cout (mol dm-3)

0.8 0.7

[A]

0.6 0.5 0.4 0.3 0.2 0.1 0

0

x=0 t=0

5

10 t/s

15

20 x=L t=τ

FIGURE 4.38 Change in the concentration of a compound A in a plug ﬂow reactor subject to a ﬁrst-order reaction.

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incidentally, can be determined experimentally by introducing a tracer at the entrance of the reactor and following its concentration at the exit. It is instructive to study the performance of the various reactor types after reaching steady-state conditions. The general kinetic equation for a CSTR at stationary conditions is t CSTR ¼

cA,out cA,in rA

(4:180)

The general kinetic equation for plug ﬂow is cA,out ð

t PF ¼ cA,in

dcA rA

(4:181)

This last equation is also valid for a batch reactor, as shown above. For a zero-order reaction (rA ¼ k) in a CSTR, it follows that under the condition that tCSTR < cA,in=k: cA,out kt CSTR ¼1 cA,in cA,in

(4:182)

(for the condition that tCSTR cA,in=k, cA,out=cA,in ¼ 0). For a ﬁrst-order reaction (r ¼ kcA) in a CSTR, it follows from Equation 4.177 that: cA,out 1 ¼ cA,in 1 þ kt CSTR

(4:183)

For a zero-order reaction in a plug ﬂow reactor the equation is cA,out kt PF ¼1 cA,in cA,in

(4:184)

Note that this equation is similar to the one for CSTR. And for a ﬁrst-order reaction in a plug ﬂow reactor it follows: cA,out ¼ exp (kt PF ) cA,in

(4:185)

Since Equations 4.182 and 4.184 are the same it follows that the type of reactor does not matter for a zero-order reaction. This is not so for a ﬁrst-order reaction, as shown in Figure 4.39. A plug ﬂow reactor gives a higher degree of conversion than a CSTR for equivalent residence times t, provided that we deal with ﬁrst-order kinetics. One can of course exploit this phenomenon. It makes immediately clear that if one wants to reach a high conversion rate, that a plug ﬂow reactor is much more efﬁcient than a CSTR. This is especially relevant for the killing of microorganisms (assuming for the moment that this can be described by a ﬁrst-order reaction), which would be very inefﬁcient in a CSTR. Furthermore, one can design a reactor and its operating conditions (thereby effectively setting the residence time t) in such a

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1

cA,out/cA,in

0.8

0.6 CSTR 0.4

0.2

Plug flow reactor 0 0

FIGURE 4.39 reactor.

5

10

15 t (s)

20

25

30

Comparison of a ﬁrst-order reaction having the same rate constant in a CSTR and a plug ﬂow

way that the desired effect is maximum. It is perhaps worth mentioning that a series of CSTRs is going to approach the performance of a plug ﬂow reactor (for the same equivalent residence time). There is much more to be said about kinetics in open systems. Here we only showed some basic principles to show the differences. The reader who is interested in more details is referred to some selected literature references at the end of this chapter. It is especially important to be aware of the phenomenon of residence time distribution. As a ﬁnal remark we would like to point out that a very important difference between open and closed systems is that in closed systems thermodynamic equilibrium is the time-invariant condition, while for open, continuous systems the steady state is the time-invariant condition, and the reader should appreciate the differences between these two conditions as they are very relevant for foods and food processing.

4.9

Concluding Remarks

In this chapter kinetic models have been introduced and a connection has been made with thermodynamics. The treatment has been kept very general, but at the same time limited to cases that are of interest for foods. Therefore, we have illustrated the material as much as possible with relevant examples from food science. It has been discussed how rates can be expressed and it has been shown when reaction rates may become diffusion controlled. For reactions in solutions this does not seem to be the case very often, except perhaps for radical and photochemical reactions. Furthermore, the basis has been shown how to tackle kinetics in the case of transport of material in addition to a chemical reaction. This chapter forms the basis for further discussion of kinetic models throughout the book. In the next chapter, we will continue the discussion by focusing on the effects of temperature and pressure.

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Appendix 4.1

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Datasets Used for Examples in This Chapter

TABLE A.4.1 Degradation of 1-Methyladenosine in Heated Milk (Figure 4.11) Concentration (mmol L1)

Heating Time (s) at 1358C 0 2

0.41 0.38

12

0.36

31.7

0.35

45

0.29

60.1

0.32

70

0.26

128

0.24

156

0.15 Initial Rate (mmol L1 s1) at 1358C

Initial Concentration (mmol L1) 0.41

0.011041

0.53

0.014196

0.64

0.01735

0.71

0.019243

1.18

0.031861

1.19

0.032177

2.35 5.92

0.063407 0.159306

Source: From Schlimme E., Ott F.G., and Kiesner C. Reaction kinetics of the heat-induced formation of N6-methyladenosine in milk. Int Dairy J 4:617–627, 1994. Note: The author would like to thank Prof. Schlimme for providing the data.

TABLE A.4.2 Nonenzymatic Browning of Whey Powder during Storage (Figure 4.12) Optical Density Exp. 1, 258C

Optical Density Exp. 2, 258C

0

1.8

1.9

30

4.3

4.1

60

6.3

6.1

90 120

7.4 9.6

7.6 9.8

150

12.0

11.8

180

12.5

12.7

210

14.5

14.8

Time (days)

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TABLE A.4.2 (continued) Nonenzymatic Browning of Whey Powder during Storage (Figure 4.12) Optical Density Exp. 1, 358C

Optical Density Exp. 2, 358C

0

1.7

1.6

10

5.2

5.0

20

7.9

7.9

30

10.6

10.5

40

13.7

13.8

50 60

16.3 20.2

16.5 20.1

70

23.2

23.4

95

27.8

27.7

Optical Density Exp. 1, 458C

Optical Density Exp. 2, 458C

0

1.7

1.7

2

5.2

5.2

4

7.1

7.0

7 11

22.4 25.2

22.4 25.3

Time (days)

Time (days)

18

31.7

31.7

28

44.4

44.2

35

50.7

50.9

Source: From Labuza Th.P. Reaction kinetics and accelerated tests. Simulation as a function of temperature. In: Computer Aided Techniques in Food Technology, Vol. I, Saguy (ed.). New York: Marcel Dekker, 1983, pp. 71–115.

TABLE A.4.3 Degradation of Betanin in Aqueous Solution at 758C (Figure 4.13) Time (min)

Betanin mg L1

0 10

4.29 3.65

20

3.04

25

2.83

35

2.30

40

2.20

50

1.83

60

1.49

90 100

0.96 0.70

Source: From Saguy I., Kopelman I.J., and Mizrahi S. Thermal kinetic degradation of betanin and betalamic acid. J Agric Food Chem 26:360–362, 1978.

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TABLE A.4.4 Lysine Loss in Milk Heated at 1608C (Figure 4.14) Time (s)

Lysine (mg L1 in Milk)

0

2.93

50

2.62

100

2.40

200

2.12

350

1.98

600

1.54

1000

1.01

1500 2000

0.84 0.62

Source: From Horak F.P. Über die Reaktionskinetik der Sporenabtötung und chemischer Veränderungen bei der thermischen Haltbarmachung von Milch zur Optimierung von Erhitzungsverfahren, PhD thesis. Technical University of Munich, Germany, 1980.

TABLE A.4.5 Reduction of Hexacyanoferrate (III) ([B]) by Ascorbic Acid ([A]) at Ionic Strength of 0.0384 M (Figure 4.15) Time

ln(A=B)

0 1.02

3.26967

1.87

3.33875

3.05

3.43085

3.96 4.98

3.54598 3.61506

5.99

3.73019

7.12

3.82229

8.03

3.89137

9.1

3.98347

10.06

4.05255

11.19

4.14465

12.15 13.22

4.25978 4.35189

14.08

4.39794

15.2

4.51307

Source: From Watkins K.W. and Olson J.A. Ionic strength effect on the rate of reduction of hexacyanoferrate (III) by ascorbic acid. J Chem Ed 57:158–159, 1980.

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TABLE A.4.6 Heat-Induced Denaturation= Aggregation of b-Lactoglobulin at 658C (Figure 4.16) Initial Concentration (g L1)

Initial Rate

1.485

9.096e-6

4.0771

3.094e-5

7.902

8.113e-5

1.62

3.262e-4

2.407

5.0e-4

3.554

8.954e-4

4.483

1.221e-3

5.505 7.128

1.64e-3 2.342e-3

8.373

2.646e-3

Time (s)

c=c0

0

1

5,000

0.9

10,000

0.85

20,000

0.78

20,000

0.78

30,000

0.72

40,000

0.7

50,000 90,000

0.55 0.4

125,000

0.35

180,000

0.25

Source: From Roefs S.P.F.M. and de Kruif C.G. A model for the denaturation and aggregation of b-lactoglobulin. Eur J Biochem 226:883–889, 1994. Note: The author would like to thank Dr. Roefs for supplying the data.

TABLE A.4.7 Degradation of Chlorophyll A at 1158C (Figure 4.17) Time (min)

c=c0

0

1

0.5 1 1.5 2 2.5

0.71

4

0.55

5

0.27

6

0.21

7 8 9

0.11

10

0.08

Source: From Canjura F.L., Schwarz S.J., and Nunes R.V. Degradation kinetics of chlorophylls and chlorophyllides. J Food Sci 56:1639–1643, 1991.

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TABLE A.4.8 Sucrose Hydrolysis at 708C and pH 2.5 (Figure 4.18) Sucrose in g L1, exp. 1

Sucrose in g L1, exp. 2

0

0.76

0.76

20

0.72

0.71

30

0.68

0.66

50

0.61

0.59

70

0.55

0.54

85

0.49

0.50

115

0.48

0.44

140 175

0.37 0.5

0.37 0.32

215

0.27

0.26

260

0.22

0.19

320

0.14

0.12

400

0.08

0.07

535

0.03

0.04

Time (min)

Source: From Pinheiro Torres A., Oliveira R., Silva C.L.M., and Fortuna S.P. The inﬂuence of pH on the kinetics of acid hydrolysis of sucrose. J Food Process Eng 17:191–208, 1994.

TABLE A.4.9 Effect of pH on Acid Hydrolysis of a Fructo-Oligomer (Figure 4.25) pH

Log (rate) for Raftilose P95

2 2.6

1.6 2

3

2.7

3.5

3

4

3.7

4.2

4

Source: From Blecker C., Fougnies C., van Herck J.C., Chevalier J.P., and Paquot M. Kinetic study of the acid hydrolysis of various oligofructose samples. J Agric Food Chem 50:1602–1607, 2002.

TABLE A.4.10 Demethylation Kinetics of Aspartame in aqueous solution at 258C (Figure 4.26) pH

Log k1

0.28

4.96

0.59

5.31

0.91

5.67

1.33

6.15

1.82 2.25

6.60 7.0

5.82

Log k2

Log k3

5.96

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4-59 TABLE A.4.10 (continued) Demethylation Kinetics of Aspartame in aqueous solution at 258C (Figure 4.26) pH

Log k1

Log k2

Log k3

6.19

5.62

6.78

5.12

7.16 7.38

4.74 4.40 4.30

7.66 8.92

4.66

3.69

9.58

4.19

3.63

10

3.84

3.59

10.41

3.38

3.70

10.87

3.00

3.65

11.21 11.43

2.59 2.40

3.57 3.65

Source: From Skwierczynski R.D. and Connors K.A. Demethylation kinetics of aspartame and L-phenylalanine methyl ester in aqueous solution. Pharm Res 10:1174–1180, 1993.

TABLE A.4.11 Tocopherol Time (h) 0

Effect of a-Tocopherol on b-Carotene (Figure 4.28) 7.5 mmol dm3

3.8 mmol dm3

0 mmol dm3

b-Carotene (mmol dm3)

b-Carotene (mmol dm3)

b-Carotene (mmol dm3)

0.76

1

0.71

1.5

0.66

2.5

0.6

3.75

0.53

6

0.72

8 10

0.39 0.26 0.15

12

0.09

13

0.7

14 20

0.65

24

0.66

39 44 47

0.63 0.63 0.62

52

0.56

55 56

0.47 0.62

60

0.18

63 81

0.09 0.57 (continued )

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4-60 TABLE A.4.11 (continued) Tocopherol Time (h)

Effect of a-Tocopherol on b-Carotene (Figure 4.28)

7.5 mmol dm3

3.8 mmol dm3

0 mmol dm3

b-Carotene (mmol dm3)

b-Carotene (mmol dm3)

b-Carotene (mmol dm3)

90

0.39

92

0.29

94

0.18

99

0.06

Source: From Takahashi, A., Shibasaki-Kitakawa, N., and Yonemoto, T., J. Am. Oil Chem. Soc., 80, 1241, 2003.

TABLE A.4.12 Photodecomposition of Riboﬂavin in Water and Milk Serum (Figure 4.29) Time (days)

Log c=c0 in Water

0

2

0.1

1.92

0.15

1.83

0.17

1.77

0.42

1.67

0.44

1.57

0.59 0.78

1.48 1.41

0.79

1.34

0.95

1.3

0.94

1.22

1.19

1.17

1.13

1.11

1.38

1.05

1.3 1.57

0.99 0.95

1.55

0.89

Time (days)

Log c=c0 in Milk Serum

0

2

0.06

1.97

0.13

1.9

0.34

1.79

0.42

1.72

0.59

1.66

0.63 0.95

1.58 1.56

0.8

1.5

1.1

1.48

1.39

1.4

1.58

1.34

2.06

1.28

2.34

1.22

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TABLE A.4.12 (continued) Photodecomposition of Riboﬂavin in Water and Milk Serum (Figure 4.29) Time (days)

Log c=c0 in Water

2.68

1.15

3.18

1.12

3.39

1.07

3.81

1.05

3.9

0.98

Source: From Toyosaki T., Yamamoto A., and Mineshita T. Kinetics of photolysis of milk riboﬂavin. Milchwiss 43: 143–146, 1988.

TABLE A.4.13

Photodecomposition of Aspartame (Figure 4.30) 0 lx

1100 lx

Time (days)

3300 lx

5500 lx

(30.001 M)

0

1.2

1.2

1.2

1.2

2 4

1.18 1.16

1.18 1.14

1.16 1.09

1.12 1.03

6

1.15

1.13

1.02

0.9

8

1.12

1.1

0.96

0.81

10

1.09

1.04

0.92

0.73

Source: From Kim, S.K., Jung, M.Y., and Kim, S.Y., Food Chem., 59, 273, 1997.

TABLE A.4.14 Riboﬂavin

Photosensitization of Ascorbic Acid by Riboﬂavin (Figure 4.31) 0 ppm

1.2 ppm

0

1.20

1.20

1.21

1.21

1.20

3 6

1.18 1.18

1.00 0.76

0.87 0.50

0.74 0.31

0.60 0.13

Time (min)

2.4 ppm

3.6 ppm

6.0 ppm

(30.0001 M)

9

1.18

0.54

0.23

0.032

0.02

12

1.17

0.35

0.04

0.003

0

15

1.17

0.22

0.004

0

0

Source: From Jung, M.Y., Kim, S.K., and Kim, S.Y., Food Chem., 53, 397, 1995.

Bibliography and Suggested Further Reading About Kinetics Andraos J. A streamlined approach to solving simple and complex kinetic systems analytically. J Chem Ed 76:1578–1583, 1999. Baird J.K. A generalized statement of the law of mass action. J Chem Ed 76:1146–1150, 1999. Clegg R.M. Derivation of diffusion controlled chemical rate constants with the help of Einstein’s original derivation of the diffusion constant. J Chem Ed 63:571–574, 1986. Corradini M.G. and Peleg M. A model of non-isothermal degradation of nutrients, pigments and enzymes. J Sci Food Agric 84:217–226, 2004. Croce A.E. The application of the concept of extent of reaction. J Chem Ed 79:506–509, 2002.

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Cvitaš T. A new look at reaction rates. J Chem Ed 76:1574–1577, 1999. Cvitaš T. and Kallay N. A mole of chemical transformations. Ed Chem 17:166–168, 1980. Davis J.S. Practical aspects of kinetic analysis. Methods Enzymol 210:374–390, 1992. De Levie R. Stochastics, the basis of chemical dynamics. J Chem Ed 77:771–774, 2000. Diemente D. A closer look at the addition of equations and reactions. J Chem Ed 75:319–321, 1998. Dyson R., Maeder M., Puxty G., and Neuhold Y.M. Simulation of complex chemical kinetics. Inorganic Reaction Mechanisms 5:1–8, 2003. Eberhart J.G. and Levin E. A simpliﬁed integration technique for reaction rate laws of integral order in several substances. J Chem Ed 72:193–195, 1995. Gellene G.I. Application of kinetic approximations to the A Ð B ! C reaction system. J Chem Ed 72:196–199, 1995. Laidler K.J. Rate-controlling step: A necessary or useful concept? J Chem Ed 65:250–254, 1988. Le Vent S. Rate of reaction and rate equations. J Chem Ed 80:89–91, 386, 2003. Lee J.Y. The relationship between stoichiometry and kinetics revisited. J Chem Ed 78:1283–1284, 2001. Lin K.C. Understanding product optimization: Kinetic versus thermodynamic control. J Chem Ed 65:857–860, 1988. Loudon G.M. Mechanistic interpretation of pH-rate proﬁles. J Chem Ed 68:973–984, 1991. Puxty G., Maeder M., and Hungerbühler K. Tutorial on the ﬁtting of kinetics models to multivariate spectroscopic measurements with non-linear least-squares regression. Chemometr Intell Lab Sys 81:149–164, 2006. Pyun C.W. Steady-state and equilibrium approximations in chemical kinetics. J Chem Ed 48:194–196, 1971. Quisenberry K.T. and Tellinghuisen J. Textbook deﬁciencies: Ambiguities in chemical kinetics rates and rate constants. J Chem Ed 83:510–512, 2006. Ramachandran B.R. and Halpern A.M. A novel experiment in kinetics: The A Ð B ! C reaction system. J Chem Ed 74:975, 1997. Schmitz G. What is a reaction rate? J Chem Ed 82:1091–1093, 2005. Snadden B. A new perspective on kinetic and thermodynamic control of reactions. J Chem Ed 62:653–655, 1985. Stumm W. (Ed.) Aquatic Chemical Kinetics. Reaction Rates of Processes in Natural Waters. New York: Wiley-Interscience, 1990. Toby S. The relationship between stoichiometry and kinetics. J Chem Ed 77:188–190, 2000. Villota R. and Hawkes J.G. Reaction kinetics in food systems. Handbook of Food Engineering, Heldman D.R. and Lund D.B. (Eds.), New York: Marcel Dekker, 1992. Viossat V. and Ben-Aim R.I. Validity of the quasi-stationary-state approximation in the case of two successive reversible ﬁrst-order reactions. J Chem Ed 75:1165–1169, 1998. About Radical Reactions Antunes F., Salvador A., Marinho H.S., Alves R., and Pinto R.E. Lipid peroxidation in mitochondrial inner membranes. 1. An integrative kinetic model. Free Radic Biol Med 21:917–943, 1996. Brimberg U.I. and Kamal-Eldin A. On the kinetics of the autoxidation of fats: Inﬂuence of pro-oxidants, antioxidants and synergists. Eur J Lipid Sci Technol 105:83–91, 2003. Brimberg U.I. and Kamal-Eldin A. On the kinetics of the autoxidation of fats: Substrates with conjugated double bonds. Eur J Lipid Sci Technol 105:17–22, 2003. Chen B.H., Chen T.M., and Chien J.T. Kinetic model for studying the isomerization of alpha- and betacarotene during heating and illumination. J Agric Food Chem, 42:2391–2397, 1994. Ingold K.U. Kinetic and mechanistic studies of free radical reactions in the 21st century. Pure Appl Chem 69:241–243, 1997. Karel M. Kinetics of lipid oxidation. In: Physical Chemistry of Foods, Schwartzberg H.G. and Hartel R.W. (Eds.), pp. 651–668, New York: Marcel Dekker, 1992.

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Labuza T.P. Kinetics of lipid oxidation. Crit Rev Food Sci Nutr 2:355–404, 1971. Minn D.B. and Boff J.M. Chemistry and reaction of singlet oxygen in foods. Comp Rev Food Sci Food Safety 1:58–72, 2002 Takahashi A., Shibasaki-Kitakawa N., and Yonemoto T. A rigorous kinetic model for b-carotene oxidation in the presence of an antioxidant, a-tocopherol. J Am Oil Chem Soc 80:1241–1247, 2003. Yanishlieva N.V., Kamal-Eldin A., Marinova E.M., and Toneva A.G. Kinetics of antioxidant action of aand g-tocopherols in sunﬂower and soybean triacylglycerols. Eur J Lipid Sci Technol 104:262–270, 2002. About Photochemistry Hippler M. Photochemical kinetics: Reaction orders and analogies with molecular beam scattering and cavity ring-down experiments. J Chem Educ 80:1074–1077, 2003. Jung M.Y., Kim S.K., and Kim S.Y. Riboﬂavin-sensitized photooxidation of ascorbic acid: Kinetics and amino acid effects. Food Chem 53:397–403, 1995. Kim S.K., Jung M.Y., and Kim S.Y. Photodecomposition of aspartame in aqueous solutions. Food Chem 59:273–278, 1997. Logan S.R. Does a photochemical reaction have a reaction order? J Chem Educ 74:1303, 1997. Rae M. and Berberan Santos M.N. A generalized pre-equilibrium approximation in chemical and photophysical kinetics. J Chem Educ 81:436–440, 2004. Toby S. Does a photochemical reaction have a kinetic order? J Chem Educ 82:37–38, 2005. About Kinetics and Mass Transport Cutler A.H., Antal M.J., and Maitland J. Jr. A critical evaluation of the plug ﬂow idealization of tubular ﬂow reactor data. Ind Eng Chem Res 27:691–697, 1988. Haas C.N., Joffe J., Heath M.S., and Jacangelo J. Continuous ﬂow residence time distribution function characterization. J Environ Eng ASCE 123:107–114, 1997. Heppell N.J. Comparison of the residence time distributions of water and milk in an experimental UHT sterilizer. J Food Eng 4:71–84, 1985. Hill C.G. An Introduction to Chemical Engineering Kinetics and Reactor Design. New York: Wiley, 1977. Jung A. and Fryer P.J. Optimising the quality of safe food: Computational modelling of a continuous sterilisation process. Chem Eng Sci 54:717–730, 1999. Levenspiel O. Chemical Reaction Engineering, 2nd ed., New York: Wiley, 1972. Malcata F.X. Modelling of a series of continuously stirred tank reactors for thermal processing of liquid foods. Int J Food Sci Technol 26:535–546, 1991. Ramaya S.V. and Antal M.J. Evaluation of systematic error incurred in the plug ﬂow idealization of tubular ﬂow reactor data. J Energy Fuels 3:105–108, 1989. Sancho M.F. and Rao M.A. Residence time distribution in a holding tube. J Food Eng 15:1–21, 1992. Missen R.W., Mims C.A., and Saville B.A. Introduction to Chemical Reaction Engineering and Kinetics. New York: John Wiley & Sons Inc., 1999. General Textbooks Atkins P.W. Physical Chemistry, 6th edn. Oxford: Oxford University Press, 1999. Cussler E.L. and Moggridge G.D. Chemical Product Design. Cambridge, UK: Cambridge University Press, 2001. Damodaran S., Parkin K.L., and Fennema O.R. Fennema’s Food Chemistry, 4th edn. Food Science and Technology, p. 1144. Taylor & Francis, Boca Raton, FL, 2008. Gardiner W.C. Rates and Mechanisms of Chemical Reactions. New York: WA Benjamin Inc., 1969. Fennema O.R. (Ed.) Food Chemistry, 3rd edn. New York: Marcel Dekker, 1996. Hill C.G. An Introduction to Chemical Engineering Kinetics and Reactor Design. New York: John Wiley & Sons, 1977.

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Laidler K.J. Chemical Kinetics, 3rd edn. New York: Harper & Row, 1987. Marangoni A.G. Enzyme Kinetics. A Modern Approach. Hoboken, NJ: Wiley Interscience, 2003. Maskill H. The Physical Basis of Organic Chemistry. Oxford: Oxford University Press, 1985. Moore W.J. Physical Chemistry. London: Longman, 1972. Olmsted III J. and Williams G.M. Chemistry, the Molecular Science. Dubuque, IA: Wm.C. Brown, 1997. Raff L.M. Principles of Physical Chemistry. Upper Saddle River, NJ: Prentice-Hall Inc., 2001. Tinoco T., Sauer K., and Wang J.C. Physical Chemistry: Principles and Applications in Biological Sciences, 3rd edn. Englewood Cliffs, NJ: Prentice-Hall International, 1995. Sutton R., Rockett B., and Swindells P. Chemistry for the Life Sciences. London: Taylor & Francis, 2000. Voet D. and Voet J.G. Biochemistry, 2nd edn. New York: Wiley & Sons, 1995. Walstra P. Physical Chemistry of Foods. New York: Marcel Dekker, 2003. Westphal G., Buhr H., and Otto H. Reaktionskinetik in Lebensmitteln (Reaction Kinetics in Foods, in German). Berlin: Springer-Verlag, 1996.

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5 Temperature and Pressure Effects

5.1

Introduction

The effect of temperature is one of the most important effects to consider when discussing kinetics in any branch of science and food science is no exception. Many foods are heat treated, for various reasons, and the effect of heat treatment on food quality is a very important issue. Shelf life of food is also strongly affected by temperature. Therefore, knowledge on how the kinetics of reactions is affected by temperature changes is essential. An emerging technological option to preserve foods is by high pressure. Many research papers are currently published on this topic, and it is for this reason that we spend some attention to kinetics of reactions induced by high-pressure treatment. Most food scientists would tend to use Arrhenius’ law and derive an activation energy from it. Arrhenius derived his famous equation empirically, and it was later put in a theoretical perspective, especially for gas reactions, based on the collision theory, which incorporates time via molecular velocities and the number of favorably oriented high-energy collisions. Arrhenius’ equation appears to ﬁt many reactions and is therefore frequently used. Although it may be a perfectly good choice in many cases (but not in all), it seems appropriate to start the discussion at a somewhat more fundamental level by explaining relevant aspects of transition state theory, also referred to as the activated complex theory or absolute rate theory. This theory bridges the gap between thermodynamics and kinetics by postulating an equivalence between energy E and frequency of atomic motions n, making it possible to deduce rate data from energy data (using the Planck expression E ¼ hPn, with hP Planck’s constant). This book is, however, not the appropriate place to discuss the transition state theory in great detail, but the basics will be addressed. The objective of this chapter is to explain the basic principles (mainly transition state theory) behind the effects of temperature and pressure and then to discuss the implications for kinetics as studied in foods.

5.2

van’t Hoff Equation

We begin the discussion with the effect of temperature on the thermodynamic equilibrium constant Keq, deﬁned in Chapter 3. It was mentioned there already several times that temperature is important when considering the thermodynamics of chemical reactions. The effect of temperature on the equilibrium constant Keq is expressed in the so-called van’t Hoff equation. Combining Equation 3.113 with Equation 3.38 results in 5-1

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ln Keq ¼

DH DS þ RT R

(5:1)

reminding the reader that the position of equilibrium (and hence the value of Keq) depends on both the standard enthalpy and entropy changes of the reaction. The temperature dependence of Keq is found by differentiating Equation 5.1 with respect to temperature d ln Keq DH ¼ dT RT 2

(5:2a)

d ln Keq DH ¼ d(1=T) R

(5:2b)

or

These expressions are known as the van’t Hoff equation and show that the temperature dependence of the equilibrium constant is determined by the difference in enthalpy between the products and reactants in their standard states for 1 mol of reaction. Thus, in contrast to the magnitude of Keq, the change in Keq with temperature is only determined by enthalpy, not by entropy (on the assumption that DH and DS are temperature independent; this assumption is not justiﬁed for hydrophobic reactions where hydration=dehydration is of importance). A most important result is that the equilibrium constant decreases with temperature for an exothermic reaction (negative DH), while it increases for an endothermic reaction (positive DH), a manifestation of Le Chatelier’s principle, discussed in Chapter 3. As an example, Figure 5.1 shows a van’t Hoff plot for the mutarotation of lactose: a-lactose ! b-lactose The enthalpy change calculated from the slope of the plot turns out to be 1250 J=mol. The mutarotation is therefore an exothermic reaction and the equilibrium shifts toward a-lactose when the temperature increases. The enthalpy value derived is in line with the value reported in handbooks.

0.6 y = 150.3x –0.05 0.5

ln Keq

0.4 0.3 0.2 0.1 0 0.0024 0.0026 0.0028 0.0030 0.0032 0.0034 0.0036 1/T (K−1)

FIGURE 5.1 Van’t Hoff plot for the mutarotation of lactose: a-lactose ! b-lactose between 108C and 908C. The equation shown is the regression equation. Dataset in Appendix 5.1, Table A.5.1.

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0.45 y = 400.0x–0.9

ln Keq

0.4

0.35

0.3 0.0031

0.00315

0.0032 1/T

0.00325

0.0033

0.00335

(K−1)

FIGURE 5.2 Van’t Hoff plot for the mutarotation of glucose: a-glucose ! b-glucose between 308C and 458C. The equation shown is the regression equation. Dataset in Appendix 5.1, Table A.5.2.

A similar analysis for glucose is shown in Figure 5.2. The enthalpy change calculated from the slope of the plot turns out to be 3322 J=mol. Like with lactose, it is an exothermic reaction and the equilibrium shifts toward a-glucose with increasing temperature. However, the reader is advised that enthalpies derived from van’t Hoff plots can be quite inaccurate, especially when there is only a limited number of data points as in Figure 5.2 and when the values of the equilibrium constants are not much different. Consequently, the value of 3322 J=mol derived for the standard enthalpy from the regression line in Figure 5.2 is substantially different from the value reported in Table 3.12 at 258C, namely 1150 J=mol. This is probably due to the fact that the temperature range used to construct Figure 5.2 is very limited, namely 308C–458C, hence there is very little difference in the equilibrium constants at the various temperatures, and this leads to very inaccurate estimates, as shown. Enthalpy values derived via calorimetry are to be preferred. For a reversible reaction it was shown that the forward and reverse rate equal each other at equilibrium and that Keq ¼ kf=kr, (Equation 4.14). As follows from Equation 5.2, both kf and kr should have an exponential dependence on 1=T, or one of them has and the other is independent of T. We will discuss next the temperature dependence of rate constants as derived via transition state theory and subsequently Arrhenius’ law.

5.3

Transition State Theory

The transition state theory forms a theoretical basis on which more practical equations (such as Arrhenius’ law) can be based. Reaction kinetics in solution has some important aspects that are worth discussing. Reactions in a gas are due to isolated encounters between individual molecules, but this is not possible in solution. The reason is that reactant molecules interact continuously with solvent molecules. If a reaction has taken place, the products will tend to diffuse away, but because of the surrounding solvent molecules, this will take some time, and perhaps in the meantime something can happen to the products or reactants. This is called the ‘‘cage effect.’’ A typical consequence is that encounters last longer in solution (say 1011 s) than in the gas phase (typically about 1013 s).

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Transition state theory is well suited for reactions in solutions and is not concerned with rates of encounters (like in gas reactions) but considers thermodynamic and statistical mechanics principles to predict how many combinations of reactants will be present in the so-called transition state. This is a type of high-energy state in which molecules can be present in an unstable but activated condition, in which they will undergo some molecular change to yield products. Consider the reactants A, B that are transformed into products P, Q via a transition state ABz, as follows: A þ B Ð ABz ! P þ Q

(5:3)

Figure 5.3A shows schematically how the potential energy of a pair of molecules changes with the reaction coordinate, i.e., the path along the potential energy curve. The reaction coordinate indicates the state of the molecules in the transition from reactants to products. At this point it is perhaps useful to remark that the reaction coordinate does not refer to a state of the macroscopic system, it only refers to the behavior of a pair of molecules. It therefore does not make sense to plot free energy as a function of the reaction coordinate, as is sometimes done (remember that thermodynamic parameters such as

Potential energy

AB ‡: transition state

Reactants A, B

Products P, Q (A)

Reaction coordinate

G

G AB‡

AB‡

∆G ‡

∆G ‡ A+B

A+B

P+Q ∆G > 0

∆G < 0 P+Q

(B)

(C)

FIGURE 5.3 Schematic presentation of the potential energy of reactants A, B, transition state ABz, and products P, Q along the reaction coordinate (A). Standard free energies G for various possible states of reactants, products, and intermediates, where the standard free energy difference DG between reactants and products is either negative (B), or positive (C). The activation standard free energies DGz are always positive. Note that there is no label on the x-axis in the case of the diagrams in B and C.

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free energy refer to huge ensembles of molecules); it should be potential energy. Plotting free energy would violate the property that it cannot show a maximum as the reaction progresses, as shown in Figure 3.16. In other words, the reaction coordinate should not be confused with the extent of reaction j discussed in Chapter 3. An alternative way of expressing energy proﬁles is to plot the standard free energy referring to 1 mol of the states in which molecules or complexes are in, but then without a label on the x-axis so as to avoid the suggestion that free energy is plotted as a function of reaction progress. Figure 5.3B and C is such alternative ways of expressing energy proﬁles. Note that DG can be positive: it implies that the equilibrium constant for that particular change is smaller than unity. It can of course also be negative, implying that the equilibrium constant is higher than unity. In contrast, a reaction can never have a positive DrG, or rather a positive dG=dj; if it does the reaction will run in the reverse way. This is extensively discussed before and the reader is referred to Chapter 3 for more details. The standard free energies for activation DGz are always positive, meaning that the equilibrium constant that describes the position of the equilibrium is always much smaller than unity, expressing that the number of molecules in the activated state is very low compared to the molecules in the nonactivated state. It is, however, possible that some molecules pass over an energy barrier when they have acquired sufﬁcient kinetic energy. This becomes clear, at least qualitatively, from the Maxwell–Boltzmann distribution, showing the distribution of the average kinetic energy of molecules (Figure 5.4). An important point is that the shape of this distribution becomes wider at higher temperature, so that more molecules are able to overcome an energy barrier at higher temperature. Coming back to Figure 5.3A, when molecules start to interact, their potential energy increases, and a maximum is reached in the activated, or transition, state. It decreases again when products are formed. The population of molecules in the transition state is very small as compared to the number of reactant or product molecules. It is assumed that there is some type of equilibrium between reactants and the activated complex (as shown in Equation 5.3), and also between products and activated complex (not shown in Equation 5.3). However, if the activated complex is formed from reactants it is assumed that the activated complex must move on to form products; similarly, the activated complex formed from products must turn into reactants. At complete equilibrium, the forward rate and the reverse rate are equal, as discussed in the previous chapter, and then the concentration of activated complex formed from reactants equals that of the activated complex formed from products. If we next consider the condition that the equilibrium is

Number frequency

T1

T2

Velocity

FIGURE 5.4 Schematic picture of the Maxwell–Boltzmann distribution of the number frequency of molecules having a certain velocity at two temperatures, T1 < T2.

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disturbed, e.g., by removing products, the concentration of activated complex molecules formed from products will be reduced but not the concentration of activated complex formed from reactants. Clearly, this is not an equilibrium as usual, and therefore it is called quasiequilibrium. This is one of the postulates of transition state theory and many modiﬁcations have been proposed. We will not go into these details, but it is remarkable to note that the basic tenets of the original transition state theory are still valid. The rate at which the equilibrium depicted in Equation 5.3 is established is assumed to be fast compared with the rate of conversion of ABz to P, Q, so the position of the equilibrium is not perturbed signiﬁcantly. To be sure: ABz is not an intermediate that can be isolated readily; rather, it represents the conﬁguration of molecules at the moment of collision. The interesting aspect of transition state theory is that it connects kinetics and thermodynamics. On the one hand, the quasiequilibrium between the transition state and the reactant molecules is considered as a thermodynamic equilibrium, so that we can postulate a dimensionless thermodynamic equilibrium constant Kz based upon activities (cf. Equation 3.100): [ABz ] a yz ¼ c Kz ¼ aA aB [A] [B] yA yB c c z

(5:4)

As discussed in Chapter 3, aA ¼ yA[A]=c, with yA the activity coefﬁcient for the molar scale. (A thermodynamic constant is dimensionless, as discussed in Chapter 3, so the concentrations are made dimensionless by dividing by the standard concentration c.) On the other hand, the formation of products out of activated complex is treated using the law of mass action (discussed in Chapter 4) and the rate is considered accordingly to be proportional to the concentration of activated complex r / [ABz ] ¼ kz [ABz ]

(5:5)

Thus, the formation of products is considered to be unimolecular characterized by a rate constant kz. Incidentally, one may wonder whether we should not use the activity of the activated complex rather than the concentration in Equation 5.5, but because of the postulate of the quasiequilibrium between reactants and activated complex it should be concentration indeed. As a reminder, this postulate is that all activated complex molecules, once formed out of reactants, should move to products, and therefore it is the number of molecules that is important, not their activity. Considerations based on statistical mechanics result in kz ¼

kB T hP

(5:6)

so that, by combining all this with Equation 5.4, the rate of product formation r is r¼

d[P] kB T kB T z yA yB [ABz ] ¼ K [A] [B](c )1 z ¼ kz [ABz ] ¼ y dt hP hP

(5:7)

where kB is the Boltzmann’s constant (1.3807 3 1023 J K1), hP is the Planck’s constant (6.626 3 1034 J s), and T as usual the absolute temperature (K). kz has dimension of frequency (s1). An interesting feature is the appearance of the activity coefﬁcients of reactants and that of the activated complex in the rate equation. This is very relevant for reactions in solutions, as we shall see in later chapters, much less so for reactions in the gas phase where the activity coefﬁcients are usually close to 1. In solutions this would be true only for ideal solutions, which are in practice only very diluted solutions Comparing Equation 5.7 with a ‘‘normal’’ rate equation for a bimolecular reaction, such as the one in Equation 4.50, shows that for ideal systems the rate constant kid can be expressed as

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kid ¼

kB T z 1 K (c ) hP

(5:8)

However, for nonideal solutions the relation becomes the observed bimolecular rate constant kobs kobs ¼

kB T z 1 yA yB yA yB K (c ) ¼ kid z z y y hP

(5:9)

As shown in Equation 3.113, the thermodynamic equilibrium constant relates to the standard Gibbs energy change, and consequently the enthalpy and entropy, of activation, as follows: z DGz DS DH z ¼ exp exp K ¼ exp RT R RT z

(5:10)

Combining Equation 5.8 with Equation 5.10 gives then as a general equation kid ¼

z kB T DS DH z exp (c )1Dm exp R RT hP

(5:11)

A similar equation would be found for the nonideal case by combining Equation 5.9 with Equation 5.10. The factor (c)1Dm is necessary to obtain the right units for rate constants. c is the concentration in the standard state, chosen as 1 mol dm3 as discussed in Chapter 3, and Dm is the molecularity (in this case (D)m ¼ 2, see Equation 5.3). The above derivation is based on a bimolecular reaction mechanism. For monomolecular reactions, the same reasoning can be followed, by assuming that the activated complex is formed because of frequent collisions with solvent molecules, related to the above mentioned cage effect. This is called the Lindemann postulate. Equation 5.11 has the correct units for a rate constant of any order because of the factor (c)1Dm, the concentration in the standard state to which the thermodynamic parameters are referred. Coming back again to the discussion on thermodynamics in Chapter 3, DGz should be seen as the standard Gibbs free energy change which would occur if 1 mol of reactants is completely converted into 1 mol of transition state (at the speciﬁed temperature and standard state). In other words, it is the change in the value of free energy between these two states at the extreme ends of a possible process; it does not vary with the progress of the reaction as the free energy of a reaction does (which does decrease!). Equation 5.11 is sometimes referred to as the Eyring equation, after one of the developers of the transition state theory. The importance of this equation is that it relates the effect of temperature on the reaction rate constant to fundamental terms of enthalpy and entropy changes. If, for instance, a high enthalpy of activation exists, this would make the reaction quite slow at moderate temperatures, but this may be compensated by a large activation entropy, whereby the reaction can still proceed at a measurable rate. A striking example of such a phenomenon is the unfolding of proteins, to be discussed in more detail in Chapter 10. This indeed requires a high activation enthalpy because of the high number of bonds being broken simultaneously upon unfolding but, at the same time, the entropy of the unfolded chain increases enormously. In other words, high activation enthalpies and entropies are characteristic for protein unfolding. On the other hand, bimolecular reactions usually have a negative activation entropy (entropy of the two reactants is lost because of bond rearrangements and bond formation). The energy released and needed in breaking and forming bonds results usually in a moderate activation enthalpy. Monomolecular reactions are usually characterized by a moderate activation entropy (either slightly negative or positive, depending on intramolecular changes, the exception being protein unfolding) and an activation enthalpy depending on the type of mechanism.

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5-8 −34 −34.5

ln (khp / kBT )

−35 −35.5 −36 −36.5 −37 −37.5 2.35

2.4

2.45

2.5 103/T

2.55

2.6

2.65

(K–1)

FIGURE 5.5 Eyring plot depicting the temperature dependence of deamidation reactions of caseinate solutions; results are shown for three different initial concentrations. Dataset is given in Appendix 5.1, Table A.5.3.

As it happens, most chemical reactions, though not all, will have a moderate activation entropy so that differences in rate constants between chemical reactions are for the most part determined by differences in activation enthalpy. As an example, reaction rate constants were determined for the heat-induced deamidation in aqueous solutions of caseinate in the range between 1108C and 1458C. The resulting logarithmic plot (according to the natural logarithm of Equation 5.11) is shown in Figure 5.5. Usually, the activation parameters are estimated via linear regression of the logarithmic plot as given in Figure 5.5. For statistical reasons to be discussed in Chapter 7, it is better to use nonlinear regression. An estimate of the activation enthalpy from these data via nonlinear regression is DHz ¼ 92.0 13.6 kJ mol1 and DSz ¼ 69.9 13.8 J mol1 K1 (95% conﬁdence interval). The negative activation entropy is consistent with a bimolecular reaction of hydrolysis of amides. The activation enthalpy and entropy are usually assumed to be independent of temperature, which in general is probably not true, but for the heat treatment of foods the temperature range is mostly not so large on the absolute temperature scale, so the approximation may then hold. A notable exception is, again, protein unfolding in an aqueous environment, because interaction with water comes into play. Upon unfolding, hydrophobic groups are exposed and cause increased structuring of water. There is thus also a contribution of enthalpy and entropy changes of the solvent water that may oppose the positive enthalpy and entropy for protein unfolding. The difference in heat capacity between unfolded and folded (native) proteins is quite large, resulting in temperature dependency of (activation) enthalpy and entropy. In general, when it comes to hydrophobic bonds, their temperature dependence is quite intricate. A very schematic impression is given in Figure 5.6, illustrating that hydrophobic bonds increase with temperature (i.e., free energy becomes more negative), especially between 08C and 608C, but also at higher temperature.

5.4

Arrhenius’ Law

Arrhenius’ law was derived empirically, but it has proven to be very worthwhile in chemical kinetics. Arrhenius’ law states that

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20 15

T∆S

Energy (kJ mol−1)

10 5 ∆Η 0 −5 −10

∆G

−15 −20

0

50 T ⬘(⬚C)

100

FIGURE 5.6 Highly schematic drawing of the change in free energy (DG), the change in enthalpy (DH), and the change in entropy (TDS) with temperature for hydrophobic bond formation.

Ea k ¼ A exp RT

(5:12)

where A is a so-called preexponential factor (sometimes called the frequency factor), and Ea is the activation energy. The dimension of A should be the same as that of the rate constant k; it therefore does have units of frequency only in the case of a ﬁrst-order reaction. It is very instructive to compare Arrhenius’ law, Equation 5.12, with the expression derived from transition state theory Equation 5.11. Obviously, Ea relates to the activation enthalpy DHz and the exact relationship is found as follows. From Equation 5.8 it follows that ln k ¼ ln

kB Kz þ ln T þ ln 1m hP (c )

(5:13)

hence d ln k d ln K z =(c )1m ¼ T þ d(1=T) d(1=T)

(5:14)

and combining the temperature effect on Kz as displayed in Equation 5.2 gives d ln k DH z ¼ T R d(1=T)

(5:15)

From the Arrhenius Equation 5.12 it follows that d ln k Ea ¼ R d(1=T)

(5:16)

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and consequently combining Equation 5.15 with Equation 5.16 results in DH z ¼ Ea RT

(5:17)

The observant reader may ﬁnd this result unexpected because of the deﬁnition of enthalpy given in Equation 3.17 H ¼ E þ PV

(5:18)

PDV ¼ RTDn

(5:19)

From the general gas law it follows that

with Dn the change in number of moles, so that it would follow that DH z ¼ Ea þ DnRT

(5:20)

However, Dn should be seen here as the change in number of moles in going from reactants to the activated state. Hence, for bimolecular reactions Dn ¼ 1, so that Equation 5.17 is indeed seen to be correct. For monomolecular reactions, Dn ¼ 0, so that in that case the activation energy equals the activation enthalpy. As mentioned, the activation energy can be seen as the energy barrier that molecules need to overcome in order to be able to react. As shown in Figure 5.4, the proportion of molecules able to do that increases with temperature, which qualitatively explains the effect of temperature on rates. The Eyring and Arrhenius equation give a quantitative account. The preexponential factor A is seen to be related to z kB T DS exp the activation entropy DSz A ¼ and this comparison makes the preexponential factor R R A much more comprehensible. The physical meaning of A as such seems to be experienced as somewhat vague, which probably accounts for the fact that the factor A very often is not reported as a result in food science literature. It gives, however, as much useful information as Ea does. An interpretation of A is that it represents the rate constant at which all molecules have sufﬁcient energy to react (i.e., Ea ¼ 0). In principle one could even envisage a negative activation energy, namely if molecules attract each other. This could be the case for positively and negatively charged reactants. Another difference between Arrhenius’ and Eyring’s expressions is that the temperature T appears in the preexponential factor in Eyring’s equation (Equation 5.11). This has a consequence in the way results are presented and analyzed. Very often, Arrhenius law is presented as a plot of ln k versus 1=T, which should result in a straight line (if the relationship holds). With Eyring’s relationship, ln(k=T) versus 1=T should be plotted (as is done in Figure 5.5). We would like to remark here that it is not a good idea to derive the activation energy parameters from linear regression of ln k or ln(k=T) versus 1=T because of the weighting of data points through logarithmic transformation; rather, nonlinear regression should be used, as is discussed in Chapter 7 on kinetics and statistics. Another remark in this respect is that the twostep procedure of ﬁrst deriving rate constants and then regressing them versus temperature results usually in very wide conﬁdence intervals if only three to four temperatures have been studied, as is frequently the case. A better approach is to substitute the rate constant in the appropriate rate equations using Equation 5.11 or Equation 5.12 and perform a nonlinear regression. In this way, all data are used to estimate the activation parameters at once and an estimate of these parameters of much higher precision is obtained. This is called global ﬁtting and will also be demonstrated in Chapter 7. It probably remains a good idea to present Arrhenius’ or Eyring’s expression in the form of a plot of ln k or ln(k=T) versus 1=T because any deviation of the data from these expressions becomes immediately apparent. In doing so,

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however, the values of the parameters estimated by nonlinear regression should be used to construct the plot. Discrepancies between experimental data and Arrhenius’ and Eyring’s relationship are indeed possible and it is the responsibility of the researcher to check this. In the case that they are not applicable (for instance, because an undetected change in mechanism occurs at the higher temperatures), the resulting parameter estimates are worthless. So, the ﬁrst step should always be to check the validity of the laws of Arrhenius=Eyring, and only if they appear to be correct the next step would be the estimation of the activation parameters. Obvious as this may seem, this rule is not always obeyed. Temperature dependence of complex reactions. It is essential to realize that the concept of transition state theory or Arrhenius law is strictly speaking only valid for elementary reactions. In the case of a complex reaction in which an observed rate constant is actually composed of several elementary ones the meaning of an activation energy determined from such rate constants becomes a bit blurred. Let us consider, for instance, the reaction k1

A þ B ! C k2

(5:21)

k3

C þ D ! P If this is the correct mechanism, the rate of formation of product P, d[P]=dt is d[P] ¼ k3 [C] [D] dt

(5:22)

However, if we can determine only component P, we are not in the position to estimate rate constant k3 because then we would need to determine compounds C and D as well (if we can do that we can apply multiresponse modeling to be discussed in Chapter 8). Let us suppose for the moment that we can only observe the formation of component P, and we could try to model it with a pseudo-ﬁrst-order reaction, for instance. The observed rate constant will then be a composite of the three elementary rate constants k1, k2, and k3. By postulating that the intermediate compound C is following a steady state after the initial start up of the reaction, we can investigate how the observed rate constant is related to the elementary ones. The steady state implies that d[C]=dt 0 d[C] ¼ k1 [A] [B] k2 [C] k3 [C] [D] ¼ 0 dt

(5:23)

Hence [C] ¼

k1 [A] [B] k2 þ k3 [D]

(5:24)

Substituting Equation 5.24 in Equation 5.22 yields d[P] k1 k3 [A] [B] [D] ¼ dt k2 þ k3 [D]

(5:25)

Now we can investigate some possibilities to see how that works out for the rate equation. First, let us assume that k2 k3[D]. It then follows that d[P] ¼ k1 [A] [B] dt

(5:26)

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We then have a ‘‘normal’’ second-order rate equation, and the rate of formation of P is completely determined by the rate of disappearance of components A and B. The observed rate constant will correspond to the elementary rate constant k1 and the temperature dependence should be Arrheniuslike. Another possibility could be that k2 >> k3[D]. It then follows that d[P] k1 k3 [A] [B] [D] ¼ k0 [A] [B] [D] ¼ dt k2 k1 k3 k0 ¼ k2

(5:27)

and we end up with a third-order rate equation where the rate constant k0 is clearly a composite one. The question is now how temperature affects this composite rate constant. If the three elementary rate constants k1, k2, k3 each obey Arrhenius law, we can apply Equation 5.16 to ﬁnd

d ln k0 E0 d ln (k1 k3 =k2 ) d( ln k1 þ ln k3 ln k2 ) Ea1 þ Ea3 Ea2 ¼ a¼ ¼ ¼ d(1=T) R R d(1=T) d(1=T)

(5:28)

In words, the activation energy determined from the observed rate constant is composed of the activation energies for the underlying elementary reactions. This can lead to unexpected results such as that a reaction decreases in rate with increasing temperature. A negative activation energy is not well conceivable for an elementary reaction (unless it concerns two oppositely charged reactants), but for a composite reaction this can happen if Ea2 > Ea1 þ Ea3 according to Equation 5.28. The question is, of course, when it will happen that a situation occurs as in the above example that k2 >> k3 [D]. In order to investigate this, we performed some simulations based on the mechanism depicted in Equation 5.21. The requirement Ea2 > Ea1 þ Ea3 implies that the preexponential factor must be very high to compensate for the higher activation energy (otherwise k2 will be of no signiﬁcance). Table 5.1 shows a possible condition where this would happen. The reaction was simulated via numerical integration of the differential equations representing the rate equations, using the values shown in Table 5.1 to calculate the rate constants via Arrhenius’ equation. The initial concentrations of A and B were arbitrarily set at 100 units and that of component D at 150 units. It was indeed found that the intermediate C showed steady-state behavior and with the numerical values chosen it was also true that k2 >> k3 [D]. It was then assumed that only compound P was experimentally accessible, so that the simulated experiment consisted of concentration–time proﬁles for various temperatures; Figure 5.7 shows some results. These proﬁles were ﬁtted to a ﬁrst-order rate equation; a reasonable ﬁt was obtained (but not perfect because the actual mechanism is more complex than a ﬁrst-order reaction as shown in Equation 5.21). The so-derived rate constants were then plotted according to the Arrhenius equation (see Figure 5.8). It is indeed found that the rate constants decrease with increasing temperature at the higher temperatures. Admittedly, this may be a rather extreme example because of the requirement that k2 >> k3 [D]. The purpose is, however, to show that the activation energy determined from observed reaction rate constants may not reﬂect a single energy barrier. Thus, one should be aware of this when determining activation parameters from observed rate constants that could be composed of several elementary rate constants, certainly in foods where it is difﬁcult to study elementary reactions. TABLE 5.1

Values Used in the Arrhenius Equation for Simulation of the Complex Reaction in Equation 5.21 Reaction Step 1

Reaction Step 2

Reaction Step 3

A

1 3 1010 dm3 mol1 s1

1 3 1026 s1

1 3 1010 dm3 mol1 s1

Ea

80 kJ mol1

150 kJ mol1

60 kJ mol1

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90

T = 353 K

80

T = 413 K

[P] (arbitrary units)

70

T = 333 K

60 50 40

T = 313 K

30 20 T = 293 K

10 0 0

5

10

15

20

25

Time (arbitrary units)

FIGURE 5.7 Simulated concentration proﬁles for compound P in the reaction mechanism displayed in Equation 5.21 using the numerical values shown in Table 5.1.

Reparameterization. It is possible to reparameterize the Arrhenius or Eyring equation; it is actually desired from a statistical point of view as will be discussed in Chapter 7. A very simple reparameterization is to introduce a reference temperature Tref. The basis for this arises from the application of Equation 5.12 at two temperatures T1 and T2, assuming that the preexponential factor and Ea do not depend on temperature

0 −1 −2

ln k

−3 −4 −5 −6 −7 −8 −9 0.002

0.0025

0.003 1/T

0.0035

0.004

(K −1)

FIGURE 5.8 Arrhenius plot of the ﬁrst-order rate constants found from the simulated concentration proﬁles shown in Figure 5.7 using the numerical values displayed in Table 5.1.

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Ea k1 ¼ A exp RT1 Ea k2 ¼ A exp RT2 If one arbitrarily chooses a reference temperature, say T2 ¼ Tref, one can combine these two equations: k Ea 1 1 ¼ exp R T Tref kref

(5:29)

The actual result of this is that the preexponential factor is replaced by a rate constant at some reference temperature. The reference temperature should preferably be chosen in the middle of the studied temperature regime. Magnitudes of rate constants. It is perhaps instructive to return brieﬂy to the actual values that rate constants can take. Table 5.2 shows orders of magnitude for rate constants, depending on conditions. The very large effect of activation energy on the rate of a reaction is apparent. In fact, without activation barriers, reactions would be so fast that foods would spoil immediately, and there would be no such thing as food technology, or life for that matter. It is just another way of saying how important kinetics is for processes that are studied by life sciences, including of course food science. Generally, the temperature effects as laid down in the Arrhenius and Eyring relationships seem to ﬁt quite well to chemical reactions in foods. As mentioned above, it should be realized, however, that observed rate constants are more often than not reﬂecting more than one elementary reaction, and one has to be careful with interpretation. Some typical values for activation enthalpies=energies and entropies for possible reactions in foods are given in Table 5.3. Several physical reactions are less temperature-dependent and often diffusion controlled; diffusioncontrolled reactions are discussed in Chapter 4. Transition state theory does not apply actually to physical reactions (such as coalescence, aggregation) because there are no molecular rearrangements. However, physical phenomena do usually have an energy barrier (due to, for instance, electrostatic repulsion), which provide stability to colloidal systems. Hence the concept of a kind of activation energy does apply but not with a temperature dependence as occurs for chemical reactions. The effect of temperature will be for the larger part on the rate of encounters. Sometimes, activation energies are reported for physical phenomena such as the temperature dependence of diffusion or viscosity. This seems to be impossible,

TABLE 5.2 Orders of Magnitude for Rate Constants of Bimolecular Reactions in Aqueous Solutions at 258C Order of Magnitude of Rate Constant (dm3 mol1 s1)

Conditions No diffusion limit and no barriera Diffusion limit, no activation energy

1014 1010

No diffusion limit Activation energy 25 kJ mol1 1

Activation energy 50 kJ mol

1

Activation energy 100 kJ mol a

1010 105 104

This is in fact the value of the preexponential factor in the Arrhenius equation, corresponding to the hypothetical situation that T ! 1.

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TABLE 5.3 Order of Magnitudes for Some Typical Food Reactions for Activation Enthalpy (DHz), Activation Energy (Ea), Activation Entropy (DSz), and the Preexponential Factor (A)

Type of Reaction

DHz=Ea (kJ mol1)

DSz (J mol1 K1)

A s1 (First Order) dm3 mol1 s1 (Second Order)

First-order chemical reaction

50–150

0

1013 to 1014

Second-order chemical reaction

50–150

50

1011 to 1012

200–500

500

1030 to 1040

Protein denaturation Microbial inactivation (Enzyme) catalyzed reactions

a

200–500 10–50

a

500 100 to þ50

1030 to 1040 1011 to 1012

Photochemical reactions

0–10?

?

?

Radical reactions

0–10?

?

?

Note: ? indicates not really known. a Not really an activation energy=enthalpy=entropy but a temperature coefﬁcient (see text).

since there is nothing to activate and there is no reaction. As discussed earlier, the point is that the temperature dependence of, for instance, diffusion apparently obeys Arrhenius’ law in several systems, but the parameter that comes out of it does not have the physical meaning of an activation energy.

5.5

Empirical Relations to Describe Temperature Dependence

As mentioned in the previous section, the relations derived from transition state theory, and to some extent the Arrhenius equation as well, are only valid for elementary reactions. In cases where temperature dependences need to be described for nonelementary reactions, one could use empirical relationships. There are also other Arrhenius-like equations proposed in the literature that could be used just as well, but which are not commonly used. The following equation would do equally well as the Arrhenius equation: B k ¼ A exp T

(5:30)

with A and B as ﬁt parameters without a physical meaning. Although this seems undesirable, one has to realize that sometimes parameters do not really have a physical meaning. For instance, if one determines an activation energy for microbial inactivation, what does it mean if an activation energy of, say 300 kJ mol1, has been derived? A mole of bacteria is somewhat hard to envisage. It would actually be better to use Equation 5.30 for phenomena that do show Arrhenius-like behavior but do not really reﬂect a deﬁned chemical reaction. For instance, as mentioned in the previous section, the effect of temperature on diffusivity can often be described using the Arrhenius equation, but there is no activation energy for molecular mobility (though there may be barriers), and therefore it does not make much sense to report an activation energy for diffusion; temperature coefﬁcients A and B like in Equation 5.30 seem more appropriate. Another example of an alternative for the Arrhenius equation is a log–logistic relationship k ¼ m0 ln (1 þ exp [c(T Tc )])

(5:31)

c (8C1), m0 (-), and Tc (8C1) are empirical ﬁt constants, and in many cases it can be assumed that m0 ¼ 1. Such an equation accounts for phenomena with a high rate at high temperature and a slow one at low temperature. The Arrhenius equation, of course, does the same, but an equation such as Equation 5.31 does not need the concept of activation energy. Figure 5.9 shows an example of the ﬁt of the log– logistic equation (Equation 5.31) for degradation of riboﬂavin in heated spinach.

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0.00008 0.00007 0.00006

k (s−1)

0.00005

0.00004 0.00003 0.00002 0.00001 0

0

20

40

60

80

100

120

140

T ⬘(⬚C)

FIGURE 5.9 Fit of the log–logistic equation (Equation 5.31) to rate constants describing the degradation of riboﬂavin in heated spinach. The ﬁt parameters are m0 ¼ 1, c ¼ 0.020, Tc ¼ 6038C. Dataset in Appendix 5.1, Table A.5.4.

There are also empirical models that describe temperature dependence of viscosity in foods undergoing a glass transition. The Williams–Landel–Ferry (WLF) model is often used to model such dependence. It is reported that Arrhenius=Eyring models do not function well for such cases. The WLF model is developed for polymers but nowadays also applied to foods that undergo glass transitions. The WLF model is actually not comparable to the Eyring equation, because it does not consider transition states; it is of empirical nature and attempts to model viscosity as a function of temperature. The WLF model is hv ln hv,g

! ¼

C1 (T 0 Tg0 ) C2 þ (T 0 Tg0 )

(5:32)

C1 (dimensionless) and C2 (8C) are parameters of the WLF model, Tg0 is the glass transition temperature (8C), and hv,g is the viscosity at the glass transition temperature. The WLF model will be discussed in more detail in Chapter 14.

5.6

Activation Energy and Catalysis

The important phenomenon of catalysis was discussed in Chapter 4. It was indicated that a catalyzed reaction does not differ in terms of the equilibrium position with respect to the uncatalyzed reaction (hence the change in standard Gibbs energy remains the same), but it increases the rate (which was loosely translated as the activation energy barrier being lowered). Now that we have discussed the concept of activation energy in terms of transition state theory and the Arrhenius equation, it is perhaps

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Standard Gibbs energy

Ea, uncatalyzed

Activation energy

Ea, catalyzed A ∆G

B

FIGURE 5.10 Incorrect representation of the lowering of the activation energy of a catalyzed reaction. A þ C ! B þ C (C represents catalyst).

useful to reconsider the effect of catalysis on activation energy in more detail. A general misconception is that a catalyst just lowers the activation energy barrier of the same elementary step as that of the uncatalyzed reaction as depicted in Figure 5.10. This is deﬁnitely not the case because a catalyzed reaction always consists of more than one step, namely the reaction of the catalyst with the reactants and the release of the catalyst from the activated complex (as a results of which products are formed and the catalyst is regenerated). So the situation becomes more complex, and could be something like the one depicted in Figure 5.11. A catalyzed reaction has thus different reaction proﬁles than its uncatalyzed counterpart. A catalyst not only catalyses the forward reaction but also the reverse reaction. It may be useful to come back brieﬂy to the concept of a rate-limiting step as discussed in Chapter 4. Sometimes, it is

Standard Gibbs energy

Standard Gibbs energy

A‡

AC ‡1

AC ‡2

∆G ‡ ∆G ‡1 A

∆G ‡2

A ∆G

B

Uncatalyzed reaction

B

Catalyzed reaction

FIGURE 5.11 Likely representation of the change in activation energy due to catalysis. The uncatalyzed reaction is A ! B, the catalyzed reaction A þ C ! B þ C (C represents catalyst).

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attempted to identify rate-limiting steps from the steps having the highest activation energy. This does not always lead to correct conclusions. Activation energy relates to a rate constant, not to a rate. It should be remembered that a rate is determined by the product of rate constant and concentration, and not just by the height of an activation energy barrier.

5.7

Parameters Used in Food Science

The parameters that have been discussed so far, orders, rate constants, activation parameters, etc., are actually all that is needed in (chemical) kinetics. However, it has become the habit to use several other kinetic parameters in food science. They originate from days gone by when it was necessary to derive parameters and models to describe (mainly microbial) changes in foods during processing and storage when no use was made of modern reaction kinetics. All these parameters can be related to the more fundamental parameters that we have discussed so far. We give a brief overview of these parameters so that the reader can see how they relate to the fundamental parameters discussed above. The parameter Q10 describes the temperature dependence of a reaction as the factor by which the reaction rate is changed when the temperature is increased by 108C. If we link that to the reaction rate constant, it can be expressed as Q10 ¼

kTþ5 kTþ10 kT5 kT

(5:33)

If the Arrhenius equation holds, it can be shown that 10Ea Q10 ¼ exp RT 2

(5:34)

The parameter is thus seen to be a rather poor measure of the activation energy, and a serious drawback is its quite strong temperature dependence especially at higher Ea (Figure 5.12). Another drawback of the Q10 parameter when linked to the activation energy is that it does not incorporate the preexponential factor or the activation entropy. So, it makes only sense to use the Q10 parameter for reactions that do not differ too much in activation entropy=preexponential factor. 80 70 Ea = 300 kJ mol–1 60

Q10

50 40 30 Ea = 200 kJ mol–1

20 10

Ea = 100 kJ mol–1

0 0

50

100

150

T⬘(⬚C)

FIGURE 5.12

Temperature dependence of the parameter Q10 as a function of the underlying activation energy.

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40 35 Ea = 100 kJ mol–1 30

Z (⬚C)

25 20

Ea = 200 kJ mol–1

15

Ea = 300 kJ mol–1

10 5 0 0

50

100

150

200

T ⬘(⬚C)

FIGURE 5.13 Temperature dependence of the parameter Z as a function of the underlying activation energy.

Another parameter to describe temperature dependence is Z, which expresses the increase in temperature that would produce an increase in rate by a factor of 10. Z is deﬁned as Z¼

2:303RT 2 10 ¼ Ea log Q10

(5:35)

Like the parameter Q10, Z is temperature-dependent which restricts its use (Figure 5.13). Z is frequently used in bacteriology to describe inactivation of microorganisms. Also used is the parameter D, especially in thermobacteriology. It is the decimal reduction value, the time needed to reduce a concentration by a factor of 10. D is nothing else than an inverse rate constant. For a ﬁrst-order reaction: D¼

2:303 k

(5:36)

9 c0 k

(5:37)

and for a second-order reaction: D¼

A plot of log D versus T 0 (in 8C) is usually taken to be a straight line (for a limited temperature range), see Figure 5.14. D relates to the Z value, like k is related to Ea: log D ¼ log Dref

0 T 0 Tref Z

(5:38)

0 (often chosen as 2508F for historical Dref is the reference value of D at the reference temperature Tref reasons, which is equal to 121.18C). Equation 5.38 is referred to as the TDT curve (thermal death time curve) or the Bigelow model.

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2 log D

1 0 −1 −2

Z T ⬘(⬚C)

FIGURE 5.14

Schematic example of a TDT curve and interpretation of the Z-value.

As shown, all these parameters can be linked to the more fundamental kinetic parameters. They still serve a purpose. On the one hand, they are usually estimated in real foods and as such reﬂect a time– temperature dependence characteristic (not pretending it is something like an activation energy) that can be used for engineering purposes; less so, however, for understanding behavior at the molecular level. On the other hand, the parameters are used commonly by regulatory agents in food safety programs in relation to thermal treatments. We come back to them brieﬂy in Chapter 13 when we discuss thermobacteriology. Typical temperature effects for reactions in foods. When the effect of temperature on reactions in foods has been established, preferably in the form of the parameters discussed if it concerns elementary reactions, i.e., activation energy=enthalpy and activation entropy=preexponential factor, the value of the parameters needs some discussion. Occasionally, there seems to be some misunderstanding regarding interpretation of activation parameters. For instance, if a high activation energy is found, the conclusion is sometimes drawn that the reaction will proceed slowly or difﬁcult. This is not necessarily true, because the reaction may proceed quite fast at very high temperature if the value of the preexponential factor A or, equivalently, DSz is high. Furthermore, if a high activation energy goes along with a high preexponential factor, the reaction may still proceed at a noticeable rate. The point is that a high activation energy indicates a strong temperature dependence, that is to say it will run very slowly at low temperature, but relatively fast at high temperature. Relevant for foods is that chemical reactions (e.g., the Maillard reaction) have a ‘‘normal’’ activation energy of about 100 kJ mol1, whereas inactivation of microorganisms can be characterized by a high activation energy, say, 300 kJ mol1 (even though, as already mentioned, it is incorrect to express it in this way; we will come back to this in Chapter 13). Figure 5.15 illustrates this. Such differences in activation energy are exploited in processes such as high-temperature short-time heating (HTST) and ultrahigh-temperature treatment (UHT). These processes are designed by choosing such time–temperature combinations that desired changes are achieved (microbial inactivation) while undesired changes (chemical reactions leading to quality loss) are minimized as much as possible. Another important consequence for foods is that reactions with relatively low activation energy will continue at a measurable rate at low temperatures, for instance during storage, leading to a limited shelf life. Quite different results are obtained with protein denaturation and microbial inactivation. (Microbial inactivation is according to some authors due to enzyme, i.e., protein, denaturation. It is questionable whether this is the sole cause of inactivation. We will come back to this in Chapter 13 on kinetics of microbial inactivation.) Protein denaturation is characterized by a high activation enthalpy=energy and

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ln k

Microbial inactivation

Chemical reaction

UHT region

1/T

FIGURE 5.15 Schematic presentation of the temperature dependence of a chemical reaction and microbial inactivation. The UHT region is characterized by time–temperature combinations that induce enough microbial inactivation and as little as possible chemical reactions.

this is compensated for by a high activation entropy=preexponential factor. As a result, the temperature dependence of such reactions is very high, much higher than that of chemical reactions. A potential pitfall in the study of denaturation of proteins is the way in which protein denaturation is studied, as discussed in Chapter 10. Frequently, for instance, the protein aggregation that is measured results from protein denaturation. The resulting kinetic parameters are then a combination of unfolding and aggregation. It may happen that at the lower temperatures unfolding is the rate-limiting step, while at the higher temperatures aggregation becomes the rate limiting step. As a consequence, a change in temperature dependence will be seen. It may even happen that the rate of aggregation is the rate-limiting step throughout and then the kinetics of aggregation is established rather than kinetics of denaturation. With biochemical reactions, i.e., enzyme-catalyzed reactions, moderate temperature dependence is found, as is to be expected for catalyzed reactions. However, with enzyme-catalyzed reactions, enzymes become inactivated above a certain temperature, and the catalyzed reaction comes effectively to an end. Most enzymes relevant in food tend to become inactivated between 508C and 808C, though some notably heat-resistant enzymes are known. The same goes for microbial growth: ﬁrst there is an increase with temperature but eventually microbes start to die. A highly schematic picture of the effect of temperature on microbial growth and enzyme action alike is in Figure 5.16; it should be noted that the actual response to temperature can be time-dependent. In the case of microorganisms, there is also a minimum temperature below which there is no growth. Photochemical reactions and radical reactions are not or only weakly temperature-dependent because the changes at the molecular level do hardly depend on thermal energy. Both types of reactions are of importance in foods, as discussed to some extent in Chapter 4. Photochemical reactions cause for instance oxidation of vitamins, they may activate certain enzymes, and they may cause ﬂavor defects. Radical reactions are most notable for oxidation reactions (of unsaturated fats, of vitamins).

5.8

Enthalpy=Entropy Compensation

An effect called enthalpy=entropy compensation, or isokinetic effect, is claimed to exist for some reactions in foods. The phenomenon was ﬁrst described in organic chemistry and would arise from a systematic variation in solvent composition, reactant molecule change (as in homologous series), pH, or

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Temperature optimum

Ln (rate)

Activity

Inactivation

1/T

FIGURE 5.16 Schematic presentation of the effect of temperature on growth and activity of enzymes and microorganisms and their inactivation.

dielectric strength. If DHz is plotted versus DSz in such situations, a straight line through the origin is found with a slope that has the dimension of absolute temperature, and this temperature is called the isokinetic temperature. The relationship is called enthalpy=entropy compensation. The mechanistic signiﬁcance is that at the isokinetic temperature all medium effects (such as pH and solvent effects) on the free energy change would vanish. This is so because a straight line through the origin implies that DH z ¼ Tiso DSz

(5:39)

with Tiso the isokinetic temperature and since DGz ¼ DH z TDSz

(5:40)

it follows that DGz ¼ 0 at T ¼ Tiso. Examples in food for which enthalpy=entropy compensation is claimed are protein denaturation, disaccharide hydrolysis at various pH values, ascorbic acid degradation at various water activities, degradation of chlorophylls, and chlorophyllides and carotenoids stability with change in solvent. There is, however, considerable debate in the literature on the question whether a compensation effect actually exists or merely is a statistical artefact. The problem is that when enthalpy and entropy values are measured from real data (i.e., containing experimental error) these parameters are highly correlated in a statistical sense, resulting from the propagation of errors (see also Chapter 7). The compensation effect does thus not necessarily imply a chemical causation. It has been shown that estimates of DGz and DHz do not suffer from this error correlation problem if evaluated at the harmonic mean of the range of experimental temperatures. The harmonic mean Thm is the reciprocal of the mean of the reciprocal temperatures Thm ¼

1 1 T

(5:41)

So, if there still appears to be correlation between DGz and DHz evaluated at Thm, a true chemical effect is found. We will not further go into details here, but the reader should be warned, when encountering reports on enthalpy=entropy compensation effects, that it could be a statistical artefact rather than a real chemical phenomenon, unless the researcher has worked with Thm. The problem of statistical correlation between parameters such as DHz and DSz is further discussed in Chapter 7.

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Variable Temperature Kinetics

In most of the experiments reported in literature, temperature was kept constant, in other words, these are isothermal experiments (in practice, it is difﬁcult to achieve this because heating-up and coolingdown times play a role). The temperature dependence of the rate constant is then estimated by ﬁtting the obtained rate constants at various temperatures using Equation 5.11 or Equation 5.12. However, it is also possible to vary the temperature in a controlled way, and to follow the concentration of the component of interest as a function of the varying temperature, which is in turn a function of time. Such experiments form the basis for variable temperature kinetics, also called nonisothermal kinetics. Let us take the general rate law equation (Equation 4.58) as a starting point

dc ¼ k(t)cn dt

(5:42)

We now have to write k(t) because the rate constant is a function of the varying temperature and hence a function of time. In other words, the Arrhenius equation becomes Ea k(t) ¼ A exp RT(t)

(5:43)

Integration of Equation 5.42, after some rearrangement, now results in ðc

1 dc ¼ A cn

c0

ðt

Ea dt exp RT(t)

(5:44)

0

One has to know the dependence of T on t, T(t), in order to be able to solve this equation. This dependency could be a temperature varying linearly with time T(t) ¼ T0 þ at

(5:45)

where the coefﬁcient a gives the rate of temperature change, or a quadratic change T(t) ¼ T0 þ at þ bT 2

(5:46)

or a sinusoidal change a 2pt 3p sin þ 2 L 2

(5:47)

T(t) ¼ Th þ (T0 Th ) exp (Jt)

(5:48)

T(t) ¼ Tm þ or an increase described by an exponential

In these equations T(t) represents temperature as a function of time, T0 the starting temperature, Tm the average temperature, and Th is the ﬁnal temperature. The parameter J in Equation 5.48 accounts for the heat transfer coefﬁcient and speciﬁc heat and mass of the product ﬂowing through a heat exchanger. To be sure, other equations than the ones given here are possible, as long as they describe the T-t proﬁle correctly. It is important to realize that the change in concentration is quite different for isothermal kinetics and nonisothermal kinetics. For a ﬁrst-order isothermal reaction it looks as depicted in Figure 5.17A, but for a linearly changing temperature the same ﬁrst-order reaction would look like the curve in Figure 5.17B. In the latter case, the reaction is slow at ﬁrst because the temperature is low but as the temperature rises the reaction rate increases, until the end when it decreases because the reactant becomes depleted (Figure 5.17C).

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1

0.8 0.6 0.4 0.2

0

250

C 0.8

200

T 0.6

150

0.4

100

0.2

50 0

0 0

5 10 Time (arbitrary units)

15

⬚C

Concentration (arbitrary units)

Concentration (arbitrary units)

1

(A)

21.10.2008 10:40pm Compositor Name: MSubramanian

0 (B)

5 10 Time (arbitrary units)

15

5

Rate (arbitrary units)

4 3 2 1 0 0 (C)

5 10 Time (arbitrary units)

15

FIGURE 5.17 Simulation of an isothermal ﬁrst-order reaction at 1438C (A), a nonisothermal ﬁrst-order reaction for Ea ¼ 100 kJ mol1, A ¼ 1 3 1010 s1 and a linear temperature increase from T 0 0 ¼ 1008C at a rate of 68C h1 (B), and the reaction rate at the nonisothermal conditions (C).

On the assumption that Ea and A do not depend on temperature and that one knows how temperature varies with time, one can derive the activation parameters directly from the concentration curve obtained from a nonisothermal experiment. An example of such a study is given in Figure 5.18 for sucrose hydrolysis at high temperature and pH 3.8, which is assumed to be a ﬁrst-order reaction. It is thus not really necessary to do isothermal studies in order to obtain activation parameters from the Eyring or Arrhenius equation, even though this is almost always done. It is true though that one needs numerical procedures and nonlinear regression techniques in order to obtain the desired estimates. This will be discussed in Chapter 7. Varying temperature kinetics is signiﬁcant for heat exchange processes such as pasteurization and sterilization. As an example, two heating proﬁles are depicted in Figure 5.19 for a UHT heating process. Incidentally, such a heating proﬁle can be described with Equation 5.48, typical for heating via an isothermal heat source. Usually one only accounts for the holding time, but the heating up and cooling down periods can sometimes be considerable so that reactions take place at a noticeable rate during these periods and should not be neglected. To account for this, it is convenient to introduce a so-called equivalent time teq to take heating-up and cooling down periods into account. This can be done by numerically integrating the right-hand side of

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10

140

9

T⬘

[Sucrose]

7

120

6 ⬚C

Sucrose (g dm−3)

8

5 4 100

3 2 1

80 200

0 0

50

100

150

Time (min)

FIGURE 5.18 Hydrolysis of sucrose (~) at pH 3.8 with varying temperature. The solid line indicates the ﬁt for a ﬁrst-order reaction in which the Arrhenius equation is incorporated; the dashed line indicates the temperature change. The estimates of the parameters via nonlinear regression are c0 ¼ 9.8 g dm3, A ¼ 2.9 3 1010 s1, and Ea ¼ 93.4 kJ mol1. Dataset in Appendix 5.1, Table A.5.5.

Equation 5.44, for instance by the trapezoid rule or Simpson’s rule, or using mathematical software. The numerical outcome is then equated to the same change that would have occurred in an isothermal process at a constant holding temperature Th ðt A 0

Ea Ea exp teq dt ¼ A exp RT(t) RTh

(5:49)

160 140 120

T ⬘ (⬚C)

100 80 Holding time

60 40 20 0 0

10

20

30

40

Time (s)

FIGURE 5.19 Table A.5.6.

Heating proﬁle during UHT heating up to 1208C (~) and 1408C (*). Dataset in Appendix 5.1,

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TABLE 5.4 Equivalent Times (teq), Holding Time (th), and Effective Heating Time (teff ¼ teq þ th) as a Function of Activation Energy Ea for the Heating Proﬁle and Holding Time at 1408C as Shown in Figure 5.19 Ea (kJ mol1)

teq (s)

th (s)

teff (s)

50

10.4

10

20.4

100

7.6

10

17.6

150

6.0

10

16.0

200

5.9

10

15.9

250 300

5.2 3.5

10 10

15.2 13.5

350

3.1

10

13.1

Thus, the equation can be solved for teq. As indicated by Equation 5.49 the equivalent time does neither depend on the preexponential factor nor on the order of the reaction, but it is essential to realize that the outcome does depend on the value of the activation energy. It is thus not allowed to generalize equivalent times for other reactions than the one that is studied. To illustrate this fact, equivalent times were calculated for the heating proﬁle as depicted in Figure 5.19 for 1408C (Table 5.4). As can be seen, the effective heating time is more than doubled for a low activation energy of 50 kJ mol1, reﬂecting the fact that reactions with a low activation energy can progress at a measurable rate at low temperature, i.e., at the heating-up and cooling-down periods. The effect becomes less for a higher activation energy, but cannot be neglected for a heating proﬁle such as the one in Figure 5.19. Table 5.4 demonstrates that chemical reactions, which have an activation energy between 50 and 100 kJ=mol, are more sensitive to nonisothermal treatments than reactions having a higher activation energy (such as protein denaturation). Another possible complication in heat exchangers is residence time distribution. If the ﬂow inside the exchanger is not turbulent, there may be considerable spread in the time during which elements of the heated material are subject to the heat treatment. In the case of turbulent ﬂow however, one can in most cases assume plug ﬂow, i.e., a more or less constant heating time for every part of the heated material. This is the preferred condition for heated foods, otherwise some parts of the heated foods are underheated, while other parts are possibly overheated. A serious limitation of variable temperature kinetics discussed hitherto is that the analysis is only valid for a single reaction. If other reactions start to interfere above a certain temperature, which is quite normal when working with foods, the above given analysis is no longer valid because it assumes only one reaction. A possible solution to this problem is to take these interfering reactions into account, and model them simultaneously. This is the domain of multiresponse modeling, to be discussed in Chapter 8. Still, for real foods this may be too complicated, not so much because of the fact that more reactions take place, but because these reactions may change the conditions and thus the course of the reaction. In other words, the occurrence of the next phase in the reaction may have been inﬂuenced by what has happened before. What is left then for engineering purposes is an empirical approach. Nonisothermal kinetics for empirical models. In the previous chapter, empirical models were introduced as an alternative to mechanistic (or presumed mechanistic) models. It is possible to model variable temperature situations also with these kinds of models without the use of the Arrhenius or Eyring equation, following a series of papers of Dr. Peleg and coworkers. As an example, the Weibull model was introduced as an empirical model in Equation 4.76 ct ¼ exp (bW t aW ) c0

(5:50)

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ct/c0

Slope = dc/dt T1

T2

t∗

t∗

t∗

Time

FIGURE 5.20 Schematic picture showing the relation between the time t*, momentary rate ct=c0, and isothermal rate at temperature T1 and T2.

This equation describes the change in ct=c0 as a function of time at otherwise constant conditions, such as a constant temperature. What we are looking for is an expression that describes the change in ct=c0 when the temperature is not constant but changing. As a ﬁrst step we can ﬁnd an expression for the rate of change of ct=c0 by differentiating Equation 5.50 at constant temperature

dct c0 dt

¼ bW aW t aW 1 exp (bW t aW )

(5:51)

T

By making now the parameters aW and bW temperature dependent we ﬁnd for the change in the ratio ct=c0 as a function of temperature dct ¼ bW (T(t)) aW (T(t)) t aW (T(t))1 exp (bW (T(t))t aW (T(t)) ) c0 dt

(5:52)

In order to be able to solve this differential equation for ct=c0 we need an expression for ct=c0 in the righthand side of Equation 5.52. This can be found as follows. Consider the time t* that corresponds to the momentary ratio ct=c0 which can be found from Equation 5.50 1 ct 1aW B c0 C t* ¼ @ A bW

0

ln

(5:53)

This relation is schematically depicted in Figure 5.20 and, of course, this is valid at any temperature but since bW is temperature dependent, t* will be different at a different temperature for the same ct=c0, or put differently, for the same value of t* ct=c0 will be different at different temperatures. By substituting Equation 5.53 in Equation 5.52, the latter equation becomes an ordinary differential equation in ct=c0 that can be solved, in principle, by numerical integration using appropriate software* * The software used for the calculations shown here was Athena Visual Studio v.11. See www.athenavisual.com. The calculations can also be done in a spreadsheet: http:==people.umass.edu=mgcorrad=RealTimeNutrientDegradation.xls

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700

[Ascorbic acid] (mg/100 g)

650

600

550

500

450

400

0

10

20

30 40 Time (min)

50

60

70

FIGURE 5.21 Degradation of ascorbic acid in the fruit amla, described by the Weibull model with ﬁxed parameter aW ¼ 0.5 at 508C (^), 608C (D), 708C (&), 808C (X), 908C (*), 1008C (&), 1208C (*). Dataset in Appendix 5.1, Table A.5.7.

aW (T(t)) 1 ct 1 aW (T(t)) dct ct c0 C B ¼ bW (T(t)) aW (T(t)) @ exp ln A c0 dt bW (T(t)) c0 0

ln

(5:54)

The dependence of the parameters aW and bW on temperature can be described by any ad hoc empirical model that is able to capture the relation, and the same goes for the dependence of temperature on time. The logistic model introduced in Equation 5.31 (with m0 ¼ 1) appears to perform well to describe the temperature dependence of bW, but to be sure: any other model may be used. Let us see how this approach works out with an example. Equation 5.50 was applied to the heatinduced degradation of ascorbic acid in the tropical fruit amla; the shape factor had the same value of aW ¼ 0.5 at all temperatures studied: see Figure 5.21. Consequently, parameter aW can be assumed temperature independent in this case and was ﬁxed at its value aW ¼ 0.5, leaving the parameter bW to be estimated at each temperature. Its temperature dependence was subsequently modeled according to Equation 5.31 (Figure 5.22). The parameter estimates found for the ﬁt in Figure 5.22 were used to model the fate of ascorbic acid in amla subject to three different nonisothermal proﬁles (representing three different heating methods, namely open pan cooking, pressure cooking, and a fuel-efﬁcient ‘‘ecocooker’’), following Equation 5.54 (Figure 5.23). In order to give a feel for the accuracy of the predictions the prediction bands for the model are also indicated; these prediction bands were estimated from the imprecision in the temperature dependence of parameter bW; since we are using nonlinear models here, these prediction bands are only approximate. Prediction bands are further discussed in Chapter 7. The model predictions shown in Figure 5.23 are certainly not perfect but they show the right trend; only for the open pan cooking results, the predictions are less than actually observed. Incidentally, note that these are true model predictions (as opposed to model ﬁts) because the model is based on independent isothermal measurements while the data points in Figure 5.23 are obtained from different nonisothermal measurements. The irregular shape of the concentration curves is due to the fact that the Weibullian shape factor aW had the value of 0.5 in this particular case. Another example links to Figure 5.9 where the temperature dependence for the ﬁrst-order

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0.06

0.05

bW

0.04

0.03

0.02

0.01

0

40

60

80

100 T ⬘ (⬚C)

120

140

FIGURE 5.22 Fit of the model in Equation 5.31 to the parameter bW for the data displayed in Figure 5.21. Fit parameters are k ¼ 0.017 8C1, Tc ¼ 291.2 8C (m0 ¼ 1, ﬁxed).

rate constant was shown for the heat-induced degradation of riboﬂavin in spinach; since this was a ﬁrstorder reaction, the Weibullian shape factor aW ¼ 1 and the rate constant corresponds to the parameter bW. The results of nonisothermal predictions and experiments (having the same temperature proﬁle as in Figure 5.23) are given in Figure 5.24. As in the previous example, the predictions look reasonable, though not perfect. In this case, there is a mismatch between the prediction for the ecocooker and the experiment. Similar equations can be set up for the other empirical models introduced in Chapter 4. Equation 4.73, for instance, is a hyperbolic model and Equation 4.74 a limited exponential model. Suppose we have the hyperbolic model displayed in Equation 4.73a for a formation reaction: c ¼ c0 þ

k1 t k2 þ t

(5:55)

The isothermal rate of the formation of a compound expressed as concentration c can be found by differentiating the model with respect to t dc k1 k2 ¼ dt T (k2 þ t)2

(5:56)

To model nonisothermal conditions, we can look at the state of c at time t* and determine from that the isothermal rate; t* is calculated from the model equation (Equation 5.55) t* ¼

k2 (T) [c c0 ] k1 (T) [c c0 ]

(5:57)

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120

740

140

700

100

700

120

660

80

620

60

580

40

500

5

10

15

60 580

500

40 20

0

5

10 Time (min)

(B)

740

120

700

100

660

80

620

60

580

40

540

20

500

15

0 20

0 0

(C)

T ⬘(⬚C)

80 620

540

0 25

20

Time (min)

[Ascorbic acid] (mg/100 g)

0

100

660

T⬘ (⬚C)

20

540

[Ascorbic acid] (mg/100 g)

740

T ⬘(⬚C)

[Ascorbic acid] (mg/100 g)

5-30

(A)

21.10.2008 10:40pm Compositor Name: MSubramanian

5

10

15 20 25 Time (min)

30

35

FIGURE 5.23 Nonisothermal predictions for the degradation of ascorbic acid (.) in amla in an open pan (A), in a pressure cooker (B), and in an ecocooker (C). The solid lines are the model predictions for the concentration proﬁle, the dashed lines the approximate 95% prediction bands. The temperature proﬁle refers to the right y-axis. Dataset in Appendix 5.1, Table A.5.7.

Equation 5.56 can be converted to a rate at nonisothermal condition by substituting the expression for t* dc k1 (T(t)) k2 (T(t)) ¼ dt k2 (T(t)) [c c0 ] 2 k2 (T(t)) þ k1 (T(t)) [c c0 ]

(5:58)

We have now an expression for the rate as a function of changing temperature based on an isothermal model. This equation can be solved numerically for any desired temperature proﬁle to obtain the proﬁle of c if we know the dependence of the parameters k1 and k2 on temperature. Another empirical model is the limited exponential model equation (Equation 4.74) c ¼ c0 þ (c1 c0 ) [1 exp (k1 t)]

(5:59)

c1 is the asymptotic value of c when t ! 1 and k1 the ‘‘rate constant.’’ For this model, the isothermal rate is

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120

0.208 40 0.204

T ⬘(⬚C)

0.212

[Riboflavin] (mg/100 g)

0.216

80

60

(A)

5

10 15 Time (min)

20

0.212

80 60

0.208

40 20 0.200

0 25

0

10 Time (min)

(B)

15

0 20

120

0.22 0.215

[Riboflavin] (mg/100 g)

5

100

0.21 80 0.205 60 0.2

T ⬘(⬚C)

0

100

0.204

20

0.2

140 120

100

0.216 [Riboflavin] (mg/100 g)

0.220

T ⬘(⬚C)

0.22

40 0.195 20

0.19 0.185 0

10

(C)

20 Time (min)

30

0 40

FIGURE 5.24 Nonisothermal predictions for the degradation of riboﬂavin (.) in spinach in an open pan (A), in a pressure cooker (B) and in an ecocooker (C). The solid lines are the model predictions for the concentration proﬁle, the dashed lines the approximate 95% prediction bands. The temperature proﬁle refers to the right y-axis. Dataset in Appendix 5.1, Table A.5.8.

dc ¼ k1 (c1 c0 ) [ exp (k1 t)] dt T

(5:60)

and t* is t* ¼

1 c0 c1 ln c c1 k1

(5:61)

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and the nonisothermal rate equation becomes dc ¼ k1 (T)(c1 c0 ) dt

(5:62)

Again, if we know how temperature T changes with time t, as well as the dependence of k1 on T, the concentration change can be calculated for any temperature proﬁle using numerical integration. So, in conclusion, variable temperature kinetics can be done for every model, provided we know the time–temperature proﬁle and the temperature dependence of the parameters, either via Arrhenius or Eyring relations, or via empirical relations such as Equation 5.31. This is especially very useful for accelerated shelf life testing, and to study the effect of varying temperature in a food chain during distribution and storage. The same approach can be applied to variable temperature kinetics for microbial kinetics; in such cases the Arrhenius=Eyring equation does not make sense because the processes studied in microbial growth and death are not single reaction events like a chemical reaction is. This is further discussed in Chapters 12 and 13.

5.10

Effect of Pressure

High-pressure treatment of foods is an emerging technology. It implies that foods are brought under high pressure (up to 1000 MPa), and as a result bacteria can be killed and some enzymes inactivated. Because this can be done at relatively low temperature, there is little or no heat damage and the preserved foods retain their freshness. This research area is strongly in development, and here we state only some basic principles. From a thermodynamic point of view, the effect of pressure can be dealt with as follows. First, we start with considering a reaction in the gaseous phase, and we state the general reaction as in Equation 3.1 nA A þ nB B ! nP P þ nQ Q The equilibrium constant KP expressed via partial pressures is (cf. Equation 3.102) P n P P n Q P

Q

Po Po KP ¼ nA nB PA PB Po Po

(5:63)

Po is the standard state pressure of 1 bar. As a reminder, based on Equation 3.113 the following relation holds Dr G ¼ RT ln KP

(5:64)

This equation shows that KP refers to standard states at Po ¼ 1 bar, so it is by deﬁnition independent of pressure. This does not mean, however, that pressure has no effect on equilibria. This should become clear from the following reasoning. We can express the equilibrium constant also in terms of mole fractions, KX KX ¼

(XP )nP (XQ )nQ (XA )nA (XB )nB

(5:65)

For a perfect gas the following relation holds between partial pressure Pi and total pressure P: Pi P ¼ Xi o Po P

(5:66)

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Combining Equation 5.65 with Equation 5.66 gives P nP P nQ P

Q

Po Po K X ¼ n A n B PA PB o P Po

nA þnB nP nQ P Po

(5:67)

Combining Equation 5.63 with Equation 5.67 results in (see also Equation 3.104) KX ¼ KP

Dn P Po

(5:68)

Dn ¼ nP þ nQ nA nB

(5:69)

By increasing the pressure, KX must change by a factor (P=Po)Dn in order to keep KP constant. So, the result of this exercise is to show that the mole fractions of the components in the equilibrium mixture do depend on the total pressure P, despite the fact that KP does not. Just another way of expressing this is by differentiating Equation 5.68 with respect to pressure

@ ln KX @P

¼ T

@ ln KP @P

Dn T

@ ln P @P

(5:70) T

Using the perfect gas law PDV ¼ DnRT, and using the fact that KP is independent of pressure so that the ﬁrst term on the right-hand side of Equation 5.67 equals zero, Equation 5.70 can also be written as

@ ln KX @P

¼ T

Dn DV ¼ P RT

(5:71)

This equation is not only valid for equilibria in the gas phase but also for equilibria in solution. DV is the volume change accompanying 1 mol of reaction with all substances in the standard state. This is actually a statement of Le Châtelier’s principle: equilibrium phenomena that are subject to a decrease in volume upon reaction will move from left to right upon an increase in pressure and vice versa. If DV can be considered pressure independent, it can be estimated from the dependence of KX on pressure P according to Equation 5.71, analogous to the van’t Hoff equation for temperature effects (Equation 5.2). High-pressure treatment thus favors reactions that result in a volume decrease. It affects mainly noncovalent bonds, which implies that low-molecular weight compounds are not really sensitive to pressure but high-molecular weight components (biopolymers) are because they are stabilized by noncovalent bonds. Also dissociation of weak acids in water is enhanced by pressure. Consequently, pressurization of acetate, citrate, and phosphate buffers is accompanied by a large negative volume change upon dissociation and this leads to a signiﬁcant acidiﬁcation of such buffer solutions. Furthermore, crystallization phenomena are also affected by pressure. Especially hydrophobic bonds and ionic bonds are very sensitive to pressure, hydrogen bonds less so. This means that the quaternary and tertiary structure of proteins will be disrupted but not so much the secondary structure when subject to high pressure. In other words, proteins (and hence enzymes) will denature, which could also be the basis for microbial inactivation as a result of high-pressure treatment. Also, aggregation and gelation of proteins may occur as a result of pressure-induced denaturation. When applying pressure, its transmission is uniform and virtually instantaneous, independent of vessel size and geometry (the isostatic principle), unless gas is present because then the gas can be compressed. Furthermore, an increase in pressure (at constant T) results in an increase in the degree of ordering of molecules (the microscopic ordering principle).

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For this book, we are mainly interested in describing the effect of high pressure on the resulting kinetics. By analogy with the effect of temperature, the activation volume DV# (the difference in molar volume of the activated complex and the reactant) and the preexponential factor AP are introduced DV # P k ¼ AP exp RT

! (5:72)

Analogous to the Arrhenius equation, a reference rate constant at a reference pressure can be chosen to eliminate the preexponential factor DV # (P Pref ) k ¼ Kref exp R

! (5:73)

The terms DV#, DH#, DS#, DG# are related DV # @T

! ¼ P

@DS# @P

!

@DH # @DV # ¼ DV # T T @T

(5:74)

! P

Currently, most studies are done on inactivation of microorganisms via high pressure. We will come back to this in Chapter 13. Furthermore, the effect of high pressure on proteins and enzymes has been studied extensively. As an example, Figure 5.25 shows pressure-induced inactivation of an enzyme,

1

0.8

[A]/[A]0

0.6

0.4

0.2

0 0

50

100

150

Time (min)

FIGURE 5.25 Inactivation of polygalacturonase in tomato juice at 208C at 400 MPa (^) and 450 MPa (D) according to a ﬁrst-order model (solid lines). Dataset in Appendix 5.1, Table A.5.9.

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−5

ln k

−6

−7

−8

−9 300

400

500

600

P (MPa)

FIGURE 5.26 Example of the use of (the logarithm of) Equation 5.73 applied to inactivation of polygalacturonase in tomato juice at 158C and Pref ¼ 525 MPa. The activation volume is estimated at 43.5 cm3 mol1, and kref at 0.002 s1. Dataset in Appendix 5.1, Table A.5.10.

polygalacturonase (PG) in tomato juice. Figure 5.26 shows how Equation 5.73 can be used to estimate the activation volume DV# for the same example. Since both pressure and temperature have an effect on protein conformation, it has also been studied how the two conditions combine. A mathematical model that takes this into account is the following, based on thermodynamic and kinetic considerations: DV0# DS# Da# (P Pref ) þ 0 (T Tref ) (P Pref )(T Tref ) RT RT RT DCp DB# T 2 T ln 1 þ Tref (P Pref ) þ RT 2RT Tref

ln k ¼ ln kref

(5:75)

In this equation Da# is the thermal expansivity factor (cm3 mol1 K1), DB# the compressibility factor (cm6 J1 mol1), DV0# (cm3 mol1), and DS#0 (J mol1 K1) the volume and entropy change between native and denatured states at Tref and Pref, respectively, DCp the heat capacity at constant pressure (J mol1 K1). These parameters need to be estimated from experimental data. A possible problem is the large number of parameters to be estimated and therefore a simpliﬁed version of Equation 5.75 has been proposed (by simply omitting DB# and DCp)

ln k ¼ ln kref

DV0# DS# Da# (P Pref ) þ 0 (T Tref ) (P Pref )(T Tref ) RT RT RT

(5:76)

With this model it can be predicted what the combined effects of temperature and pressure are. Figure 5.27 gives a so-called isorate contour plot, again for the example of inactivation of polygalacturonase. It shows the temperature–pressure combinations that lead to the same extent of inactivation of the enzyme.

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600

500

P (MPa)

400

300

200

100

0 273

293

313 T (K)

333

353

FIGURE 5.27 Isorate contour plot for inactivation of polygalacturonase in tomato juice (characterized by a rate constant k ¼ 0.00144 s1) as a function of temperature and pressure. Dataset in Appendix 5.1, Table A.5.11.

5.11

Concluding Remarks

This chapter has discussed the effects of temperature and pressure on kinetics, as both these conditions are of major importance for food technology. Parameters such as activation energy, enthalpy, and entropy are frequently used. One should, however, realize that these parameters pertain to elementary reactions. The concept of transition state theory and activation energy becomes somewhat blurred when applied to the complex reactions taking place in foods. One should be careful with interpretation of such derived parameters, and it would probably be better to use at least the term apparent activation energy. Empirical models are available to handle situations in which the concept of transition state theory is not applicable. Nonisothermal cases can be handled both with Arrhenius=Eyring models as well as empirical models. In any case, knowledge on temperature and pressure coefﬁcients is extremely important when modeling food quality attributes, and this chapter has described the tools for this.

Appendix 5.1 TABLE A.5.1

Datasets Used for Examples in This Chapter Mutarotation of Lactose (Figure 5.1)

0

T (8C)

Keq

1=T (K1)

ln Keq

10

1.62

0.003534

0.48

15

1.6

0.003472

0.47

20

1.59

0.003413

0.46

25 30

1.58 1.56

0.003356 0.0033

0.46 0.45

35

1.55

0.003247

0.44

40

1.56

0.003195

0.44

50

1.52

0.003096

0.42

60

1.5

0.003003

0.40

75

1.46

0.002874

0.38

90

1.44

0.002755

0.36

Source: From Roetman, K. and Buma T.J. Temperature dependence of the equilibrium b=a ratio of lactose in aqueous solution. Neth Milk Dairy J 28:155–165, 1974.

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TABLE A.5.2

Mutarotation of Glucose (Figure 5.2) 0

Temperature T (8C)

Keq

30

1.503

35

1.467

40

1.447

45

1.409

Source: From Le Barc H.N., Grossel J.M., Looten P., and Mathlouthi M. Kinetic study of the mutarotation of D-glucose in concentrated aqueous solution by gas-liquid chromatography. Food Chem 74:119–124, 2001.

TABLE A.5.3 Reaction Rate Constants for Deamidation of Caseinate during Heating from 1108C to 1458C (Figure 5.5) T 0 (8C)

103=T

145

2.392

35.705

2.392

35.815

ln(khP=kBT)

2.392

35.97

140

2.421 2.421

35.199 35.272

2.421

35.336

130

2.481

35.73

120

110

2.481

35.803

2.481

35.895

2.545

36.444

2.545

36.499

2.545 2.611

36.609 37.011

2.611

37.103

2.611

37.149

Source: From Metwalli A.A.M. and Van Boekel M.A.J.S. On the kinetics of heat-induced deamidation and breakdown of caseinate. Food Chem 61:53–61, 1998.

TABLE A.5.4 Rate Constants for the Degradation of Riboﬂavin in Heated Spinach (Figure 5.9) T (8C)

k 3 105 (s1)

50

1.33E þ 00

60

1.83E þ 00

70 80

2.66E þ 00 3.17E þ 00

90

5.17E þ 00

100

5.33E þ 00

120

6.67E þ 00

Source: From Nisha P., Singhal R.S., and Pandit A.B. A study on the degradation kinetics of riboﬂavin in spinach (Spinacea oleracea L.). J Food Eng 67:407–412, 2005.

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TABLE A.5.5 Results for Nonisothermal Kinetics of Sucrose Hydrolysis, pH 3.8 (Figure 5.18) Time (min)

T (8C)

Sucrose Concentration (g L1)

0

100.0

0

100.0

10.0 9.833

5

100.5

9.772

5

100.5

9.603

20

102.3

9.18

20

102.3

9.011

30

105.568

8.923

30 40

105.568 106.494

8.517 9.004

40

106.494

8.396

55

108.761

8.614

55

108.761

8.275

65

111.869

7.917

65

111.869

7.545

80

115.322

6.715

80 90

115.322 117.754

6.546 6.594

90

117.754

6.019

105

120.186

5.188

105

120.186

5.189

120

123.466

5.121

120

123.466

3.952

130

126.409

5.034

130 140

126.409 128.849

3.83 2.561

140

128.849

2.358

155

131.788

1.73

155

131.788

1.527

165

135.726

2.521

165

135.726

2.318

180

137.166

0.643

180

137.166

0.474

Source: From Silva, C.L.M., Oliveira, F.A.R., Lamb, J., Torres, A.P., and Hendrickx, M., Int. J. Food Sci. Technol., 29, 227, 1995.

TABLE A.5.6 (Figure 5.19) Time (s)

Dataset for Heating Proﬁles in a UHT Heat Exchanger

T (8C) Final temperature 1208C

T (8C) Final temperature 1408C

0

20

20.563

4

77.465

80

5.429

88.169

102.254

6.714

93.803

115

8

101.127

123.099

9.286

105.634

126.479

10.571

109.577

129.859

11.857 13.143

112.394 115.085

133.239 135.366

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TABLE A.5.6 (continued) Exchanger (Figure 5.19) Time (s)

Dataset for Heating Proﬁles in a UHT Heat

T (8C) Final temperature 1208C

T (8C) Final temperature 1408C

15.714

115.211

137.183

15.714

116.901

137.183

17.143

118.028

138

18.286

118.028

138.873

19.571

118.592

139.437

20.714

118.592

139.437

22

118.592

139.437

23.286 25.429

118.592 119.155

139.437 140

25.571

119.155

139.437

27

119.155

140

28.429

119.155

139.437

29.857

119.155

140

33

61.127

61.127

33.286

55.493

55.493

33.571 35.429

49.859 42

49.859 42

36

30

30

37

23.38

22.817

Source: From Pompei C. and Rossi M. Use of a model solution for the evaluation of heat damage in milk treated in an ultra high-temperature heat exchanger. J Agric Food Chem 42:360–365, 1995.

TABLE A.5.7 Isothermal and Nonisothermal Heating Effect on Ascorbic Acid Content of the Fruit Amla (Figures 5.21 and 5.23) Isothermal, mg=100 g: Time (min)

508C

608C

708C

808C

908C

1008C

1208C

0 5

686

686

686

686

686

686

686 629

10

660

654

647

643

610

595

15

589 569

20

647

632

625

616

582

560

30

632

622

613

600

573

546

40

629

616

602

585

562

539

50

623

609

595

578

556

532

60

619

604

592

573

528

504

551

Nonisothermal, mg=100 g standard deviation (n ¼ 3): Time (min)

Open Pan

Pressure Cooking

Ecocooker 686 5

0

686 5

686 5

5

636 3

667 6

10

612 5

20 30

634 3

676 9

580 5

15 590 11

641 5 605 4

Source: From Nisha P., Singhal R.S., and Pandit A.B. A study on degradation kinetics of ascorbic acid in amla (Phyllanthus emblica L.) during cooking. Int J Food Sci Nutr 55:415–422, 2004.

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TABLE A.5.8 Isothermal and Nonisothermal Heating on Riboﬂavin Content of Spinach (Figure 5.24) Isothermal, mg=100 g: Time

508C

608C

708C

808C

908C

1008C

1208C

0

0.215

0.215

0.215

0.215

0.215

0.215

0.215

10

0.213

0.213

0.215

0.214

0.214

0.206

0.202

20

0.212

0.21

0.213

0.213

0.209

0.201

0.195

30

0.210

0.208

0.21

0.209

0.203

0.194

0.187

40

0.209

0.207

0.207

0.204

0.197

0.186

0.178

50

0.207

0.204

0.203

0.201

0.194

0.182

0.172

60

0.205

0.201

0.198

0.196

0.189

0.177

0.166

Nonisothermal, mg=100 g standard deviation (n ¼ 3): Time (min)

Open Pan

Pressure Cooker

Ecocooker

0

0.215 0.002

0.215 0.002

0.215 0.002

5

0.215 0.002

0.215 0.002

10 15

0.211 0.001

0.208 0.003 0.205 0.000

20

0.206 0.004

0.214 0.004 0.205 0.002 0.190 0.005

30

Source: From Nisha P., Singhal R.S., and Pandit A.B. A study on degradation kinetics of riboﬂavin in spinach (Spinacea oleracea L.). J Food Eng 67:407–412, 2005.

TABLE A.5.9 Results for High-Pressure Inactivation of Polygalacturonase in Tomato Juice (Figure 5.25) t (min)

C=CO at 400 MPa

C=CO at 450 MPa

0

1.0

1.0

5 10

0.800737

0.67781 0.367879

15

0.311403

20

0.64118

30

0.136847 0.037712

40

0.573753

60

0.211072

80

0.15988

100 120

0.178669 0.114559

0.033746

Source: From Fachin, D., Van Loey, A.M., Nguyen, B.L., Verlent, I., Indrawatti, and Hendrickx, M.E., Innovative Food Sci. Emerg. Technol., 4, 135, 2003.

TABLE A.5.10 Rate Constant for Inactivation of Polygalacturonase at 158C in Tomato Juice as a Function of Pressure (Figure 5.26) P (MPa)

k (s1)

400

0.00019

450

0.000532

500

0.001502

550

0.002782

Source: From Fachin, D. and Van Loey, A.M., Innovative Food Sci. Emerg. Technol., 4, 135, 2003.

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TABLE A.5.11 Parameter Values for Isorate Contour Plot of Inactivation of Polygalacturonase (Figure 5.27) Parameter DV0# (cm3 mol1 ) DS#0 (J mol1 K1 ) Da# (cm3 mol1 K1 ) 0

kref (s1)

Value 39.55 175.8 0.849 0.000343

Source: From Fachin, D. and Van Loey, A.M., Innovative Food Sci. Emerg. Technol., 4, 135, 2003. Note: Rate constant is kept constant at 0.00144 s1, temperature and pressure varied to achieve this ﬁxed value using Equation 5.63.

Bibliography and Suggested Further Reading General References Atkins P.W. Physical Chemistry, 6th ed. Oxford: Oxford University Press, 1999. Carlos Aledo J. Metabolic pathways: Does the actual Gibbs free-energy change affect the ﬂux rate? Biochem Mol Biol Ed 29:142–143, 2001. Cruickshank F.R., Hyde A.J., and Pugh D. Free energy surfaces and transition state theory. J Chem Ed 54:288–291, 1977. Hammes G.G. Thermodynamics and Kinetics for the Biological Sciences. New York: Wiley Interscience, 2000. Hamori E. Building a foundation for bioenergetics. Biochem Mol Biol Ed 30:296–302, 2002. Harris D.A. A uniﬁed approach to enzyme catalysis. Biochem Ed 14:2–6, 1986. Kost D. and Pross A. The misuse of energy proﬁles. Educ Chem 14:87–88, 1977. Laidler K.J. Chemical Kinetics, 3rd ed., New York: Harper & Row, 1987. Laidler K.J. Just what is a transition state. J Chem Ed 65:540–542, 1988. Logan S.R. The meaning and signiﬁcance of ‘‘the activation energy’’ of a chemical reaction. Educ Chem 23:148–150, 1986. Núnez de Castro I. and Alonso F.J. Energy diagrams for enzyme catalyzed reactions: A confusing point in the textbooks. Biochem Ed 25:87–89, 1997. Missen R.W., Mims ChA., and Saville B.A. Introduction to Chemical Reaction Engineering and Kinetics. New York: John Wiley & Sons, Inc., 1999. Peleg M., Engel R., Gonzalez-Martinez C., and Corradini M.G. Non-Arrhenius and non-WLF kinetics in food systems. J Sci Food Agric 82:1346–1355, 2002. Robinson P.J. Dimensions and standard states in the activated complex theory of reaction rates. J Chem Ed 55:509–510, 1978. Walstra P. Physical Chemistry of Foods. New York: Marcel Dekker, 2003. Winzor D.J. and Jackson C.M. Interpretation of the temperature dependence of equilibrium and rate constants. J Mol Recognit 19:389–407, 2006. Wolfenden R., Snider M., Ridgway C., and Miller B. The temperature dependence of enzyme rate enhancements. J Am Chem Soc 121:7419–7420, 1999. High Pressure Balny C. and Masson P. Effects of high pressure on proteins. Food Rev Int 4:611–628, 1993. Fachin D., Van Loey A.M., Nguyen B.L., Verlent I., Indrawatti O., Hendrickx, M.E. Inactivation kinetics of polygalacturonase in tomato juice. Innovative Food Sci Emerg Technol 4:135–142, 2003.

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Hendrickx M., Ludikhuyze L., Van den Broeck I., and Weemaes C. Effects of high pressure on enzymes related to food quality. Trends Food Sci Technol 9:197–203, 1998. Hinrichs J., Rademacher B., and Kessler H.G. Reaction kinetics of pressure-induced denaturation of whey proteins. Milchwissenschaft 51:504–509, 1996. Indrawati O., Van Loey A.M., Ludikhuyze, L.R., and Hendrickx, M.E. Soybean lipoxygenase inactivation by pressure at subzero and elevated temperatures. J Agric Food Chem 47:2468–2474, 1999. Matser A.M., Krebbers B., Van den Berg P., and Bartels P.V. Advantages of high pressure sterilization on quality of food products. Trends Food Sci Technol 15:79–85, 2004. Isokinetic Effect Banks B.E.C., Damjanovic V., and Vernon C.A. The so-called thermodynamic compensation law and thermal death. Nature (London) 240:147–148, 1972. Connors K.A. Chemical Kinetics. The Study of Reaction Rates in Solution. New York: VCH Publishers, 1990. Krug R.R., Hunter W.G., and Grieger-Block R.A. 1. Some fundamental statistical problems associated with the analysis of van ’t Hoff and Arrhenius data. J Phys Chem 80:2335–2341, 1976. Krug R.R., Hunter W.G., and Grieger-Block R.A. Enthalpy-entropy compensation. 2. Separation of the chemical from the statistical effect. J Phys Chem 80:2341–2351, 1976. Labuza T.P. Enthalpy=entropy compensation in food reactions. Food Techn 34(2):67–77, 1980. Madamba P.S., Driscoll R.H., and Buckle K.A. Enthalpy-entropy compensation models for sorption and browning of garlic. J Food Eng 28:109–119, 1996. McBane G.C. Chemistry from telephone numbers: The false isokinetic relationship. J Chem Ed 75:919– 922, 1998. Ozilgen M., Durukan A., and Ulgen N. Enthalpy-entropy and frequency factor-activation energy compensation relations for microbial death in fruit juices. Lebensmittel Wissenschaft und Technologie 24:378–381, 1991. Rhim J.W., Nunes R.V., Jones V.A., and Swartzel K.R. Appearance of a kinetic compensation effect in the acid-catalyzed hydrolysis of disaccharides. J Food Sci 54:222–223, 1989. Rhim J.W., Jones V.A., and Swartzel K.R. Kinetic compensation effect in the heat denaturation of whey protein. J Food Sci 55:589–592, 1990. Ulgen N. and Ozilgen M. Kinetic compensation relations for ascorbic acid degradation and pectinesterase inactivation during orange juice pasteurisation. J Sci Food Agric 57:93–100, 1991. Nonisothermal Reaction Kinetics Alibrandi G., D’Aliberti S., and Pedicini R. Computer simulation of variable-parameter kinetic experiments. Chem Educator 6:185–191, 2001. Corradini M.G. and Peleg M. A model of non-isothermal degradation of nutrients, pigments and enzymes. J Sci Food Agric 84:217–226, 2004. Corradini M.G. and Peleg M. Prediction of vitamins loss during non-isothermal heat processes and storage with non-linear kinetic models. Trends Food Sci Technol 17:24, 2006. Corradini M.G. and Peleg M. Shelf-life estimation from accelerated storage data. Trends Food Sci Technol 18:37–47, 2007. Deindoerfer F.H. and Humphrey A.E. Analytical method for calculating heat sterilization times. Appl Microbiol 7:256–264, 1959. Deindoerfer F.H. and Humphrey A.E. Principles in the design of continuous sterilizers. Appl Microbiol 7:264–270, 1959. Dolan K.D. Estimation of kinetic parameters for nonisothermal food processes. J Food Sci 68:728–741, 2003. Labuza T.P. A theoretical comparison of losses in foods under ﬂuctuating temperature sequences. J Food Sci 44:1162–1168, 1979.

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Nunes R.V., Rhim J.W., and Swartzel K.R. Kinetic parameter evaluation with linearly increasing temperature proﬁles-integral methods. J Food Sci 56:1433–1437, 1991. Peleg M., Engel R., Gonzalez-Martinez C., and Corradini M.G. Non-Arrhenius and non-WLF kinetics in food systems. J Sci Food Agric 82:1346–1355, 2002. Peleg M., Corradini M.G., and Normand M.D. Kinetic models of complex biochemical reactions and biological processes. Chemie Ingenieur Technik 76:413–423, 2004. Rhim J.W., Nunes R.V., Jones V.A., and Swartzel K.R. Determination of kinetic parameters using linearly increasing temperature. J Food Sci 54:446–450, 1989. Silva C.L.M., Oliveira F.A.R., Lamb J., Torres A.P., and Hendrickx M. Experimental validation of models for predicting optimal surface quality sterilization temperatures. Int J Food Sci Technol 29:227–241, 1995. WLF Models Sopade P.A., Halley P., Bhandari N., D’Arcy B., Doebler C., and Cafﬁn N. Application of the WilliamsLandel-Ferry model to the viscosity-temperature relationship of Australian honeys. J Food Eng 56: 67–75, 2003. Peleg M. On the use of the WLF model in polymers and foods. Crit Rev Food Sci Nutr 32:59–66, 1992. Roos Y.H. Phase Transitions in Foods. New York: Academic Press, 1995.

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6 Charge Effects

6.1

Introduction

The presence of charged particles can have large implications for the systems in which these particles are present, and this is certainly also true for foods. Foods contain dissociated salts, acids, and bases; biopolymers such as proteins and polysaccharides carry charge as well, depending on pH. Reactions between charged particles depend on a particular way on the charge but also on the properties of the medium in which they are present. Many fundamental investigations have been carried out in the ﬁrst half of the twentieth century by famous scientists. It is essential to understand the behavior of ions to understand properties of foods, and this includes kinetics where ions can exert a major effect. The activity concept is indispensable to understand the behavior of ions, so that is what we start with. Thus, the present chapter builds further upon the activity concept introduced in Chapter 3. After discussing the sometimes peculiar aspects of ion activities, we will take a look at the consequences for kinetics between charged particles.

6.2

Models for Ion Activities

The activities of ions have been studied in much detail and remain a topic of interest. It is an important topic for foods too, but it is not a simple one. We need to pay detailed attention to it in order to understand the intricacies involved. To start with, Figure 6.1 shows an example of the experimentally determined molal activity coefﬁcient of three simple salt solutions as function of their molality. Foods, of course, will usually not contain just one salt but a mixture of salts. The complications with mixtures of salts have been dealt with in the literature for the case of milk and, incidentally, also for nonfood systems such as seawater. First, however, we start with considering a solution of just one salt. Two phenomena are apparent from Figure 6.1, namely that salt solutions show strong deviation from ideality even at low concentrations, the more so for higher valency and, second, that the activity coefﬁcient can increase again at high concentration, and can actually become larger than unity at higher molalities (not shown here). An important phenomenon with species like salts is that they dissociate in aqueous solution, be it partly or completely. This means that we should be looking at the activity coefﬁcients of the ionic species rather than the salt as a whole. How can we do that? As a reminder, the chemical potential of a component is unique for a given solution at a given temperature and pressure, but the chemical potential in the standard state depends on the concentration scale adopted, and so does the activity. This is extensively discussed in Chapter 3. If we have electrolytes it would be desirable to refer to the chemical potentials of the separate 6-1

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1 0.9 0.8 Molal activity coefficient, g

NaCl 0.7 0.6 0.5 CaCl2 0.4 0.3 0.2 CuSO4 0.1 0 0

0.2

0.4

0.6

0.8

1

Molality (mol kg−1)

FIGURE 6.1 Experimentally determined molal activity coefﬁcients of aqueous NaCl (^), CaCl2 (&), and CuSO4 (~) solutions as a function of molality at 258C. The lines are just to guide the eye. Dataset in Appendix 6.1, Table A.6.1.

ionic constituents, but this is hypothetical because we cannot add only one type of ion to a solution because of the requirement of electroneutrality. We therefore need to go into somewhat more detail for the case of electrolytes. We start very general. Suppose we have 1 mol of a salt CpAq that can dissociate into p moles of cation C having valency zþ and q moles of anion A having valency z: Cp Aq Ð pCzþ þ qAz

(6:1)

For reasons of electrical neutrality, it is required that p zþ ¼ jq z j

(6:2)

So, for NaCl, p ¼ q ¼ 1, and zþ ¼ 1, z ¼ 1, and for Ca3(PO4)2, p ¼ 3, q ¼ 2, zþ ¼ 2, z ¼ 3. The chemical potential of the salt CpAq can be written as z þ RT ln aCzþ þ q m mCp Aq ¼ p m Az þ RT ln aA Czþ p q z þ qm ¼ pm Az þ RT ln ((aCzþ ) (aA ) ) Czþ

(6:3)

Thus, we can write for the activity of salt CpAq: aCp Aq ¼ (aCzþ )p (aAz )q

(6:4)

Using the molality scale, a ¼ g m, we can transform this equation into aCp Aq ¼ (g Czþ )p (gAz )q (mCzþ )p (mAz )q

(6:5)

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Remembering that upon complete dissociation: mCzþ ¼ pmCp Aq

(6:6)

mAz ¼ qmCp Aq it follows that aCp Aq ¼ (gCzþ )p (gAz )q (pmCp Aq )p (qmCp Aq )q

(6:7)

¼ (gCzþ )p (gAz )q pp qq (mCp Aq )pþq

Since we cannot measure the activity coefﬁcients of separate ions, the so-called mean ion activity coefﬁcient g is introduced: (g )pþq ¼ (gCzþ )p (gAz )q

(6:8)

p 1=(pþq) g ¼ g þ g q

(6:9)

aCp Aq ¼ pp qq (g mCp Aq )pþq

(6:10)

or

Equation 6.7 then changes into

Next to the salt activity we can also deﬁne a mean salt activity: 1

p

q

a ¼ (aCP Aq )pþq ¼ (aCzþ aAz )1=(pþq) ¼ (pp qq )1=(pþq) (g mCp Aq )

(6:11)

and a mean ionic molality: m ¼ (pq qq )1=(pþq) mCp Aq

(6:12)

These equations are needed for further analysis: it relates the (mean) activity of a salt to its concentration. As mentioned, this was a general derivation. To make it somewhat more comprehensible, Table 6.1 shows some examples for various types of salts using Equations 6.10 and 6.11. The experimental activity coefﬁcients (as shown, for example, in Figure 6.1) reﬂect actually the mean ion activity coefﬁcients of the salt; they are calculated from the measured activity of a salt solution and the known stoichiometric concentration. TABLE 6.1 Examples of Activity Coefﬁcients and Activities of Various Electrolytes Assuming Complete Dissociation as Compared to the Neutral Solute Sucrose Type

Example

Nonelectrolyte

Sucrose

1:1, 2:2

NaCl, CaSO4

2:1 1:2 1:3

CaCl2 Na2SO4 Na3PO4

g

Mean Salt Activity

— ðg Czþ g Az Þ1=2 1=3 g zþ g 2 z C2 A 1=3 g zþ g z C3 A 1=4 gCzþ g Az

— gm 4

1=3

gm

4 gm 271=4gm 1=3

Activity a of Solute Dissolved at Molality m gm g2 m2 4g 3 m3 4g 3 m3 27g 4 m4

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6-4

0.8 0.7 0.6

g

0.5 0.4 0.3 0.2 0.1 0 250

350

450

550

T (K)

FIGURE 6.2 Effect of temperature on ion activity coefﬁcient for aqueous solutions of NaCl at mole fraction 0.026 (^), of Na2SO4 at mole fraction 0.026 (~), and of MgSO4 at mole fraction 0.011 (&). The lines are just to guide the eye. Dataset in Appendix 6.1, Table A.6.2.

Incidentally, ion activity coefﬁcients depend on temperature. Figure 6.2 shows an example for the ion activity coefﬁcients of NaCl, Na2SO4, and MgSO4. In the temperature range relevant for foods, say 250–400 K, the recorded changes are noticeable but not drastic. Empirical models have been developed to account for the temperature dependence. We will not discuss them any further here. Now that we have deﬁned activity equations for electrolytes we can take a closer look at theories for activities of charged species.

6.2.1

Debye–Hückel Type Models

Obviously, ionic activity coefﬁcients refer to positively or negatively charged ions but these are not experimentally accessible, as mentioned. Theory has been developed to calculate ionic activity coefﬁcients and this has resulted in the famous Debye–Hückel (DH) theory. In this theory, deviation from ideality is accounted for by calculating electrostatic effects with reference to a central ion while treating all other ions as point charges. The resulting expression is a relation between the natural logarithm of the activity coefﬁcient (ln g) and the charges z, a function called the Debye screening parameter k, the distance of closest approach dR, and several fundamental parameters: ln g ¼

jzþ z je20 k 8p«0 «r kB T 1 þ kdR

(6:13)

The distance of closest approach dR is the sum of the effective radii of a pair of interacting ions. In the DH theory the ions are taken to be all of the same size and then dR equals the diameter of the ions. It is usually estimated to be between 0.3 and 0.6 nm. Furthermore, e0 is the elementary charge, «0 is the permittivity of

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vacuum, and «r is the relative permittivity of the medium.* Dielectric properties are seen to have an effect on activity coefﬁcients. For foods, this is relevant if other solvents such as ethanol are present; the relative permittivity of ethanol is 24.3 as opposed to 78.4 for water at room temperature. The parameter k (dimension m1) is deﬁned as k¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ X ri zi2 4pLB

(6:14)

i

where ri is the number density of ion i (number of particles per m3) and the sum is over all ionic species in solution. The relation between ri and concentration ci (in mol dm3) is of course: ri ¼ 103 ci NAV

(6:15)

The parameter LB (which is twice the so-called Bjerrum distance, dimension length) is deﬁned as LB ¼

e20 4p«0 «r kB T

(6:16)

In water at room temperature LB 0.7 nm. Next, the total ionic strength IT is introduced, deﬁned as IT ¼

1X 2 ci z i 2 i

(6:17)

Its dimension is the same here as that of the concentration ci (mol dm3); it can also be expressed in molality (mol kg1 solvent), or in number densities. The summation in this equation is taken over all ions present. It is assumed that salts are completely dissociated into ions, which is why it is called total ionic strength. The reason why we stress this becomes apparent below when we discuss the phenomenon of ion association. In the equation shown, ionic strength has units of molarity. For instance, a solution of 0.01 M CaCl2 has an ionic strength of 1 IT ¼ (0:01 22 þ 2 0:01 12 ) ¼ 0:03 mol dm3 2

(6:18)

It is perhaps instructive to explain very brieﬂy where the concept of ionic strength originates from. The role of ionic strength is that it accounts for the local charge density near an ion in solution; the mean P charge density is, of course, zero for a solution as a whole for reasons of electroneutrality, i.e., i ci zi ¼ 0. The expression displayed in Equation 6.17 derives from an expansion of the Poisson–Boltzmann equation in the D–H theory. Ionic strength does not appear directly in the fundamental thermodynamic equation that describes how free energy varies with, for instance, temperature and pressure. Therefore, it needs to be speciﬁed, for instance when using stoichiometric equilibrium constants. These do depend on ionic strength because activity coefﬁcients depend on ionic strength. Combining all these expressions yields a relation between k (m1) and ionic strength IT (mol dm3): k¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8pLB 1000NAV IT

(6:19)

* According to IUPAC, the relative permittivity of a medium «r is deﬁned as «m=«0 with «m the permittivity of the medium. For water, for instance, «r is 78.4 at 208C. The dielectric constant is the old name for relative permittivity.

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By combining Equation 6.19 with Equation 6.13 we can ﬁnd an expression in which ion activity coefﬁcients are a function of ionic strength, f(I). The natural logarithm in Equation 6.13 is converted to the base-10 logarithm in most textbooks. To avoid confusion we will also do that; the conversion factor, ln(10) ¼ 2.303, is included in the constants to follow. For an anion we ﬁnd 2 f (I) log g ¼ z

(6:20)

2 f (I) log gþ ¼ zþ

(6:21)

For a cation, a similar equation holds

For the mean ionic activity coefﬁcient the equation becomes log g ¼ jzþ z j f (I)

(6:22)

The function f(I) can take on various forms. The simplest one is used in the so-called limiting DH law: pﬃﬃﬃﬃ f (I) ¼ ADH IT

(6:23)

while the function for the extended DH law is: f (I) ¼

pﬃﬃﬃﬃ ADH IT pﬃﬃﬃﬃ 1 þ BDH dR IT

(6:24)

Two new parameters are introduced here. ADH and BDH are temperature-dependent constants. The parameter ADH is given by the following equation (if ionic strength is expressed in mol dm3):

ADH

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2pNAV 1000L3B 1 1 (dm3 :mol )2 ¼ ln 10

(6:25)

The numerical value for ADH at 258C for aqueous systems is 0.51, a value that is frequently used in the DH equation, but it should not be forgotten that the parameter does have units, as indicated. The parameter BDH is k BDH ¼ pﬃﬃﬃﬃ (kg1=2 mol1=2 m1 ) IT

(6:26)

The numerator in Equation 6.24 accounts for the effect of long-range Coulomb forces while the denominator signiﬁes how these are modiﬁed by the short-range interactions between ions. Table 6.2 lists numerical values of ADH and BDH as a function of temperature for aqueous solutions (note that the relative permittivity is also temperature dependent: it decreases with temperature). The value of the parameter dR is uncertain and acts as an adjustable parameter. Scatchard has proposed to take the product BDHdR as 1.5 dm3=2 mol1=2 (hence, taking dR as 0.46 nm) so that: pﬃﬃﬃﬃ ADH IT pﬃﬃﬃﬃ f (I) ¼ 1 þ 1:5 IT

(6:27)

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TABLE 6.2 D–H Constants, Relative Permittivity «r, and Bjerrum Length LB as a Function of Temperature for Aqueous Solutions at a Pressure of 1 bar «r

LB (nm)

ADH (dm3 mol1)1=2

BDH 3 1010 (m1 mol1=2 (dm3)1=2)

0

87.74

0.696

0.491

0.325

5

85.76

0.700

0.494

0.325

10

83.83

0.703

0.498

0.326

15

81.94

0.707

0.502

0.327

20

80.1

0.711

0.506

0.328

25 30

78.3 76.54

0.715 0.719

0.510 0.515

0.329 0.330

35

74.82

0.724

0.520

0.331

40

73.15

0.729

0.525

0.332

50

69.91

0.739

0.536

0.334

75

62.42

0.768

0.568

0.341

T 0 (8C)

Others have proposed unity for BDHdR. This results in the equation of Güntelberg: pﬃﬃﬃﬃ ADH IT pﬃﬃﬃﬃ f (I) ¼ 1 þ IT

(6:28)

The last two equations are a bit awkward in the sense that the numerical values BDHdR ¼ 1.5 and 1, respectively, have a dimension of (dm3)1=2 mol1=2 and this is not clear by looking at the equation. A modiﬁcation of the Güntelberg equation was suggested by Guggenheim: f (I) ¼

pﬃﬃﬃﬃ ADH IT b0 I T pﬃﬃﬃﬃ 1 þ IT zþ z

(6:29)

where b0 is an adjustable empirical parameter. Again, there seems to be a dimensional problem here. The DH constants have units that actually depend on the unit of the ionic strength because the right-hand side terms in Equations 6.20 through 6.22 need be dimensionless. If ionic strength is expressed in molality, the units change to (kg mol1)1=2 for ADH and to (kg1=2 mol1=2 m1) for BDH. If concentration would be expressed in the SI unit mol m3, the parameter ADH changes numerically by a factor of 10001=2 from 0.510 to 0.0161 at 258C. In fact, it would be better to deﬁne ionic strength as a dimensionless parameter: IT ¼

1 X ci 2 z 2 i c i

(6:30)

where ci is molarity (mol dm3) and c the standard state molarity of 1 mol dm3, as a result of which ADH and BDH would have no units. However, this is not done in most cases in the literature and to avoid confusion we will conform to the current practice. However, the reader should be aware of this possible problem of units. For foods, it would make more sense actually to use molality than molarity because it is somewhat hard to imagine what a liter solution means in products like, say, cheese. For real solutions, molarity can be used as well, of course. For example, the ionic strength of milk is about 0.08 M, and in general, the ionic strength for most foods will be between 0.001 and 1 M or m. When using buffers in model systems that attempt to mimic a speciﬁc food, it should be realized that two different buffers having the same

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0.5

0

0

−0.2

−0.5 Log g ±

Log g ±

6-8

−0.4

−1

−0.6

−1.5

−0.8

−2

−1 (A)

0

0.5

1

1.5 2 IT (M)

2.5

3

3.5

−2.5

0

(B)

1

2

3

4

5

IT (M)

FIGURE 6.3 Experimental mean ion activity coefﬁcients as a function of total ionic strength IT in mol dm3 for NaCl (A) and CaCl2 (B). Dotted line: limiting DH, Equation 6.21; hyphenated line: Güntelberg, Equation 6.28; solid line: Davies, Equation 6.31. Dataset in Appendix 6.1, Table A.6.3.

concentration and pH could have a markedly different ionic strength. This phenomenon can have a large kinetic effect as is discussed below. It can have also a strong effect on enzyme activity, for instance. Equation 6.23 is the so-called limiting DH theory because it is only valid in the limit of IT ! 0, i.e., for very diluted solutions (say IT < 1–10 mM). The extended DH equation, the Güntelberg and Guggenheim equations are valid for, say, IT < 100 mM. A frequently used equation in the literature is the so-called Davies equation, which is an empirical relation based on the DH theory and is in fact a modiﬁcation of the Guggenheim equation (Equation 6.29): f (I) ¼ ADH

pﬃﬃﬃﬃ IT pﬃﬃﬃﬃ 0:2IT 1 þ IT

(6:31)

The Davies equation seems to give a good ﬁt to experimental values, also at higher ionic strength. Figure 6.3 shows experimental mean activity coefﬁcients for NaCl and CaCl2 and the ﬁt by the limiting DH equation, the extended DH equation, and the Davies equation. Indeed, the Davies equation performs best of these three equations. Table 6.3 summarizes the various DH equations. However, the reader is advised that the equations displayed in Table 6.3 do not reﬂect the actual ion activity coefﬁcients if ion pairs are present. However, before we discuss this, we ﬁrst dive into the phenomenon that is also shown in Figure 6.3, namely that the ion activity coefﬁcient increases again at higher ionic strengths. This can be explained by a more recent theory called the mean spherical approximation.

6.2.2

Mean Spherical Approximation Theory

The MSA approach is better suited to describe nonideality of electrolyte solutions in concentrated systems (i.e., >0.1 M) than the DH equations can; it can, for instance, explain the increase in mean activity coefﬁcient at the higher concentrations. The DH theory neglects the ﬁnite size of ions and this becomes a problem in more concentrated solutions because it leads to an overestimation of the effects of

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TABLE 6.3

Overview of the Various D–H Expressions Based on f (I) ¼

pﬃﬃﬃﬃ ADH IT b0 IT pﬃﬃﬃﬃ 1 þ BDH dR IT zþ z

Equation pﬃﬃﬃﬃ f (I) ¼ ADH IT pﬃﬃﬃﬃ ADH IT pﬃﬃﬃﬃ f (I) ¼ 1 þ BDH dR IT pﬃﬃﬃﬃ ADH IT pﬃﬃﬃﬃ f (I) ¼ 1 þ 1:5 IT pﬃﬃﬃﬃ ADH IT pﬃﬃﬃﬃ f (I) ¼ 1 þ IT pﬃﬃﬃﬃ ADH IT b0 IT pﬃﬃﬃﬃ f (I) ¼ 1 þ IT zþ z pﬃﬃﬃﬃ IT pﬃﬃﬃﬃ 0:2IT f (I) ¼ ADH 1 þ IT

0

dR ¼ 0, b ¼ 0 dR 6¼ 0, b0 ¼ 0 BDHdR ¼ 1.5 kg1=2 mol1=2, b0 ¼ 0 BDHdR ¼ 1 kg1=2 mol1=2, b0 ¼ 0 BDHdR ¼ 1 kg1=2 mol1=2, b0 6¼ 0 BDHdR ¼ 1 kg1=2 mol1=2, b0 ¼ 0.2jzþzj

Name D–H limiting law (DHLL) Extended D–H equation Scatchard equation Güntelberg equation Guggenheim equation Davies equation

ion–ion interactions, especially for polyvalent ions. In the MSA approach, the ﬁnite size of all solute species, charged or uncharged, is taken into account. This accounts for excluded volume effects, usually called the hard sphere contribution, which is basically an entropic effect and accounts for how space can be occupied by spheres. Also, the decrease in dielectric permittivity of the solution with increasing concentration of solute needs to be considered; this latter effect leads to an increase in the strength of ion–ion interactions with increasing volume fraction of solutes. MSA calculations need to be done in the molarity scale, so if comparisons are to be made with experimental ionic activity coefﬁcients, which are usually tabulated in the molality scale, recalculations are necessary. How to do this has been discussed in Chapter 3. The resulting expression for the single ionactivity coefﬁcient is made up of two contributions; for instance, for the molar activity coefﬁcient yi: ln yi ¼ ln yies þ ln yihs

(6:32)

The advantage of the MSA is that it leads to relatively simple, though lengthy, analytical expressions. The electrostatic part due to the ionic atmosphere is indicated by the superscript ‘‘es,’’ the hard sphere part due to the ﬁnite size of solute species is indicated by the superscript ‘‘hs.’’ We will not list the complete equations here; some are given in Appendix E. Simpliﬁed expressions do exist, however. The most basic one for the electrostatic part, for equal ion sizes, reads ln yies ¼ LB

zi2 G 1 þ GdR

(6:33)

with G¼

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 2kdR 1 2dR

(6:34)

The screening parameter G in the MSA thus replaces the screening parameter k in the DH theory. However, (2G)1 is greater than k1 because the ﬁnite diameter accounted for in the MSA results in the

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fact that ions cannot approach each other so closely as point charges can in the DH theory. In the limit of zero ionic strength I, G ! 0.5k. In the framework of the MSA theory, ionic strength is deﬁned as 1X 2 rz 2 i i i

I¼

(6:35)

One equation for the hard sphere contribution for equal-sized spheres is 2

hs ln yihs ¼ ln y ¼

6J 3J 2J þ 2þ 1 J (1 J) (1 J)3

J¼

(6:36)

pdi3 X rj 6 j

(6:37)

The summation in Equation 6.37 is over all solute species, including neutral species, and Equation 6.37 represents in fact the volume fraction of solutes. The single ion-activity coefﬁcients can be expressed, of course, as mean ion activity coefﬁcients using Equation 6.9. The central parameter in the electrostatic part of the MSA model is thus the screening parameter G. The ionic strength parameter I is enclosed indirectly (via k) in this screening parameter. The interested reader is referred to some selected references given at the end of this chapter for more detailed information. The above given MSA equations are the simplest possible. They constitute the so-called restrictive, primitive MSA; restrictive because all ion sizes are assumed equal and primitive because the solvent is considered a dielectric continuum that only manifests itself via its permittivity, it ignores the discrete nature of the solvent molecules (this is the so-called McMillan–Mayer framework, see Appendix E). Figure 6.4 shows a calculation of the electrostatic and hard sphere contribution for a 1:1 electrolyte and 2:2 electrolyte using Equations 6.33 and 6.36, assuming aqueous solutions at 258C, ion sizes of 0.5 nm, and complete dissociation. The ﬁgure shows nicely that in the range 0.1–0.5 M the contribution of the 1.6

1.5 ln(y hs)

1.2

0.5 ln(y hs)

0.8

ln ( y)

ln (y)

−0.5 0.4 ln(y es)+ln(y hs)

0

ln(y es) + ln(y hs)

−2.5

ln(y es)

−0.4

−1.5

ln(y es) −0.8 0 (A)

0.5 1 c (mol dm−3)

1.5

−3.5 (B)

0

0.5 1 c (mol dm−3)

1.5

FIGURE 6.4 Hard sphere (hs) and electrostatic (es) contribution to the mean ionic activity coefﬁcient on the molar scale for a 1:1 electrolyte (A) and a 2:2 electrolyte solution (B), calculated with the restricted MSA theory. Note the difference in the y-axis scale in A and B.

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hard sphere effect starts to become noticeable, and it is for this reason that the MSA theory is superior to the extended DH theory and related equations for ionic strengths above 0.1 M. In the DH model the presence of a salt is characterized only by its charge, whereas in the MSA model a salt is distinguished by its diameter and its charge. However, the results shown in Figure 6.4 are only meant to illustrate trends. Several improvements can and should be made: .

. .

. .

Dependence of the permittivity of the solvent on concentration must be taken into account. The dielectric constant of a solution decreases when the salt concentration increases by reducing the density of solvent dipole moments. Allowance should be made for different sizes of the various cations and anions. Ion sizes depend on salt concentration. It should be realized that the relevant ion diameter in the MSA theory includes the hydration sphere. This hydration sphere decreases with increasing salt concentration. The effect of the solvent should be taken into account, resulting in the nonprimitive MSA. A very important aspect that needs to be considered is the phenomenon of ion association. This should also be done, in principle at least, when using the DH equations but the ion association effect is less pronounced at IT < 0.1 M, so neglecting it in these cases has relatively less impact (though not negligible anymore for high valency ions) than in the regions above IT > 0.1 M where the MSA is very useful. The phenomenon of ion association is discussed in Section 6.3.

For 1:1 electrolytes, the primitive MSA is applicable to an ionic strength of about 0.3 M. The nonprimitive MSA can be valid up to a few molar. The adjustable parameter in MSA is the ionic diameter. A relatively new way to calculate activity coefﬁcients is via Monte Carlo simulations based on statistical thermodynamics. Like the DH model ions are treated as hard, charged spheres in a continuum with a certain permittivity. Unlike the DH theory where the distribution of particles is approximated with a continuous charge distribution, the distribution of the particles is simulated by moving them randomly in a box, and all interionic particle interactions are taken into account. This topic is, however, beyond the scope of this book; a few references are given at the end of this chapter. Table 6.4 summarizes some guidelines on when to use the various models for activity coefﬁcients.

6.2.3

Pitzer Equations

In order to overcome the limitations of the DH equations for higher ionic strengths than 0.1 M, Pitzer proposed to use virial expansions of the DH equations, as was basically also proposed in the Guggenheim equation shown above. The virial coefﬁcients are supposed to account for speciﬁc interactions between ions that occur at the higher ionic strengths (say above 0.1 M). These equations are therefore also referred to as speciﬁc interaction theory (SIT). Because they account for interactions between ions, these equations should be useful for multicomponent electrolyte solutions. A general equation is: pﬃﬃﬃﬃ X X pﬃﬃﬃﬃ 2 IT g pﬃﬃﬃﬃ þ mi mj Bgij þ mi mj mk Cijk ln 1 þ 1:2 IT þ 1 þ 1:2 IT 1:2 ij ijk

ln g i ¼ z2 A

TABLE 6.4 Overview of Models Describing the Relations between Ionic Activity Coefﬁcients and Ionic Strength IT Ionic Strength IT < 0.01 M

Activity Coefﬁcient Estimation via DHLL equation

IT < 0.1 M

Estimation via extended DH, Güntelberg, or Guggenheim equation

0.1 M < IT < 0.3 M IT > 0.3 M

Primitive mean spherical approximation, Davies equation Nonprimitive mean spherical approximation

(6:38)

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At 258C, A ¼ 0.392, and the Bij and Cijk parameters are related to binary ions i and j and to ternary ions i, j, k. respectively. These parameters account for speciﬁc ion interactions, as mentioned, and are based on experimental measurements made at the same ionic strength. There are many variants of Pitzer equations. They all contain a lot of parameters that do not really refer to physical properties. Moreover, the large number of parameters is a drawback from a modeling point of view. Nevertheless, they are used extensively in practice because they offer a solution to the practical problem of how to estimate activity coefﬁcients when theoretical models are not really valid. Some references on the use of Pitzer equations are given at the end of this chapter.

6.3

Ion Pairing Models

Quantitative studies of electrolyte solutions have shown that at very low concentrations nonideality can be accounted for satisfactorily by the mean activity coefﬁcient and the cause of this nonideality is the existence of long-range electrostatic interactions. The DH theory as well as the MSA assumes complete dissociation of electrolytes into solvated ions, and this assumption appears to be valid only at strongly diluted solutions. At higher concentrations much stronger deviations come into play due to short range Coulombic interactions. These interactions can only be explained by assuming ion association, also called ion pairing. In contrast to common belief it is thus not true that salts are completely dissociated into ions. The reason why it is important to consider these aspects is that we are interested in activities of free ionic species, and the mean ionic activity coefﬁcients do not reﬂect these when ion pairs are formed. This is so because activity coefﬁcients are calculated from measured activities on the one hand and from concentrations that are calculated assuming complete dissociation on the other hand. If ion pairs are formed this assumed concentration is no longer correct. Fortunately, we can correct for this if we are able to estimate the extent of ion association. Cations and anions can associate, for instance, Naþ and SO42 can associate into the ion pair NaSO4 and Naþ and Cl into NaCl0. Such behavior is expected especially for polyvalent electrolytes in aqueous solutions (but also for monovalent ions as we shall see) and in nonaqueous solutions of lower dielectric permittivity. Ion pairs are physically associated entities, moving together and having a ﬁnite but noticeable lifetime. An ion pair lifetime is of the order of 2 ns, which is about 20 times larger than the decay of a diffusion-controlled encounter (Chapter 4). Ion pairs can be measured experimentally via advanced spectroscopic techniques and conductivity measurements, for instance. The result is that at any instant a given fraction of ions is paired and although an individual ion pair does not last long, the timeaveraged concentration of ion pairs will remain constant (at constant conditions such as temperature, ionic strength, pressure). Consequently, we can apply concepts from equilibrium thermodynamics to the phenomenon of ion pairs. Suppose we have a solid salt CpAq(s) that we add to water; we also assume that the salt will dissolve completely in the water so that there is no CpAq(s) left after a while. We can then write Cp Aq (s) ! pCzþ þ qAz

(6:39a)

Suppose then that part of the dissolved ions form ion pairs as Czþ þ Az Ð CAzþ þz

(6:39b)

We thus have free ions as well as ion pairs in solution. We assume here only association into doublets and not into triplets or higher associations; so in the case of CaCl2 we assume CaClþ ion pairs but not CaCl20 ion pairs. This is not to say that the latter are not formed, but we omit it here for simplicity. If ion pairs are formed, the association into doublets will occur the most abundantly. The activity of an electrolyte

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forming ion pairs can be expressed in two equivalent ways (expressed here as molarity, but this could be equally well molality): aCp Aq ¼ (yC,T cC,T )p (yA,T cA,T )q ¼ (yC,F cC,F )p (yA,F cA,F )q

(6:40)

The subscripts ‘‘T’’ and ‘‘F’’ indicate the total (stoichiometric) and the free ion concentration, respectively. The validity of Equation 6.40 can be proven by a thermodynamic reasoning, which we do not repeat here. Equation 6.40 allows us to calculate the free ion activity coefﬁcient if we know the total (or mean) activity coefﬁcient and the concentration of ion pairs. Of course, the following mass balance relation holds: cC,T ¼ cC,F þ cC,ip cA,T ¼ cA,F þ cA,ip

(6:41)

The subscript ‘‘ip’’ refers to ion pairs. A simple example may be illustrative. Suppose that we have a solution of CA (p ¼ q ¼ 1, zþ ¼ z ¼ 1) of 1 M, and that the mean ion activity coefﬁcient of this solution 2 is 0.60 (remembering that gþ C g A ¼ g , Equation 6.8). The question is now what the activity coefﬁcient of the free ions is if ion pairs are formed. The activity is then: aCp Aq ¼ (0:60 1:0)2 ¼ 0:36

(6:42)

Suppose we know that the concentration of ion pairs CA0 is 0.1 M, and consequently the concentration of free ions is 0.9 M as follows from Equation 6.41. Using Equation 6.40 we then can calculate that: (yC,F 0:9)(yA,F 0:9) ¼ 0:36 0:36 ¼ 0:44 yC,F yA,F ¼ (y,F )2 ¼ 0:81 y,F ¼ 0:67

(6:43)

The mean free ion activity coefﬁcient g,F is deﬁned, of course, following Equation 6.9. We thus ﬁnd that the activity coefﬁcient of 1 M solution of CA would be 0.60 if no ion pairs are present while that of the free ions is 0.67 if 10% ion pairs would be formed. In other words, the activity coefﬁcient of the free ions is higher than that of the mean activity coefﬁcient of the solution as a whole. Equation 6.40 shows that the total activity coefﬁcient is a function of the free activity coefﬁcient as well as the ratio of free to total ion concentration. The problem is thus that, if ion pairs are formed, the actual mean free ion activity coefﬁcient is not directly comparable to the ones that are estimated from experiments. This is so because experimentally the activity of an electrolyte solution is measured (not the activity coefﬁcient!), and subsequently the mean ion activity coefﬁcient is calculated from this activity and the known stoichiometric concentration. However, if the concentration of free ions is lower because of the presence of ion pairs then this calculation is in error. Put in a different way, when ion pairs are present, calculations based on the stoichiometric electrolyte concentration overestimate the effects of nonideality due to ions. Furthermore, the ionic strength is lower than calculated. Of course, we can correct for this if we are able to estimate the amount of ion pairs present. Amount of ion pairs. Thus, we are left with the question how to calculate the amount of ion pairs formed in electrolyte solutions. Since the ion pairs are in equilibrium with the free ions as depicted in Equation 6.39b, the association of free ions into ion pairs can be described by a thermodynamic equilibrium constant, in this case called an association constant Kipth : Kipth ¼

aCA(zþ þz ) aCzþ aAz

(6:44)

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TABLE 6.5 Some Association Constants for Ion Pairs at 258C. Dataset in Appendix 6.1, Table A.6.4 Salt

Ion Pair

Association Constant Kipth

0

NaCl

NaCl

0.6

NaNO3

NaNO30

0.3

Na-acetate

Na-acetate0

0.5

Na2SO4

NaSO4

5.0

Na2S2O3

NaS2 O3

4.0

KCl

KCl0

0.6

KNO3

KNO30

0.6

K2SO4 CaCl2

AppKSO41 CaClþ

Ca(NO3)2

CaNO3þ

Ca(ðH2 PO4 )2

CaH2 PO4þ

CaHPO4

CaHPO40

Ca3(PO4)2

CaPO4

CaSO4

CaSO40

10 6 2 10 500 106 190

Ca(lactate)2

Ca(lactate)þ

Ca(butyrate)2 Ca-oxalate

Ca(butyrate)þ Ca(oxalate)0

30 4 1000

Kipth is a thermodynamic constant because it is deﬁned in terms of activities. The magnitude of this association constant is to a ﬁrst approximation determined by the valences z and it becomes quite high for valences higher than 1 (in the order of 102 to 103). In other words, ions with valences >1 tend to associate strongly into ion pairs. Table 6.5 shows some examples to give an impression of the order of magnitude of the effect. We can also consider activity coefﬁcients of ion pairs. If they carry no charge, the activity coefﬁcient will be close to unity in dilute solution. However, if they carry a charge, the activity coefﬁcient will be, most likely, different from unity. Values for thermodynamic association constants can be more or less generalized, as shown in Table 6.6. Equation 6.44 can be rewritten in terms of a stoichiometric constant Kipst ; the reason to do this is that we normally work with concentrations (molarity or molality), so Kipst is the parameter that is experimentally accessible. To make this step, we need to work with activity coefﬁcients: Kipth ¼

yip [CA(zþ þz ) ] yCA(zþ þz ) ¼ Kipst 2 ¼ Kipst K y y [Czþ ] [Az ] yCzþ yAz

TABLE 6.6 Orders of Magnitudes for Thermodynamic Association Constants as a Function of the Valencies of the Ions Constituting a Salt at Room Temperature Valencies

Example

Kipth

zþ ¼ 1, z ¼ 1

NaCl, KCl

0.1–1

zþ ¼ 1, z ¼ 2

Na2SO4

1–10

zþ ¼ 2, z ¼ 1

Ca(NO3)2, CaCl2

1–10

zþ ¼ 2, z ¼ 2

CaHPO4

100–500

zþ ¼ 1, z ¼ 3

Na3PO4

500–1000

zþ ¼ 2, z ¼ 3

Ca3(PO4)2

106

(6:45)

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where we have taken molarity as the concentration scale. This equation shows that the extent of ion pair formation in terms of concentration expressed via Kipst depends on the thermodynamic constant Kipth as well as on the activity coefﬁcients. Note that the stoichiometric association constant is not constant, in contrast to its thermodynamic counterpart, because it depends on the activity coefﬁcients that depend in turn on the value of the concentration. So, if activity coefﬁcients change, Kipst has to change as well because Kipth is a true constant. As an example from the literature, stoichiometric ion association constants were derived from dielectric relaxation spectra. The stoichiometric association constant was related to ionic strength via a Guggenheim type function: log Kipst ¼ log Kipth

pﬃﬃﬃﬃ 2ADH jzþ z j IT 3=2 pﬃﬃﬃﬃ þ Bb IT þ Cb IT 1 þ IT

(6:46)

Bb and Cb are empirical ﬁt constants. Figure 6.5 shows the ﬁt of this equation to measured data for the ion pair sodium malonate. Such measurements can be used to estimate Kipth . Figure 6.5 shows very clearly that Kipst is strongly dependent on ionic strength; in other words, Kipst is only a constant at constant ionic strength. In the literature, three ways are offered to calculate the amount of ion pairs formed, via the mass action law (MAL), via a method developed by Pytkowicz and coworkers, and via a special form of the MSA theory, called the binding MSA (BIMSA) or associating MSA (AMSA).

6.3.1

Mass Action Law

The ﬁrst method is as follows. Let us deﬁne a parameter aip such that a fraction (1 aip) of the cations forms ion pairs. Consequently, aip ¼ 1 corresponds to the absence of ion pairs and aip ¼ 0 to complete association into ion pairs. Then, a solution of a salt CpAq at a molar concentration c will result in a concentration of

1.5 1 0.5

log K stip

0 −0.5 −1 −1.5 −2 −2.5 −3

0

0.5

1

1.5

2

2.5

3

IT (mol dm−3)

FIGURE 6.5 The (logarithm of the) stoichiometric association constant Kipst as a function of ionic strength for the ion pair sodium malonate. The line indicates the ﬁt of Equation 6.46 to the data. Dataset in Appendix 6.1, Table A.6.5.

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TABLE 6.7 Relation between the Free Ionic Activity Coefﬁcient y,f and the Stoichiometric Activity Coefﬁcient y

Type of Salt

Concentration of Ion Pairs in a Salt Solution of Stoichiometric Concentration c

Example

(1 aip)c

Formula y ¼ aip

p¼q¼1

NaCl, MgSO4

p ¼ 1, q ¼ 2

CaCl2

(1 aip)c

y,f ¼ y

4 aip (1 þ aip )2

p ¼ 2, q ¼ 1

Na2SO4

2(1 aip)c

y,f ¼ y

!1=3 1 (aip 6¼ 0:5) a2ip (2aip 1)

y,f

!1=3

Cations: paipc Ion pairs: p(1 aip)c Anions: qc p(1 aip)c As a next step, the chemical potential of all the solute species (free ions and ion pairs) is considered, from which eventually an expression for the activity coefﬁcient can be deduced. We give here the result of this exercise without derivation; it relates the mean free ionic activity coefﬁcient y,f to the mean stoichiometric (or total) activity coefﬁcient y as follows: y,f ¼ y

pp qq p (paip ) [q p(1 aip )]q

1=(pþq) (6:47)

Table 6.7 shows examples for some typical salts. This table shows clearly that the free ion activity coefﬁcient is higher than the mean stoichiometric ionic activity coefﬁcient (which is the one that is deduced from experiments) if ion pairs are formed (aip < 1). This important point is frequently neglected, but it can lead to considerable error as we shall see. Recall that the DH and MSA type equations given above assume that complete dissociation takes place, so that these equations do not differentiate between a free and a stoichiometric activity coefﬁcient. Equation 6.47 and Table 6.7 show that if the degree of ion pair association is known, the free ion activity coefﬁcient can be calculated from the experimentally determined y, and, in reverse, if the free ion activity coefﬁcient would be known, the degree of ion association can be determined. The relation between aip and the stoichiometric association constant Kipst for a salt CpAq is Kipst ¼

[CAip ] p(1 aip )c ¼ [Czþ ] [Az ] (paip c)(qc p(1 aip )c)

(6:48)

For symmetrical electrolytes, such as NaCl and MgSO4 (p ¼ q ¼ 1), the relation between the thermodynamic association constant, the parameter aip, the mean ion activity coefﬁcient, and the concentration becomes (cf. Equations 6.45 and 6.48) Kipth ¼

(1 aip )yip (1 aip )yip ¼ 2 a2ip cyþ y a2ip cy

(6:49)

Usually, the activity coefﬁcient for the ion pair yip is taken as unity if the ion pair is uncharged. In more concentrated solutions this approximation may not hold. For unsymmetrical electrolytes, such as CaCl2,

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the activity coefﬁcient of the ion pair, e.g., CaClþ, can be taken, to a ﬁrst approximation, as that of the anion, e.g., Cl. In the case of a 2:1 electrolyte as CaCl2, the relation becomes Kipth ¼

(1 aip ) a2ip cyþ

(6:50a)

In the case of a 1:2 electrolyte such as Na2SO4 with ion pair NaSO4 it becomes Kipth ¼

(1 aip ) a2ip cy

(6:50b)

The interesting phenomenon is that ion association depends on ionic strength because activity coefﬁcients depend on ionic strength, while ionic strength depends in turn on ion association. If ion pairs are formed, ionic strength decreases; to account for that, the effective ionic strength Ie is introduced X 1 X 2 2 zf ,i cf ,i þ zip,j cip,j Ie ¼ 2 i j

! (6:51)

(on the molar scale). The subscript ‘‘f ’’ refers to free ions. This can be rewritten for the salt CpAq forming ion pairs as Ie ¼

1 2 2 2 [qc p(1 aip )ci ] þ zip p(1 aip )ci zþ paip ci þ z 2

(6:52)

Equation 6.52 shows that the effective ionic strength will be less than the total ionic strength if ion association occurs, the more so when the ion pair carries no charge. In order to calculate the effective ionic strength we need to know the concentration of ion pairs and the charge on the ion pairs, and to be able to calculate this we need activity coefﬁcients (via DH-related equations or the MSA approach). This calls for an iterative calculation. A pseudo-algorithm is given in Figure 6.6. A worked example may help to illustrate the implications. Suppose we have a 1:1 univalent salt, say NaCl, at a concentration of 0.1 mol dm3. From Table 6.5 we assume the association constant Kipth to be 0.6. The ionic strength equals the molar concentration in the case of a completely dissociated 1:1 electrolyte, so as a ﬁrst guess we take IT ¼ 0.1 mol dm3. The mean ionic activity coefﬁcient is then calculated using, for instance, the extended DH equation (Equation 6.24, which is valid up to 0.1 M). From these data, the stoichiometric association constant Kipst can be calculated using Equation 6.45. Using Equation 6.47 next, this results in an estimate of the extent of ion pair formation via the parameter aip. Then we can calculate the concentration of free ions and from that the effective ionic strength Ie. This gives the starting point for the next iteration and usually after four to ﬁve iterations the calculation converges to a ﬁnal value. Figure 6.7 shows some calculated results for the extent of ion pair formation where we have used the extended DH equation (actually the Scatchard equation (Equation 6.27), but using the effective ionic strength rather than total ionic strength). This equation should be valid up until an effective ionic strength of 0.1 M, while for higher concentrations the MSA model should be used. It shows that the effect of ion pair formation deﬁnitely cannot be neglected, especially not for polyvalent ions. It also implies that the effective ionic strength Ie can be substantially lower than calculated as if all ions are dissociated, i.e., total ionic strength IT. Figure 6.8 shows what can happen at higher electrolyte concentration; this is calculated via the MSA model because the ionic strength is higher than 0.1 M. It is seen that the extent of ion pairing decreases

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Electrolyte concentration

Calculate total ion concentration

Calculate total ionic strength IT

Calculate y±,f

Kipth

st

Calculate Kip

Calculate degree of ion association aip

Calculate free ion and ion pair concentration

Calculate effective ionic strength Ie Difference previous iteration k3 so that KM ¼

k2 k1

(9:13)

From Scheme 9.2 it follows that k2 [E] [S] ¼ ¼ Ks k1 [ES]

(9:14)

and so it appears that KM equals the dissociation constant of the ES complex Ks if k3 is much smaller than k2. In that case, a high value of KM indicates a weak binding of E to S and a low value of KM the opposite. It is important to realize that, in this case, KM does not give information about the catalysis process because the actual chemical change happens only after the formation of the ES complex. If the ﬁrst step in Scheme 9.2 is assumed to be irreversible (i.e., the Van Slyke equation, k2 ¼ 0), the parameter KM is a ratio of forward rate constants rather than a (pseudo) equilibrium constant. It thus appears that the original Van Slyke and Michaelis–Menten equations are in fact limiting cases of the steady-state assumption introduced by Briggs and Haldane. The maximal rate vmax is related to the turnover number of an enzyme, which is by deﬁnition the number of substrate moles converted into product per unit time per mole of enzyme when the enzyme is fully saturated with enzyme. From Equation 9.9 it follows that the turnover number equals k3, also called kcat. Turnover numbers of most enzymes are usually in between 1 and 104 per second. The rate is then determined by the actual catalyzed chemical reaction; kcat is a constant, while vmax depends on the total enzyme concentration ET. Note, however, that k3, or kcat, the turnover number, is not the only rate constant involved in determining the maximal rate of an enzyme. This becomes apparent when combining Equations 9.1 and 9.5:

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v¼

kcat [E] [S] KM

(9:15)

This shows that the ratio kcat=KM is an apparent second-order rate constant. This quantity is therefore a measure of the catalytic or enzyme efﬁciency. Combining Equation 9.15 with Equation 9.13 shows that: kcat k3 k1 ¼ KM k2 þ k3

(9:16)

This equation reveals that the ratio kcat=KM is limited by the rate constant k1, which cannot be higher than the diffusion-controlled limit. As discussed in Chapter 4, the value of bimolecular diffusion-limited rate constants is in the order of 108 to 1010 dm3 mol1 s1 (because enzymes are large molecules, k1 is usually estimated to be 108 dm3 mol1 s1) and this is then also the upper limit for kcat=KM. Michaelis–Menten kinetics for reversible reactions. So far, we have only considered reactions in one direction, from reactant to product. This implies in thermodynamic terms that the equations given are suitable for reactions which are strongly exergonic, i.e., have a large negative DrG or, equivalently, a large Keq (>>1). However, reactions for which DrG is close to zero or positive may behave differently in the sense that the reverse reaction becomes prominent. When initially only reactant is present, the reaction will take off in the direction of P, but as soon as product is formed this will have an effect on the rate. When the reaction is started in the presence of product, the rate will depend on the initial concentrations of reactants and products on the one hand and the value of DrG on the other hand and the rate may actually be zero. For foods, the situation is quite realistic that reaction products are already present when the enzyme reaction takes off. As a result, the reaction rate may become zero (or even negative from the product formation point of view). An example to illustrate this effect is the isomerization of glucose-6-P into fructose-6-P, catalyzed by the enzyme phosphoglucoisomerase. The DrG for this reaction is þ1840 J mol1. Figure 9.8 shows how dG=dar (see Equation 3.98) is affected by the concentrations of reactants and products. The reader is reminded that reactions will take place only when dG=dar < 0, as discussed in Chapter 3; this requirement is, of course, also true for enzyme reactions. The Michaelis–Menten equation as displayed in Equation 9.10 does not take this into account and would predict a ﬁnite rate for the situation that dG=dar > 0 and this is physically impossible. It is therefore essential to take reversible reactions into account as shown in Scheme 9.3. The Michaelis–Menten equation for the rate of product formation under steady-state conditions as in Scheme 9.3 can be derived as v¼

[ET ](k1 k3 [S] k2 k4 [P]) k2 þ k3 þ k1 [S] þ k4 [P]

(9:17)

The maximum forward (from reactant to product) and maximum reverse rate (from product back to reactant) leads to the following corresponding maximum rates:

k1 E+S

k3 E+P

ES k2

SCHEME 9.3 Reversible enzymatic reaction mechanism.

k4

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10,000

10,000 dG/dar (J mol−1)

15,000

dG/dar (J mol−1)

15,000

5,000

5,000

0

−5,000 −10,000 (A)

0

−5,000

0

0.1

0.2

0.3

0.4

0.5

[Glucose-6-P] (M)

−10,000

0

0.1

(B)

0.2 0.3 [Glucose-6-P] (M)

0.4

0.5

FIGURE 9.8 dG=dar for the transformation of glucose-6-P to fructose-6-P as a function of the substrate concentration glucose-6-P in the presence of reaction product concentration [fructose-6-P] ¼ 0.01 M (A) and [fructose6-P] ¼ 0.1 M (B).

vmax,f ¼ k3 [ET ]

(9:18)

vmax,r ¼ k2 [ET ]

(9:19)

The Michaelis constant KM is as in Equation 9.4, and the one for the reverse reaction is k3 þ k2 k4

(9:20)

(vmax,f =KM )[S] (vmax,r =KP )[P] 1 þ [S]=KM þ [P]=KP

(9:21)

Kp ¼

Combining the previous expressions results in v¼

This expression reduces to the common Michaelis–Menten equation (Equation 9.10) if we set [P] ¼ 0. In fact, Equation 9.21 appears to be the general form of the Michaelis–Menten equation and the commonly used one (Equation 9.10) is a special case in which the reverse reaction is neglected. Nevertheless, Equation 9.10 appears to be applicable to many enzyme systems, which then must imply that vmax,f >> vmax,r and KP >> KM. A more elaborate analysis about reversible reactions is to consider the possibility that was already shown in Scheme 9.1. This mechanism leads to an expression for the rate similar to one in Equation 9.21 but with different ones for the maximum rates and Michaelis constants: vmax,f ¼

k3 k5 [ET ] k3 þ k4 þ k5

(9:22)

vmax,r ¼

k2 k4 [ET ] k2 þ k3 þ k4

(9:23)

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KM ¼

k2 k4 þ k2 k5 þ k3 k5 k1 (k3 þ k4 þ k5 )

(9:24)

KP ¼

k2 k4 þ k2 k5 þ k3 k5 k6 (k2 þ k3 þ k4 )

(9:25)

When equilibrium is reached, the forward and reverse rates are balanced, the net rate becomes zero, and the concentrations do not change anymore (though the actual reaction itself, of course, does not stop). For reaction Scheme 9.3, the equilibrium constant can be expressed as (compare Equations 3.100 and 4.14) Keq ¼

[P]eq k1 k3 ¼ [S]eq k2 k4

(9:26)

For the reaction depicted in Scheme 9.1 the expression becomes Keq ¼

[P]eq k1 k3 k5 ¼ [S]eq k2 k4 k6

(9:27)

Combining the previous two equations with the expression for the rate, we ﬁnd [P] vmax,f vmax,r [S] Keq v¼ vmax,f [P] KM vmax,f þ vmax,r [S] þ Keq

(9:28)

Combining several previous equations leads to the following expression for the equilibrium constant Keq ¼

vmax,f KP vmax,r KM

(9:29)

This equation is called the Haldane relationship. Its importance is that it shows that the kinetic parameters are not independent of each other and that they are linked via the equilibrium constant. Here we can make a connection with the afﬁnity of a reaction as deﬁned in Chapter 3 (Equation 3.127, see also Equation 4.26) [S] vmax,f KP [S] Af ¼ RT ln Keq ¼ RT ln vmax,r KM [P] [P]

(9:30)

This equation can be useful if one wants to make a connection with irreversible thermodynamics, as discussed in Chapter 3. Enzymatic reactions in the framework of irreversible thermodynamics become increasingly important in the ﬁeld of metabolomics and systems biology. Equation 9.29 can be used to ﬁnd an expression for the forward rate in the case of a reversible reaction:

[P] vmax,f [S] K eq v¼ [P] KM 1 þ þ [S] KP

(9:31)

It is instructive to see how the value of the equilibrium constant affects the rate plot: see Figure 9.9 for a simulation. It clearly shows the importance of considering the equilibrium position when Keq is smaller than, say, 10. A completely wrong rate proﬁle would be predicted if reversibility would be neglected in

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1 0.9

Keq = 50

0.8

K eq = 3

v (arbitrary units)

0.7 Keq = 1

0.6

Keq = 0.5

0.5 0.4

Keq = 0.3

0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

[S] (arbitrary units)

FIGURE 9.9 Simulation of the dependence of rate v on substrate concentration [S] using Equation 9.31 with the following parameters: KM ¼ 0.02, KP ¼ 0.01, [P] ¼ 0.1, vmax,f ¼ 1 (arbitrary units). Keq is varied as indicated in the graph.

such cases. This example shows clearly the importance of considering both thermodynamics (via the equilibrium constant) and kinetics (enzyme kinetic parameters) for the resulting rate equations. A real example of a reversible enzyme-catalyzed reaction is the following. The enzyme enolase catalyses the hydration of phosphoenolpyruvate (PEP) to 2-phospho-D-glycerate (2PG). Initial rates for the conversion of PEP were determined in the presence of the product 2PG at various concentrations. Equation 9.21 was ﬁtted to all the data at once; incidentally, this is an example of global ﬁtting as discussed in Chapter 7: the datasets share common parameters. The parameter estimates are shown in Table 9.3 and the ﬁts in Figure 9.10. Note that the 95% conﬁdence intervals are quite narrow. This good precision is achieved due to the global ﬁtting procedure. Also note that the rate for conversion of PEP can become negative in the presence of 2PG; this means that 2PG is converted to PEP, in other words that the reverse reaction rate is positive. Equation 9.29 allows to calculate the equilibrium constant for this reaction, which happens to be 0.21 and is also obtained with good precision (see Table 9.3). Incidentally, a Lineweaver–Burke plot (discussed in the next section) of the data shown in Figure 9.10 would result in curved plots rather than straight lines (except for the experiment in which [2PG] ¼ 0).

TABLE 9.3 Parameter Estimates for the Parameters in Equation 9.21 Fitted to the Data Shown in Figure 9.10 Estimate

95% Conﬁdence Interval

vmax,f (mM min1)

0.036

0.0004

KM (mM) vmax,r (mM min1)

0.128 0.191

0.007 0.072

KP (mM)

0.142

0.051

Keq (via Equation 9.29)

0.21

0.004

Parameter

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Enzyme Kinetics

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0.04

v (µM min−1)

0.03

0.02

0.01

0

0

1

2

3 PEP (mM)

4

5

6

FIGURE 9.10 Effect of the presence of the product 2PG on the conversion of PEP by enolase. Concentrations of 2PG: ^: 0 mM, ~: 0.05 mM, ~: 0.1 mM, þ : 0.2 mM, : 0.4 mM. The lines are the ﬁts of Equation 9.21 with parameter values as shown in Table 9.3. Dataset in Appendix 9.1, Table A.9.2.

It may be useful to recapitulate the meaning of the various kinetic parameters that we encountered so far. Table 9.4 gives an overview. The discussion so far has been limited to one-substrate cases. Equally important are so-called bi–bi reactions in which two substrates are involved. It concerns especially transferases (catalyzing the transfer of a speciﬁc functional group from one substrate to another) and oxidoreductases (catalyzing oxidation– reduction reactions). Such mechanisms are discussed in Section 9.5.

9.2.1

Linearized Plots

The characteristic constants vmax and KM can be derived from experimentally determined initial rates (v0) as a function of substrate concentration [S] via nonlinear regression, as indicated above. Before the advent of computers this was (and unfortunately still is) done by inverting the Michaelis–Menten equation (Equation 9.10): 1 1 KM 1 þ ¼ v vmax vmax [S]

(9:32)

TABLE 9.4 Overview of the Relation between Various Kinetic Parameters Used in Michaelis–Menten Type Kinetics Mechanism

vmax

kcat=KM

KM

Michaelis–Menten (rapid equilibrium assumption)

k3[ET]

k2=k1

k1k3=k2

Briggs–Haldane (steady-state assumption)

k3[ET]

(k2 þ k3)=k1

(k1k3)=(k2 þ k3)

Reversible Michaelis–Menten Scheme 9.3

k3[ET]

(k2 þ k3)=k1

(k1k3)=(k2 þ k3)

Reversible Michaelis–Menten Scheme 9.1

k3k5[ET]=(k3 þ k4 þ k5)

(k2k4 þ k2k5 þ k3k5)= (k1(k3 þ k4 þ k5)

(k1k3k5)=(k2k4 þ k2k5 þ k3k5)

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1/v

1/vmax Slope = KM/vmax

1/[S] −1/KM

FIGURE 9.11

Schematic example of a Lineweaver–Burke plot.

If Michaelis–Menten kinetics is obeyed, a plot of 1=v versus 1=[S] gives a straight line with an intercept on the y-axis 1=vmax, an intercept on the x-axis of –1=KM, and a slope of KM=vmax. Such a plot is called a Lineweaver– Burke plot and the values of KM and vmax are usually estimated by linear regression, see Figure 9.11. The major objection against this procedure is of a statistical nature: by inverting v and [S], the error structure of the data is disturbed, as a result of which the conditions for linear regression are violated (as discussed in Chapter 7). Consequently, the resulting estimates of vmax and KM can be seriously biased (the seriousness depends on the actual data and the experimental design). Interestingly, Lineweaver and Burke were among the ﬁrst to appreciate this problem and advised statistical weighting to correct for the bias; weighting of the velocity data proportional to v4 should remedy the problem, but unfortunately this is very rarely done. Weighting of the substrate data would not be necessary if they are error-free but they never are, of course. Having nonlinear regression procedures available it is not necessary anymore to invert data; nonlinear regression of Equation 9.10 gives directly the estimates of vmax and KM, as shown above in Figure 9.5. An example is given to illustrate the point. Figure 9.12 shows a Lineweaver–Burke plot for the action of the protease papain on casein. The kinetic values derived from this plot via linear regression of the Lineweaver–Burke plot are KM ¼ 57.2 mM and vmax ¼ 13.0 mM min1. However, nonlinear regression on the untransformed data gives KM ¼ 27.0 mM and vmax ¼ 9.7 mM min1. Figure 9.13 shows the results for the untransformed data, and it is clear that a large bias exists in the parameters obtained via the unweighted Lineweaver–Burke plot. Admittedly, this is a rather extreme example. Apart from the distortion of the error structure of the data due to inverting them, the large bias is also due to a wrong experimental design in this case. More measurement points should have been taken at lower substrate concentrations (well below KM ¼ 27 mM). Other results may show less large deviations, but nevertheless parameters obtained via linear regression of transformed data are in principle biased! Other transformations are used as well besides the Lineweaver–Burke plot. We mention them for the sake of completeness. They were developed to facilitate estimation of kinetic parameters in days gone by when computers were not yet found on every corner of the lab bench. Strangely enough they are still used for this purpose even though that is no longer needed. The reason they are still used extensively is probably because they help to visualize the meaning of kinetic parameters. The Hanes–Woolf plot is obtained by plotting [S]=v versus [S]: [S] 1 KM [S] þ ¼ vmax v vmax

(9:33)

The Eadie–Scatchard plot is obtained by plotting v=[S] versus v: v 1 vmax vþ ¼ KM [S] KM

(9:34)

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0.7 0.6

1/v (M−1 min)

0.5 0.4 0.3 0.2 0.1 0 −0.05

0

0.05 1/[S]

0.1

0.15

(µM−1

)

FIGURE 9.12 Lineweaver–Burke plot for action of papain on casein. Dataset in Appendix 9.1, Table A.9.3.

The Eadie–Hofstee plot is v¼

KM þ vmax [S]

(9:35)

Incidentally, Scatchard plots, much used in physical chemistry, are of the same mathematical form as the Eadie–Hofstee plot, and suffer the same disadvantage for parameter estimation via transformed data. 12

10

v (M/min)

8

6

4

2

0 0

50

100

150

200

[S] (µM)

FIGURE 9.13 Michaelis–Menten plot for the action of papain on casein. The solid line reﬂects the Michaelis– Menten equation with parameters obtained via nonlinear regression, the dotted line reﬂects the Michaelis–Menten equation with parameters obtained via linear regression of the Lineweaver–Burke plot shown in Figure 9.12.

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In conclusion, it should be clear that linearized plots cannot be used for parameter estimation. However, they may be useful to present results (after having estimated the parameters from nontransformed data) because they show whether or not deviations from the expected relationship occur: the human eye is well capable to detect this in a glance for straight lines.

9.3

Enzyme Inhibition

One of the typical difﬁculties of enzyme kinetics is that enzymes are easily inhibited in various ways. Inhibitors can act such that inhibition is competitive, uncompetitive or noncompetitive. Competitive inhibition means that an inhibitor I competes with the substrate for the enzyme, so we have to expand in Scheme 9.4 the uninhibited case depicted in Scheme 9.2: The inhibition constant KI is KI ¼

[E] [I] [EI]

(9:36)

Ks is as given in Equation 9.14. The resulting equation in Michaelis–Menten terminology assuming steady-state=rapid equilibrium condition is v¼

v [S] vmax [S] max ¼ [I] aKs þ [S] þ [S] Ks 1 þ KI

(9:37)

with a¼

1þ

[I] KI

(9:38)

The other possibility is that the inhibitor binds to the ES complex (but not at the active site) such that the enzyme does not convert the substrate anymore, and this is called uncompetitive inhibition (Scheme 9.5): Ks is as given in Equation 9.14 and the inhibition constant KI0 is KI0 ¼

k1 E+S + I k5 KI

[ES] [I] [ESI]

Ks ES

(9:39)

k3

E+P

k2

k4 EI

SCHEME 9.4 Competitive enzyme inhibition mechanism. Ks and KI are the equilibrium dissociation constants for the ES and EI complex, respectively.

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Enzyme Kinetics

9-17 Ks

k1 E+S

k3 E+P

ES k2

+ I k5

K⬘I

k4 ESI

SCHEME 9.5 Uncompetitive enzyme inhibition mechanism. Ks and KI0 are the equilibrium dissociation constants for the ES and ESI complex, respectively.

The resulting Michaelis–Menten type equation is v¼

vmax [S] v [S] ¼ max 0 [I] Ks þ a [S] Ks þ [S] 1 þ Ki [I] 0 a ¼ 1þ 0 KI

(9:40)

(9:41)

The equilibrium arrows in Schemes 9.4 and 9.5 indicate that inhibition is considered reversible in both cases. In the competitive case, inhibition is maximal at low substrate concentration, and the effect of inhibition disappears at high [S] when the number of substrate molecules is much higher than those of the inhibitor molecules, and so the substrate molecules win the competition. In other words, vmax is not affected by competitive inhibition. The opposite is the case for uncompetitive inhibition. At high [S], all enzyme is bound in the ES form and this complex is then maximally inhibited by I; at low [S], enough enzyme is left to form the ES complex (remember that the reactions are reversible). In fact, competitive and uncompetitive inhibition are two extremes where in the one extreme the inhibitor binds only the free enzyme form and in the other only the enzyme–substrate complex. A mix of both effects can occur also, and this is appropriately called mixed inhibition, and less appropriately noncompetitive inhibition. It should not come as a surprise that the resulting Michaelis–Menten equation is v¼

vmax S aKs þ a0 [S]

(9:42)

A special case of inhibition is product inhibition: this can be competitive as well as uncompetitive; see Scheme 9.6, which is an extension of Scheme 9.2. It implies that the product formed associates with the enzyme and thus hinders its own formation. In the above equations one has then to replace [I] by [P]; the difference is of course that product inhibition can only occur when the reaction is well on its way. It should not be noticeable with initial rate studies. Product inhibition can be quite important in foods,

k1 E+S

ES

k3

k2

SCHEME 9.6 Mechanism to describe product inhibition.

k4 E+P

EP k5

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9-18

Increasing [I]

1/v

[I] = 0

1/[S]

FIGURE 9.14 Lineweaver–Burke plot typical for competitive enzyme inhibition. Such a plot should only be used as a diagnosis plot of enzyme inhibition.

which is one reason why initial rate studies are not always relevant for more practical situations. Below we will show a few cases of product inhibition. It is still common practice to show enzyme inhibition in the form of linearized plots, as well as to derive the parameter estimates from such plots. In addition to the already mentioned Lineweaver–Burke plot, a so-called Dixon plot is presented in which 1=v is plotted as a function of several concentrations of I, at ﬁxed substrate concentrations. However, as the reader may have noticed, the author is of the opinion that linearized plots should not be used anymore, at least not for parameter estimation. They are still useful as diagnostic plots for a ﬁrst impression of the mechanism, or to obtain starting values for subsequent nonlinear regression. A Lineweaver–Burke plot for competitive inhibition looks as displayed in Figure 9.14, for noncompetitive inhibition as in Figure 9.15 and for uncompetitive inhibition as in Figure 9.16. A Dixon plot for noncompetitive inhibition looks as shown in Figure 9.17A, and for competitive and mixed inhibition as in Figure 9.17B. For uncompetitive inhibition the lines run parallel as in Figure 9.17C. Another phenomenon that can occur is substrate inhibition. This would become noticeable in the hyperbolic Michaelis–Menten plot if the initial rate ﬁrst increases with substrate concentration but then decreases at higher substrate concentrations (if this is observed, it is of course no proof that it is indeed substrate inhibition, there may also be other reasons for such a behavior). The kinetic scheme 00 for substrate inhibition is shown in Scheme 9.7. The KI accounts for the reversible formation of an unproductive enzyme–substrate complex ESS: 00

KI ¼

[E] [S]2 [ESS]

(9:43)

Increasing [I]

1/v

[I] = 0

1/[S]

FIGURE 9.15 Lineweaver–Burke plot typical for noncompetitive enzyme inhibition. Such a plot should only be used as a diagnosis plot of enzyme inhibition.

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Enzyme Kinetics

9-19

Increasing [I] 1/v

[I] = 0

1/[S]

FIGURE 9.16 Lineweaver–Burke plot typical for uncompetitive enzyme inhibition. Such a plot should only be used as a diagnosis plot of enzyme inhibition.

Increasing [S] 1/v

1/v

[I]

[I]

(B)

Increasing [S]

1/v

(A)

Increasing [S]

(C)

[I]

FIGURE 9.17 Dixon plots for noncompetitive inhibition (A), competitive and mixed inhibition (B) and uncompetitive inhibition (C). Such plots should only be used as diagnosis plots of enzyme inhibition.

E+S + 2S

Ks

ES

kcat

E+P+Q

K ⬙I ESS 00

SCHEME 9.7 Kinetic scheme for substrate inhibition. Ks and KI are the equilibrium dissociation constants for the ES and ESS complex, respectively.

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9-20

The rate equation for such a situation is v¼

vmax [S] Ks [S]2 Ks þ [S] þ KI

(9:44)

It is quite likely that every type of inhibition does occur in foods with its many components. Finally, we would like to remind the reader that a situation could occur in which the reaction comes effectively to an end because thermodynamic equilibrium is achieved, as indicated above in Figures 9.9 and 9.10. This may be mistaken as inhibition but it is not, of course. The equations discussed above do not take this effect into account because one studies usually situations far from equilibrium. However, should one want to study situations close to equilibrium one should be aware of this thermodynamic limitation on rates. We now show an example of enzyme inhibition. It concerns polyphenoloxidase (PPO). The literature contains a number of examples of inhibition of this enzyme, as it is very important in relation to enzymatic browning in fruits and vegetables. Unfortunately, most of these reports still use linearized plots for parameter estimation. We would recommend to use nonlinear regression to derive the relevant parameters, for instance as in this example. It opens up more possibilities for model checking and may give better parameter estimates. This example is about the inhibition of polyphenoloxidase by oxalic acid; oxalic acid could be used as an inhibitor of enzymatic browning in foods. Initial rates were determined for a range of substrate concentrations as a function of oxalic acid concentration. The next question was to determine the kinetic mechanism by which oxalic acid inhibits PPO. Three models were tested, namely competitive inhibition, uncompetitive inhibition, and mixed inhibition, and parameters were estimated from all data available (global ﬁtting). The results in Figure 9.18 show clearly that it is deﬁnitely not uncompetitive inhibition, but the difference between a mixed inhibition model and the competitive mechanism model is not immediately clear from the graphs. The model discrimination test via the Akaike criterion shown in Table 9.5 gives preference to the mixed inhibition model, despite the fact that it has one parameter more. Unfortunately, there are no replicates available, so a goodness-of-ﬁt test cannot be performed. The precision of the parameter estimates is not very good; this is because we are trying to estimate four parameters from a limited number of data points. The ﬁts are certainly not perfect. The conclusion from this exercise would be that it is most likely that the inhibition mechanism is mixed inhibition, and the next phase of the modeling step should be to direct experimental design in such a way that the parameters can be estimated more precisely.

9.4

Progress Curves

Concentrations are usually easier to measure than rates, which is one reason why progress curves are of interest. The reaction velocity and substrate concentration vary both continuously during a reaction, and therefore a series of progress curves could contain in principle enough information for the determination of vmax and KM, but also to see whether perhaps another mechanism is active. Furthermore, it is then not necessary anymore to determine initial velocities, which is not trivial. Progress curves can be described by solving differential equations that can be written based on a postulated kinetic scheme. These differential equations can be numerically solved and ﬁtted to experimental progress curves to ﬁnd estimates for the relevant parameters. This is actually nothing more than we have advocated before, particularly in Chapters 4 and 8, but now applied to enzyme reactions. Software to do this is widely available, both commercially but also as freeware or shareware.* The familiar enzymatic * A number of calculations described in this chapter were done using the software program DynaFit (www.biokin. com=dynaﬁt). See references at the end of this chapter. This program derives differential equations based on proposed kinetic schemes. In addition, it performs the usual enzyme kinetic analyses, such as initial reaction rate versus substrate concentration and inhibition. Lineweaver and Dixon plots can be asked as diagnosis plots.

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0.8

0.9

0.7

0.8 0.7 0.6

0.5

v (AU min−1)

v (AU min−1)

0.6

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1 0 (A)

0.1

0

2

4 6 [S] (mM)

8

0

10

0

2

4 6 [S] (mM)

(B)

8

10

0.8 0.7

v (AU min−1)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

2

4

6

8

10

[S] (mM)

(C)

FIGURE 9.18 Enzyme inhibition mechanisms for the effect of oxalic acid on polyphenoloxidase. The lines describe competitive inhibition (A), uncompetitive inhibition (B), mixed inhibition (C). ^: 0 mM oxalic acid, D: 10 mM oxalic acid, þ: 30 mM oxalic acid, &: 50 mM oxalic acid. Dataset in Appendix 9.1, Table A.9.4.

parameters can still be derived if so desired. In fact, the Michaelis–Menten equation (Equation 9.10) is a rate equation that can be integrated. Integrated Michaelis–Menten equation. Recalling that the Michaelis–Menten equation in its simplest form is a rate equation describing the disappearance of substrate S and formation of product P as a function of the substrate concentration S: v¼

d[S] d[P] vmax [S] ¼ ¼ dt dt KM þ [S]

This equation can be integrated and this yields

(9:45)

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Kinetic Modeling of Reactions in Foods

9-22 TABLE 9.5

Parameter Estimates for the Inhibition of Polyphenoloxidase by Oxalic Acid

Model Competitive inhibition

Uncompetitive inhibition

Mixed inhibition

Parameter

Estimate 95% CI

vmax

0.75 0.06

Ks

0.39 0.15

KI

3.85 1.55

vmax

0.89 0.19

Ks

1.10 0.63

KI0 vmax Ks

30.9 18.1 0.78 0.06

SSr

AIC

DAIC

0.0389

144.1

3

0.170

108.7

38.4

0.0301

147.1

0

0.45 16

KI

5.48 2.73

KI0

135.9 127.6

Note: SSr, residual sums of squares; AIC, Akaike information criterion.

KM ln

[S0 ] þ [S0 ] [St ] ¼ vmax t [St ]

(9:46)

Unfortunately, this equation does not give an explicit expression for [St] or [Pt] as a function of time, while the experimental progress curve is, of course, a plot of concentration of reactant and=or product versus time. By rearranging Equation 9.46 one could ﬁnd the parameters KM and vmax from a linear plot: 1 [S0 ] [S0 ] [St ] vmax ¼ þ ln KM t [St ] KM t

(9:47)

This indicates that in principle KM and vmax can be estimated from a single progress curve. However, the earlier objections against estimating parameters from transformed data are also valid here, and use of Equation 9.47 is not a recommended method to ﬁnd KM and vmax. However, it could be an idea to use this method to ﬁnd initial estimates for KM and vmax from a single progress curve, to be used in later estimation procedures. There is, however, a possible pitfall in analyzing progress curves in this way: one has to be sure that (1) the enzyme is stable over the whole of the progress curve, (2) the reversible reaction must be negligible, (3) there must not be product inhibition. This is in contrast with the method of initial rates where one can safely assume that enzyme inactivation or product inhibition will not be noticeable at the very beginning of the reaction. If one of the conditions mentioned is not met, one will ﬁnd parameters that are confounded with other effects. It is therefore advisable to ﬁrst check the validity of the assumptions. A possible check is Selwyn’s test. Selwyn’s test. The so-called Selwyn’s test is useful to check whether the observed time course in change in substrate and=or product concentration is entirely due to the catalyzed reaction and nothing else. If the system is in steady state and the enzyme is stable and there is no inhibition, the product kcat[E0]t should be constant. This implies that the time courses for reactions with various enzyme concentrations [E0] should be superimposable when the time axis is multiplied by [E0]. Figure 9.19 gives an example of such an analysis; this clearly shows for this case that there are no confounding effects taking place. Figure 9.20 shows another case where Selwyn’s test shows that the mechanism is not a just a catalyzed reaction from reactant to product; rather, there could be product inhibition, or inactivation of the enzyme in the course of the reaction. What the complication is needs to be further investigated, of course (this is done below for this particular case).

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9-23

250

250

200

200 [Malate] (µM)

[Malate] (µM)

Enzyme Kinetics

150

100

150

100

50

50

0

0 0

5

(A)

10

15 20 t (min)

25

30

0

2

(B)

4 6 8 [E0]∗ t (nM min)

10

12

FIGURE 9.19 Progress curves for the hydration of fumarate to malate by the enzyme fumarase (A) and Selwyn’s test for this reaction (B). : [E0] ¼ 108 pM, &: [E0] ¼ 216 pM, ~: [E0] ¼ 432 pM; [S] ¼ 253 mM. Dataset in Appendix 9.1, Table A.9.5.

9

9

8

8

7

7

6

6

5 4

5 4

3

3

2

2

1

1

0 (A)

[P] (mM)

[P] (mM)

Analysis of progress curves. The analysis of progress curves is best illustrated by some examples. The ﬁrst case concerns the action of b-galactosidase on a synthetic substrate, for which ﬁve progress curves are available. We take Scheme 9.2 as starting point and assume rapid equilibrium, which implies a diffusion-limited rate constant k1 ¼ 6 3 106 mM min1 and we are left with the task to estimate

0

20

40

60 t (min)

80

100

120

0 (B)

0

20

40

60 80 ∗ −1 (ET) t (µg mL min)

100

120

FIGURE 9.20 Progress curves (A) and Selwyn’s plot (B) for the hydrolysis of 4-nitrophenyl phosphate by alkaline phosphatase. ^: 141 mg mL1 enzyme, &: 282 mg mL1 enzyme, D: 564 mg mL1 enzyme, : 846 mg mL1 enzyme, j : 1128 mg mL1 enzyme. Dataset in Appendix 9.1, Table A.9.6. 3

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9-24

k2 and k3. (Estimation of k1 was not well possible with the present dataset.) The resulting differential equations are d[E] ¼ k1 [E] [S] þ k2 [ES] þ k3 [ES] dt d[S] ¼ k1 [E] [S] þ k2 [ES] dt d[ES] ¼ k1 [E] [S] k2 [ES] k3 [ES] dt d[P] ¼ k3 [ES] dt

(9:48)

These equations were solved numerically and ﬁtted to the data. Note that we do not need a steady-state assumption here because we do not need an analytical expression for the rate if we solve the equations numerically. The result of solving the equations in Equation 9.48 and ﬁtting them to the data via nonlinear regression is in Figure 9.21. The results are not very good overall, there is clearly a misﬁt. The mechanism of product inhibition was therefore introduced, Scheme 9.6, yielding the following set of differential equations: d[E] ¼ k1 [E] [S] þ k2 [ES] þ k3 [ES] k4 [E] [P] þ k5 [EP] dt d[S] ¼ k1 [E] [S] þ k2 [ES] dt d[ES] ¼ k1 [E] [S] k2 [ES] k3 [ES] dt d[P] ¼ k3 [ES] k4 [E] [P] þ k5 [EP] dt d[EP] ¼ k4 [E] [P] k5 [EP] dt

(9:49)

4.5 4 3.5

[PNPG] (mM)

3 2.5 2 1.5 1 0.5 0

0

100

200 300 Time (min)

400

500

FIGURE 9.21 Progress curves for the action of galactosidase on PNPG and ﬁt of the Michaelis–Menten mechanism according to Equation 9.48 (solid lines). Dataset in Appendix 9.1, Table A.9.7.

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Enzyme Kinetics

9-25

4.5 4 3.5

[PNPG] (mM)

3 2.5 2 1.5 1 0.5 0

0

100

200 300 Time (min)

400

500

FIGURE 9.22 Progress curves for the action of galactosidase on PNPG and global ﬁt of the Michaelis–Menten mechanism þ product inhibition according to Equation 9.48 (solid lines). Dataset in Appendix 9.1, Table A.9.7.

The same rate constant was assumed for formation of the ES and EP complex, i.e., k1 ¼ k4 in Scheme 9.5, and this rate constant was ﬁxed at its diffusion limit value of 6 3 106 mM1 min1. This left three rate constants to be estimated from this scheme. The results are in Figure 9.22, and they are much better than the ones displayed in Figure 9.21, though certainly not perfect when looking at the residuals. Since replicate data were not available we cannot test the goodness of ﬁt, unfortunately. Table 9.6 shows the parameter estimates and model discrimination test via the Akaike criterion. The model with product inhibition is clearly favored despite the presence of one parameter more. TABLE 9.6

Parameter Estimates for the Action of Galactosidase on the Synthetic Substrate PNPG

Model

Estimate 95% CI

Parameter

Rapid equilibrium

Rapid equilibrium þ product inhibition

k1 (mM1 min1)

6 3 106 (ﬁxed)

k2 (min1)

1.25 3 107 5.5 3 106

k3 (min1)

1.29 3 104 2.84 3 103

Ks ¼ k2=k1 (mM)

2.08 0.9

vmax (k33[E]) (mM min1)

0.013 0.003

k1 (mM1 min1)

6 3 106 (ﬁxed)

k2 (min1) k3 (min1)

1.30 3 107 4.55 3 106 2.42 3 104 5.0 3 103

k4 (mM1 min1)

6 3 106 (ﬁxed)

1

Ks (mM)

2.17 0.77

vmax (k33[E]) (mM min1)

0.024 0.005

KI (mM)

0.59 0.17

KM (mM)

2.6 0.45

vmax (mM min1) 1

Note: k1 and k4 ﬁxed at 6 3 10 mM criterion. 6

1

AICc

DAIC

260

107

0.115

367

0

3.52 3 106 1.04 3 106

k5 (min )

Michaelis–Menten via initial rates

SSr 0.705

0.024 0.002 6

min . [E] ¼ 10

mM. SSr, residual sums of squares; AIC, Akaike information

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9-26

160 140

v (mM min-1) ⫻ 104

120 100 80 60 40 20 0

0

1

2

3

4

5

[S] (mM)

FIGURE 9.23 Michaelis–Menten plot for initial rates of galactosidase acting on PNPG, as shown in Figures 9.21 and 9.22. The line is obtained via nonlinear regression. Dataset in Appendix 9.1, Table A.9.7.

For comparison, we show in Figure 9.23 the Michaelis–Menten plot obtained from initial rates that were available for the same case. There is, of course, no effect of product inhibition because only initial velocities are used. The KM and vmax estimates are also given in Table 9.6. They compare favorably to the estimates found from the global ﬁt to the progress curves, where Ks and vmax can be calculated indirectly from the rate constants, at least if we assume the product inhibition mechanism. For the plain rapid equilibrium scheme the result for vmax does not match very well, which is another indication that the plain mechanism is not the correct one. The next example is a further analysis of the plots shown in Figure 9.19 about the action of a phosphatase. Selwyn’s test already indicated that this was not a straightforward case of one-substrate irreversible catalysis without any disturbance. Here we analyze two possible causes for this behavior, namely ﬁrst-order enzyme inactivation and product inhibition according to Scheme 9.8 (compare Scheme 9.6). The model to describe the experimental results was in terms of a Michaelis–Menten equation derived by applying the steady-state assumption: d[P] k3 [E] [S]=KM ¼ [S] [P] dt 1þ þ KM KP d[E] ¼ dt

E+S

k1 k2

ES

(9:50)

ki [E] [S] [P] þ 1þ KM KP k3

E+P

k4 k5

EP

ki E⬘

SCHEME 9.8 Kinetic scheme describing enzyme inactivation and product inhibition. E0 symbolizes inactivated enzyme.

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Enzyme Kinetics

9-27

9 8 7

[P] (mM)

6 5 4 3 2 1 0 0

20

40

60 Time (min)

80

100

120

FIGURE 9.24 Global ﬁt of Equation 9.50 (solid lines) to progress plots for the hydrolysis of 4-nitrophenyl phosphate by alkaline phosphate. ^: 141 mg mL1 enzyme, &: 282 mg mL1 enzyme, D: 564 mg mL1 enzyme, : 846 mg mL1 enzyme, 3j : 1128 mg mL1 enzyme. Dataset in Appendix 9.1, Table A.9.6.

Introduction of only the step of enzyme inactivation with rate constant ki without product inhibition (by putting [P] ¼ 0 in Equation 9.50) was unsuccessful; product inhibition without enzyme inactivation (ki ¼ 0) was also unsuccessful: it was impossible to ﬁnd a suitable ﬁt of the model to the data. Only when product inhibition and enzyme inactivation were combined a ﬁt was found: see Figure 9.24 and Table 9.7. The ﬁt shown in Figure 9.24 is reasonable but certainly not perfect, especially not for the higher enzyme concentration. There appeared also to be a statistical problem since the conﬁdence interval for KM could not be estimated. Probably, there is not enough information in the data to extract all parameter values. Nevertheless, the analysis indicates that the mechanism in Scheme 9.8 is not unreasonable and a better experimental design for further analysis would probably solve the statistical problem of estimating the precision of KM. The next example shows how inhibition can be estimated from progress curves. It concerns the action of pepsin on a chromogenic substrate in the presence of various amounts of a pepstatin analog inhibitor. TABLE 9.7 Parameter Estimates Obtained by Fitting Equation 9.50 to the Data Displayed in Figure 9.24 Parameter

Estimate

95% Conﬁdence Interval

k3 (mmol min1 mg1)

0.946

0.035

KM (mM)

2.61

Indeterminate

KP (mM)

1.587

0.191

ki (min1)

0.218

0.008

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Kinetic Modeling of Reactions in Foods

The task is to ﬁnd the kinetic parameters that describe the inhibition. The substrate concentration was not varied, so a plot like in Figure 9.14 through 9.17 could not be made. The mechanisms of competitive and uncompetitive inhibition were investigated, as displayed in Schemes 9.4 and 9.5. For the competitive mechanism the following differential equations can be derived. d[E] ¼ k1 [E] [S] þ k2 [ES] þ k3 [ES] k4 [E] [I] þ k5 [EI] dt d[S] ¼ k1 [E] [S] þ k2 [ES] dt d[ES] ¼ k1 [E] [S] k2 [ES] k3 [ES] dt d[P] ¼ k3 [ES] dt d[I] ¼ k4 [E] [I] þ k5 [EI] dt d[EI] ¼ k4 [E] [I] k5 [EI] dt

(9:51)

It was not possible to estimate all the rate constants from the available data and so the following assumptions were made: .

.

.

The rate constant for the binding of substrate and inhibitor was assumed to be the same (k1 ¼ k4). Rapid equilibrium was assumed so that we can put k1 equal to the diffusion limit, estimated to be 108 M1 s1 for enzymes. The inhibition constant KI in Scheme 9.4 is written in terms of the rate constants k5 and k4: KI ¼ k4=k5. By making these assumptions, these rate equations could be simultaneously solved using numerical integration and ﬁtted to the available datasets and resulted in parameter estimates for the rate constants k2, k3, and k5.

The ﬁts are shown in Figure 9.25 and the parameter estimates in Table 9.8. Similarly, kinetic equations were derived for the uncompetitive inhibition Scheme 9.5 with k1 ¼ k4: d[E] ¼ k1 [E] [S] þ k2 [ES] þ k3 [ES] dt d[S] ¼ k1 [E] [S] þ k2 [ES] dt d[ES] ¼ k1 [E] [S] k2 [ES] k3 [ES] k4 [ES] [I] þ k5 [ESI] dt d[P] ¼ k3 [ES] dt d[I] ¼ k4 [ES] [I] þ k5 [ESI] dt d[ESI] ¼ k4 [ES] [I] k5 [ESI] dt

(9:52)

This model did not ﬁt well (plots are not shown); model comparison and goodness-of-ﬁt analyses are reported in Table 9.9. It is beyond doubt that the uncompetitive mechanism is not valid at all, while the goodness of ﬁt for the competitive mechanism is quite acceptable. This is perhaps not unexpected because the pepstatin analog is supposed to work according to a competitive mechanism, but the model comparison is shown here to make clear how it can be used to discriminate between models based on progress curve analysis.

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Enzyme Kinetics

9-29

100 90 80 70

[S] (µM)

60 50 40 30 20 10 0

0

1

2 3 Time (min)

4

5

FIGURE 9.25 Global ﬁts of the competitive inhibition kinetic scheme to progress curves of pepsin activity in the presence of a pepstatin analog. [E] ¼ 0.04 mM, [S] ¼ 100 mM, pH 4, T’ ¼ 37 8C, ^: [I] ¼ 0 mM, ~: [I] ¼ 0.05 mM, : [I] ¼ 0.1 mM, þ : [I] ¼ 0.2 mM, *: [I] ¼ 0.3 mM, &: [I] ¼ 0.5 mM, ~: [I] ¼ 1.0 mM, &: [I] ¼ 2.0 mM. Dataset in Appendix 9.1, Table A.9.8.

There are many more things to be said about the analysis of progress curves, but we leave it here. It is hoped that the reader has obtained enough inside information to handle the abundant literature on this topic.

9.5

Kinetics of Two-Substrate Reactions

So far, we have only considered enzyme-catalyzed reactions in which one substrate is converted to one product. However, Table 9.1 showed already that there are more possibilities. Next to one-substrate mechanisms, reactions in which two substrates are converted into products are equally important. It concerns the bi–bi and bi–uni reactions in Table 9.1. One makes usually the distinction between so-called ping-pong mechanisms and sequential mechanisms. A ping-pong mechanism means that one or more products need to TABLE 9.8 Parameter Estimates for the Kinetic Competitive Inhibition Scheme Applied to the Inhibition of Pepsin by a Pepstatin Analog (Figure 9.25) Parameter k4 ¼ k4 (mM1 s1) k2 (s1) k3 or kcat (s1) k5 (s1)

Estimate 100 (ﬁxed) 2113 7514 9.2

95% Conﬁdence Interval — 263 24.5 0.6

Derived Parameters Ks ¼ k2=k1 ¼ 21 mM KI ¼ k4=k5 ¼ 0.09 mM vmax ¼ k3 [E] ¼ 300 mM s1

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9-30

TABLE 9.9 Goodness of Fit and Model Discrimination for Competitive and Noncompetitive Inhibition of Pepsin by a Pepstatin Analog F-Value for Lack of Fit

Mechanism

Sampling Probability

AIC

DAIC

Competitive inhibition

1.00

0.491

162

0

Uncompetitive inhibition

9.427

0.000

230

392

be released before all substrates can react, while a sequential mechanism means that ﬁrst all substrates need to combine with the enzyme before products can be formed. A further distinction can be made into ordered and random sequential mechanisms. Ordered sequential mechanisms imply that there is a speciﬁc order in going from substrates to products, while this is not so for random sequential mechanisms. Let us ﬁrst consider the random sequential bi–bi mechanism. Scheme 9.9 shows the model for this situation. The resulting rate equation for this scheme is v [A] [B] ¼ vmax KsA K AB þ K AB [A] þ K BA [B] þ [A] [B]

(9:53)

vmax ¼ kcat [ET ] With two substrates, one can hold one of the reactants constant and vary the other. The response of the system in such a case helps to establish the kinetic parameters. So, if [A] is kept constant, Equation 9.53 becomes v [B] ¼ 0 0 vmax K þ [B] vmax [A] 0 ¼ BA vmax K þ [A] K A K AB þ K AB [A] K 0 ¼ s BA K þ [A]

(9:54)

Similar expressions hold when [B] is kept constant v

[S] þ [A] v [B] max 0 ¼ AB vmax K þ [B] K A K AB þ K BA [B] K 0 ¼ s AB K þ [B]

0 vmax

E+A + B

¼

KA s

K0

(9:55)

EA + B KAB

K Bs

EAB

EB + A K BA

kcat

E+P+Q

SCHEME 9.9 Scheme for the random sequential bi–bi mechanism for the two substrates A and B. The K symbols represents the equilibrium dissociation constants for the enzyme complexes in the reaction indicated.

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KAB

KA s E+A

EA + B

kcat

EAB

E+P+Q

SCHEME 9.10 The ordered sequential bi–bi mechanism for the two substrates A and B. The K symbols represents the equilibrium dissociation constant for the enzyme complexes in the reaction indicated.

The case for an ordered sequential bi–bi mechanism is depicted in Scheme 9.10. Examples of enzymes acting like this are lipoxygenase and polyphenoloxidase. v [A] [B] ¼ vmax KsA K AB þ K AB [A] þ [A] [B]

(9:56)

vmax ¼ kcat [ET ] When [B] is kept constant, Equation 9.56 becomes v [A] ¼ 0 0 vmax K þ [A] 0 vmax [B] 0 vmax ¼ AB K þ [B] KsAB K AB K 0 ¼ AB K þ [B]

(9:57)

For constant [A] it becomes v [B] ¼ 0 0 vmax K þ [B] 0 vmax ¼ vmax

K0 ¼

KsA K AB [A]

(9:58) þ K AB

The kinetic scheme for a ping-pong bi–bi mechanism is shown in Scheme 9.11. Xanthine-oxidase is an enzyme that shows this mechanism. The resulting rate equation is v [A] [B] ¼ vmax (kcat =k2 )KmA [B] þ KmB [A] þ [A] [B](1 þ kcat =k2 )

(9:59)

vmax ¼ kcat [ET ]

A Km

E+A

k1 k−1

B Km

EA

k2

E⬘+ B

k3 k−3

E⬘B

kcat

E+Q

P

SCHEME 9.11 The ping-pong bi–bi mechanism for the two substrates A and B. The K symbols represents the equilibrium dissociation constant for the enzyme complexes in the reaction indicated.

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Random

Random Ping-pong K⬘

K⬘

Ping-pong

Ordered (A)

Ordered

(B)

FIGURE 9.26 Dependence of the apparent Michaelis constant K0 on concentration of (A) when (B) is held constant (A) and on (B) when (A) is held constant (B) for random-sequential, ordered-sequential, and ping-pong mechanisms.

When [B] is kept constant, Equation 9.59 becomes v [A] ¼ 0 0 vmax K þ [A] v0 [B] 0 ¼ Bmax vmax Km þ [B] (kcat =k2 )KmA [B] K0 ¼ KmB þ [B]

(9:60)

When [A] is kept constant, Equation 9.59 becomes v [A] ¼ 0 0 vmax K þ [A] vmax [A] 0 ¼ vmax (kcat k2 )KmA þ [A] KmA [B] K0 ¼ (kcat =k2 )KmA þ [A]

(9:61)

A relevant example of bi–bi mechanisms for foods is, once again, the oxidation of polyphenols by polyphenoloxidase, where oxygen and the polyphenol are the two substrates. It is not an easy task to ﬁnd out which mechanism is operative. One can get an impression of the kind of mechanism by studying the behavior of K0 as a function of the two substrates A and B for a ﬁrst impression as is very schematically depicted in Figure 9.26. Likely models can then be tested via model discrimination as discussed above for one substrate mechanisms.

9.6

Other Types of Enzyme Kinetics

Cooperative and allosteric enzymes. In so-called cooperative enzymes, there are low- and high-afﬁnity substrate binding sites. If the binding of one substrate molecule results in altered substrate binding afﬁnity, this is called cooperativity. The binding afﬁnity can increase or decrease in principle, but usually it is an increase. If enzyme activity is affected by binding to other sites than the active site, we speak of allosteric enzymes. This binding can be substrate binding (homotropic allosteric response) as well as

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other ligands (heterotropic allosteric response). It is not always possible to distinguish between allosteric effects and cooperativity. Cooperative effects result in sigmoidal curves when v is plotted versus [S] rather than hyperbolic Michaelis–Menten type curves. Michaelis–Menten kinetics is therefore not suitable for cooperative and allosteric effects. A general equation that appears to be suitable to describe cooperativity is the so-called Hill equation: v¼

vmax [S]h KHh þ [S]h

(9:62)

The exponent h is called the Hill coefﬁcient, and expresses the degree of cooperativity in the substrate binding process; the higher h, the higher the cooperativity. KHh is called the Hill constant and is a measure for the afﬁnity of the enzyme for its substrate; however, it does not reﬂect the enzyme–substrate dissociation constant. For h ¼ 1, the Hill constant equals the Michaelis constant. Figure 9.27 shows some simulated curves described by the Hill equation. Note that if h ¼ 1, the Hill equation reduces to the Michaelis–Menten equation (Equation 9.10). If one ﬁnds experimentally sigmoid curves, one could derive the usual parameters plus the Hill coefﬁcient via nonlinear regression. Figure 9.28 shows a real life example. The initial rate–concentration plot is deﬁnitely not hyperbolic, but sigmoidal and therefore the Hill equation was ﬁtted to the data, which gives a reasonable ﬁt, also shown in Figure 9.28. Immobilized enzyme kinetics. Immobilized enzymes are increasingly used, also in food technology. Enzymes can then be used much more efﬁciently, and usually their stability is much enhanced. The reaction is carried out in enzyme reactors. The enzyme is bound on a carrier, which is kept in a reactor, the substrate is fed to the reactor and converted by the enzyme, and the product is removed. This can be done in batch reactors, in plug ﬂow reactors and in continuous-stirred reactors. In addition, membrane

1.2

1

h=4 h=3

v (arbitrary units)

0.8

h=2 h=1

0.6

0.4

0.2

0

0

0.5

1 [S] (arbitrary units)

1.5

2

FIGURE 9.27 Simulated curves using the Hill equation to describe cooperativity. vmax ¼ 1 (arbitrary units) and KM ¼ 0.5 (arbitrary units), the Hill coefﬁcient h is varied (Equation 9.62).

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0.25

v (mM min−1)

0.2

0.15

0.1

0.05

0

0

5

10 15 Cholesterol (mM)

20

FIGURE 9.28 Initial rate of cholesterol oxidase as a function of cholesterol concentration. The line is the ﬁt of the Hill equation with parameters vmax ¼ 0.298 mM min1, KH ¼ 11.8 mM, h ¼ 2.6. Dataset in Appendix 9.1, Table A.9.9.

reactors are also possible. The kinetics of immobilized enzymes do not fundamentally change relative to the nonimmobilized ones, but there may be shifts in parameter values when compared to the nonimmobilized enzyme. Interfacial enzyme kinetics. When there are different phases present in a food, an interfacial area will exist. The most noticeable example is, of course, the presence of fat droplets in oil-in-water emulsions or water droplets in water-in-oil emulsions. Enzyme catalysis can take place at such interfaces, such as lipases hydrolyzing triglycerides. That implies that before the actual catalysis takes place enzymes must bind to the interface, and, incidentally, it may then undergo a conformational change due to the adsorption process. This way of substrate binding is of course different from the enzyme-substrate binding discussed hitherto. One way to tackle interfacial kinetics is via the simple Scheme 9.12. The adsorption of the enzyme to the interface can be described by a Langmuir type equation, with the enzyme coverage parameter q and at equilibrium this results in q¼

E* [E] ¼ * Kd* þ [E] Emax

(9:63)

This equation thus describes the surface coverage by interfacial enzyme as a function of the enzyme concentration in solution. Emax * is the saturation surface concentration of interfacial enzyme. The equilibrium constant describing dissociation from the interface Kd* is (compare Scheme 9.12)

ka E kd

E∗ + S

k1 k2

ES

k3

E+P

SCHEME 9.12 Simpliﬁed scheme representing interfacial kinetics. E symbolizes free enzyme (mol dm3), E* the enzyme bound to the interface (mol m2).

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Kd* ¼

kd ka

(9:64)

The change in interfacial enzyme coverage is described as dq * E*)As kd (E*)As ¼ ka [E](Emax dt

(9:65)

As is the surface area per unit volume (m2 dm3). We can now set up the following equations: v ¼ k3 (E*)As

(9:66)

[E](Emax * E*) E*

(9:67)

[ET ] ¼ [E] þ (E*)As

(9:68)

Kd* ¼

Rearranging and normalization by [ET] leads to v¼

vmax aint Kd* þ aint

(9:69)

* )(1 q)As aint ¼ (Emax

(9:70)

vmax ¼ k3 [ET ]

(9:71)

Thus, we end up again with a Michaelis–Menten type equation (Equation 9.69) describing a hyperbola: see Figure 9.29. Figure 9.30 shows some simulations for various values of the parameters involved to see how these parameters inﬂuence the resulting rate plot. King–Altman procedure. The King–Altman procedure is an ingenious graphically based procedure developed to derive manageable rate expressions for complicated enzymatic reaction schemes. The

12 vmax

10

v (M s−1)

8 6 4 2 0

0 K ∗d

50 As (m2 dm−3)

100

FIGURE 9.29 Initial rate v versus interfacial area per unit volume As describing interfacial enzyme kinetics.

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12 10

12

K ∗d (M)

J = 0.1 Ε ∗max = 0.00001

10−6

Ε *max = 0.00001 K ∗d = 0.00001

10

10−5

0.1 0.4 0.6

8 v (M s−1)

v (M s−1)

8 6 10−4

J=

0.8

6 4

4

2

2 −3

10 0 (A)

0

2

4

6

8

10

0

12

As (m2 dm−3)

0

2

4

(B)

12

10

12

Ε ∗max (mol m−2) 10−4

J = 0.1 K ∗d = 0.00001

10

6 8 As (m2 dm−3)

10−5

v (M s−1)

8

6 10−6

4 2 0

0 (C)

2

4

6

8

10

12

As (m2 dm−3)

FIGURE 9.30 Effect of the parameters Kd* (A), q(B), and E*max (C) on initial rate v versus surface area As for the interfacial kinetics model.

procedure is based on graph theory and mainly used in biochemical engineering. The graphs consist of vertices, representing enzyme species, and arcs, representing interconversion between any two enzyme species; pseudo ﬁrst-order reversible reactions are postulated between the various enzyme forms. Weights are given by the appropriate rate constant. If a rate constant ¼ 0, then the arc from one enzyme to another is not given. The results lead to steady-state rate equations. We do not discuss it further; the interested reader can ﬁnd some references at the end of this chapter.

9.7

Temperature Effects

As was discussed in Chapter 5, temperature has a large effect on reaction rates. This is also true for enzyme-catalyzed reactions, but there are differences. Enzymes being proteins, they may denature and

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10

2

9

1.8

8

1.6

7

1.4

6

1.2

5

1

4

0.8

3

0.6

2

0.4

1

0.2

0 (A)

9-37

ln k

v (µM−1 s−1)

Enzyme Kinetics

0

10

20

30 40 T ⬘(⬚C)

50

60

0

70

0

20

(B)

40 T ⬘(⬚C)

60

80

FIGURE 9.31 Effect of temperature on the activity of the enzyme galactosidase (A) and on the rate constant for the action of invertase on sucrose (B). Dataset in Appendix 9.1, Tables A.9.10 and A.9.11.

loose their enzyme activity. Also the activation energy and entropy is different from the uncatalyzed reaction. Figure 9.31 gives two experimental examples. This ﬁgure clearly indicates that, starting from low temperature, increasing temperature increases the reaction rate but only up to a maximum; after a certain temperature is reached, the activity rapidly declines, due to denaturation (unfolding) of the enzyme. The resulting temperature-activity plot is the balance of higher activity at higher temperature and the loss of active enzyme because of denaturation. If we consider the condition that the enzyme is saturated with substrate (i.e., [S] >> KM) the rate is as expressed in Equation 9.9, v ¼ kcat [ET]. The concentration of active enzyme is the total amount minus the inactive, i.e., denatured form. An equilibrium constant can be deﬁned for the ratio between the concentration of native [EN] and denatured enzyme [ED] in the scheme EN Ð ED: Keq ¼

[ED ] [EN ]

(9:72)

(The intricacies of protein unfolding are discussed in the next chapter.) The mass balance for the enzyme is [ET ] ¼ [EN ] þ [ED ]

(9:73)

In order to describe experimental results as shown in Figure 9.31 we can write the rate as a function of enzyme concentration v(T) ¼ kcat (T)[EN ] ¼

kcat (T)[ET ] 1 þ Keq (T)

(9:74)

In this last equation, both Keq and kcat are made temperature dependent. For the dependence of kcat we can use the Arrhenius equation (Equation 5.12) Ea kcat (T) ¼ A exp RT

(9:75)

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0.6

2.5

0.5

2

In k

ln (activity)

0.4 1.5 0.3

1 0.2

0.5

0 0.0031

0.1

0.0032

0.0033

(A)

1/T

0.0034

(K −1)

0.0035

0 0.0033 0.0034 0.0034 0.0035 0.0035 0.0036 0.0036 0.0037

0.0036

(B)

1/T (K −1)

FIGURE 9.32 Arrhenius plot for the activity of the enzyme galactosidase (A) and on the rate constant for the action of invertase on sucrose (B). Dataset in Appendix 9.1, Tables A.9.10 and A.9.11.

Alternatively the Eyring equation (Equation 5.11) could be used. The temperature dependence of Keq is more complicated 0 0 DHd DCP0 DCP Td DHd0 1 DCP0 (1 þ ln Td ) þ ln T þ Keq (T) ¼ exp RTd R R R R T

(9:76)

This equation combines basically the van’t Hoff equation (Equation 5.2) and the relation between heat capacity DCp0 and enthalpy, Equation 3.16. It is assumed in this derivation that DCp0 is temperature independent over the range where unfolding takes place. Td and DHd0 are the temperature and enthalpy change at the denaturation temperature. Equations 9.75 and 9.76 can be used to ﬁt proﬁles such as the ones in Figure 9.31, provided that the parameter values in Equation 9.76 are known. We now focus on the increase in activity well before it reaches the maximum; to be sure, the position of such a maximum is, of course, enzyme speciﬁc. Figure 9.32 shows an Arrhenius plot of the data shown in Figure 9.31. Table 9.10 shows the parameter estimates obtained via nonlinear regression of the Eyring equation (see Chapter 5). The activation enthalpy is rather low compared to the one for a ‘‘normal’’ chemical reactions (usually around 100 kJ mol1), and this is expected because a low activation energy is a typical consequence of catalysis. The activation entropy is negative because the formation of the enzyme–substrate complex leads to less conformational possibilities. In other words, the magnitude of the rate constant is much higher and the temperature dependence of catalyzed reactions, including

TABLE 9.10 Parameter Estimates Resulting from Nonlinear Regression of the Eyring Equation on the Data Shown in Figure 9.32 Parameter

Galactosidase (95% CI)

Activation enthalpy (kJ mol1)

37 2.7

Activation entropy (J mol1 K1)

112 9

Invertase (95% CI) 9.2 1.4 210 5

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kcatalyzed ln k

Rate enhancement

kuncatalyzed 1/T

FIGURE 9.33 reaction.

Schematic drawing of rate enhancement accomplished by enzymes as compared to the uncatalyzed

enzyme reactions, is substantially lower than for an uncatalyzed reaction. The rate enhancement of enzymes is particularly high at lower temperatures. This is schematically shown in Figure 9.33. A word of caution, however, is necessary here. When it comes to interpretation of the estimates as in Table 9.10, it is not straightforward to relate the parameters to the basic rate constants in, for instance, the Michaelis–Menten Scheme 9.2. The resulting activity is a combination of the various reaction rate constants and concentrations, and the temperature dependence found from an Arrhenius or Eyring plot of enzyme activity cannot be attributed to one single reaction rate constant, or a single elementary step. This situation has been discussed basically in Equations 5.21 through 5.28, and the analysis given there is perfectly applicable to the temperature dependence of enzyme kinetic constants.

9.8

pH Effects

Enzymes being proteins are sensitive to pH and it is quite common in literature to report a pH proﬁle, showing the dependence of enzyme activity on pH. Figure 9.34 gives one of the many examples available, in this case liver enzymes acting on metmyoglobin being much more active in the lower pH range, and a pectinesterase, which seems to have its pH optimum between pH 7 and 9. Many models are available to describe the effect of pH. We will consider only a simple case (proposed by Michaelis in 1911) which shows the general line of thinking. The enzyme is considered to be a dibasic acid with two nonidentical acidic groups, and dissociation can take place as depicted in Scheme 9.13. The dissociation constants Ki are called group dissociation constants. The total enzyme concentration is [ET ] ¼ [HEH] þ [ EH] þ [HE ] þ [E ]

(9:77)

The following equations can be derived for the various enzyme species: [HEH] ¼

[ET ] K1 þ K3 K1 K2 þ þ2 1þ [Hþ ] [H ]

[ET ]K1 [Hþ ] [ EH] ¼ K1 þ K3 K1 K2 þ þ2 1þ [Hþ ] [H ]

(9:78)

(9:79)

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3.5

120

3

100

80 % Activity

v ⫻ 108/M−1 s−1

2.5 2 1.5

60

40 1 20

0.5 0

0 5

(A)

5.5

6

6.5

7

5

7.5

pH

6

7 pH

(B)

8

9

FIGURE 9.34 pH–activity plot for liver enzymes on metmyoglobin reduction (A) and papaya pectinesterase on pectin (B). Dataset in Appendix 9.1, Tables A.9.12 and A.9.13.

[ET ]K3 [Hþ ] [HE ] ¼ K1 þ K3 K1 K2 þ þ2 1þ [Hþ ] [H ]

(9:80)

[ET ]K1 K2 [Hþ ]2 [E ] ¼ K1 þ K3 K1 K2 þ þ 2 1þ [Hþ ] [H ]

(9:81)

K1 K 2 ¼ K 3 K 4

(9:82)

Furthermore,

The above relations are shown graphically in Figure 9.35. −

EH

K1

K2

HEH

E− −

K4

K3 HE−

SCHEME 9.13 Dissociation possibilities for a dibasic acid of an enzyme E. The K-values represent dissociation constants.

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1 0.9

E− −

HEH

Relative concentration

0.8

−

EH

0.7 0.6 0.5 0.4 0.3 0.2 HE−

0.1 0

3

4

5

6

7

8

9

10

pH

FIGURE 9.35 Relative concentrations of enzyme forms with a dibasic acid as a function of pH. Numerical values for dissociation constants are K1 ¼ 13106, K2 ¼ 13108, K3 ¼ 13106.8, K4 ¼ 13107.2.

One can imagine certain pH proﬁles assuming that enzyme activity is reﬂected by concentrations of one of the forms depicted in Figure 9.35. The plot shown for HEH is reﬂected in the behavior shown in Figure 9.34A. If EH is the active form, for instance, the pH dependency will be bell shaped as the curve for [EH]; such behavior is reminiscent of what is shown in Figure 9.34B. The initial rate will in such a case vary according to v¼

kcat [ET ] [Hþ ] K1 K2 =(K1 þ K3 ) þ 1þ K1 þ K3 [Hþ ]

(9:83)

The effect of pH on enzyme activity is often plotted as the logarithm of a kinetic constant versus pH; this is known as a Dixon–Webb plot. Let us consider as a relatively simple case a Michaelis–Menten type equation in which the enzyme as well as the enzyme–substrate complex is protonated (Scheme 9.14). Applying the steady-state assumption results in

vmax [S] 1 þ ([Hþ ]=K2 ) v¼ 1 þ ([Hþ ]=K1 ) þ [S] KM 1 þ ([Hþ ]=K2 )

(9:84)

We thus have an apparent vmax and KM as a function of pH 0 vmax ¼

vmax [Hþ ] 1þ K2

(9:85)

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9-42 k1 E+S

H+

K1

K2

HE

SCHEME 9.14

k3

ES

E+P

H+ HES

Protonation of free enzyme and of the enzyme–substrate complex.

1 0 [Hþ ] 1 þ B K1 C 0 C KM ¼ KM B @ [Hþ ]A 1þ K2

(9:86)

Figure 9.36 shows a Dixon–Webb plot for these apparent kinetic parameters. More complicated expressions can be derived, but we leave it here because the general approach is as given here. Some selected references on this topic are given at the end of this chapter.

9.9

Experimental Design for Enzyme Kinetics

2.5

2.5

2

2

1.5

1.5

log (K ⬘M)

log (v ⬘max)

Experimental design was already discussed in Chapter 7 and the general rules given there also apply, of course, to enzyme experiments. As a reminder, proper experimental design is about how to set up experiments such that most information is extracted from the data. There is considerable literature on experimental design speciﬁcally for enzyme experiments; some references are listed below. As discussed in Chapter 7, an important consideration is what the goal of the experiment is ﬁnding out a particular mechanism (model discrimination), ﬁnding the most precise estimates (parameter estimation), or making predictions. Depending on that goal one can choose a particular design. As shown before, it is

1

0.5

0

1 pK1 = 5

0.5

5

(A)

FIGURE 9.36

6

7

8 pH

9

0

10 (B)

pK2 = 7

3

4

5

6

7

8

9

10

pH

Dixon–Webb plot for the apparent vmax (A) and KM (B) according to the mechanism in Scheme 9.14.

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necessary to have replicate results so that experimental error can be estimated and modeled. As a rule of thumb, for simple Michaelis–Menten kinetics, substrate concentrations should be chosen around the KM value, but not necessarily equally spaced around KM. Prior knowledge about the magnitude of KM is needed, obviously, in order to make such designs. One can also approach the design problem as a decision problem: one has to decide about what are the most useful settings to obtain information. In this sense, this becomes a Bayesian inspired design in maximizing a utility function. This is an interesting new approach; some references are given at the end of this chapter.

9.10

Enzyme Kinetics in Foods

The complex matrix of food is quite a challenge when studying enzyme kinetics: initially enzymes may be separated from substrate and inhibitors due to compartmentalization. Upon processing or because of ripening processes, enzymes may encounter substrate but also inhibitors and activators. However, it is difﬁcult to predict how and what will happen. That means that the only way to ﬁnd out is via experimentation. An interesting discussion arises as to whether such studies should be done in ‘‘clean’’ model systems, or in the foods themselves. When done in clean systems, one learns probably more about the enzyme itself, but less about what the effect in the food will be. In terms of food quality prediction it does not seem to make much sense to work with puriﬁed enzymes and substrates, except when the interest is purely in the mechanistic effect. Having said that, the food matrix is probably too complex to study in its whole, so some sort of model system is needed. On the other hand, when studies are carried out in the food, there are so many possible complications that it will be hard to draw conclusions. The author is of the opinion that both types of studies should be done, starting at the mechanistic level, i.e., with puriﬁed systems, but working towards the food situation. One should mimic the food situation as closely as possible eventually. Typical food problems that can ‘‘disturb’’ enzyme kinetics are . . . . .

. .

Compartmentalization of enzyme and substrate Limiting substrates Mass transfer limitations Presence of inhibitors Conditions for Michaelis–Menten kinetics may not apply in foods: no clean initial velocities, no saturating substrate, possibility of product inhibition and enzyme inactivation Enzymes as well as substrates and products are unstable Effects of ionic strength, pH

One particularly important effect in foods is the presence of seemingly inert molecules. Even though they are inert, they can have a large effect on kinetics. This is the topic of molecular crowding. Molecular crowding effects on enzyme activity. The phenomenon of molecular crowding, to be discussed further in Chapter 14 on reactions in the food matrix, is basically a volume exclusion effect. In a crowded solution, less volume is available to molecules than in a dilute solution and as a result molecules behave highly nonideal in the thermodynamic sense. The effect is obviously much larger for macromolecules. The effect of nonideal behavior can be expressed in activity coefﬁcients. Molecular crowding effects are currently studied in detail for biochemical reactions in cells. No doubt, these effects are equally important in foods. To be sure, the consequence of molecular crowding is that activities of reactants (including enzymes) can be strongly inﬂuenced by cosolutes, also if these cosolutes are inert with respect to the reactants themselves. For proteins the main consequences of molecular crowding are that association of macromolecules is favored. When this happens, there is a reduction in volume and this is thermodynamically favorable in a crowded environment. Crowding also enhances the stability of folded proteins with respect to the unfolded state; this is relevant for stability of enzymes: the denaturation temperature will go up, as is discussed in Chapter 10. If an enzymatic reaction is depending on enzymes that are active as

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9-44

K⬘a

Ka E+S

ES

ES∗

K‡ ES‡

kcat

E+P

SCHEME 9.15 Kinetic scheme showing the various intermediates in enzymatic catalysis in terms of transition state theory. K represents the thermodynamic equilibrium constant for the complexes shown.

associated oligomers, crowding will have a direct effect on enzymatic activity because association is promoted. In contrast, enzymes acting as monomers will lose activity if they become associated. If the association of enzyme and substrate leads to a more compact ES complex, molecular crowding will enhance the formation of the ES complex (i.e., lower the Michaelis constant KM). Then there is also the effect of crowding on diffusion, which is relevant for the enzyme–substrate encounter rate: the more crowded, the more difﬁcult diffusion becomes. A semiquantitative explanation for crowding effects is as follows. The analysis is based on transition state theory discussed in Chapter 5; see also Figure 9.1. We consider a kinetic scheme as displayed in Scheme 9.15. Assuming molar concentrations with activity coefﬁcients y (see Chapter 3) we can write for each thermodynamic equilibrium constant yES [ES] [ES] ¼ Ya yE yS [E] [S] [E] [S]

(9:87)

Ka0 ¼

y * [ES* ] [ES* ] ES ¼ Ya0 yES [ES] [ES]

(9:88)

Kz ¼

yESz [ESz ] [ESz ] ¼ Yz y * [ES* ] [ES* ] ES

(9:89)

Ka ¼

We will concentrate on an initial rate expression and after some algebraic manipulation it can be shown that v ¼ kcat [ESz ] ¼

kcat Ka Ka0 K z [E] [S] Ya Ya0 Y z

(9:90)

This can in turn be written as v¼

kcat Ka Ka0 K z [E] [S] h i K0 Ya þ Ka [S] 1 þ Ya0 Ya0 Y z

(9:91)

a

The vmax,id in ideal conditions (all activity coefﬁcients equal unity) is vmax,,id ¼

kcat Ka0 K z [E] 1 þ Ka0

(9:92)

and this leads to the expression v¼

vmax,id Ka0 K z [S](1 þ Ka0 ) h i K0 Ya þ Ka [S] 1 þ Ya0 Ya0 Y z a

(9:93)

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Enzyme Kinetics

9-45

We now have an expression for the initial rate as function of substrate concentration with an explicit dependence on activity coefﬁcients. For the enzyme rate expressions given in previous sections it has been assumed tacitly that activity coefﬁcients are unity, i.e., thermodynamic ideal behavior was assumed. The task now becomes to ﬁnd expressions for the activity coefﬁcients. A general expression for an activity coefﬁcient is 2 yi ¼ exp42Bi,i [i] þ

X

3 Bi,j [j] þ 5

(9:94)

j6¼i

Bi,i and Bi,j are osmotic second virial coefﬁcients describing physical interactions between pairs of molecules i and between pairs of molecules i and j, respectively. For instance, the activity coefﬁcient for free enzyme is * [ES*] þ BE,ESz [ESz ] þ ] yE ¼ exp [2BE,E [E] þ BE,S [S] þ BE,ES [ES] þ BE,ES

(9:95)

In dilute systems the molar concentrations of the enzyme species are very small, which implies that the term in the exponent in Equation 9.95 is virtually zero, so that the activity coefﬁcient for the enzyme yE is unity. However, when high concentrations of an inert cosolute C are present, the term BE,C[C] cannot be neglected anymore and as a result yE will be different from unity. Similarly, the interaction between the cosolute C and other species cannot be neglected anymore and these terms become dominant to the extent that the following approximation is valid: Ya expb(BES,C BE,C BS,C )[C]c

(9:96)

* BES,C )[C]c Ya0 expb(BES,C

(9:97)

Y z exp [(BESz ,C B*

(9:98)

ES ,C

)[C]]

In the case of impermeable spheres, the second virial coefﬁcient describing physical interaction between components i and j having radii ri and rj, respectively, is Bi,j ¼

4pNAV (ri þ rj )3 zi zj (1 þ kri þ krj ) þ 3 2I(1 þ kri )(1 þ krj )

(9:99)

The distance of closest approach is, obviously, ri þ rj, and the ﬁrst term in the right hand side of this equation describes the volume from which the two molecules exclude each other mutually, the so-called covolume. The second term in the right hand side of the equation describes electrostatic interaction in Debye–Hückel terms (discussed in Chapter 6): z is the charge, k the Debye screening length and I the ionic strength. For uncharged species, the last term disappears, of course. For species that cannot be described as impenetrable spheres, such as random coil polymers, the following expression approximates the second virial coefﬁcient: Bi,C

" 1=2 # 2 2 4 2 2 ¼ Nav r i lC þ p r i lC þ 4 p p ri3 3 3 3

(9:100)

Overlooking all this, several phenomena related to molecular crowding can be explained, at least qualitatively, but perhaps even quantitatively to the extent that the given expressions are reasonable approximations. If we ﬁrst consider the case that the cosolute affects only the interaction between enzyme and substrate (Ya 6¼ 1, Ya0 ¼ 1, Y z ¼ 1), Equation 9.93 results in

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9-46

K0 Vmax Ka (1 þ a [S] Ya v¼ Ka0 [S] 1 þ Ka (1 þ Ya

(9:101)

For KM in the presence of cosolute C can be written KM,C ¼ KM expb(BES,C BE,C BS,C )[C]c

(9:102)

The effect of crowding in this case of thermodynamic nonideality between enzyme and substrate is that there is tighter binding of substrate because (BE,C þ BS,C) > BES,C. vmax is not effected by the cosolute in such a case. If, however, the cosolute has an effect on subsequent steps in the catalysis process, there is an effect on vmax. The effect of cosolutes on maximal velocities is vmax ¼

vmax ,id (1 þ Ya0 ) (Ya0 þ Ka0 )Y z

(9:103)

Hence, vmax will be enhanced in crowding conditions if the formation of ESz goes along with a volume decrease because BES*,C > BESz,C . On the other hand, if the activated ESz complex has the larger volume such that BES*,C < BESz,C , vmax will decrease. With the aid of this semiquantitative analysis of molecular crowding effects can be predicted. Indeed, the predicted effects have also been observed. An example is shown in Figure 9.37 where the activity of the enzyme glyceraldehyde-3-phosphate dehydrogenase is shown as a function of the enzyme concentration. The activity of this enzyme decreases with its concentration. This is explained by the fact that this enzyme is subject to self-association into tetramers and the monomer has a much larger speciﬁc activity than the tetramer; with increasing concentration more monomer associates and therefore the activity 3.5 3 2.5

log v

2 1.5 1 0.5 0 −9

−8

−7 log (cGAPD)

−6

−5

FIGURE 9.37 Activity of the enzyme glyceraldehyde-3-phosphate dehydrogenase (GAPD) as a function of enzyme concentration (in M) in the absence (.) and in the presence of 18 g dm3 of the macromolecular crowder ribonuclease (). Dataset in Appendix 9.1, Table A.9.14.

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Enzyme Kinetics

9-47

2.5

v (mM min−1)

2

1.5

1

0.5

0

0

0.2

0.4 0.6 Sucrose (M)

0.8

1

FIGURE 9.38 Effect of molecular crowding: inﬂuence of sucrose on enzymatic activity of invertase. The solid line indicates the Michaelis–Menten equation with parameters vmax ¼ 2.3 mM min1 and KM ¼ 0.025 M. Dataset in Appendix 9.1, Table A.9.15.

decreases. Addition of macromolecules that do not take part in the enzyme reaction decreases enzyme activity nevertheless as shown in Figure 9.37. It was not only observed for the crowder shown here (ribonuclease) but also with b-lactoglobulin, bovine serum albumin, polyethyleneglycol. This is a typical effect due to macromolecular crowding: addition of inert macromolecules decreases the available volume and induces therefore the self association of the enzyme. Figure 9.38 shows just another simple example where the activity of invertase is inﬂuence by the presence of the substrate sucrose. Of course, the results displayed in Figure 9.38 could also be due to substrate inhibition (since sucrose is the substrate for invertase) as well as of an effect of viscosity increase affecting the encounter rate of enzyme and substrate, but it was ascertained in this case that these factors are not the cause for the effect that enzyme activity markedly decreased at high solute concentration, and the sole explanation remaining is the effect of molecular crowding by sucrose. The main message here is that the food scientist working with enzymes should be prepared that when he=she moves from test tube experiments to real foods that effects of molecular crowding can make orders of magnitudes difference on kinetics for reasons indicated.

9.11

Concluding Remarks

It is clear that the topic of enzymatic reactions in foods is an important but complicated one and the kinetics can be quite intricate. Rate equations for complete reversible reaction mechanisms can be very lengthy and tedious to handle. It is essential to reduce such equations to more simple ones, be it that these simpler equations should still give an adequate prediction at the conditions that the reaction is studied. The most relevant conditions for enzymes are the presence of activators and inhibitors, product and substrate inhibitions, temperature, pH, ionic strength, and the degree of molecular crowding. This chapter has given some tools to handle these situations. However, the reader is advised that the topic is vast and much more can be said about enzyme kinetics, and indeed many textbooks and reviews are available. Some of them are listed in the ‘‘Bibliography and suggested further reading’’ section.

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9-48

Appendix 9.1

Datasets Used for Examples in This Chapter

TABLE A.9.1 Initial Rates of Sucrose Hydrolysis by Invertase as a Function of Substrate Concentration (Figures 9.5 and 9.6) Initial Rate (min1)

Sucrose (M) 0.0292

0.182

0.058

0.258

0.0584

0.265

0.0876

0.311

0.117 0.117

0.33 0.342

0.146

0.349

0.175

0.372

0.205

0.347

0.234

0.371

Source: From Chase A.M., Meier H.Cv., and Menna V.J. The non-competitive inhibition and irreversible inactivation of yeast invertase by urea. J Cell Comp Physiol 59:1–13, 1962.

TABLE A.9.2 [E] ¼ 14.7 nM

Product Inhibition by 2-Phospho-D-Glycerate (2PG) on Enolase Action on PEP (Figure 9.10) PG ¼ 0

PG ¼ 50 mM

PG ¼ 200 mM

PG ¼ 400 mM

1

Rate (mM min )

PEP (mM) 0

0

0.05 0.06

0.0084 0.0125

0.09

0.0154

0.15

0.019

0.25

0.0235

0.35 0.5

PG ¼ 100 mM

0

0.0077 0.0284

0.8

0.0143

0.0012

0.0208

0.012

1 1.2

0.0323

0.0252

0.0182

0.0018 0.0052

2

0.0333

0.0292

0.0246

2.5

0.0338

0.0298

0.0268

0.0195

0.0067

3

0.0341

0.0316

0.0282

0.022

0.0108

3.5

0.0344

0.0319

0.0293

0.0238

0.0141

4

0.0346

0.0298

0.0258

2.2

4.23 4.53

0.0035

0.0261 0.0264

0.0178

5

0.02

5.5

0.0211

Source: From Duggleby R.G. Product inhibition of reversible enzyme-catalysed reactions. Biochim Biophys Acta Protein Struct Mol Enzymol 1209:238–240, 1994.

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Enzyme Kinetics

9-49 TABLE A.9.3 Action of Papain on Casein (Figures 9.12 and 9.13) v (mM min1)

S (mM) 9.5

1.72

19

3.91

28.5

5.2

38

5.28

57

7.75

85.5

7.61

95

7.5

114.5 133

8.03 7.93

171.5

7.7

190

8.465

Source: Data taken from Ritchie, R.J. and Prvan, T. Biochem. Educ., 24, 196, 1996; Tayyab, S. and Quamar, S., Biochem. Educ., 20, 116, 1992.

TABLE A.9.4

Effect of Oxalic Acid on Polyphenoloxidase (Figure 9.18) 0 mM Oxalic Acid

10 mM Oxalic Acid

30 mM Oxalic Acid

50 mM Oxalic Acid

v (AU min1)

v (AU min1)

v (AU min1)

v (AU min1)

Substrate Catechol (mM)

0

0

0

0

0

0.3

0.255

0.1125

0.08125

0.0275

0.5

0.41

0.19

0.125

0.04625

1

0.6125

0.325

0.2375

0.11375

4

0.7375

0.555

0.48125

0.27875

10

0.73125

0.57375

0.5325

0.42875

Source: From Son S.M., Moon K.D., and Lee C.Y. Kinetic study of oxalic acid inhibition on enzymatic browning. J Agric Food Chem 48:2071–2074, 2000.

TABLE A.9.5 Hydration of Fumarate to Malate by Fumarase (Figure 9.19). t (min)

E (pM)

P (mM)

0.5

1.08E-04

6.32

4

1.08E-04

30.04

7

1.08E-04

49.8

10

1.08E-04

67.19

13

1.08E-04

83

16

1.08E-04

98.02

19 22

1.08E-04 1.08E-04

110.67 122.53 133.6

25

1.08E-04

3

2.16E-04

43.48

6

2.16E-04

76.68 (continued )

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9-50

TABLE A.9.5 (continued) Hydration of Fumarate to Malate by Fumarase (Figure 9.19). t (min)

E (pM)

P (mM)

9

2.16E-04

105.14

12

2.16E-04

128.06

15

2.16E-04

147.04

18

2.16E-04

163.64

21

2.16E-04

174.7

24

2.16E-04

184.98

2

4.32E-04

53.75

5 8

4.32E-04 4.32E-04

111.46 150.99

11

4.32E-04

175.49

14

4.32E-04

191.3

17

4.32E-04

200.79

20

4.32E-04

204.74

23

4.32E-04

207.11

Source: Duggleby, R.G., Methods Enzymol., 249, 61, 1995.

TABLE A.9.6 Hydrolysis of 4-Nitrophenyl Phosphate by Alkaline Phosphatase (Figure 9.20 and 9.24) Time (min)

[E] ¼ 141 mg mL1

[E] ¼ 282 mg mL1

[E] ¼ 564 mg mL1

[E] ¼ 846 mg mL1

[E] ¼ 1128 mg mL1

[P] in mM 0 2.0

1.639

2.3 2.35

0.521 0.894

2.36 2.64

1.341 0.297

4.24

2.533

6.11 6.35

3.389 0.594

7.0

1.19

7.6 8.6

2.308 3.463

9.2 10.4

4.357 0.854

10.8

3.909

11.0

1.599

11.4

4.953

12.0

3.089

13.5 14.1

3.312

14.7 15.3 15.4

4.505 1.076 1.97 5.473

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Enzyme Kinetics

9-51

TABLE A.9.6 (continued) Hydrolysis of 4-Nitrophenyl Phosphate by Alkaline Phosphatase (Figure 9.20 and 9.24) Time (min)

[E] ¼ 141 mg mL1

[E] ¼ 282 mg mL1

[E] ¼ 564 mg mL1

16.3 3.721

17.5

2.193 1.223

18.8

6.031

19.1 19.7

5.323 3.981

20.6 21.5

2.341 1.297

22.2

5.694

22.5

4.241

22.55

6.439

23.0 24.0 26.2

2.489 1.371 4.538

26.23

5.954

26.5 26.74

2.674 1.444

28.4 29.8

6.959 1.48

29.8

2.822

29.9 32.1 33.2

4.76 6.399 1.591

33.3

2.933

35.8 36.6

5.056 1.664

38.6

7.403

39.1 42.3 42.5

3.117 6.88 1.737

45.7

5.388

49.6 53.3

3.374 1.845

56.5

7.918

60.2

7.321

63.9

5.68

67.5 70.1 70.9

3.554 8.1 1.913

74.1

7.428

78.1

5.899

81.4 85.1 85.2

[E] ¼ 1128 mg mL1

4.914

16.9 18.7

[E] ¼ 846 mg mL1

3.661 1.983 8.132

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9-52

TABLE A.9.6 (continued) Hydrolysis of 4-Nitrophenyl Phosphate by Alkaline Phosphatase (Figure 9.20 and 9.24) Time (min)

[E] ¼ 141 mg mL1

[E] ¼ 282 mg mL1

[E] ¼ 564 mg mL1

[E] ¼ 846 mg mL1

89.2

[E] ¼ 1128 mg mL1

7.535

92.9

5.968

96.0

8.203

96.2 99.6

3.694 1.978

99.7 100.7

7.568 8.164

102.8

6.002

104.7

7.566

106.4

3.727

108.0 109.8

6.037 2.012

111.4 115.0

3.726 2.01

Source: Duggleby, R.G., Biochim. Biophys. Acta Protein Struct. Mol. Enzymol., 1205, 268, 1993.

TABLE A.9.7 Galactosidase Action on the Synthetic Substrate PNPG (p-Nitrophenyl a-D-Galactopyranoside) (Figures 9.21 and 9.22) Time (min)

PNPG(mM)

0

0.24

0.48

1

2

4

5

0.231

0.462

0.97

1.951

3.936

10

0.221

0.445

0.94

1.895

3.863

20

0.203

0.413

0.885

1.803

3.741

40

0.168

0.355

0.786

1.616

3.506

60

0.144

0.313

0.7

1.473

3.3

120 180

0.09 0.056

0.226 0.158

0.537 0.405

1.115 0.82

2.782 2.355

240

0.036

0.115

0.312

0.631

2.037

305

0.026

0.08

0.249

0.534

1.87

360

0.012

0.066

0.189

0.44

1.703

420

0.012

0.061

0.183

0.422

1.616

Initial velocities (Figure 9.23) S (mM) 0.24 0.48

v0 mM min1 20.1 37.7

1 2

65 105.6

4

144.6

Source: From Durance T.D., Makhijani S.B.J., and Nakai S. Determination of reaction order by linearization and its application to estimation of Michaelis-Menten parameters. J Food Biochem 10:107–115, 1986.

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Enzyme Kinetics TABLE A.9.8 Time (min)

9-53

Inhibition of Pepsin by a Pepstatin Analog (Figure 9.25)

I¼0 mM

I ¼ 0.05 mM

I ¼ 0.1 mM

I ¼ 0.2 mM

I ¼ 0.3 mM

I ¼ 0.5 mM

I ¼ 1 mM

I ¼ 2 mM

[S] in mM 0

100

100 99.26

100 98.6

100 99.33333

100

100

0.05

98.66

98.66

99.8

0.05

98.13

99.06

98.33

0.05

98

98.6

99.3

98.73333

99.2

99.8

98.66667

98.86

0.1

97.33

97.06

98

98.4

97.8

0.1 0.1

97.46 96.73

97.26 97.93

97.33 98.06

98.53333 98.06667

98.06 98.73

0.5

87.8

89.26

91.4

91.93333

0.5

87.66

88.86

89.13

92.33333

0.5

87.53

88.86

91.13

91.06667

93.06

1

77.13

78.46

81.93

84.6

85.93

1

76.06

78.73

80.26

84.53333

85.66

1

75.66

79.73

82.13

83.86667

87.06

1.5 1.5

66.2 65.66

69.33 69.06

73.06 71.5

77.73333 76.66667

80.2 79.33

1.5

65.46

69.8

73.66

76.6

80.66

2

55.8

60.73

64.66

70.8

74.33

79.2

2

55.46

60.06

63.26

71

73.2

79.2

2

54.93

61.33

65.26

69.73333

74.46

2.5

46.53

52.46

57.06

64.8

68.2

2.5

45.93

51

55.93

64.06667

67.33

2.5 3

45.26 37.93

52.06 44.66

58.13 50.13

63 58.4

3

37.33

43.93

48.06

3

36.67

44.53

51.66

3.5

29.87

37.26

3.5

29.067

36.33

3.5

29.33

4 4 4

100 99.6 99.6

100 100.13 99.33

99

100.13

98.33

99

99.4

98.73

98.33 98.8

99 99.33

92.33

94.2

95.93

98.13

93.06

93.73

95.93

96.86

95.73

97.33

89.06

92.8

96.6

89.46

92.6

94.4

92.13

95.33

89.93 89.8

94.8 92.73

89.46

94.06

86.4

91.86

86.06

91.06

85.66

92

75.06

83

90

75

83.46

88.33

68.66 63.06

70.8

82.86 80.06

89.8 88.13

58.73333

62.06

70.53

57.26667

62.73

43.2

51.8

57.8

41.13

52.33333

56.2

37.73

44.46

50.86667

58.4

23.26

30.73

37.06

46.66667

53.06

22.33333 21.73

30.2 31.33

35.2 39

47.2 45.73333

51.73 52.86

4.5

16.06

24.8

30.93

41.33333

48.53

58.26

71.73

83.13

4.5

16.06

23.8

29.66

41.46667

47

57.86

71.33

80.93

4.5

15.66

25.33

32.66

40.33333

48

72.06

81.6

5

10.4

19.4

26.2

36.86667

43.8

54.26

80.93

5

11.06

18.26

24.06

37.2

42.26

54.06

79.46

5

10.33

19.53

27.26

36

44.4

Source: From examples given in www.biokin.com/dynaﬁt.

84 84.33

79.73

86.86

79.66

87.46

66.06

77.2

86.4

66.2

77

85.06

77.06

85.46

62.06

74.26

85.13

62.06

73.46 73.6

82.53 83.73

80.26

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TABLE A.9.9 Cholesterol Oxidase Action on Cholesterol (Figure 9.28) [Cholesterol] (mM)

Initial Rate (mM min1)

0

0

1.43

0.005

1.43

0.01

2.72

0.02

2.72

0.015

3.26

0.02

3.26

0.02

3.95 3.95

0.02 0.03

4.35

0.03

4.35

0.03

4.89

0.03

4.84

0.04

5.28

0.03

5.19

0.04

6.62 6.57

0.04 0.05

7.85

0.06

7.8

0.07

8.25

0.07

8.25

0.08

9.68

0.1

9.68

0.11

10.32 10.32

0.12 0.13

12.94

0.17

12.94

0.18

14.42

0.2

14.42

0.2

15.41

0.2

15.41

0.21

16 16

0.2 0.21

17.88

0.22

17.83

0.22

19.11

0.22

19.11

0.22

19.11

0.23

Source: From Vasudevan P.T. and Zhou T. Kinetics of cholesterol oxidation by cholesterol oxidase. Appl Biochem Biotechnol 60:63–72, 1996.

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Enzyme Kinetics

9-55 TABLE A.9.10 Effect of Temperature on Activity of Galactosidase (Figures 9.31A and 9.32A) Temperature (8C)

Activity (mM s1)

11.1

1.202

16.1

1.549

21.1

2.042

25.5

2.951

30.9

3.548

35.6

4.571

40.5

5.623

45.5 47.5

7.413 6.761

50.6

8.511

52.7

9.120

55.9

8.511

57.0

7.413

59.2

4.786

60.3

2.951

Source: From Hei D.J. and Clark D.S. Estimation of melting curves from enzymatic activity temperature proﬁles. Biotechnol Bioeng 42:1245–1251, 1993.

TABLE A.9.11 Effect of Temperature on Invertase Activity (Figures 9.31B and 9.32B) T=K

k (min1)

276

1.05

277

1.06

284

1.09

290

1.14

293

1.16

298

1.22

304

1.31

310 315

1.45 1.51

320

1.84

325

1.93

330

1.86

333

1.68

337

1.52

338

1.28

339 339

1.22 1.12

339

1.08

341

1.05

342

1.04

342

1.02

Source: From Westphal G., Vogel J., and Pusch D. Prozessberechnung der enzymatischen Saccharosehydrolyse in Abhangigkeit von der Temperatur. Acta Biotechnol 8:357–365, 1988.

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9-56

TABLE A.9.12 Effect of pH on Enzymatic Reduction of Metmyoglobin (Figure 9.34A) pH

vmax (3108 M s1)

5.06

3.28

5.26

3.01

5.36

2.99

5.39

2.52

5.6

2.1

5.74

2.03

5.8

2.68

5.84 5.9

2.43 2.27

5.91

1.64

5.95

2.08

6.03

1.71

6.08

1.87

6.1

1.76

6.2

1.44

6.5 6.54

0.85 0.85

6.7

0.83

6.75

0.7

6.8

0.67

6.94

0.55

7.19

0.47

7.31

0.53

7.31

0.41

Source: From Mikkelsen A. and Skibsted L.H. Kinetics of enzymatic reduction of metmyoglobin in relation to oxygen activation in meat products. Z Lebensm Unters Forsch 194:9–16, 1992.

TABLE A.9.13 Effect of pH on Papaya Pectinesterase (Figure 9.34B) pH

% Activity

5.0

6.2

6.0 7.0

54.3 95.4

7.5 8.0

99.0 100

8.5

91.8

9.0

88.3

Source: From Fayyaz A., Asbi B.A., Ghazali H.M., Che Man Y.B., and Jinap S. Kinetics of papaya pectinesterase. Food Chem 53:129–135, 1995.

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Enzyme Kinetics

9-57 TABLE A.9.14 Effect of Macromolecular Crowding due to Ribonuclease on Activity of Glyceraldehyde-3Phosphate Dehydrogenase (Figure 9.37) log v (AU min1)

log v with 18 g dm3 Ribonuclease

8.96

2.91

2.86

8.8

2.91

8.64

2.91

log c (M)

8.44

2.87

8.44

2.8

8.26 8.06

2.77 2.69

7.95

2.9

2.54

7.77

2.39

7.64

2.81

7.5

2.77

2.15

7.28

2.83

2.03

6.97

2.71

6.81 6.56

2.36

6.27

2.16

1.81

6.12 6.01

1.57 1.99

5.51

1.47

Source: Minton, A.P. and Wilf, J., Biochemistry, 20, 4821, 1981.

TABLE A.9.15 Effect of Molecular Crowding by Sucrose on Invertase Activity (Figure 9.38) Sucrose [S] (M)

v (mM min1)

0

0

0.01

0.34

0.02 0.04

0.73 1.35

0.06

1.64

0.08

1.73

0.1

1.8

0.13

1.73

0.16

1.67

0.21

1.6

0.26 0.32

1.53 1.44

0.41

1.33

0.49

1.26

0.61

1.24

0.75

1.06

0.9

0.99

Source: Shearwin, K.E. and Winzor, D.J., Arch. Biochem. Biophys., 260, 532, 1988.

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9-58

Kinetic Modeling of Reactions in Foods

Bibliography and Suggested Further Reading General Textbooks Alberty R.A. Thermodynamics of Biochemical Reactions. Hoboken, NJ: Wiley Interscience, 2003. Cornish-Bowden A. Fundamental enzyme kinetics. 3rd ed., London: Portland Press, 2004. Leskovac V. Comprehensive Enzyme Kinetics. New York: Kluwer Academic=Plenum Publishers, 2003. Marangoni A.G. Enzyme Kinetics. A Modern Approach. Hoboken, NY: Wiley Interscience, 2003. Segel L.A. Biological Kinetics. Cambridge Studies in Mathematical Biology, Cambridge University Press, 1991. Whitaker J.R. Principles of Enzymology for the Food Sciences. New York: Marcel Dekker, 1994. Whitaker J.R., Voragen A.G.J., and Wong D.W.S. Handbook of Food Enzymology. New York: Marcel Dekker, 2003. About Michaelis–Menten Kinetics Heinrich R., Melendez-Hevia E., and Cabezas H. Optimization of kinetic parameters of enzymes. Biochem Mol Biol Educ 30:184–188, 2002. Hoh C.Y. and Cord-Ruwisch R. A practical kinetic model that considers endproduct inhibition in anaerobic digestion processes by including the equilibrium constant. Biotechnol Bioeng 51:597– 604, 1996. Keleti T. Two rules of enzyme kinetics for reversible Michaelis-Menten mechanisms. FEBS 208:109–112, 1986. Michaelis L. and Menten M.L. Die Kinetik der Invertinwirkung. Biochem Z 49:334–369, 1913. Northrop D.B. On the meaning of KM and V=KM in enzyme kinetics. J Chem Educ 75:1153–1157, 1998. Núnez de Castro I. and Alonso F.J. Energy diagrams for enzyme catalysed reactions: A confusing point in the textbooks. Biochem Educ 25:87–89, 1997. Ratkowsky D.A. A suitable parameterization of the Michaelis-Menten enzyme reaction. Biochem J 240:357–360, 1986. Ritchie R.J. and Prvan T. Current statistical methods for estimating the Km and Vmax of MichaelisMenten kinetics. Biochem Educ 24:196–206, 1996. Schnell S. and Hanson S.M. A test for measuring the effects of enzyme inactivation. Biophys Chem 125:269–274, 2007. Selwyn M.J. Michaelis-Menten data: misleading textbook examples. Biochem Educ 23:138–141, 1995. Tenekedjiev K. and Kolev K. Introduction to interpretation of stochastic parameters. Biochem Mol Biol Educ 30:414–418, 2002. Wachsstock D.H. and Pollard T.D. Transient state kinetics tutorial using the kinetics simulation program KINSIM. Biophys J 67:1260–1273, 1994. About Lineweaver–Burke Plots Chong D.P. On the use of least squares to ﬁt data in linear form. J Chem Educ 71:489–490, 1994. Tayyab S. and Quamar S. A look into enzyme kinetics: Some introductory experiments. Biochem Educ 20:116–118, 1992. Ritchie R.J. and Prvan T. Current statistical methods for estimating the Km and Vmax of MichaelisMenten kinetics. Biochem Educ 24:196–206, 1996. About Progress Curves Duggleby R.G. Analysis of progress curves for enzyme-catalyzed reactions—application to unstable enzymes, coupled reactions and transient-state kinetics. Biochim Biophys Acta Protein Struct Mol Enzymol 1205:268–274, 1993. Duggleby R.G. Analysis of enzyme progress curves by nonlinear regression. Meth Enzymol 249:61–90, 1995.

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Duggleby R.G. Quantitative analysis of the time courses of enzyme-catalyzed reactions. Methods 24:168– 174, 2001. Straathof A.J.J. Development of a computer program for analysis of enzyme kinetics by progress curve ﬁtting. J Mol Catal B: Enzym 11:991–998, 2001. Willeman W.F., Hanefeld U., Straathof A.J.J., and Heijnen J.J. Estimation of kinetic parameters by progress curve analysis for the synthesis of (R)-mandelonitrile by Prunus amygdalus hydroxynitrile lyase. Enzyme Microbial Technol 27:423–433, 2000. About the King–Altman Method Chou K.C. Graphic rules in steady and non-steady state enzyme kinetics. J Biol Chem 264:12074–12079, 1989. King E.L. and Altman C. A systematic method of deriving rate laws for enzyme-catalysed reactions. J Phys Chem 60:1375–1378, 1956. Cornish-Bowden A. Fundamental Enzyme Kinetics. 3rd ed., Chapter 4. London: Portland Press, 2004. Leskovac V. Comprehensive Enzyme Kinetics, Chapter 4. New York: Kluwer Academic=Plenum Publishers, 2003. Software Programs for Enzyme Kinetics Analysis Kuzmic P. Program DYNAFIT for the analysis of enzyme kinetic data: application to HIV proteinase. Anal Biochem 237:260–273, 1996. Web Site: www.biokin.com=dynaﬁt Straathof A.J.J. Development of a computer program for analysis of enzyme kinetics by progress curve ﬁtting. Journal of Molecular Catalysis B: Enzymatic 11:991–998, 2001. Web Site for the computer program Encora: http:==www.tnw.tudelft.nl=live=pagina.jsp?id ¼ ca553f24–64c7–4d5ea344-ce30830894ab&lang ¼ en About Experimental Design for Enzyme Kinetics Dette H. and Biedermann S. Robust and efﬁcient designs for the Michaelis-Menten model. J Am Stat Assoc 98:679–686, 2003. Duggleby R.G. Experimental designs for the distribution-free analysis of enzyme kinetic data. In: Kinetic Data Analysis, Endrenyi L. (Ed.), pp. 169–179. New York: Plenum Press 1981. Lindner P.F.O. and Hitzmann B. Experimental design for optimal parameter estimation of an enzyme kinetic process based on the analysis of the Fisher information matrix. J Theor Biol 238:111–123, 2006. López-Fidalgo J. and Wong W.K. Design issues for the Michaelis-Menten model. J Theor Biol 215:1–11, 2002. Murphy E.F., Gilmour S.G., and Crabbe M.J.C. Effective experimental design: Enzyme kinetics in the bioinformatics era. Drug Discov Today 7:S187-S191, 2002. Murphy E.F., Gilmour S.G., and Crabbe M.J.C. Efﬁcient and accurate experimental design for enzyme kinetics: Bayesian studies reveal a systematic approach. J Biochem Biophys Methods 55:155–178, 2003. Murphy E.F., Gilmour S.G., and Crabbe M.J.C. Efﬁcient and cost-effective experimental determination of kinetic constants and data: The success of a Bayesian systematic approach to drug transport, receptor binding, continuous culture and cell transport kinetics. FEBS Lett 556:193– 198, 2004. About Enzyme Inhibition Hoh C.Y. and Cord-Ruwisch R. A practical kinetic model that considers endproduct inhibition in anaerobic digestion processes by including the equilibrium constant. Biotechonol Bioeng 51:597– 604, 1996. Ochs R.S. Understanding enzyme inhibition. J Chem Educ 77:1453–1455, 2000. Whiteley C.G. Enzyme kinetics: Partial and complete non-competitive inhibition. Biochem Educ 27:15– 18, 1999.

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About Thermodynamics and Enzyme Reactions Alberty R.A. Calculating apparent equilibrium constants of enzyme-catalyzed reactions at pH 7. Biochem Educ 28:12–17, 2000. Alberty R.A. Thermodynamics of systems of biochemical reactions. J Theor Biol 215:491–501, 2002. Alberty R.A. A short history of the thermodynamics of enzyme-catalyzed reactions. J Biol Chem 279:27831–27836, 2004. Alberty R.A. Relations between biochemical thermodynamics and biochemical kinetics. Biophys Chem 124:11–17, 2006. Aledo J.C. Metabolic pathways: Does the actual Gibbs free-energy change affect the ﬂux rate? Biochem Mol Biol Educ 29:142–143, 2001. Aledo J.C. Coupled reactions versus connected reactions. Biochem Mol Biol Educ 35:85–88, 2007. Aledo J.C. and Esteban del Valle A. Glycolysis in wonderland: The importance of energy dissipation in metabolic pathways. J Chem Educ 79:1336–1339, 2002. Bozlee B.J. Reformulation of the Michaelis-Menten equation: How enzyme-catalyzed reactions depend on Gibbs energy. J Chem Educ 84:106–107, 2007. Cannon W.R., Singleton S.F., and Benkovic S.J. A perspective on biological catalysis. Nat Struct Biol 3:821–833, 1996. Cornish-Bowden A. and Hofmeyer J.H.S. Enzymes in context. Kinetic characterization of enzymes for systems biology. The Biochemist April: 11–14, 2005. Edwards R.A. The free energies of metabolic reactions (DG) are not positive. Biochem Mol Biol Educ 29:101–103, 2001. Harris D.A. A uniﬁed approach to enzyme catalysis. Biochem Educ 14:2–6, 1986. Heinrich R., Melendez-Hevia E., and Cabezas H. Optimization of kinetic parameters of enzymes. Biochem Mol Biol Educ 30:184–188, 2002. Hoh C.Y. and Cord-Ruwisch R. A practical kinetic model that considers endproduct inhibition in anaerobic digestion processes by including the equilibrium constant. Biotechonol Bioeng 51:597– 604, 1996. Kjelstrup S., Rubi J.M., and Bedeaux D. Active transport: A kinetic description based on thermodynamic grounds. J Theor Biol 234:7–12, 2005. Rottenberg H. The thermodynamic description of enzyme-catalysed reactions. Biophys J 13:503–511, 1973. Weiss H.M. The roles of acids and bases in enzyme catalysis. J Chem Educ 84:440–442, 2007. Winzor D.J. and Jackson C.M. Interpretation of the temperature dependence of equilibrium and rate constants. J Mol Recognit 19:389–407, 2006. Wolfenden R. Thermodynamic and extrathermodynamic requirements of enzyme catalysis. Biophys Chem 105:559–572, 2003. About Macromolecular Crowding Asaad N. and Engberts J.B.F.N. Cytosol-mimetic chemistry: Kinetics of the trypsin-catalyzed hydrolysis of p-nitrophenyl acetate upon addition of polyethylene glycol and N-ter-butyl acetoacetamide. J Am Chem Soc 125:6874–6875, 2003. Asaad N., Den Otter M.J., and Engberts J.B.F.N. Aqueous solutions that model the cytosol: Studies on polarity, chemical reactivity and enzyme kinetics. Org Biomol Chem 2:1404–1412, 2004. Chebotareva N.A., Kurganov B.I., and Livanova N.B. Biochemical effects of molecular crowding. Biochemistry (Moscow) 69:1522–1536, 2004. Ellis R. and Minton A.P. Join the crowd. Nature 425:27–28, 2003. Hu Z., Jiang J., and Rajagopalan R. Effects of macromolecular crowding on biochemical reaction equilibria: A molecular thermodynamic perspective. Biophys J 93:1464–1473, 2007.

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Lonhienne T.G.A., Jackson C.M., and Winzor D.J. Thermodynamic non-ideality as an alternative source of the effect of sucrose on the thrombin-catalyzed hydrolysis of peptide p-nitroanilide substrates. Biophys Chem 103:259–269, 2003. Lonhienne T.G.A. and Winzor D.J. A potential role for isothermal calorimetry in studies of the effects of thermodynamic non-ideality in enzyme-catalyzed reactions. J Mol Recognit 17:351–361, 2004. Minton A.P. Molecular crowding: Analysis of effects of high concentrations of inert cosolutes on biochemical equilibria and rates in terms of volume exclusion. Methods Enzym 295:127–149, 1998. Minton A.P. The inﬂuence of macromolecular crowding and macromolecular conﬁnement on biochemical reactions in physiological media. J Biol Chem 276:10577–10580, 2001. Minton A.P. How can biochemical reactions within cells differ from those in test tubes? J Cell Sci 119:2863–2869, 2006. Minton A.P. and Wilf J. Effect of macromolecular crowding upon the structure and function of an enzyme: Glyceraldehyde-3-phosphate dehydrogenase. Biochemistry 20:4821–4826, 1981. Patel C.N., Noble S.M., Weatherly G., Tripathy A., Winzor D.J., and Pielak G.J. Effects of molecular crowding by saccharides on a-chymotrypsin dimerization. Protein Sci 11:997–1003, 2002. Ralston G.B. Effects of ‘‘crowding’’ in protein solutions. J Chem Educ 67:857–860, 1990. Shearwin K.E. and Winzor D.J. Substrate as a source of thermodynamic nonideality in enzyme kinetic studies: Invertase-catalyzed hydrolysis of sucrose. Arch Biochem Biophys 260:532–539, 1988.

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10 Kinetics of Protein and Enzyme Denaturation

10.1

Introduction

It goes without saying that proteins are very important in foods. Apart from their nutritional value they play an important role in texture of foods, they can act as emulsiﬁers, foaming agents, and gelling agents. Proteins are characterized by their primary (basically the amino acid sequence), secondary (a-helix and b-sheet), tertiary (3D structure), and quaternary structures (association of subunits). Processing can alter the secondary, primary, and quaternary structure, and this can have large consequences. Enzymes are, of course, also proteins. As shown in the previous chapter, enzymes can have a large effect on food quality for the very reason that they act as catalysts. In the cases that such reactions are undesirable it is, obviously, important to prevent their action. Fortunately, enzymes can be inactivated relatively easily. Traditionally this is done by a heat treatment, but there are also other ways, such as high pressure, change in chemical environment (pH, solvent quality, ionic strength) change. In any case, knowledge of the kinetics of enzyme inactivation is essential, in order to be able to optimize the treatment. Since enzymes are proteins, the mechanism of inactivation is basically protein denaturation, and therefore it makes sense to discuss protein and enzyme denaturation in one chapter.

10.2

Protein Stability

The stability of proteins is so important in foods that a separate section on this topic is warranted. With stability is meant the resistance against denaturation, or unfolding. Denaturation can result in a change of functional properties of proteins, in a loss of enzymatic activity, it may play a part in inactivation of microorganisms (Chapter 13), in an increase in digestibility, in inactivation of antinutritional factors, to name some examples. There is some ambiguity about the question what denaturation is. Denaturation can be described as a change in the native conformation of a (globular) protein due to a change in conditions. Causes for denaturation are heat, a change in pH, solvent quality, adsorption to surfaces, high pressure, to name the most important ones for foods. The most relevant cause for foods is, of course, heat. Denaturation does not result in a change in primary structure, but it may result in a change in secondary, tertiary, and quaternary structure. Denaturation does not necessarily result in complete destruction of all native structures, but it will result in an (partial) unfolding of the protein. Such unfolding is in principle reversible, i.e., upon removal of the cause of the unfolding (in the case of heat 10-1

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10-2

by cooling) refolding into the original native structure is possible but also into another conformation. This is the very cause of ambiguity: the term denaturation is also frequently used to indicate irreversible changes in proteins (such as aggregation), but strictly speaking Intermediates this is not correct. What happens after unfolding is that amino acid residues (buried in the interior of the protein hitherto) become exposed and consequently Random coil Folded conformation their reactivity increases and this may result in irreversible changes. Let us take a closer look at unfolding Folding and further reactions. Unfolding As indicated above, denaturation is in principle a reversible process, and equilibrium is postulated FIGURE 10.1 Highly schematic drawing meant to between the native and denatured state (assuming illustrate the principle of cooperativity in protein this is a one step reaction). This implies that both conformation. native and unfolded molecules are present but no unstable intermediate structures. This is plausible because of presumed cooperative effects in bonds that are involved in stabilizing a certain protein conformation. An individual noncovalent bond involved in protein conformation is weak but together they may be stronger than the sum of the individual interactions. A highly schematic picture of this is shown in Figure 10.1. The explanation for such a cooperative effect is that intramolecular internal bonds can be formed much easier than when they would have to be formed intermolecularly between individual molecules. In other words, the thermodynamic activity of intramolecular groups is effectively much higher than their concentration would indicate. Conversely, this means that if such bonds are disrupted in the process of denaturation then the loss of a few bonds will result very easily in the disruption of many bonds. So, we assume equilibrium between native and denatured molecules: k1

N ÐD k2

(10:1)

N symbolizes native protein molecules and D denatured protein molecules (these species can also be partly denatured in the sense that certain protein domains may be unfolded and others not). This equilibrium is considered to be fast such that both states are populated with amounts that allow a thermodynamic treatment. Equilibrium can thus be characterized by a proper thermodynamic equilibrium constant Keq ¼ k1=k2 (compare Equation 4.14). Focusing on temperature effects, the denaturation temperature is by deﬁnition the temperature at which Keq ¼

[D] k1 ¼ ¼1 [N] k2

(10:2)

It is thus the temperature at which the number of unfolded molecules equals that of folded molecules. At temperatures below the denaturation temperature the equilibrium lies toward the left in Equation 10.1, i.e., mainly native molecules, and above the denaturation temperature it is the opposite, i.e., mainly unfolded molecules. How much of both species are present at a certain temperature is governed by the standard Gibbs energy at that particular temperature. As a reminder, the equilibrium constant is related to the standard Gibbs energy, as displayed in Equation 3.113, DrG ¼ RT ln Keq. It is interesting to note that the conformational stability of most proteins (in terms of DG N!D ) is not very high, say in the order of 10–50 kJ mol1, far smaller than for one covalent bond. Figure 10.2 gives a hypothetical but realistic example of DG N!D as function of temperature. If DGN!D < 0, unfolded molecules predominate. The effect of a changing temperature is thus to change the position of the equilibrium.

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Kinetics of Protein and Enzyme Denaturation

10-3

∆GN→D (kJ mol−1)

30 20

10

0 Tden

−10

Tden

−20

0

20

40

60

80

T (C)

FIGURE 10.2 Conformational stability expressed as the change in Gibbs energy for the reaction folded ! unfolded of a hypothetical protein as a function of temperature.

An interesting feature following from Figure 10.2 is that at low temperatures also so-called cold denaturation can occur; the reason for this is the disappearance of hydrophobic bonding at low temperatures. We will not discuss this phenomenon further. Equation 10.1 indicates thus a two-state population with unfolded and folded molecules. The unfolding process is believed to be a cooperative transition: the breaking of one or more bonds destabilizes others. Because it concerns many bonds, this requires a large energy input per mole protein DH. If this were the only factor, unfolding would not happen because of this required high energy input. However, unfolding causes at the same time a very large entropy increase. It should be remarked here that this pertains to the entropy of the whole system. Thus, for a protein in an aqueous environment it concerns the entropy of the protein itself (very much increased upon unfolding) and the entropy of the water molecules surrounding the protein (decreased because of the exposure of hydrophobic amino acid residues). On the whole, the entropy gain is quite high for unfolding proteins in water. This then compensates for the high positive enthalpy, so that the overall DG remains moderate, remembering that DG ¼ DH TDS. Figure 10.3 shows a schematic illustration of both conformational and activation standard enthalpy and entropy changes. N‡ N‡, D

J mol−1

D

∆H

+

0

N

‡

∆H T∆S ≈ T∆S ‡ N

N‡ ∆G

‡

∆G N→D

N D

_

FIGURE 10.3 Schematic drawing of standard enthalpy and entropy changes upon protein unfolding from native molecules N to denatured molecules D via the activated complex. The x-axis is not labeled (Chapter 5).

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Kinetic Modeling of Reactions in Foods

10-4

Keq = [D]/[N] = k1/k2

ln k or ln K

k2 (refolding)

k1 (unfolding)

1/T

FIGURE 10.4 Schematic drawing of the temperature dependence of the reaction rate constant describing unfolding (k1) and refolding (k2) and the equilibrium constant for unfolding (Keq).

With regard to unfolding, it is generally accepted that there is a single rate-limiting step and a single transition state that is considered to be a distorted high energy form of the native state. The activation enthalpy of unfolding is considered to be independent of temperature. The heat capacities of the folded and transition state are assumed to be nearly the same. The temperature dependence of the reaction rate constant k1 describing unfolding follows usually the Arrhenius=Eyring equation quite well, i.e., a plot of ln k1 versus 1=T is linear. This is, however, not so for the reaction rate constant k2 describing refolding; it may decrease with temperature. The reason for this is twofold. First, the heat capacity of the unfolded molecule is much higher than that of the folded molecule due to hydrophobic amino acids that are exposed upon unfolding. Second, the metastable intermediates in the prefolded state are destabilized easier with increasing temperature. Consequently, the temperature dependence of the equilibrium constant is not according to the van’t Hoff equation (Equation 5.2). Figure 10.4 shows a schematic picture. Prior to refolding there will be some transient prefolded state (not a random coil); sometimes these intermediate states are described as molten globules or compact intermediates, but there is no consensus about the actual existence of such species. A rate-limiting step for refolding may be the occurrence in unfolded molecules of trans–cis isomerization of peptide bonds involving proline residues. Peptide bonds in native proteins are in the trans-conformation and isomerization does not easily occur except for bonds involving proline. If trans–cis isomerization occurs, refolding in the native state is obviously hindered. Concerning thermodynamics of unfolding, the following analysis may be helpful. Protein unfolding goes along with a change in heat capacity DCP: the heat capacity of the unfolded protein is higher than that of the folded protein, which is, as mentioned, due to exposure of hydrophobic amino acids residues. The thermodynamic parameters DCP, DG, DH are temperature dependent, so we should write DG(T) ¼ DH(T) TDS(T)

(10:3)

The temperature dependence of the enthalpy change can be written as ðT DCP dT DH(T) ¼ DH(Tden ) þ

(10:4)

Tden

Tden indicates the denaturation temperature. Similarly, for the entropy change we can write: ðT DCP dT DS(T) ¼ DS(Tden ) þ Tden T

(10:5)

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Kinetics of Protein and Enzyme Denaturation

10-5

As a ﬁrst approximation we can assume DCP to be temperature independent, so that integration of the previous equations yields DH(T) ¼ DH(Tden ) þ DCP (T Tden ) DS(T) ¼ DS(Tden ) DCP ln

Tden T

(10:6) (10:7)

By deﬁnition, DG(Tden) ¼ 0 and consequently DS(Tden) ¼ DH(Tden)=T, so that we can write Equation 10.3 as TDH(Tden ) Tden DG(T) ¼ DH(Tden ) þ DCP (T Tden ) þ TDCP ln T Tden T Tden ¼ DH(Tden ) 1 þ DCP (T Tden ) þ TDCP ln T Tden

(10:8)

This equation was already used in Chapter 9 (Equation 9.76) while discussing the temperature dependence of enzyme activity. This equation can be rewritten in terms of the equilibrium constant via the relation DG ¼ RT ln Keq and describes therefore the temperature dependence of the equilibrium constant of unfolding. It is an approximate relation because we have assumed that DCP is temperature independent, which is not really true, but it is usually not too bad an approximation if the temperature range is not too large. For a food technologist it is usually more interesting what happens after unfolding because that may have a large impact on food quality. Nevertheless, it is essential to understand unfolding processes in order to understand what happens next. Expanding Equation 10.1 we can envisage a few scenarios. The ﬁrst scenario is to assume equilibrium between N and D and to introduce a further reaction step describing the formation of an irreversibly unfolded molecule I out of the unfolded molecule D, ﬁrst proposed by Lumry and Eyring in the 1950s: k1 N

k3

D

I

k2

SCHEME 10.1

Model 1 (Lumry and Eyring) for protein=enzyme denaturation.

Alternatively, the second reaction may be a bimolecular one: two unfolded molecules react together to form an aggregate: k1 N

k2

D+D

SCHEME 10.2

D

k3

I

Model 2 for protein=enzyme denaturation.

Some proteins are notably heat stable. There may be two reasons for that. The ﬁrst is that the protein has a high conformational stability, in other words that it starts to unfold only at high temperature. The second possibility is that unfolding does occur at relatively low temperature but that the subsequent reaction occurs only very slowly (i.e., very low k3), so that upon cooling the molecule refolds into the native state and it appears as if nothing has happened. Thermal instability of proteins on the other hand implies a relatively fast reaction (i.e., very high k3) leading to an irreversibly unfolded molecule, which may have a large impact on food quality. Table 10.1 gives an overview of possible reactions leading to irreversible changes.

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10-6

TABLE 10.1 Possible Reactions Occurring after Unfolding Causing Irreversible Changes in Unfolded Protein Molecules Reaction

Amino Acid Residue Involved

SH=SS exchange

Cysteine, cystine

Deamidation

Asparagine, glutamine

Oxidation

Tryptophan, methionine

Hydrolysis of peptide bonds

All

Trans–cis isomerization

Proline

Aggregation=coagulation

Hydrophobic residues

Dissociation of subunits

Cyst(e)ine, hydrophobic residues

Chemical cross-links (lysinoalanine, lanthionine, histidinoalanine, Maillard-type cross-links)

Phosphoserine, cysteine, lysine, histidine

When it comes to kinetics of denaturation, there are many conﬂicting results in literature. Even for the same protein reported orders may vary from 1 to 2. These conﬂicting results are most likely due to different conditions under which denaturation is studied; denaturation is not a property of the protein alone but also of the environment it is in. However, conﬂicting results may remain even if the conditions are the same because of a nonoptimal experimental design. This leads to estimates of the reaction order with very large conﬁdence intervals so that almost any order would give a reasonable ﬁt. An example of this was shown in Chapter 7 on the denaturation of a-lactalbumin. It is the author’s experience that for many results published in literature about protein (and enzyme denaturation as well) the imprecision in parameter estimates is disappointingly large. Also, all too often, authors force a ﬁrst-order plot through their data without exploring the possibility that another order may be more appropriate. A striking example is given in Figure 10.5 for denaturation of apo-lactoferrin. Clearly, there is a misﬁt for nt ¼ 1. Some authors have reported the order with respect to concentration, nc, rather than the one with respect to time, nt, in relation to protein denaturation; an example was given in Figure 4.16 for the denaturation of b-lactoglobulin. It appears that then more consistent results are obtained because this 200

c (mg dm−3)

160

120

80

40

0 0

1000

2000 Time (s)

3000

4000

FIGURE 10.5 Denaturation of apo-lactoferrin in phosphate buffer at 728C. Solid line is the optimal ﬁt obtained for an order nt ¼ 2.2, the broken line is for a forced ﬁrst-order plot (nt ¼ 1). Dataset in Appendix 10.1, Table A.10.1.

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10-7

method uses only initial rates and possible interfering reactions in the course of the reaction are avoided. The difference between the two orders is discussed in Chapter 4. Just another reason for the discrepancies reported in literature is due to the method of analysis used. If one analyzes the remaining native molecules after a heat treatment (by whatever method), one not only determines the remaining native molecules that were not unfolded but also the ones that were actually unfolded but were able to refold in the native state after cooling. The following analysis may illustrate this. Taking Scheme 10.1 as a starting point, the rate of irreversible change is r¼

d[I] ¼ k3 [D] dt

(10:9)

The problem in using this equation is that usually neither [D] nor [I] is measured, but the remaining [N] after cooling; this will include then refolded molecules if [D] refolds back easily upon cooling (large k2 upon cooling). So, the change detected is actually [N þ D], but after cooling we have no way of distinguishing between [N] and [D]. However, if we assume rapid equilibrium between [N] and [D], we can use the equilibrium constant to state something about [D]: Keq ¼

k1 [D] 1 x ¼ ¼ k2 [N] x

(10:10)

where x represents the fraction of [N]=([N] þ [D]). So we can write [D] ¼

Keq ([N] þ [D]) 1 þ Keq

(10:11)

This equation signiﬁes that at temperatures below the denaturation temperature Tden the fraction of [D] of the total protein concentration is low; for instance, the fraction of [D] is 0.09 for Keq ¼ 0.1. At the denaturation temperature Tden, Keq ¼ 1, and the fraction of [D] is half of the total protein concentration, while at temperatures well above Tden, the fraction of [D] is virtually unity. Equation 10.9 can now be written as r¼

Keq d[I] ([N] þ [D]) ¼ kobs ([N] þ [D]) ¼ k3 1 þ Keq dt

(10:12)

To recapitulate, if T >> Tden then Keq >> 1, hence kobs k3, in other words, the observed rate is completely determined by the reaction leading to irreversible changes. Figure 10.6 gives a simulated example of such a situation. If T ¼ Tden, Keq ¼ 1, hence kobs ¼ 0.5 k3. Such behavior is simulated in Figure 10.7. At temperatures well below the denaturation temperature, Keq 1, hence kobs ¼ k3Keq 1, in other words nothing happens because almost all molecules are in the native state and the reaction leading to irreversible changes is very slow anyway at the lower temperature. The same trend would be observed for the reaction depicted in Scheme 10.2, albeit with slightly different kinetics due to the bimolecular reaction. Another scenario is to assume that refolding does not occur (k2 0). In fact, this is the situation discussed in Chapter 4 for a consecutive reaction, Equation 4.48: [N] ¼ [N]0 exp (k1 t) k1 [N]0 [D] ¼ (exp (k1 t) exp ( k3 t)) k3 k1 k1 exp (k3 t) k3 exp (k1 t) [I] ¼ [N]0 1 þ k3 k1

(10:13)

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Kinetic Modeling of Reactions in Foods

10-8

1

c (arbitrary units)

0.8

[I]

0.6

[N+D]

0.4

[D] 0.2 [N]

0 0

20

40 60 80 t (arbitrary units)

100

120

FIGURE 10.6 Simulation of protein denaturation and subsequent reaction leading to irreversible unfolding. Referring to Scheme 10.1, k1 ¼ 1, k2 ¼ 0.01 (a situation similar to T >>Tden) and k3 ¼ 0.01, [N]0 ¼ 1 (arbitrary units). It is assumed that all remaining D refolds back into N after cooling.

Figure 10.8 gives a simulated example of such a situation; if one assumes that renaturation is not possible, also not after cooling, and one can only measure N, no information on reaction rate constants other than k1 can be obtained from such experiments. Only if one would be able to measure D and I, this would give information on k1 and k3. Sometimes, one is able to measure more than one response, for instance by measuring peak areas in gel permeation chromatography. One example of this is shown in Figure 10.9, a study on denaturation and

1

c (arbitrary units)

0.8

[N]

0.6

[N+D]

0.4

[I] [D]

0.2

0 0

20

40 60 80 t (arbitrary units)

100

120

FIGURE 10.7 Simulation of protein denaturation and subsequent reaction leading to irreversible unfolding. Referring to Scheme 10.1, k1 ¼ 1, k2 ¼ 1, and k3 ¼ 0.01 (a situation similar to T ¼ Tden), [N]0 ¼ 1 (arbitrary units). It is assumed that all remaining D refolds back into N after cooling.

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Kinetics of Protein and Enzyme Denaturation

10-9

1 [N] 0.8 c (arbitrary units)

[D] [I]

0.6

0.4

0.2

0

0

20

40 60 80 t (arbitrary units)

100

120

FIGURE 10.8 Simulation of protein denaturation and subsequent reactions leading to irreversible unfolding. Referring to Scheme 10.1, k1 ¼ 0.1, k2 ¼ 0, and k3 ¼ 0.01, [N]0 ¼ 1 (arbitrary units). It is assumed that all remaining D does not refold back into N after cooling.

subsequent aggregation of the potato protein patatin. Another example is given in Figure 10.10 for the denaturation and aggregation of the enzyme b-galactosidase, where it was even possible to measure an intermediate; incidentally, the decrease in native protein coincided with the decrease in enzymatic activity. The simultaneous analysis of residual protein as well as of aggregated protein makes it possible to analyze the rate constants for disappearance of native protein as well as the formation of aggregated

1

0.8

c/c0

0.6

0.4

0.2

0 0

0.2

0.4 0.6 Time (h)

0.8

1

FIGURE 10.9 Denaturation=aggregation of patatin as a function of time after heating at its denaturation temperature 558C. Native protein (*), aggregated protein (*), measured as fraction of total peak area from gel permeation chromatography. Dataset in Appendix 10.1, Table A.10.2.

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Kinetic Modeling of Reactions in Foods

10-10

100 90 80

Peak area (%)

70 60 50 40 30 20 10 0

0

20

40 Time (min)

60

80

FIGURE 10.10 Denaturation=aggregation of b-galactosidase. Native protein (*), denatured protein (&), aggregated protein (~) as percentage of peak areas from gel permeation chromatography. Dataset in Appendix 10.1, Table A.10.3.

protein, in principle at least. In fact, such analyses can be done via multiresponse modeling as discussed in Chapter 8. However, there is an additional difﬁculty when using peak areas as measure for concentration. As a ﬁrst check, of course, the total peak area of all fractions measured should remain constant; if it is not, some component is apparently missing. In the two examples displayed in Figures 10.9 and 10.10 the peak area remains approximately constant. However, if it is assumed that two molecules merge into one aggregated molecule as in a bimolecular reaction, one has to account for this in the kinetic analysis: one aggregated molecule gives the same response in peak area as two nonaggregated molecules. This means that the interpretation of the measured response depends on the assumed mechanism, and that is a bit tricky. For the data in Figure 10.9 several models were tested and the one that came closest to the data was one in which denatured protein D had to be transformed to a reactive form D*: k1 N

k2 k3

D

D∗ + D∗

SCHEME 10.3

D

D∗

k4 I

Kinetic model for denaturation and aggregation of patatin.

The experimental results were obtained at the denaturation temperature of patatin (558C) and rapid equilibrium was assumed between N and D, and consequently high values for k1 and k2 were given, and ﬁxed (i.e., not estimated). The modeling result is shown in Figure 10.11, using the multiresponse

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Kinetics of Protein and Enzyme Denaturation

10-11

1 0.9 0.8 0.7

c/c0

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4 0.6 Time (h)

0.8

1

FIGURE 10.11 Fit of the model in Scheme 10.3 to the denaturation and aggregation data for patatin at 558C. The same data as displayed in Figure 10.9.

modeling technique (Chapter 8). Although the trend in the data is followed by the model, the ﬁt is certainly not perfect. Another model was tried with the b-galactosidase data. First, the most simple model N ! D ! I was used (Equation 10.13), resulting in the ﬁt shown in Figure 10.12A. The model is overestimating the ﬁt for the aggregated protein. Therefore, it was tested whether a second-order aggregation model gave a better ﬁt using the model shown in Scheme 10.2; the ﬁt is shown in Figure 10.12B. There is not too much difference with Figure 10.12A. Therefore, Scheme 10.3 used for patatin was also applied to the b-galactosidase data (Figure 10.12C). Though there are slight differences, it is remarkable that more or less the same ﬁts are obtained with different models. Most likely, the models are still too simple to catch all the intricacies of protein denaturation. A kinetic analysis of protein denaturation is thus seen to be quite complicated, and the possible kinetic analysis depends strongly on what one can measure experimentally. As a result of all this, departures from simple kinetics readily occur with proteins. This can also become apparent from temperature studies. Figure 10.13 shows the temperature dependence of the observed rate constant kobs for denaturation of a milk protein. Such behavior clearly reﬂects the shift of rate controlling steps if temperature changes. Other reasons for departure from simple kinetics are heterogeneity of proteins, multiple rate-limiting steps in unfolding and subsequent reactions, the reaction leading to irreversibility is neither ﬁrst nor second order (e.g., in the case of aggregation). Since proteins are macromolecules they are sensitive to volume exclusion, or in other words, macromolecular crowding. This phenomenon was already touched upon in Chapter 9 on enzyme kinetics. Qualitatively, it implies that if the surroundings of the protein are crowded, that the denaturation temperature will increase, because it becomes energetically unfavorable to unfold in a crowded environment due to volume exclusion effects. An example is given in Figure 10.14 where the denaturation temperature of the enzyme a-chymotrypsin is given as a function of sucrose concentration. Sucrose is inert with respect to unfolding of the protein molecule and acts purely as a ‘‘crowder,’’ that is to say the observed effect on denaturation temperature is due to volume exclusion effects. The effect is seen to be considerable. It is to be expected that such effects play a role in foods. In other words, if the denaturation temperature of an isolated and puriﬁed protein is studied, it may well be that this does not correspond to

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Kinetic Modeling of Reactions in Foods

10-12

90

90

80

80

70

70

60 50 40

60 50 40

30

30

20

20

10

10

0 (A)

Peak area (%)

100

Peak area (%)

100

0

20

40 Time (min)

60

0

80

0

20

(B)

40 Time (min)

60

80

100 90 80

Peak area (%)

70 60 50 40 30 20 10 0 (C)

0

20

40 Time (min)

60

80

FIGURE 10.12 Kinetic models applied to the b-galactosidase data. First-order consecutive reaction (A), ﬁrst-order denaturation followed by a second-order aggregation (B), and ﬁt of the model shown in Scheme 10.3 (C). Same data as in Figure 10.10.

the actual denaturation temperature of the same enzyme in the food matrix where potentially many crowders may be present. As discussed in Chapter 3, nonideal behavior can be captured conveniently in activity coefﬁcients. This can also be done for proteins in crowded conditions. Crowding effects will manifest themselves in activity coefﬁcients. To give just an example of this, Figure 10.15 shows the activity coefﬁcient of a solution of hemoglobin as a function of its concentration. The effect is seen to be enormous: the activity coefﬁcient at c ¼ 100 g dm3 is about twice its value than that at 1 g dm3 while it is up to about 100 times higher at c ¼ 300 g dm3. Enzymes are, of course, also proteins and therefore the above analysis is also valid for enzymes. Nevertheless, enzymes show additional features that make it worth to discuss these in a little bit more detail, and this is the subject of Section 10.3.

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Kinetics of Protein and Enzyme Denaturation

−1

10-13

Moderate T-dependence (irreversible inactivation)

ln k

−3

Strong T-dependence (unfolding)

−5

−7

−9

−11 0.0022 0.0023 0.0024 0.0025 0.0026 0.0027 0.0028 0.0029 0.003 1/T (K −1)

FIGURE 10.13 Arrhenius plot for denaturation of b-lactoglobulin A (~) and a-lactalbumin (.) in heated milk. Dataset in Appendix 10.1, Table A.10.4.

10.3

General Kinetic Schemes Describing Enzyme Inactivation

Being able to inactivate enzymes is an important goal in food technology because the presence of enzymes can have a large effect on food quality. Traditionally, inactivation is achieved via a heat treatment, but newer methods such as high-pressure treatment and pulsed electric ﬁelds are increasingly 100

65

55

y

T den

60

10

50

45

0

0.2

0.4 0.6 0.8 Sucrose (M)

1

1.2

FIGURE 10.14 Effect of sucrose on the denatur0 ation temperature Tden of a-chymotrypsin in an aqueous solution. Dataset in Appendix 10.1, Table A.10.5.

1

0

50

100

150 200 c (g dm−3)

250

300

350

FIGURE 10.15 Molar activity coefﬁcient y of hemoglobin as a function of its concentration. Dataset in Appendix 10.1, Table A.10.6.

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10-14

log (activity)

studied. Whatever the technology applied, Activation/inactivation knowledge of inactivation kinetics is indispensable because that opens the possibilities Firs t-or to optimize. When it comes to the mechander ism of enzyme inactivation, it is basically Grace period due to protein denaturation (unfolding) followed by some inactivation reaction that makes refolding impossible, as discussed in Section 10.2. However, refolding may occur when the cause for unfolding is removed, that is to say if the inactivation reaction did Biphasic inactivation not go to completion during the treatment. Several types of inactivation curves are possible. Figure 10.16 shows some schematic Time examples. In the following we discuss some of the mechanisms underlying such FIGURE 10.16 Some types of possible enzyme inactivation behavior. curves. The simplest scheme for protein=enzyme denaturation was already shown in Scheme 10.1 and it is equally applicable to enzyme denaturation. In the case of an enzyme, it is usually assumed that the denatured form D is not active, but that it can refold back to the native, active form N after the treatment. While D can in principle refold back to an active form N, it can also transform into an inactive form I, and it is assumed that this is an irreversible process, so once the I form is reached activity is lost. Mostly, it is assumed that inactivation of enzymes can be described as a ﬁrst-order reaction. An apparent ﬁrst-order reaction, if Scheme 10.1 is applicable, implies that the step with rate constant k3 is rate determining. In many cases, if not all, remaining enzyme activity will be measured rather than inactivated enzyme and this makes sense because that is what counts. Usually, a relative activity is reported by dividing the measured activity a by the initial activity before the treatment a0: arel ¼

a a0

(10:14)

An example of apparent ﬁrst-order behavior is given in Figure 10.17 for heat inactivation and another one in Figure 10.18 for pressure-induced inactivation. Many inactivation curves reported in literature conform more or less to the ones depicted in Figures 10.17 or 10.18. However, the mechanism may be more complicated than in Schemes 10.1 or 10.2 but this may go unnoticed in graphs such as the ones in Figures 10.17 and 10.18. This is due to the fact that enzyme activity is measured, and there are three important aspects to consider. 1. First of all, the activity is measured at an assay temperature different (usually lower) from the treatment temperature. This means that we have to extrapolate from the measuring temperature to the treatment temperature if we want to infer what happens at the treatment temperature. Refolding can take place during cooling to the assay temperature, and thus the situation will be different when the activity is measured. The problem then arises as to how to couple that information to the mechanism: what happens to the enzyme at the molecular level at the treatment temperature. 2. A second possible problem is that the enzyme activity measured (a) is actually the sum of the activities of all active enzyme forms (ai), while these forms may behave differently at different conditions.

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10-15

2

1

1.8

0.9

1.6

0.8

1.4

0.7

1.2

0.6

1

0.5

arel

log (%)

Kinetics of Protein and Enzyme Denaturation

0.8

0.4

0.6

0.3

0.4

0.2

0.2

0.1

0

0 0

200

400 600 Time (s)

(A)

800

1000

0

200

400 600 Time (s)

(B)

800

1000

FIGURE 10.17 An example of apparent ﬁrst-order heat-induced inactivation kinetics of pectin methylesterase from tomato at 69.88C, presented as a logarithmic plot (A) and as relative activity plot (B). The lines represent a ﬁrst-order model. Dataset in Appendix 10.1, Table A.10.7.

a¼

X

ai

(10:15)

i

3. Third, it is quite conceivable that an enzyme is partly denatured while retaining activity, complete or partial. The fact that (partial) denaturation has occurred will then go unnoticed, unless another measurement technique is employed to study the reaction, such as circular dichroism (CD) or differential scanning calorimetry (DSC). This is advisable whenever possible when one wants to state something about an underlying mechanism.

0

1

−0.2

0.9

−0.4

0.8

−0.6

0.7

−0.8

0.6

−1

arel

ln(A/A0)

A major problem in interpretation is that several possible mechanisms all result in apparent ﬁrst-order behavior if one is only able to measure enzyme activity, and it will not be possible to distinguish between

−1.2

0.4

−1.4

0.3

−1.6

0.2

−1.8

0.1

−2

0 0

(A)

0.5

20

40 Time (min)

60

0

80 (B)

20

40 Time (min)

60

80

FIGURE 10.18 An example of apparent ﬁrst-order pressure-induced inactivation kinetics of an a-amylase from Bacillus subtilis at a pressure of 700 MPa, presented as a logarithmic plot (A) and as relative activity plot (B). The lines represent a ﬁrst-order model. Dataset in Appendix 10.1, Table A.10.8.

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Kinetic Modeling of Reactions in Foods

10-16 TABLE 10.2

Overview of Some Possible Kinetic Schemes for Enzyme Inactivation

Kinetic Scheme k1

EÐ D

Remarks

Activity at T Tden

Activity at T > Tden

Only denaturation

Partly, E present in noticeable amounts

No, only D present

Yes, if refolded correctly

Denaturation followed by inactivation

Partly, E present in noticeable amounts

No, only D and I present. Kinetic scheme at T:

Depends on k3, or k03 at T

k2

k1

k3

E Ð D ! I k2

or

k03

D þ D ! I k1

k3

D ! I, or k03

k3

E1 Ð D1 ! I1

2D ! I

k2

k4

k6

E2 Ð D2 ! I2 k5

Activity at Tassay

Isoenzymes E1, E2, biphasic inactivation

Partly, E1 and E2 present in noticeable amounts

No, only D1 and D2, I1 and I2 present Kinetic scheme at T:

Depends on k3 and k6 at T, a ¼ E1 þ E2

k3

D1 ! I1 k6

k1

EÐ E

0

k2

k3

D Ð k 4

D! I k5

Partly denatured E0 shows enzymatic activity: E0 ¼ aE grace period if a < 1, activation period if a > 1.

0

Partly, E and E present in noticeable amounts

D2 ! I2 No, only D and I present Kinetic scheme at T:

Depends on k5 at T, a ¼ E þ E0 ¼ (1 þ a) E

k5

D ! I

Note: Tassay is the assay temperature at which enzyme activity is measured, T is the treatment temperature, Tden is the denaturation temperature. Usually, Tassay T. E, active enzyme; E0 , partly denatured but still active enzyme; D, denatured and inactive enzyme; I, inactive enzyme. The assumption is that refolding has occurred rapidly at the assay temperature (any D present will turn into E). a, measured enzyme activity; a, relative activity (0, can be larger than 1).

these mechanisms, unless other measurements are applied that give additional information. Let us consider some possibilities. Table 10.2 gives an overview, which is by no means exhaustive. An important point to notice from Table 10.2 is that parameters describing unfolding cannot be derived from heating experiments above the denaturation temperature. This can be done only at temperatures around the denaturation temperature, and then the kinetic equations become quite cumbersome. We focus here on some possible inactivation models. In many cases, as mentioned, it is found that inactivation can be modeled as a ﬁrst-order equation. This is actually quite remarkable considering the complex three-dimensional structure of proteins; one would expect perhaps also stable intermediates that are still partially enzymatically active. In many cases, the actual mechanism is probably more complex but conditions lead to apparent ﬁrst-order behavior. Nonetheless, also non-ﬁrst-order inactivation can be observed, perhaps more often than a researcher is willing to see. It is impossible to consider all possible inactivation mechanisms, because the possibilities are endless. As remarked, the simplest mechanism conceivable is irreversible ﬁrst-order inactivation where the rate determining step is rate constant k3 in Scheme 10.1, and neglecting the previous steps: k1

E ! I The equation describing such inactivation is just a ‘‘normal’’ ﬁrst-order equation: arel ¼ exp (k1 t)

(10:16)

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10-17

The next possibility is that there are only two forms, active enzyme and inactivated enzyme, but they can be transformed into each other by a reversible ﬁrst-order reaction: k1

E ÐI k2

This leads eventually to equilibrium with an equilibrium activity [E]1 and the equation describing such a situation is [E] [E]1 ¼ exp ((k1 þ k2 )t) [E]0 [E]1

(10:17)

Then we come to the situation already depicted in Scheme 10.1. The equation describing this mechanism as an overall ﬁrst-order inactivation is arel ¼ exp (k0 t) k1 k3 k0 ¼ k1 þ k2

(10:18)

where the apparent ﬁrst-order rate constant is a composite one. The next possibility is that an enzyme is transformed to another conformation that still has enzymatic activity: b

k1 E

E1

SCHEME 10.4 Model 3 for change in enzyme activity leading to an enzyme form E1 that has a different activity than the native enzyme E. b is a measure for the ratio of speciﬁc activities of E and E1.

Obviously, it is not possible to distinguish the enzyme activity of E and E1 and therefore the measured activity will be a weighted average of the active species: arel ¼

E þ bE1 E0

(10:19)

with b¼

d1 d0

(10:20)

d0 and d1 are the speciﬁc activity of species E and E1, respectively. The next logical extension of this mechanism is k1 E

b E1

k2 I

SCHEME 10.5 Model 4 for enzyme inactivation via an intermediate enzyme form E1 that has a different activity than the native enzyme E.

The equation describing this mechanism is arel ¼

bk1 k1 b 1þ exp (k2 t) exp (k1 t) k2 k1 k2 k1

(10:21)

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This is thus a biexponential equation describing biphasic inactivation. A similar biexponential is found for a series–parallel mechanism: k1 E

b E1

k2 I

k3 E

SCHEME 10.6

I

Model 5 for enzyme inactivation via a parallel route.

arel

¼ 1þ

bk1 k1 b exp (k2 t) exp ((k1 þ k3 )t) k2 k1 k3 k2 k1 k3

(10:22)

These are just a few examples of inactivation mechanisms. Several of them lead to apparent ﬁrst-order behavior, sometimes indicated as disguised ﬁrst-order inactivation. Of course, there is also the possibility for second-order inactivation, for instance in the following mechanism where the enzyme dissociates into subunits:

E

k1 k2

E1 2E1

SCHEME 10.7

k3

2E1 I1 I2

Model 6 for enzyme inactivation via an intermediate following a consecutive route.

It is not so easy to ﬁnd analytical solutions for these kinds of mechanisms and it is usually simpler to resort to numerical integration of ODEs. For the mechanism depicted in Scheme 10.7 this would lead to d[E] ¼ k1 [E] dt d[E1 ] ¼ 2k1 [E] k2 [E1 ] 2k3 [E1 ]2 dt

(10:23)

Figure 10.19 shows an example of biphasic inactivation. Several mechanisms are conceivable for biphasic behavior. The simplest one is to postulate the presence of two isoenzymes showing different heat stability. Both isoenzymes are inactivated in a parallel fashion but at different rates. A model for this is arel ¼ a exp (k1 t) þ (1 a) exp (k2 t)

(10:24)

where a represents the fraction of the less resistant enzyme k1 is the rate constant for inactivation of the less resistant enzyme k2 is the rate constant for inactivation of the more resistant enzyme Another model that describes biphasic inactivation is the fractional conversion model, discussed in Chapter 4, which assumes that a constant and stable enzyme activity a1 remains (at least over the period investigated):

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1

2

0.8 0.6 arel

log arel

1.5

1

0.4 0.5

0

0.2 0 0

2

(A)

4

6

8

10

0

2

4

6

8

10

Time (min)

(B)

Time (min)

FIGURE 10.19 Biphasic inactivation of the enzyme cathepsin D in milk at 62.68C, logarithmic plot (A) and relative activity plot (B) with a ﬁt of Equation 10.24 (solid line) and Equation 10.25 (—-); parameters of model Equation 10.24: k1 ¼ 1.63 min1, k2 ¼ 0.11 min1, a ¼ 0.79, parameters of model Equation 10.25: k ¼ 1.30 min1, a1 ¼ 0.11. Dataset in Appendix 10.1, Table A.10.9.

a a1 ¼ exp (kt) a0 a1

(10:25)

Both models were ﬁtted to the data displayed in Figure 10.19B. Table 10.3 shows the results of model discrimination as discussed in Chapter 7, and the models appear to perform equally well. There are more mechanisms conceivable for biphasic inactivation. The mechanisms in Scheme 10.8 cannot be distinguished from the ones above in a ﬁrst-order plot if one only measures E (the remaining active enzyme) or I (the inactivated enzyme): (a)

E

E1

E2

I

(b)

E

E1

E2

I

(c)

E

E1

I

(d)

E

E1

E2

I

SCHEME 10.8 Various kinetic models that lead to biphasic behavior. E1 and E2 represent partly denatured but still active enzyme forms.

In these equations, E1 and E2 represent partly active enzymes that have undergone some structural changes (i.e., denaturation). TABLE 10.3 Results of a Model Discrimination Study on Inactivation of Cathepsin D in Milk at 62.68C (Figure 10.19B) SSr

log (Posterior Probability)

AICc

DAICc

Model 1 (Equation 10.24)

0.0106

4.485

71.94

0.4

Model 2 (Equation 10.25)

0.014

4.308

72.36

0

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Sometimes it may seem that enzyme activity reaches a plateau value, suggesting that an equilibrium state is attained (as in Figure 10.19B where a1 0.1). However, it may also be that the experiment was stopped too early and that further inactivation does occur if one would extend the analysis period. Another possibility is the so-called grace-period inactivation in which the enzyme appears to be stable for a period of time, followed by an inactivation period. This could occur, for instance, with the mechanism in Scheme 10.9 if the reverse reaction in the ﬁrst step cannot be neglected. E

SCHEME 10.9

E1

I

E2

Kinetic model leading to grace-period behavior.

The unfortunate message from all this is that it will not be easy to differentiate between all the possible mechanisms. As remarked before, it may help if additional structural measurements can be made. In fact, this could turn it into a multiresponse model, which allows a much better model discrimination, as we have discussed in Chapter 8. Global ﬁtting. In the quest for suitable models, an interesting approach is available in global ﬁtting of models to experiments. This can be done easily when measurements are done at various temperatures. It may be assumed that the temperature effect on enzyme inactivation can be described by Arrhenius’ or Eyring’s law. A consistent model in which the Arrhenius relation is incorporated should then be applicable at every temperature. The following three examples may demonstrate the usefulness of this approach. Figure 10.20 shows the inactivation of a peroxidase according to a ﬁrst-order reaction at various temperatures. There is a slight difference between the individual and global ﬁts: the individual ﬁts look a little bit better, but overall the differences are not large. Analyzing the Arrhenius parameters via the two-step procedure and via the global ﬁt gave the same estimates: k00 ¼ 0:155 0:011 min1 and Ea ¼ 368 28 kJ mol1. In other words, the ﬁrst-order model seems applicable at every temperature studied.

1

0.8

0.8

0.6

0.6

a/a0

a/a0

1

0.4

0.4

0.2

0.2

0

0 0

(A)

2

4

6 Time (min)

8

10

0

12 (B)

2

4

6

8

10

12

Time (min)

FIGURE 10.20 First-order inactivation model for broccoli peroxidase at 658C (^), 67.58C (~), 708C (*), and 728C (&) via individual ﬁts (A) and via global ﬁtting applying the reparameterized Arrhenius equation (B). Dataset in Appendix 10.1, Table A.10.10.

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1

0.8

0.8

0.6

0.6 arel

arel

1

0.4

0.4

0.2

0.2

0

0 0

10

20

(A)

30 40 Time (min)

50

60

0

10

20

(B)

30 40 Time (min)

50

60

FIGURE 10.21 Heat inactivation of a yeast invertase at various temperatures modeled via (A) a ﬁrst-order model (solid lines) and (B) the ﬁt of Equation 10.24 representing biphasic inactivation (solid lines). The ﬁts were derived for each temperature individually. (^) 408C, (~) 508C, (*) 558C, (&) 608C. Dataset in Appendix 10.1, Table A.10.11.

The second example of global ﬁtting is given in Figure 10.21 showing the heat inactivation of a yeast invertase. The inactivation data could not be ﬁtted satisfactorily by a ﬁrst-order model (Figure 10.21A). There is a hint of biphasic inactivation, so the model displayed in Equation 10.24 was applied for each temperature, as shown in Figure 10.21B. Judged by the ﬁts obtained the biphasic inactivation model seems to perform well. However, if we apply the global modeling approach by forcing the model in Equation 10.24 to all data at once, as advocated in Chapter 7, we obtained the ﬁt as shown in Figure 10.22 using the model in Equation 10.24.

1

0.8

arel

0.6

0.4

0.2

0

0

10

20

30 Time (min)

40

50

60

FIGURE 10.22 Heat inactivation of a yeast at various temperatures. The solid lines are the result of the global ﬁt of Equation 10.24 to all data at once. (^) 408C, (~) 508C, (*) 558C, (&) 608C. Same data as in Figure 10.21.

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TABLE 10.4 Parameter Estimates from the Iso-Enzyme Inactivation Model (Equation 10.24) for the Data Displayed in Figure 10.21B k1 95% CI

k2 95% CI

408C

0.052 0.045

0

a 95% CI 0.09 0.03

508C

0.06 0.01

0

0.211 0.019

558C

0.126 0.080

0.005 0.004

0.322 0.13

608C

0.345 0.126

0.076 0.045

0.76 0.21

Clearly, the ﬁt resulting from the global approach is not a good one, and in fact we have obtained evidence that the isoenzyme inactivation model is not the correct one, even though it performs well at each temperature separately. The reason for this apparent discrepancy becomes apparent if we take a look at the parameter estimates displayed in Table 10.4, derived from the individual ﬁts at each temperature separately. These results show ﬁrst of all that the parameters are fairly uncertain, and in some cases even inestimable. Furthermore, it appears that the parameter a is temperature dependent, something one would not expect if the isoenzyme mechanism is the correct one. In the global-ﬁt approach only one overall value for a can be found. The fact that a is not constant has its bearings on the values for the other parameters. Just another way to detect this is to make an Arrhenius plot of the rate constant k1 (Figure 10.23) (this could not be done for k2 since only two values were available). It appears from Figure 10.23 that parameter k1 does not obey Arrhenius’ law (of course, this is actually doing the same as performing a global ﬁt). So, in conclusion, this shows that the isoenzyme inactivation mechanism is not the correct one for this particular case, and one has to search for a better model. In doing so, it would be worthwhile to spend some effort in making a better experimental design to come to better model discrimination and parameter estimation. A third example of complex inactivation kinetics is the following. It concerns a proteinase from Pseudomonas ﬂuorescens, a psychrotrophic bacterium that can be found in milk. It produces rather heat stable proteinases (and lipases for that matter) that can cause problems in UHT milk in the sense that some enzymatic activity remains after UHT treatment, leading to proteolysis resulting into protein destabilization and bitter taste. The simplest model to test is the one in Equation 10.16. Figure 10.24 shows ﬁrst-order plots at three temperatures obtained via global ﬁtting using the Eyring equation. 0 −0.5

ln k1

−1 −1.5 −2 −2.5 −3 −3.5 0.00295 0.003 0.00305 0.0031 0.00315 0.0032 0.00325 1/T (K −1)

FIGURE 10.23 Arrhenius plot for the parameter k1 displayed in Table 10.4.

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Kinetics of Protein and Enzyme Denaturation

Log (residual activity)

0

−0.2

−0.4

−0.6

−0.8 0

50 Time (min)

100

FIGURE 10.24 Global ﬁrst-order plots for the inactivation of a puriﬁed proteinase from P. ﬂuorescens isolated from milk 908C (&), 1008C (^), and 1108C (D). Dataset in Appendix 10.1, Table A.10.12.

10-23

It is clear that the ﬁts are not satisfactory, there is a hint of grace-period behavior and accordingly more complicated models were tested. Two of them are shown in Scheme 10.10 and the resulting plots in Figure 10.25. In both models, the two rate constants k1 and k2 were not estimated because it was assumed that at the temperatures studied there would not be any native E present, since it was known that the denaturation temperature of this particular enzyme was near 558C. Parameter estimates are given in Table 10.5 along with the Akaike criterion, and it appears that the model in Scheme 10.10b performs much better than that in Scheme 10.10a. This is not only indicated by the Akaike criterion, but also by the large conﬁdence intervals for the parameters model in Scheme 10.10a. Although the model in Scheme 10.10b performed best, this does not mean that it is the ultimate model. It is, however, not conﬂicting with the data and it could be used as a starting point for further studies, if so desired.

Regeneration. It is possible that under certain conditions an enzyme regains its activity, partly or completely, after cooling. This must then be due to refolding taking place. Alternatively, it may be that refolding does occur rapidly but in an incorrect way. The incorrectly folded enzyme may then slowly return into a correctly folded way and thereby regain activity. Regeneration has been reported especially for the enzyme peroxidase in several cases and in several foods. Figure 10.26 shows an example for peroxidase in broccoli. It can be seen that there is considerable recovery of enzyme activity after a relatively short time, indicating that the enzyme is quite well capable of refolding back to an active form. The author is not aware of systematic studies on the kinetics of refolding, although this should be of importance to get an idea of how fast refolding can take place, and also the temperature dependence should be interesting, if one is to avoid regeneration of enzymatic activity. Autodigestion of proteases. Some bacterial proteases excreted by psychrotrophic microorganisms are extremely heat stable and their presence in UHT sterilized milk products may cause enzymatic deterioration. However, even though these enzymes are heat stable, inactivation is sometimes observed at

(a)

k3

k1 k2

(b)

k2

I

k4

k1 E

k5 D2

D1

E

k3 k4

k6 D2

D1

I

k5

E1 b

SCHEME 10.10

Two kinetic models to describe inactivation of a proteinase from P. ﬂuorescens.

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0

0

−0.2

−0.2

Log (residual activity)

Log (residual activity)

10-24

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8 0

50 Time (min)

(A)

100

0

50 Time (min)

(B)

100

FIGURE 10.25 Inactivation plots according to Scheme 10.10a (A) and Scheme 10.10b (B) of a puriﬁed proteinase from P. ﬂuorescens isolated from milk at 908C (&), 1008C (^), and 1108C (D). Same dataset as in Figure 10.24. TABLE 10.5 10.10a and b

Parameter Estimates and Akaike Criterion for the Models Shown in Scheme Eyring Parameter Estimates for Rate Constant 95% Conﬁdence Intervals k3: DHz ¼ 267.9 829.2 kJ mol1 DSz ¼ 436.8 511.2 J mol1 K1

Model in Scheme 10.10a

DAICc 12

k4: DHz ¼ 55.4 873.2 kJ mol1 DSz ¼ 161.8 1425.3 J mol1 K1 k5: DHz ¼ 62.5 724.1 kJ mol1 DSz ¼ 140.8 1951.1 J mol1 K1 k3: DHz ¼ 55.8 17.4 kJ mol1

Model in Scheme 10.10b

0

DSz ¼ 157.9 48.1 J mol1 K1 k6: DHz ¼ 221.2 44.4 kJ mol1 DSz ¼ 298.1 129.0 J mol1 K1 b: 0.61 0.17

relatively low temperatures in the range of 508C–608C (sometimes referred to as ‘‘low temperature inactivation,’’ a slightly confusing term because 508C–608C is not really low). This process is properly called autodigestion. Figure 10.27 gives an impression. The kinetic scheme describing autodigestion is

SCHEME 10.11

Step 1:

E

Step 2:

E+D

D E−D

→ E+I

Kinetic scheme describing intermolecular autodigestion of proteases.

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10-25

1

0.8

a/a0

0.6

0.4

0.2

0

0

1

2

3

4

5

Time (min)

FIGURE 10.26 Regeneration of the enzyme peroxidase from broccoli. Enzyme activity measured directly after heating at 758C (*) and enzyme activity after heating at 758C when keeping the enzyme 30 min at 208C before measuring (*). Dataset in Appendix 10.1, Table A.10.13.

The reaction in step 1 in Scheme 10.11 is the reversible unfolding process near the denaturation temperature, leading to the presence of native and denatured enzyme. The native enzymes that are still present can then hydrolyze their unfolded counterparts leading to irreversible loss of active enzyme (and concomitant formation of breakdown products I that are no longer enzymatically active). The results in Figure 10.27 suggest that the denaturation temperature should then be near 558C, the temperature at which both native and enzyme form are present in equal amounts; this was conﬁrmed by DSC. At temperatures below the denaturation temperature (say T 0 < 458C), most of the enzyme molecules will not be unfolded and therefore cannot be attacked by other enzyme molecules. Above the denaturation

1

Residual activity

0.8 0.6 0.4 0.2 0 40

60

80

100

120

140

T (C)

FIGURE 10.27 Example of the phenomenon of autodigestion for a puriﬁed protease from P. ﬂuorescens 228F in aqueous solution: Residual protease activity at 378C after holding the enzyme solution for 20 min at the indicated temperature. Dataset in Appendix 10.1, Table A.10.14.

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temperature (say T 0 > 658C), all enzymes are unfolded, and therefore there is no enzymatic activity and no hydrolysis takes place and upon cooling they refold into active enzymes. With increasing temperature (say at T 0 > 808C), the unfolded enzyme becomes subject to irreversible reactions so that refolding upon cooling becomes increasingly difﬁcult and this is actually the ‘‘normal’’ thermal inactivation.

10.4

Food Matrix Effects

It can make quite a difference whether enzyme inactivation is studied on isolated enzymes or in foods. Obviously, interpretation is much easier with isolated enzymes, but if the inactivation kinetics is much different from the behavior in a real food it becomes very dangerous, if not impossible to predict something for enzyme behavior in the food. In this section, we highlight just two examples to show the intricacies involved. The ﬁrst example shows the effect of water content on enzyme inactivation; in the second example, we compare the inactivation behavior of the milk protease plasmin in puriﬁed form and in its native environment milk. Effect of water content. Heat stability of enzymes is known to be strongly inﬂuenced by the water content of the system in which the enzyme is present. Usually, the stability increases strongly with decreasing water content. Figure 10.28 shows an example for the enzyme lipoxygenase. The explanation for such behavior is not straightforward but it is probably linked to the phenomenon of volume exclusion. With less water available as solvent, it becomes less favorable for proteins to unfold, and therefore heat stability increases in such situations (i.e., k decreases). Another effect may have to do with the interaction of water with the protein itself resulting in enthalpic and entropic effects; the 0.003

0.025

72°C

76°C

0.0025

0.02 k (s−1)

k (s−1)

0.002 0.0015

0.015 0.01

0.001 0.005

0.0005 0 (A)

0

0.2 0.4 0.6 0.8 Water content (g g–1)

0

1 (B)

0

0.2 0.4 0.6 0.8 Water content (g g–1)

1

0.12 80°C

0.1

k (s−1)

0.08 0.06 0.04 0.02 0 (C)

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Water content (g g–1)

FIGURE 10.28 Heat stability of lipoxygenase in a glucose calcium-alginate gel, expressed as the ﬁrst-order rate constant for inactivation as a function of water content at 728C (A), 768C (B), and 808C (C). Dataset in Appendix 10.1.

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6

0

5

−2

4

−4

3

−6

2

−8

1

−10

0 (A)

10-27

ln k

log D

Kinetics of Protein and Enzyme Denaturation

40

60

80

100 T (C)

120

140

160

−12 0.0024 0.0025 0.0026 0.0027 0.0028 0.0029 0.0030 0.0031 (B)

1/T (K −1)

FIGURE 10.29 Heat inactivation of plasmin in milk, expressed as the logarithm of D-values as a function of temperature (A) and as Arrhenius plot (B). Dataset in Appendix 10.1, Table A.10.16.

consequences of these effects will be difﬁcult to predict. Why there is a minimum observed in the plots shown in Figure 10.28 is unknown. In any case, the implication of this phenomenon is that in dried foods, the heat stability of enzymes may be much higher than in an aqueous environment, and this could have a large effect on the resulting quality. Behavior of plasmin. Plasmin is a milk-indigenous enzyme and is known to be very heat stable in milk. Its temperature dependence as found for inactivation in heated milk does deﬁnitely not conform to expected behavior. Figure 10.29 gives an example; it can be seen that above 1008C the inactivation rate constant (expressed as D-value in a TDT plot, or as rate constant k in an Arrhenius plot) hardly decreases, which is very atypical for protein=enzyme denaturation. The interesting fact is that plasmin, when isolated and puriﬁed, is not heat stable at all (Figure 10.30). The fraction of native enzyme as a function of temperature was estimated by subtraction of the extrapolated activity (Figure 10.30A) from the measured activity. The enzyme is seen to loose its activity at a temperature near 558C; to be sure that unfolding indeed occurs at that temperature, unfolding was conﬁrmed by CD studies. However, the enzyme was not irreversibly inactivated because enzyme activity was regained after holding it for 10 min at 658C, or 13 h at 548C. Inactivation started to become noticeable at higher temperatures between 758C and 908C. Another remarkable effect is that the inactivation of the enzyme is strongly enhanced in the presence of sulfhydryl (SH) compounds, such as cysteine and b-lactoglobulin; Figure 10.31 gives an impression. This is relevant because SH groups are present in milk. However, as shown in Figure 10.29, in milk the enzyme is much more stable and is deﬁnitely not inactivated between 758C and 908C. Therefore, there must be protecting agents present in milk, though there is also b-lactoglobulin present in milk, which should enhance inactivation when its sulfhydryl groups are exposed upon heating. As it happens, the major milk protein casein appears to protect plasmin from thermal inactivation, by a mechanism that is not yet understood; it is likely that plasmin associates with casein molecules. A qualitative model that attempts to explain the inactivation behavior of plasmin in milk is given in Scheme 10.12.

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10-28

1

0

0.9

−1

0.8 Fraction native enzyme

−0.5

ln (activity)

−1.5 −2 −2.5 −3 −3.5

(A)

0.6 0.5 0.4 0.3 0.2

−4 −4.5 2.9

0.7

0.1

3

3.1

3.2

3.3

3.4

0 30

3.5

1000/T (K −1)

40

50

60

70

80

T (C)

(B)

FIGURE 10.30 Activity of puriﬁed plasmin in aqueous solution as function of temperature (A) and calculated fraction of native enzyme as function of temperature (B). Dataset in Appendix 10.1, Table A.10.17. 100

Activity (%)

80

60

40

20

0

0

5

10 15 Time (min)

20

25

FIGURE 10.31 Heat inactivation of plasmin in the absence (*) and presence (*) of cystein (2 mmol dm3) at 708C. Dataset in Appendix 10.1, Table A.10.18. Reaction step 1:

PL

PL∗

Reaction step 2:

PL∗

iPL

Reaction step 3:

β-LG

β-LG∗

Reaction step 4:

PL ∗ + β-LG∗

iPL

Reaction step 5:

β-LG ∗ +

β-LG ∗-cas

Reaction step 6:

PL ∗ + cas

cas

PL ∗-cas

SCHEME 10.12 Qualitative scheme showing the various reactions of plasmin in milk upon thermal treatment. PL ¼ plasmin, PL* ¼ denatured plasmin, iPL ¼ inactivated plasmin, b-LG ¼ b-lactoglobulin, b-LG* ¼ denatured b-lactoglobulin with exposed SH groups, cas ¼ casein.

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This scheme shows the complexity of inactivation of plasmin in milk: Plasmin is subject to unfolding (step 1), the unfolded form can be inactivated directly (step 2) and, furthermore, it can react in its unfolded state with SH compounds (step 4), which in heated milk are delivered by denatured b-lactoglobulin (step 3); as it happens, b-lactoglobulin also reacts with casein (step 5) and is therefore in competition with casein and, ﬁnally, unfolded plasmin is protected by its association with casein (step 6). This example is meant to illustrate the phenomenon that inactivation behavior of enzymes studied after enzyme puriﬁcation can be completely different from their behavior in the food matrix. Thus, it is dangerous to extrapolate from test-tube experiments to real foods. As shown for plasmin, the enzyme is not very stable in aqueous solution, but it is very stable in its ‘‘natural’’ environment milk.

10.5

Concluding Remarks

This chapter has shown that the kinetic analysis of protein inactivation is not straightforward, especially when only the native or only the denatured form can be measured. When both forms (and possible intermediates) can be measured, the testing of kinetic models becomes easier, though still not easy. For enzymes, analysis is even more complicated due to such phenomena as disguised kinetics, autodigestion, and interaction with components in the food matrix. Notwithstanding these difﬁculties, it remains of utmost importance to be able to describe inactivation by kinetic models. In many cases, one has to be content with apparent kinetics. If one wants to make statements about molecular events, more elaborate techniques such as DSC and CD are needed, next to a kinetic analysis.

Appendix 10.1 Datasets Used for Examples in This Chapter

TABLE A.10.1 Denaturation of Apo-Lactoferrin (Figure 10.5) Time (s)

c (mg dm3)

0

181.9

320

114.8

650

93.3

900

85.1

1200

85.1

1500

66.0

1800 2100

57.5 50.1

2400

50.1

2700

38.0

3000

41.7

3400

41.7

3600

37.1

4000

26.9

Source: From Sanchez L., Peiro J.M., Castillo H., Perez M.D., Ena J.M., and Calvo M. Kinetic parameters for denaturation of bovine milk lactoferrin. J Food Sci 57:873–879, 1992.

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10-30 TABLE A.10.2 Time (h)

Denaturation of Patatin (Figure 10.9)

Fraction Native Protein

Fraction Aggregated Protein

0

1

0

0.03

0.8

0.01

0.07

0.72

0.02

0.1

0.7

0.04

0.13

0.64

0.08

0.17 0.5

0.58 0.36

0.13 0.34

1

0.32

0.52

Source: From Pots A.M., Gruppen H., De Jongh H.J.J., Van Boekel M.A.J.S., Walstra P., and Voragen A.G.J. Kinetic modeling of the thermal aggregation of patatin. J Agric Food Chem 47:4593–4599, 1999.

TABLE A.10.3 Time (min)

Denaturation of b-Galactosidase (Figure 10.10)

Fraction Native

Fraction Intermediate

Fraction Final Aggregate

0

91.5

7.2

1.2

5

64.6

7.8

24.8

10

47.7

6.6

42.3

15

37.3

5.3

54.0

20

29.5

4.4

63.0

30

16.9

2.8

74.0

45

9.1

2.2

82.8

60 75

4.4 1.6

1.6 1.6

89.0 92.2

Source: From Yoshioka S., Aso Y., Izutsu K., and Kojima S. Is stability prediction possible for protein drugs? Denaturation kinetics of beta-galactosidase in solution. Pharm Res 11:1721–1725, 1994.

TABLE A.10.4 Denaturation Rate Constants k of b-Lactoglobulin A and a-Lactalbumin as a Function of Temperature (Figure 10.13) T 0 (8C)

k for b-LG-A

k for a-LA

70 75

0.00015 0.00047

0.00012 0.00055

80

0.00274

0.00156

85

0.00811

0.00181

90

0.02184

0.00288

95

0.03599

0.00341

100

0.05253

0.00566

110

0.07468

0.00971

120 130

0.13365 0.17788

0.01574 0.02597

140

0.26009

0.04108

150

0.37953

0.0704

Source: From Dannenberg, F. and Kessler, H.G., Milchwissenschaft, 43, 3, 1988.

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Kinetics of Protein and Enzyme Denaturation

10-31

TABLE A.10.5 Example of Molecular Crowding on Denaturation: Effect of Sucrose on the Denaturation 0 of a-Chymotrypsin (Figure 10.14) Temperature Tden 0 Tden (8C)

[Sucrose] (M) 0

49.8

0.1

51.0

0.5 0.7

53.9 55.7

0.9

58.1

1

58.1

Source: From Lee J.C., and Timasheff S.N. The stabilization of proteins by sucrose. J Biol Chem 256:7193–7201, 1981.

TABLE A.10.6 Molar Activity Coefﬁcient of Hemoglobin as a Function of Its Concentration (Figure 10.15) c (g dm3)

Molar Activity Coefﬁcient y

1

1

25

1.2

50

1.5

75

2

100 150

2.5 4.7

200

10

250

25

300

90

Source: From Minton, Biochem., 55, 119, 1983.

A.P.,

Mol.

Cell.

TABLE A.10.7 Inactivation of Tomato Pectin Methylesterase (Figure 10.17) Time (s)

Log (%)

0

2

64.171

1.9

183.104

1.6

305.405

1.5

424.337

1.2

546.619

1.0

662.318 784.658

0.9 0.7

903.648

0.5

Source: From Anthon G.E., Sekine Y., Watanabe N., and Barrett D.M. Thermal inactivation of pectin methylesterase, polygalacturonase, and peroxidase in tomato juice. J Agric Food Chem 50:6153–6159, 2002.

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10-32

TABLE A.10.8 Inactivation of a-Amylase from Bacillus subtilis at High Pressure (700 MPa) (Figure 10.18) Time (min)

ln(a=a0)

0

0

10

0.257

20

0.453

30

0.765

40

1.078

50

1.34

60

1.547

70

1.748

Source: From Geeraerd A.H., Herremans C.H., Ludikhuyze L.R., Hendrickx M.E., and Van Impe J.F. Modelling the kinetics of isobaric–isothermal inactivatoin of Bacillus subtilis a-amylase with artiﬁcal neural networks. J Food Eng 36:263–279, 1998.

TABLE A.10.9 Biphasic Inactivation of the Enzyme Cathepsin D in Milk at 62.68C (Figure 10.19) Time (min)

log(% activity)

0

2

0.15

1.88

0.5 1

1.76 1.67

2

1.24

2

1.19

3

1.06

3

0.99

6

0.88

6

0.83

9 9

0.63 0.56

Source: From Hayes M.G., Hurley M.J., Larsen L.B., Heegarrd C.W., Magboul A.A.A., Oliveira J.C., McSweeney P.L.H., and Kelly A.L. Thermal inactivation kinetics of cathepsin D. J Dairy Res 68:267–276, 2001.

TABLE A.10.10 Time (min) 0

Inactivation of Broccoli Peroxidase (Figure 10.20)

a=a0 at 658C 1

a=a0 at 67.58C 1

0.5

1

0.89

0.73

0.75

0.44

0.61

0.25

0.887

3

1 0.62 0.35

4

0.17 0.08

5 6

a=a0 at 728C

0.76

1 2

a=a0 at 708C

0.75 (continued )

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Kinetics of Protein and Enzyme Denaturation TABLE A.10.10 (continued) (Figure 10.20) Time (min)

a=a0 at 658C

10-33

Inactivation of Broccoli Peroxidase a=a0 at 67.58C

7

a=a0 at 708C

a=a0 at 728C

0.54

8

0.70

9

0.45

10

0.66

11

0.38

Source: From Thongsook, T. and Barrett, D.M., J. Agric. Food Chem., 53, 3215, 2005.

TABLE A.10.11 Time (min)

Inactivation of a Yeast Invertase (Figure 10.21)

a=a0 at 408C

a=a0 at 508C

a=a0 at 558C

a=a0 at 608C

0

1

1

1

1

6

0.977

0.946

0.793

0.245

10

0.954

0.901

0.753

0.138

20

0.954

0.847

0.633

0.048

30

0.918

0.824

0.587

0.026

60

0.913

0.796

0.5

0.01

Source: From Vrabel, P., Polakovic, M., Stefuca, V., and Bales, V., Enzyme Microb. Technol., 348, 1997.

TABLE A.10.12 Time (min)

Inactivation of a Puriﬁed Proteinase from P. ﬂuorescens (Figure 10.24)

a=a0 at 908C

Time (min)

a=a0 at 1008C

Time (min)

a=a0 at 1108C

10

0.886

5

0.839

3

0.882

20

0.781

5

0.833

6

0.662

20

0.784

5

0.913

6

0.725

30 30

0.735 0.698

5 10

0.873 0.708

9 9

0.668 0.614

40

0.617

10

0.699

10

0.543

40

0.654

10

0.799

10

0.543

60

0.423

10

0.858

10

0.522

60

0.435

15

0.611

10

0.642

60

0.499

15

0.593

10

0.573

70

0.33

15

0.631

12

0.457

70 80

0.404 0.309

15 20

0.623 0.505

15 15

0.353 0.438

80

0.386

20

0.514

18

0.353

80

0.336

20

0.513

18

0.359

80

0.295

20

0.517

20

0.238

80

0.312

25

0.407

20

0.221

100

0.217

25

0.455

20

0.224 (continued )

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10-34 TABLE A.10.12 (continued) (Figure 10.24) Time (min)

Inactivation of a Puriﬁed Proteinase from P. ﬂuorescens

a=a0 at 908C

Time (min)

a=a0 at 1008C

Time (min)

a=a0 at 1108C

100

0.168

30

0.377

20

0.241

100

0.231

30

0.337

21

0.252

100

0.217

35

0.301

21

0.199

35

0.345

24

0.189

40

0.222

40

0.252

Source: From Schokker, E.P. and Van Boekel, M.A.J.S., J. Agric. Food Chem., 45, 4740, 1997.

TABLE A.10.13 Time (min)

Regeneration of Peroxidase (Figure 10.26)

% Activity, Directly Measured

% Activity, Measured after 30 min at 208C

0.2

87.9

98.0

0.5

81.8

92.9

1

63.6

83.8

2

40.4

77.8

3

30.3

69.7

4

18.2

61.6

Source: From Thongsook, T. and Barrett, D.M., J. Agric. Food Chem., 53, 3215, 2005.

TABLE A.10.14 Autodigestion of a Puriﬁed Protease from P. ﬂuorescens (Figure 10.27) T (8C)

a=a0

40.5

1

45.6

0.98

48.4 49.3

0.83 0.65

51.6

0.13

53

0.075

53.5

0.15

56.9

0.35

57.9

0.42

59.8

0.54

62.1 65.9

0.7 0.89

70.3

0.98

80.6

0.91

90.2

0.67

100

0.155

111

0.17

120

0.05

130

0

Source: From Schokker E.P. and Van Boekel M.A.J.S. Mechanism and kinetics of inactivation at 40–708C of the extracellular proteinase from Pseudomonas ﬂuorescens 22F. J Dairy Res 65:261–272, 1998.

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Kinetics of Protein and Enzyme Denaturation TABLE A.10.15 (Figure 10.28)

10-35

Heat Stability of a Lipoxygenase

Water Content (g g1)

k at 728C

k at 768C

k at 808C

0.167

0.000864

0.00244

0.00619

0.231

0.000316

0.00155

0.00465

0.375

0.000121

0.000625

0.00617

0.5

0.000288

0.6

0.000721

0.00539

0.03

0.75

0.00271

0.0195

0.108

0.0125

Source: From Liou J.K. An approximate method for nonlinear diffusion applied to enzyme inactivation during drying. PhD thesis, Wageningen University, the Netherlands, 1982.

TABLE A.10.16 (Figure 10.29) T0 (8C)

Heat Inactivation of Plasmin D-Value (s)

60

59,520

70

11,700

75 80

660 187

85

126

90

71

100

33

110

23

120

18

140

13

Source: From Saint Denis T., Humbert G., and Gaillard J.L. Heat inactivation of native plasmin. Plasminogen and plasminogen activators in bovine milk: A revisited study. Lait 81:715–729, 2001.

TABLE A.10.17 Plasmin Activity as a Function of Temperature (Figure 10.30) 1000=T

ln(Activity)

2.96

3.9

3 3.05

2.53 1.56

3.09

1.26

3.15

1.18

3.2

1.35

3.25

1.54

3.31

1.84

3.36

2.07

3.42 3.47

2.44 2.84

Source: From Metwalli, A.A.M., de Jongh, H.H.J., and Van Boekel, M.A.J.S., Int. Dairy J., 8, 47, 1998.

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10-36

TABLE A.10.18 Effect of Cystein on Heat Inactivation of Plasmin (Figure 10.31) t (min) 0 0.3

% Activity (no cystein) 100

% Activity (2mM cystein) 100 72.14

0.4

68.42

0.5

45.2

0.7

28.17

1

22.6

5

0.31

10

75.23

15 20

68.73 59.75

25

53.25

Source: From Metwalli, A.A.M., de Jongh, H.H.J., and Van Boekel, M.A.J.S., Int. Dairy J., 8, 47, 1998.

Bibliography and Suggested Further Reading About Protein Stability and Protein Denaturation Busti P., Gatti C.A., and Delorenzi N.J. Thermal unfolding of bovine beta-lactoglobulin studied by UV spectroscopy and ﬂuorescence quenching. Food Res Int 38:543–550, 2005. Chaplin L.C. and Lyster R.L.J. Irreversible heat denaturation of bovine a-lactalbumin. J Dairy Res 53:249–258, 1986. Chen X.D., Chen Z.D., Nguang S.K., and Anema S.G. Exploring the reaction kinetics of whey protein denaturation=aggregation by assuming the denaturation step is reversible. Biochem Eng J 2:63–69, 1998. Creighton T.E. Protein folding. Biochem J 270:1–16, 1990. Creighton T.E. Proteins: Structures and Molecular Properties, 2nd ed. New York: Freeman, 1993. Croguennec T., O’Kennedy B.T., and Mehra R. Heat-induced denaturation=aggregation of betalactoglobulin A and B: Kinetics of the ﬁrst intermediates formed. Int Dairy J 14:399–409, 2004. Dannenberg F. and Kessler H.G. Application of reaction kinetics to the denaturation of whey proteins in heated milk. Milchwissenschaft 43:3–7, 1988. Le Bon C., Nicolai T., and Durand D. Kinetics of aggregation and gelation of globular proteins after heatinduced denaturation. Macromolecules 32:6120–6127, 1999. Liang K.K., Hayashi M., Shiu Y.J., Shao J., Yan Y.J., and Lin S.H. Thermodynamics and kinetics of protein folding: A mean ﬁeld theory. Phys Chem Chem Phys 5:5300–5308, 2003. Lumry R. and Eyring H. Conformation changes of proteins. J Phys Chem 58:110–120, 1954. Lyster R.L.J. The denaturation of a-lactalbumin and B-lactoglobulin in heated milk. J Dairy Res 37:233–244, 1970. Privalov P.L. Stability of proteins. Small globular proteins. Adv Protein Chem 33:167–241, 1979. Relkin P. Thermal unfolding of b-lactoglobulin, a-lactalbumin, and bovine serum albumin. A thermodynamic approach. Crit Rev Food Sci Nutr 36:565–601, 1996. Roefs S.P.F.M. and de Kruif C.G. A model for the denaturation and aggregation of B-lactoglobulin. Euro J Biochem 226:883–889, 1994. Seckler R. and Jaenicke R. Protein folding and protein refolding. FASEB J 6:2545–2552, 1992. Tanford C. Protein denaturation. Adv Protein Chem 23:121–282, 1968.

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About Enzyme Inactivation Ahern T.J. and Klibanov A.M. The mechanism of irreversible enzyme inactivation at 1008C. Science 228:1280–1284, 1985. Aymard C. and Belarbi A. Kinetics of thermal deactivation of enzymes; a simple three parameters phenomenological model can describe the decay of enzyme activity, irrespectively of the mechanism. Enzyme Microb Technol 27:612–618, 2000. Daniel R.M. The upper limits of enzyme thermal stability. Enz Microb Technol 19:74–79, 1996. Hei D.J. and Clark D.S. Estimation of melting curves from enzymatic activity temperature proﬁles. Biotechnol Bioeng 42:1245–1251, 1993. Hayes M.C., Hurley M.J., Larsen L.B., Heegaard C.W., Magboul A.A.A., Oliveira J.C., McSweeney P.L. H., and Kelly A.L. Thermal inactivation kinetics of bovine cathepsin D. J Dairy Res 68:267– 276, 2001. Lencki R.W., Arul J., and Neufeld R.J. Effect of subunit dissociation, denaturation, aggregation, coagulation, and decomposition on enzyme inactivation kinetics. 1. First-order behavior. Biotechnol Bioeng 40:1421–1426, 1992a. Lencki R.W., Arul J., and Neufeld R.J. Effect of subunit dissociation, denaturation, aggregation, coagulation, and decomposition on enzyme inactivation kinetics. 2. Biphasic and grace period behavior. Biotechnol Bioeng 40:1427–1434, 1992b. Makki F. and Durance T.D. Thermal inactivation of lysozyme as inﬂuenced by pH, sucrose and sodium chloride and inactivation and preservative effect in beer. Food Res Int 29:635–645, 1996. Metwalli A.A.M., de Jongh H.H.J., and Van Boekel M.A.J.S. Heat inactivation of bovine plasmin. Int Dairy J 8:47–56, 1998. Owusu R.K., Makhzoum A., and Knapp J. The thermodynamic stability of lipases and proteases from psychrotrophic bacteria. Food Chem 39:187–195, 1991. Owusu R.K., Makhzoum A., and Knapp J.S. Heat inactivation of lipase from psychrotrophic Pseudomonas ﬂuorescens P38-activation parameters and enzyme stability at low or ultra-high temperatures. Food Chem 44:261–268, 1992. Owusu R.K. and Berthalon N. A test for the 2-stage thermoinactivation model for Chymotrypsin. Food Chem 48:231–235, 1993. Owusu R.K. and Berthalon N. Determination of enzyme global thermostability from equilibrium and kinetic analysis of heat inactivation. Food Chem 51:15–20, 1994. Polakovic M. and Vrabel P. Analysis of the mechanism and kinetics of thermal inactivation of enzymes: Critical assessment of isothermal inactivation experiments. Process Biochem 31:787– 800, 1996. Rollema H.S. and Poll J.K. The alkaline milk proteinase system: Kinetics and mechanism of heat inactivation. Milchwissenschaft 41:41, 1986. Sadana A. Biocatalysis. Fundamentals of Enzyme Deactivation Kinetics. Englewood Cliffs, NJ: PrenticeHall, 1991. Schokker E.P. and Van Boekel M.A.J.S. Kinetic modeling of enzyme inactivation: Kinetics of heat inactivation at 908C–1108C of extracellular proteinase from Pseudomonas ﬂuorescens 22F. J Agric Food Chem 45:4740–4747, 1997. Schokker E.P. and Van Boekel M.A.J.S. Kinetics of thermal inactivation of the extracellular proteinase from Pseudomonas ﬂuorescens 22F: Inﬂuence of pH, calcium, and protein. J Agric Food Chem 47:1681–1686, 1999. Vrabel P., Polakovic M., Stefuca V., and Bales V. Analysis of mechanism and kinetics of thermal inactivation of enzymes: Evaluation of multitemperature data applied to inactivation of yeast invertase. Enzyme Microb Technol 20:348–354, 1997. Zale S.E. and Klibanov A.M. On the role of reversible denaturation (unfolding) in the irreversible thermal inactivation of enzymes. Biotechnol Bioeng 25:2221–2230, 1983.

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Enzyme Regeneration Adams J.B. The inactivation and regeneration of peroxidase in relation to the high-temperature-short time processing of vegetables. J Food Technol 13:281–297, 1978. Adams J.B. Review: Enzyme inactivation during heat processing of food-stuffs. Int J Food Sci Technol 26:1–20, 1991. Thongsook T. and Barrett D.M. Heat inactivation and reactivation of broccoli peroxidase. J Agric Food Chem 53:3215–3222, 2005. About the Effect of Molecular Crowding on Protein and Enzyme Stability Agena A.M., Bogle I.D.L., and Pessoas F.L.P. An activity coefﬁcient model for proteins. Biotechnol Bioeng 55:65–71, 1997. Cheung M.S., Klimov D., and Thirumalai D. Molecular crowding enhances native state stability and refolding rates of globular proteins. Proc Natl Acad Sci USA 102:4753–4758, 2005. Davis-Searles P.R., Saunders A.J., Erie D.A., Winzor D.J., and Pielak G.J. Interpreting the effects of small uncharged solutes on protein-folding equilibria. Annu Rev Biophys Biomol Struct 30:271– 306, 2001. Despa F., Orgill D.P., and Lee R.C. Molecular crowding effects on protein stability. Ann NY Acad Sci 1066:54–66, 2005. Eggers D.K. and Valentine J.S. Molecular conﬁnement inﬂuences protein structure and enhances thermal protein stability. Protein Sci 10:250–261, 2006. Ellis R. Macromolecular crowding: Obvious but underappreciated. Trends Biochem Sci 26:597–604, 2001. Ellis R. and Minton A.P. Join the crowd. Nature 425:27–28, 2003. Hall D. and Minton A.P. Macromolecular crowding: Qualitative and semiquantitative successes, quantitative challenges. Biochim Biophys Acta 1649:127–139, 2003. Hu Z., Jiang J., and Rajagopalan R. Effects of macromolecular crowding on biochemical reaction equilibria: A molecular thermodynamic perspective. Biophys J 93:1464–1473, 2007. Minton A.P. Excluded volume as a determinant of macromolecular structure and reactivity. Biopolymers 20:2093–2120, 1981. Minton A.P. The effect of volume occupancy upon the thermodynamic activity of proteins: Some biochemical consequences. Mol Cell Biochem 55:119–140, 1983. Minton A.P. Molecular crowding: Analysis of effects of high concentrations of inert cosolutes on biochemical equilibria and rates in terms of volume exclusion. Methods Enzymol 295:127–149, 1998. Minton A.P. Effect of a concentrated ‘‘inert’’ macromolecular cosolute on the stability of a globular protein with respect to denaturation by heat and by chaotropes: A statistical-thermodynamic model. Biophys J 78:101–109, 2000. Minton A.P. The inﬂuence of macromolecular crowding and macromolecular conﬁnement on biochemical reactions in physiological media. J Biol Chem 276:10577–10580, 2001. Minton A.P. Inﬂuence of macromolecular crowding upon the stability and state of association of proteins: Predictions and observations. J Pharm Sci 94:1668–1675, 2005. Minton A.P. How can biochemical reactions within cells differ from those in test tubes? J Cell Sci 119:2863–2869, 2006. Patel C.N., Noble S.M., Weatherly G., Tripathy A., Winzor D.J., and Pielak G.J. Effects of molecular crowding by saccharides on a-chymotrypsin dimerization. Protein Sci 11:997–1003, 2002. Ralston G.B. Effects of ‘‘crowding’’ in protein solutions. J Chem Educ 67:857–860, 1990. Schnell S. and Turner T.E. Reaction kinetics in intracellular environments with macromolecular crowding: Simulations and rate laws. Prog Biophys Mol Biol 85:235–260, 2004. Shimizu S. and Smith D.J. Preferential hydration and the exclusion of cosolvents from protein surfaces. J Chem Phys 121:1148–1154, 2004.

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Winzor D.J. and Wills P.R. Effects of thermodynamic nonideality on protein interactions. Equivalence of interpretations based on excluded volume and preferential solvation. Biophys Chem 25:243–251, 1986. Winzor D.J. and Wills P.R. Molecular crowding effects of linear polymers in protein solutions. Biophys Chem 119:186–195, 2006. Zimmerman S.B. and Minton A.P. Macromolecular crowding: Biochemical, biophysical and physiological consequences. Annu Rev Biophys Biomol Struct 22:27–65, 1993.

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11 Kinetics of Physical Changes

11.1

Introduction

Physical changes in foods are very important in relation to quality. Many foods are dispersions: they contain (colloidal) particles. Examples are fat globules in an aqueous phase (as in milk), water globules in a fat phase (as in margarine), foam bubbles in an aqueous phase (as in beer foam), fat crystals in an oil phase (as in butter and margarine), protein particles in an aqueous phase (as in milk), starch granules (as in bread), etc. Quality changes arise from changes in such dispersions. Most colloidal systems are inherently unstable in a thermodynamic sense, in other words the systems are not in equilibrium; they may, however, be kinetically stable. For instance, emulsions will tend to separate into an oil and water phase, which is the result of coalescence. Aggregation may occur, of emulsion droplets, of protein particles, of crystals, etc. Sometimes these processes are desirable, for instance, coalescence in the case of churning of cream, coagulation of casein micelles in cheese curd formation, formation of sugar crystals in sugar manufacture, etc. Mostly, however, such changes will result in quality loss. Examples are aggregation and sedimentation of fruit particles in a fruit juice, serum separation in tomato ketchup, creaming of fat globules in milk, sugar crystallization in candy bars, and many more examples could be given. In this chapter we will discuss how some of these changes can be described kinetically. The size scale of particles present in foods may span up to six orders of magnitude; the structural elements range from low molecular weight molecules to high molecular weight compounds to colloids to microorganisms, i.e., from nanometers to some hundred micrometers. In general, the larger the particles, the longer the time scales involved at which something happens (such as a molecular reaction or aggregation of colloidal particles). We need to remark that kinetics of physical changes is not an easy topic to discuss. Most of the available models are developed for dilute, ‘‘clean’’ systems, and foods are all but ‘‘clean’’ and dilute, as remarked before. One has to be careful therefore in applying models to describe physical changes in foods. Much active research is going on in this ﬁeld and it is beyond the scope of this book to discuss all these developments; the interested reader will ﬁnd some references at the end of this chapter. We will limit ourselves therefore to some phenomena that are important for food quality such as diffusion, aggregation and coalescence. Nevertheless, it is stressed that the length of this chapter is inversely proportional to its importance for food quality.

11-1

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Kinetic Modeling of Reactions in Foods

11-2

11.2

Kinetics of Diffusion

If there are concentration differences within a system, these differences will tend to disappear over time, and this is an entropy-driven process. This process is commonly known as diffusion, caused by the random translational and rotational motion of molecules and particles. In relation to quality of foods, the migration of water due to diffusion is particularly important. We are interested in how fast these changes occur. Two theories will be discussed here, one the conventional Fickian diffusion, and the other the Maxwell–Stefan (MS) approach which has become more popular the last decades, for reasons that will be discussed. We will start with Fick’s laws because those are probably the familiar ones, and we then move on to show the more general MS approach.

11.2.1

Fick’s Laws

The rate of migration of matter is measured by its ﬂux J (concentration of a species times its velocity, mol m2 s1). This ﬂux is found to be proportional to the concentration gradient (of a compound, or of the number of particles per unit volume): J/

dc dx

(11:1)

If we introduce a proportionality constant in Equation 11.1 called the diffusion coefﬁcient Df we ﬁnd J ¼ Df

dc dx

(11:2)

Equation 11.2 is the ﬁrst diffusion law of Fick. In fact, we introduced this equation already in Equation 3.152 in relation to irreversible thermodynamics. Einstein derived that the root-mean-square distance in a given direction for translational motion is pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ hx2 i ¼ 2Df t

(11:3)

For spherical particles (Equation 4.159), the Stokes–Einstein equation is Df ¼

kB T 6phv Rp

(11:4)

This equation shows that small molecules and particles move faster than bigger ones. Typical values for the diffusion coefﬁcient Df in water at room temperature are 1.7 3 109 m2 s1 for water molecules, 4.7 3 1010 m2 s1 for sucrose molecules in water, 6 3 1011 m2 s1 for a globular protein as serum albumin, and 4 3 1013 m2 s1 for an emulsion droplet of size 1 mm. The diffusion coefﬁcient depends on size and shape of the molecule=particle, on its interaction with the solvent, and the viscosity of the solvent. Our interest is in describing a situation where the concentration is not homogeneous and we would like to describe the rate of change in such a situation. Consider a situation of a thin slab such as depicted in Figure 11.1. The concentration at x is c at time t and there is an inﬂux J1 at the left-hand side. The amount of molecules M (in moles) that enter the slab with area Aa in an interval dt is M ¼ J1 Aa dt

(11:5)

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Kinetics of Physical Changes

11-3 Area Aa

J1

J2

dx

FIGURE 11.1 Schematic picture to show the ﬂow in and out of a volume element with length dx.

As a result of this inﬂux, there is a concentration change inside the slab: @c J1 Aa dt J1 ¼ ¼ @t Aa dx dt dx

(11:6)

There is of course also an outﬂow J2 at the right-hand side of the slab which is similarly @c J2 Aa dt J2 ¼ ¼ dx @t Aa dx dt

(11:7)

The ﬂux J1 entering the slab at x minus the ﬂux J2 leaving the slab at x þ dx equals accumulation over dx. It then follows that the concentration change is @c J1 J2 ¼ dx @t

(11:8)

Applying Fick’s ﬁrst law gives @c ¼ @t

Df

@c @c þ Df @x x @x xþdx dx

(11:9)

and this can be written as 2 @c @ c ¼ Df @t x @x2 t

(11:10)

This is Fick’s second law, giving the change in concentration c with time at any x as a function of the concentration gradient. For diffusion in all three directions it can be derived that: 2 @c @ c @2c @2c þ þ ¼ Df @t @x2 @y2 @z2

(11:11)

Fick’s second law is a second-order differential equation with respect to space and a ﬁrst-order one with respect to time (see also Appendix A). The solution depends on the boundary conditions, two for space and one for time; those boundary conditions must be speciﬁed. Many books are written with particular

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11-4

solutions of Fick’s equations. With modern software, numerical solutions are another option to solve them relatively easy. We will not discuss this in detail but refer to some references given at the end of this chapter. Two solutions are worth considering though, because they are instructive. One is the calculation of mass transfer via diffusion over a ﬂat boundary with area Aa, where the concentration of a component is maintained at a certain level. The amount transported per unit time is then: rﬃﬃﬃﬃﬃﬃﬃ Df t (11:12) M ¼ 2Aa c p This shows the typical square root relation with time for diffusion, already shown in Equation 11.3. Another equation that is useful is the concentration as a function of the distance x from a ﬂat inﬁnite surface: 3 2 ðy 2 2 c(x) ¼ c(1 erf y) ¼ c41 pﬃﬃﬃﬃ exp (z )dz 5 p (11:13) 0 x y ¼ pﬃﬃﬃﬃﬃ 2 Dt In this equation, erf symbolizes the error function, and z is an integration variable. This equation is, for instance, useful to predict the migration of molecules from a packaging material into a food. It can be derived from Equation 11.13 that x0:5 Df t0:5

(11:14)

x0.5 is the distance over which the concentration is halved, and t0.5 is the time at which that happens. Given the fact that for most solutes in water Df 1010 m2 s1, the latter equation gives an order of magnitude estimation of diffusion effects. A complication in foods is that the diffusion coefﬁcient depends quite strongly on the food matrix. In the presence of macromolecules and networks, diffusion is strongly hindered and the diffusion coefﬁcient can be orders of magnitude lower than the one found in water. This will be further discussed in Chapter 14; see, however, also the MS approach discussed in Section 11.2.2. Fick’s laws are the starting point for all kinds of rate equations, such as moisture loss or uptake, drying kinetics, sorption kinetics, rate of migration of residues from packaging materials into foods, etc. Such situations are thoroughly discussed in books dedicated to food engineering, so we refer the interested reader to these; some references are given at the end of this chapter. Diffusion through a barrier. It can be of great importance for food quality that diffusion of molecules can be inﬂuenced, for instance by packaging or by applying a coating. It concerns diffusion of water, ﬂavor molecules, colored components, oxygen, etc. This situation can be modeled in a simpliﬁed way as follows. Imagine for instance two aqueous layers (phase I) separated by a lipid layer (phase II) as shown in Figure 11.2, and we would like to model how fast a component migrates through the lipid layer into the other compartment. It is assumed that the concentration in phase a is the same everywhere. It is known that the solubility of the component differs in phases a and b, and that it dissolves better in phase a. This can be quantiﬁed in the partition coefﬁcient (to be discussed in more detail later on in this chapter, Equation 11.82): Pa=b ¼

Ca Cb

(11:15)

From Fick’s ﬁrst law it can be derived that the mass ﬂux J through an area Aa at steady state is J¼

Df Aa Pa=b Dc x2 x1

(11:16)

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Kinetics of Physical Changes

11-5

Phase β

Concentration, c

Phase α

Phase α

[c]

[c] [c]

x1

x2

Distance, x

FIGURE 11.2 Simpliﬁed representation of the diffusion of a component from the left compartment (phase a) through a layer (phase b) to the right compartment.

This equation shows how the ﬂux can be inﬂuenced. One possibility is to adjust the thickness of layer x1 x2, another to search for a material in which the effective diffusion coefﬁcient is different in phase b, and yet another to search for materials for which the partition coefﬁcient differs. Equation 11.17 also shows that the diffusion through a very thin layer is hardly hindered, unless the values of Df and Pa=b are such that diffusion is hindered strongly. This is of importance for diffusion through layers and ﬁlms that are very thin: diffusion is then hardly retarded. The ﬁlm model. The ﬁlm model is used extensively in chemical engineering, and also in food engineering. It tackles ﬂow near interfaces. Concentration gradients in the bulk are assumed to be absent because of convective currents and eddies. Near phase interfaces, eddies die out and a small ﬁlm is supposed to exist in which the only transport mechanism is diffusion (Figure 11.3). The thickness dz of such a ﬁlm is roughly 104 m in gases, 105 m in liquids, 104 to 107 m in membranes, and even in solid particles a thin ﬁlm is assumed to exist with a thickness of roughly 0.1 3 diameter of the solid particle. Film ∆z Phase a

No concentration gradients due to convection

Phase boundary

Phase b

Concentration gradient due to diffusion

FIGURE 11.3 Schematic illustration of the ﬁlm model (not to scale).

No concentration gradients due to convection

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Mass transfer via packaging materials. Food quality is very much inﬂuenced by how the food is packaged it regulates the interaction with the environment and it is all about mass transfer. Packaging may inﬂuence the mass transfer of water vapor, oxygen, volatile aroma components, and other molecules via . .

.

Permeation: transfer through the packaging material Sorption: penetration and dispersal of molecules coming from the food into the packaging material (a process sometimes referred to as ‘‘scalping’’) Migration: transfer of molecules from the packaging material into the food

Furthermore, packaging can protect against light (basically against photooxidation). If there is a headspace, the relations become even more intricate because it adds a vapor interface on the food side. As for the environment, the factors of importance with respect to food quality are the oxygen content, the light intensity and wavelength, the relative humidity, and the temperature. Assuming that the packaging material does not allow permeation of microorganisms or dirt materials, its permeability to oxygen, light, and moisture is important. A special case is modiﬁed atmosphere and controlled atmosphere packaging, in which the food is contained in a special gas composition to control metabolic changes within the food and to control growth of microorganisms. Composition varies then usually with respect to oxygen and carbon dioxide content, and the permeability with respect to CO2 and nitrogen can then also be important. If changes in the food are inﬂuenced by transfer of light, oxygen and moisture through the package, kinetics could be controlled by mass transfer effects (diffusion-limited reactions, as discussed in Chapter 4). It should be noted that this not only concerns transfer from the environment to the food, but also from the food to the environment. For instance, transport of moisture from bread to the environment may lead to a quality loss because of drying out (physical quality loss). The permeation of gases (oxygen, moisture) will be governed by diffusion and can thus be described by Fick’s ﬁrst law of diffusion. For unsteady state diffusion (concentration is a function of time), the second law of Fick is applicable. In this way, the rate of change in concentration is related to the spatial variation of the concentration at that point. The following example may illustrate. Suppose that we are interested in how fast the migration of a component from the packaging material into the food occurs, a situation that is usually not desired. The concentration change of such a component (in one dimension) can be described by the following equations: @cP @ 2 cP ¼ Df ,P 2 @t @x

@cF Aa @cP ¼ Df ,P @t VF @x x¼lP

(11:17) (11:18)

cP is the concentration of the component in the package having thickness lP cF the concentration in the food having a volume VF Aa is the contact area between the packaging material and the food The boundary conditions at t ¼ 0 are that the initial concentration of the component in the packaging material cP0 has a ﬁnite value while the concentration at time zero in the food is zero, i.e., cF0 ¼ 0; at x ¼ 0, @[email protected] ¼ 0 and at x ¼ lP, cP ¼ PP=F 3 cF with PP=F the partitioning coefﬁcient of the component between the packaging material and the food. The concentration in the food cF is calculated from this last relation. Figure 11.4 shows how the concentration in the packaging material changes as a function of x at various times, while Figure 11.5 shows how the concentration in the food increases with time; for convenience concentration and space were made dimensionless.* * The calculations were done with the software program AthenaVisualStudio v.11, a program capable of handling partial and ordinary differential equations via numerical integration. Web site: www.athenavisual.com

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Kinetics of Physical Changes

11-7 t =0

1.0

t = 10

t = 25

cP

0.6 Food

Package

0.8

0.4 t = 50 0.2

0.0 0.0

0 1 x

t = 75 0.2

0.4

0.6

0.8

1.0

x

FIGURE 11.4 Dimensionless concentration change cP of a component in a packaging material as a function of dimensionless distance x and time t (h). Parameter values are lP ¼ 104 m, PP=F ¼ 1.4, VF ¼ 104 m3, Aa ¼ 0.02 m2, cP0 ¼ 2.3 g dm3, Df,P ¼ 3 3 1014 m2 s1.

With such simulations one could explore how migration can be inﬂuenced by changing parameters as the diffusion constant and the partition coefﬁcient; these parameters can be manipulated by changing the properties of the packaging material. Packaging materials that are permeable to gases are usually polymeric ﬁlms. Diffusion of gases through such materials starts with a dissolution process of the gas molecules in the surface of the ﬁlm, followed by diffusion through the polymer and ﬁnally desorption of the molecule at the surface. In this respect, one uses frequently the permeability coefﬁcient Pc, which is the product of the diffusion coefﬁcient Df and the solubility coefﬁcient S and is a measure of the steady-state transfer rate of a permeant: Pc ¼ Df S

(11:19)

0.020

cF

0.015

0.010

0.005

0.000 0

20

40

60

80

Time

FIGURE 11.5 Dimensionless change in concentration cF of a component in a food migrating from a packaging material as a function of time (h). Same parameter values as in Figure 11.4.

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11-8

The solubility coefﬁcient S refers to the amount of the permeant molecules dissolved in the material at equilibrium conditions.

11.2.2

Maxwell–Stefan Approach

The Maxwell–Stefan (MS) equation is a more general alternative for the Fick equations; in fact, the Fick equation is a limiting case of the MS equation. MS equation is especially useful for multicomponent mass transfer and should be used in such cases because the Fick equation is not well suited for multicomponent diffusion. Here we consider only one-dimensional diffusion; the reader is referred to literature for the case of three dimensions. Why is the MS equation more general than the Fick equation? Fick’s law explains the ﬂux of component i with respect to the mixture; a species should move down to its concentration gradient (Equation 11.2). However, Fick’s law is not able to explain several common phenomena, for instance that in some situations components move against their concentration gradient, or the diffusion of ions, while this can be explained logically by the MS approach. The basic idea of the MS approach is that mass transfer is driven by the potential gradient of a species and limited by friction with the surroundings (which can include a solid matrix); the balance of these two determines the speed with which a species will move. The MS equation allows also other driving forces than just concentration gradients, such as forces due to pressure, centrifugation, and electrical forces. The approach is to take into account the friction between each pair of components (including any solid matrix), and to consider all relevant driving forces as well as to account for viscous ﬂow in heterogeneous media. The basis for this approach is that motion in a mixture is governed by forces working on individual species. In this respect there are driving forces coming from a potential gradient and friction forces arising due to velocity differences. As discussed in Chapter 3, the concept of forces and displacement is also used in thermodynamics and there is actually a link with irreversible thermodynamics. In fact, Fick’s law was already mentioned there as an example in Equation 3.152 and also the MS equation is ﬁrmly rooted in irreversible thermodynamics. In terms of irreversible thermodynamics, the rate of entropy production is a sum of the products of two quantities: the force acting on component i and the relative velocity of the movement of i with respect to the mixture. This is the dissipation due to diffusion. At equilibrium the driving forces have disappeared (are dissipated), and in the case of diffusion this implies that there are no net movements anymore. There are internal and external forces. Internal forces cause motion inside a mixture, external forces cause motion of the mixture as a whole (e.g., gravitational force). In general, the driving force Fi is the negative gradient of the total potential ci: Fi ¼

dci dz

(11:20)

Examples are gravitational, electrical, pressure, and chemical driving forces. The driving force for mass transfer due to molecular diffusion is the negative gradient of the chemical potential of component i over distance z: F¼

dmi d ln ai RT dai ¼ RT ¼ dz dz ai dz

(11:21)

which results effectively in motion of component i relative to the mixture in which it is in. Then there is friction if one component i moves with respect to another component j; for component i the friction is proportional to the concentration of component j and the velocity difference n1n2: zi,j Xj (ni nj )

(11:22)

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Kinetics of Physical Changes

11-9

zi,j is the friction coefﬁcient between components i and j (in N s mol1 m1), Xj is the mole fraction of component j. Friction coefﬁcients in liquids are of the order of 1012 to 1013 N mol1 (ms1)1; these large values arise because of the huge numbers of molecules involved. On the other hand, driving forces are also large as shown in the following example. It concerns the transfer of CO2 in a carbonated liquid such as a soda drink or beer. Use is made of the ﬁlm model introduced above, so we consider mass transfer in a ﬁlm between the bulk liquid and the bubble of thickness dz ¼ 105 m, and let us assume that the mole fraction of CO2 in the drink is 0.003, and that in the bubble 0.001. Assuming ideal behavior, the driving force for mass transfer is according to Equation 11.20 Fi ¼ 2.4 3 108 N mol1 CO2. The concept of driving force and friction force is the basis of the MS equation (after the nineteenth century scientists Maxwell and Stefan) as it expresses the balance between a driving force and the friction due to motion; for a component i surrounded by other components j it reads X zi,j Xj ni nj Fi ¼ (11:23) j6¼i

In words, the MS equation reads that the driving force on a component i in a mixture equals the sum of the friction forces between i and the other components j, while the friction exerted by j on i is proportional to the fraction of j in the mixture and the difference in velocity ni and nj between i and j, respectively. In other words, mass transfer is driven by the potential gradient of a species and limited by friction with the surroundings (which can include a solid matrix); the balance of the two determines the speed with which all this occurs. The velocity referred to here is the diffusive velocity, not the thermal velocity (which is much larger and random). The velocities in the MS equation are differences, they only say something about relative motion inside the mixture, not about the mixture as a whole. To obtain absolute velocities of components, other relations are needed, such as mass balances, stagnant matrices, reaction stoichiometry, equimolar exchange, and the like. Instead of friction coefﬁcients, one commonly uses MS diffusivities indicated by the symbol D to differentiate it from the Fickian diffusion coefﬁcient Df. In the MS equation, the diffusion coefﬁcient, as well as the friction coefﬁcient, are mutual coefﬁcients. In literature one often ﬁnds so-called selfdiffusivities: these are diffusivities of a species with the same physical properties as the solvent. The relation of the MS diffusion coefﬁcient with the friction coefﬁcient is RT Di,j ¼ § 1,2

(11:24)

For liquids, the order of magnitude for D is 109 m2 s1, for gases 105 m2 s1. Sometimes, mass transfer coefﬁcients are used rather than the diffusion coefﬁcient. The relation is D1,2 km 1,2 ¼ (in m s1 ) dz

(11:25)

The order of magnitude for kmi,j in gases is 101 to 102, and in liquids 104 to 105; in packaging materials kmi,j is several orders of magnitude lower, as it should be if the packaging material is to protect the food from, for instance, gas and vapor entering the food from the atmosphere. It is sometimes easier to work with ﬂuxes rather than with velocities, as we saw already for Fickian diffusion. The ﬂux Ji is the product of the velocity of a component and its concentration: Ji ¼ ni ci ¼ ni cXi (in mol s1 m2 )

(11:26)

The MS equation in ﬂux form is Fi cXi ¼

X i

z1,2 Xj Ji Xi Jj

(11:27)

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11-10

The MS diffusion coefﬁcient Df and the Fickian diffusion coefﬁcient D are in general not equal, but they are, obviously, related. An important difference between the MS and Fick equation is the effect of nonideality (in the thermodynamic sense). Deviations from ideality end up in the Fickian diffusion constant Df, whereas in the MS approach they are part of the driving force, since the driving force is based upon the chemical potential: Fi ¼ RT

d ln ai d ln fi Xi ¼ RT dz dz

(11:28)

Comparing this equation with Equation 11.2 results in Df 1,2 d ln f1 ¼ 1 þ X1 D dX1 1,2

(11:29)

It shows how much the Fickian diffusion coefﬁcient can deviate from its MS counterpart in terms of the activity coefﬁcient. Figure 11.6 shows an example for an ethanol–water mixture. It is seen that the MS diffusion coefﬁcient is hardly affected by the composition of the mixture, while the Fick diffusion coefﬁcient is strongly depending on it. This is, as mentioned, because the effect of nonideality is enclosed in the Fickian diffusion coefﬁcient. There are several thermodynamic models, expressing the behavior of activity coefﬁcients as a function of composition. These can be linked to Equation 11.28, if so desired, to get a quantitative expression of the difference in the two diffusion coefﬁcients.

2 Ð1,2

Ð1,2, Df1,2/10−9 m2 s−1

1.6

1.2

Df1,2

0.8

0.4

0

0

0.2

0.4

0.6

0.8

1

X1

FIGURE 11.6 MS diffusion coefﬁcient D1,2 (.) and Fick diffusion coefﬁcient Df1,2 (*) as a function of the mole fraction of ethanol X1 in a binary ethanol–water mixture. Dataset in Appendix 11.1, Table A.11.1.

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Kinetics of Physical Changes TABLE 11.1

11-11

Some Potentials and Driving Forces to be Used in the MS Equation

Type

Potential mi ¼ Constant þ ln (fi Xi ) RT ci Vi P ¼ RT RT

Driving Force d ln fi Xi Fi ¼ RT dz dP Fi ¼ Vi dz

Electrical

ci Ff zi ¼ c RT RT e

Fi ¼ Ff zi

Centrifugal

ci Mi v2 z 2 ¼ RT 2RT

Fi ¼ Mi v2 z

Gravity

ci Mi gz ¼ RT RT

Fi ¼ Mi g

Chemical potential for nonideal mixtures Pressure

dce dz

Note: Ff, Faraday constant (C mol1); g, acceleration due to gravity (m s2); v, angular frequency (s1).

As mentioned, the MS equation allows for other driving forces than chemical potential gradients. Table 11.1 gives an overview of some relevant potentials and driving forces. Such an analysis allows to calculate, for instance, that gravity becomes more important for movement of particles than chemical potential differences if the particle size is larger than about 0.3 mm. Ion mass transfer. Mass transfer of ions can show some peculiar effects. It was already discussed in Chapter 6 that ionic species show strong nonideal behavior and several theories were presented to account for this nonideality. Obviously, the most important difference with mass transfer of neutral solutes is the presence of electrical driving forces. The MS equation in terms of ﬂuxes for such a case is X d ln ai dc RT þ Ff e ¼ Xj Ji Xi Jj cXi RT D dz dz j6¼i i,j

(11:30)

In the case of very dilute solutions, friction between ions and the solvent dominates and if the velocity of the solvent is zero, we obtain the Nernst–Planck equation: Ji ¼ c Di,w

dXi Ff zi Xi dce þ dz RT dz

(11:31)

In considering the ﬁlm model for ion mass transport, it should be realized that such a ﬁlm is much thicker than the electrical double layer (say 104 nm versus a few nanometers), and consequently we can P consider the condition of electroneutrality in the ﬁlm model: i zi Xi ¼ 0 and the no-current condition, implying that nþ ¼ n. When solving the transport equations it appears that there is a small but relevant electrical potential difference across the ﬁlm, this is the so-called diffusion potential: RT Dc ¼ 2Ff

1 k,w

1 kþ,w

n

(11:32)

with Ff the Faraday constant and k,w and kþ,w the mass transfer coefﬁcient. This has the result that anion and cation move with the same velocity, in other words as a single substance: 1 1 1 dX n ¼ 2 kmþ,w km,w X

(11:33)

However, for mixed electrolytes this is no longer true; it may happen that the electrical gradient caused by an electrolyte forces another ion to move against its concentration gradient.

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11-12

7

6

Ð/10−10 m2 s−1

5

4

3

2

1

0 0

50

100

150

KCl (mol m−3)

FIGURE 11.7 MS diffusion constant D of lysozyme as a function of KCl concentration. Dataset in Appendix 11.1, Table A.11.2.

When studying diffusion of proteins, electrical effects are important because proteins are polyelectrolytes. Figure 11.7 shows the diffusion coefﬁcient of the protein (enzyme) lysozyme when it has chloride ions as counterions at 0 M KCl. When KCl is added it is seen that the diffusion coefﬁcient decreases. The explanation is that in the absence of KCl the protein and its counterions must move together in order to maintain electroneutrality and the resulting diffusivity is the average of the protein and the chloride ions. When KCl is added, the role of the protein is taken over by the more mobile potassium ions and the protein need not move anymore along with the chloride ions. Figure 11.8 shows diffusivities of NaCl, in terms of DNaþ,w, DCl,w, and DNa,Cl. It is seen that the mutual diffusivities of the separate ions and water do not depend much on the concentration of NaCl, but the mutual diffusivity of Naþ and Cl do, because there is a lot of friction between the two ions. Another noteworthy fact is that the diffusion coefﬁcients of Hþ and OH are much higher (in the order of 108 m2 s1) than those of other ions (in the order of 109 m2 s1). Also, diffusivities decrease strongly with increasing charge numbers. All in all, diffusion of charged species can be quite peculiar and we cannot do justice to all the intricacies in this book. We just mentioned a few points to draw attention to it. The reader is advised that there is substantial literature on these phenomena and some references are given at the end of this chapter. However, it is reassuring that the phenomena can be tackled by the MS equation, in principle at least. An example about salt diffusion in cheese may be illustrative to conclude this section on diffusion. Gouda type cheese is salted by putting the cheese into a brine solution. As a result, salt diffuses into the cheese while water moves out of the cheese. Attempts have been done to model this via Fickian diffusion, which was to some extent possible, but modeling via the MS approach appeared to be more powerful. For the MS approach to model salt and water diffusion, cheese can be considered as a three component system characterized by three MS diffusion constants:

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Kinetics of Physical Changes

11-13

−8

Ðw,Cl−

−9 log Đi,j (m2 s−1)

Ðw,Na+

−10

−11 −2

Ðw,NaCl

−1.5

−1

−0.5

0

0.5

1

log NaCl (mol dm−3)

FIGURE 11.8 Mutual MS diffusion coefﬁcients of water–Cl, water–Naþ, and water–NaCl in NaCl solutions. Dataset in Appendix 11.1, Table A.11.3.

D1,2: for the components NaCl (species 1)–water (species 2) D1,3: for the components NaCl–cheese matrix (species 3) D2,3: for the components water–cheese matrix The cheese matrix is assumed to consist of protein-fat-amorphous calcium phosphate. For obvious reasons, the diffusional ﬂux of the cheese matrix is assumed to be negligible, which leaves two ﬂux equations for one dimension: "

J1 J2

#

31 2 X2 X3 X1 @ ln f1 þ 6D1,2 D1,3 6 1 þ X1 @X D1,2 7 1 7 6 6 ¼ ct 6 7 6 4 4 X2 X1 X3 5 @ ln f2 þ @X1 D1,2 D1,2 D2,3 2

32

3 dX1 76 76 dz 7 7 7 @ ln f2 54 dX2 5 1 þ X2 @X2 dz @ ln f1 @x2

(11:34)

where ct is the total molar concentration (mol m3) X is the mole fraction J represents the molar diffusional ﬂux of species 1 (NaCl) and 2 (water) with respect to the cheese matrix f is the rational activity coefﬁcient The ﬁrst matrix on the right hand side accounts for the friction between the species and the second matrix for nonideal behavior of the species concerned. We do not discuss here the intricacies involved in estimating activity coefﬁcients and diffusion coefﬁcients in a complex matrix as cheese is, but only show the application of the MS modeling approach in Figure 11.9. The interested reader is referred to the original literature source for details. As shown, the MS approach is well capable to model the diffusion of NaCl and water for cheese immersed in brine, in fact better than was obtained with the Fick diffusion equations.

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Kinetic Modeling of Reactions in Foods 25

45

20

40 Water in cheese (%)

Salt in cheese moisture (%)

11-14

15

10

0

35

30

25

5

(A)

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20

0

0.01 0.02 0.03 0.04 0.05 x (m)

0.06 0.07

0

0.01 0.02 0.03 0.04 0.05 0.06 0.07

(B)

x (m)

FIGURE 11.9 Application of MS diffusion equation (solid line) to predict the salt and water content after 8.1 days due to diffusion of NaCl (A) and water (B) in cheese in contact with brine (20% salt solution); x is the distance from cheese–brine interface. Dataset in Appendix 11.1, Table A.11.4.

11.3

Kinetics of Changes in Dispersity

Many foods are in fact disperse systems. Milk is an oil-in-water emulsion, butter and margarine are an water-in-oil emulsion and also may contain fat crystals, the major milk protein in milk is in the form of aggregates (casein micelles); fruit juices contain particles; ice cream is an intricate dispersion of icecrystals, air bubbles, fat crystals, and casein micelles; bread is a sponge; and the list goes on and on. As mentioned before, such dispersions can be unstable and this will, in that case, lead frequently to quality loss. We will discuss in this section some kinetic equations describing such changes.

11.3.1

Kinetics of Aggregation of Colloids

Aggregation of colloidal particles has a large effect on properties of the system. It can lead to creaming or sedimentation, and to changes in rheological properties. Whether or not particles will aggregate depends on colloidal interaction forces, i.e., electrostatic repulsion and attraction, steric repulsion and van der Waals attraction, depletion interaction, etc. The simplest (but already quite complicated) situation is that of unhindered ﬂocculation of particles that meet each other due to diffusion and immediately stick together. Smoluchowski developed this theory in the early 1900s based on diffusion kinetics. In fact, the outcome was already given in Chapter 4 where it was used to calculate how fast diffusion-limited reactions would be (Equation 4.161). In the case of colloidal particles, this phenomenon is described as fast perikinetic aggregation. The resulting equation for equal-sized particles in a dilute system is

dN 4kB T 2 N ¼ kp N 2 ¼ 4pDf dp N 2 ¼ dt 3hv

(11:35)

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where N represents the number of particles t represents time Df is the Fickian diffusion coefﬁcient dp is the particle diameter hv is the viscosity of the suspending medium kp is the second-order perikinetic rate constant Integration of this equation leads to the following simple expression for the number of particles as a function of time: Nt ¼

N0 1 þ kp N0 t

(11:36)

This is, not surprisingly, the same expression as the second-order reaction discussed in Equation 4.53. However, Equation 11.36 expresses a behavior different from a chemical reaction, and it can be used not only to see how the number of primary particles decreases but also the course of dimers, trimers, and so on. A useful transformation in this respect is the introduction of the halving time t0.5: t0:5 ¼

phv dp3 1 1 ¼ ¼ kp N0 4pDf N0 d 8kB Tw0

(11:37)

where w0 is the volume fraction of primary particles, so that Nt 1 ¼ N0 1 þ t t0:5

(11:38)

and Ni (t=t0:5 )i1 ¼ N0 (1 þ t t0:5 )iþ1

(11:39)

This relation is depicted in Figure 11.10. Some interesting features follow from this. Table 11.2 shows some calculations for halving times in water at room temperature, demonstrating the large effect of particle size and volume fraction when encounters are dominated by Brownian motion: smaller particles are much more sensitive to aggregation than larger ones, and obviously, increasing the volume fraction leads to shorter halving times. These numbers are just to show the trends; for real food systems it may be much different, as will be shown below. Besides perikinetic aggregation, there is orthokinetic aggregation. This implies that encounters are brought about by velocity gradients. In the most simple case of simple shear with velocity gradient Gs (s1), the second-order equation becomes

dN 2 3 4 ¼ dp Gs N 2 ¼ w0 Gs N ¼ ko N 2 dt 3 p

(11:40)

where ko represents the orthokinetic second-order rate constant. It is instructive to compare the perikinetic and orthokinetic rate constants:

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1 0.9 0.8 0.7 N1

N/N0

0.6

Ni

0.5 0.4 0.3 N2

0.2

N3

0.1

N4

0 0

0.5

1

1.5

2 t/t0.5

2.5

3

3.5

4

FIGURE 11.10 The course of particle aggregation as a result of fast perikinetic aggregation according to Equation 11.39. N1 is the number of primary particles, N2 that of dimers, N3 that of trimers, N4 that of tetramers, Ni that of the total number of particles.

4kB T 3hv 2 3 ko ¼ dp Gs 3 3 ko dp hn Gs ¼ kp 2kB T kp ¼

(11:41)

There is a strong effect of particle size in the case of orthokinetic aggregation. If we have a particle size dp ¼ 3 mm and a very low shear rate, say Gs ¼ 1 s1, it can be calculated that orthokinetic aggregation is already more important than perikinetic aggregation: the ratio ko=kp is then by a factor 3.3 times larger for an aqueous system at 208C. The opposite is true for smaller particles: then perikinetic aggregation is much more important. This means that even the smallest amount of ﬂow can easily lead to aggregation if particle sizes are in the micron range, and this is very relevant in foods. Flow rates of Gs ¼ 0.1 s1 can easily arise, if only because of temperature gradients. As mentioned, the Smoluchowski equation does not take colloidal interaction forces between particles into account, nor hydrodynamic interactions. As it happens, the van der Waals attraction between particles may roughly compensate the hydrodynamic resistance, so that in the absence of repulsive forces,

TABLE 11.2 Halving Times t0.5 (s) as a Function of Volume Fraction w0 and Particle Diameter dP in Water at Room Temperature w0 ¼ 0.001 w0 ¼ 0.01 w0 ¼ 0.1

dp ¼ 0.01 mm

dp ¼ 0.1 mm

dp ¼ 1 mm

dp ¼ 10 mm

1 3 104

1 3 101

1 3 102

105

5

2

1 3 10

104

3

1 3 10

103

1 3 10

6

1 3 10

1 3 10 1 3 10

1 0

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the Smoluchoswki equation often holds reasonably well. However, when repulsion exists between particles, this will slow down aggregation. In such a situation, the aggregation rate is conveniently called slow perikinetic and orthokinetic aggregation with rate constants kSp and kSo , respectively. A so-called stability factor W is introduced: ko W kp kSp ¼ W kSo ¼

(11:42)

The question is of course if we can ﬁnd an expression for this stability factor W. This appears not to be easy, though approximate expressions may be derived for simple and well-deﬁned cases. We cannot do justice to it in this book but the following very brief introduction to colloidal stability is appropriate in order to understand some basic concepts of the stability factor W. We do however urge the reader to consult other books more dedicated to this topic, as listed at the end of this chapter. As mentioned, under very strict conditions, W can be related to colloidal interaction as given by the DLVO theory, the ﬁrst colloidal interaction theory developed by Deryagin, Landau, Verweij, and Overbeek in the 1940s, based on electrostatic repulsion and van der Waals attraction. Colloidal particles often carry a charge; in foods this will be mainly due to proteins adsorbed to such particles. Proteins have acid and basic groups that can ionize, depending on pH. The ionized groups are surrounded by ions of opposite charge, the so-called electrical double layer. If two identically charged particles approach each other they will experience repulsion when the electrical double layers start to overlap. The equation for the repulsive interaction energy (in Joules) between two globules with radius Rp as function of the interparticle distance h is VR ¼ 2p«r «0 Rp c20 ln½1 þ exp (kh) 4:5 109 Rp c20 ln½1 þ exp (kh)

(11:43)

where «r is the dielectric constant of the medium «0 is the permittivity of free space c0 is the electric surface potential 1=k is the thickness of the electrical double layer We encountered these parameters already in Chapter 6. The approximation in Equation 11.43 is for conditions in water at room temperature. The thickness 1=k is related to the ionic strength of the solution I, which was already given in Equation 6.19, and can roughly be approximated as 1 0:3 pﬃﬃ k I

(11:44)

with 1=k in nm and I in mol dm3. With respect to attraction between colloidal particles, van der Waals forces come into play. The equation for the attraction energy VA (in Joules) between two spherical particles with radii Rp is VA ¼

AH R p 12h

(11:45)

where AH represents the Hamaker constant, which is material speciﬁc and also depends on the dispersion medium. For instance, for oil emulsion droplets in water AH (4–6) 3 1021 J. According to the DLVO theory, the total interaction energy VT between two particles is the sum of VR and VA. Depending on the magnitude of the Hamaker constant and the surface charge and thickness of electrical double layer,

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300

200 VR

V/kB T

100

0

−100

VT

0

4

8

12

16

20

VA

−200 −300 h (nm)

FIGURE 11.11 Example of a DLVO plot for interaction between two spheres as a function of interparticle distance h with RP ¼ 2 mm, c0 ¼ 16 mV, I ¼ 0.01 M, AH ¼ 1020 J. VT ¼ VA þ VR . The interaction energies VA, VR, VT, are divided by kBT.

particles can attract or repel each other. The distance h over which this occurs is typically of the order of 1–20 nm. Figure 11.11 shows a theoretical example. Though the calculation is exact, the ﬁgure is only meant to indicate some general trends. It shows that particles attract each other at very small distance (the so-called primary minimum where VT < 0), repel each other at short distance (the primary maximum where VT > 0), but may be attracted to each other at slightly larger distance (the secondary minimum where VT < 0). It should be mentioned that the particles cannot easily reach the primary minimum because there may be quite a high energy barrier (in Figure 11.3 about 40 kBT), in other words the particles may be quite stable. The usefulness of the DLVO theory for food colloids is that it can explain trends that are observed experimentally, even though there are many complications in foods. The effect of pH on colloidal interaction is explained by the effect of pH on ionizable groups on the surface, hence inﬂuencing the surface charge. The effect of ionic strength is explained via its effect on the thickness of the electrical double layer; hence adding salt to a colloidal system decreases repulsion and divalent ions more so than monovalent ions. As mentioned, this much-simpliﬁed discussion does no justice to the DLVO theory, but we needed this to be able to explain at least qualitatively effects that are important for kinetics of changes in dispersity. In addition to electrostatic repulsion, there may be so called steric repulsion. This is due to macromolecules adsorbed onto particles; part of the molecules is adsorbed onto the particles, but other parts may stick out in the solution (depending on solvent quality) and when such extending layers interact a repulsive energy is felt by the particles. Again, the theory is much more complicated than this, but it serves to indicate trends. The result is that particles with adsorbed polymers are stabilized in addition to electrostatic repulsion if the polymers are charged. The distance over which steric repulsion is experienced is much larger than electrostatic repulsion. In general, therefore, particles stabilized by polymers are fairly stable, and there is no secondary minimum. Another important type of interaction relevant for foods is depletion interaction. This phenomenon is due to nonadsorbing, dissolved polymers in a system containing colloidal particles (e.g., emulsion droplets). The nonadsorbing polymer must keep a certain distance away from the colloidal particles

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(volume exclusion). Consequently, the polymer concentration is higher in the bulk than in the vicinity of the particles (where a certain volume is depleted of polymer) and this gives rise to a higher osmotic pressure of the bulk solution. If then two colloidal particles come close together (i.e., start to aggregate) part of the depleted layers overlap and the concentration of the polymer in the bulk decreases and so does the osmotic pressure. The system will strive for the lowest possible osmotic pressure, and so aggregation becomes a driving force for this lowering of osmotic pressure. This is conveniently called depletion aggregation. The interaction free energy involved in this phenomenon is roughly proportional to the concentration of the polymer, while it also depends on the interaction between the polymer and the solvent (solvent quality for the polymer). Now, we return to the phenomenon of slow coagulation and the stability factor W. As mentioned, this factor is in principle just a ﬁt factor, it lacks physical meaning in the sense that it cannot be interpreted directly in terms of energy barriers, nor can it be calculated from ﬁrst principles. However, qualitatively it is clear that W must be related somehow to the balance of repulsive and attraction interactions discussed above. Its value can be roughly estimated from experimental observations. Using Equation 11.37, the halving time t0.5 ¼ 1 s for a particle with dp ¼ 1 mm and w ¼ 0.1 (Table 11.2), which means that coagulation is very fast. If we put W ¼ 107 for the same conditions then t0.5 ¼ 4 months. Based on actual observed coagulation times, one can make rough estimates of W. For aggregation of casein micelles during renneting, for instance, the following observation was made. In their native form and in conditions as in fresh milk, casein micelles are stable against ﬂocculation because of steric and electrostatic repulsion. Due to the action of rennet (hydrolysis of k-casein) steric stabilization is lost, and the stability factor reduces considerably such that aggregation approaches fast perikinetic coagulation according to Smoluchowski. Addition of CaCl2 is another way of increasing coagulation; the action of Ca2þ ions decreases the stability factor W because of their effect on ionic strength and the thickness of the electrical double layer. Hence, the effects can at least qualitatively be explained. Based on experimentally determined ﬂocculation times, a rough estimate of the stability factor W is 106 for intact casein micelles, and 103 for renneted micelles. Overall, the above equations are, of course, not directly applicable to foods, where particles do not have equal size, and do interact, while many other kinds of complications arise. It should be realized, therefore, that such equations can only predict trends and, perhaps, an order of magnitude estimation. However, the equations show that slow coagulation has to do with interaction between particles and this is the knowledge that the food technologist could exploit, at least qualitatively.

11.3.2

Kinetics of Creaming or Settling

Creaming and settling can lead to serious quality defects. Examples are the formation of a cream layer on top of milk, or the sedimentation of cocoa particles in chocolate milk, or of fruit particles in a fruit juice. Such changes are usually reversible as they are only a result of the forces of gravity, and they do not affect food safety at all, but they make the food unattractive for the consumer. Therefore, a producer needs to know how he can slow down such processes; in other words, he needs to know the kinetics of such changes. Starting very simple, consider a single globular particle of diameter dp and density rp in a ﬂuid of density rc and viscosity hv. The gravity force Fg (with acceleration due to gravity gg ¼ 9.81 ms2) acting upon such a particle is according to Archimedes: 1 Fg ¼ pdp3 gg (rp rc ) 6

(11:46)

When such a particle moves in a ﬂuid with velocity np, a friction force acts upon the particle: Fs ¼ 3phv dp np

(11:47)

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4.5

0.35

4

0.3

3.5 3

0.25

vS (µm s−1)

vS (µm s−1)

0.4

0.2 0.15

2.5 2 1.5

0.1

1

0.05

0.5

0

0 0

(A)

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0.002

0.004

0.006

0.008

0

0.01

hv (N s m−2)

(B)

2

4

6

8

10

dp (µm)

FIGURE 11.12 Effect of ﬂuid viscosity on the creaming rate of a particle nS with dp ¼ 3 mm and Dr ¼ 60 kgm3 (A), and the effect of particle size on the creaming rate with hv ¼ 1 mN s m2 and Dr ¼ 70 kg m3 (B).

When Fg equals Fs, it follows that nS ¼

gg (rp rc )dp2 18hv

(11:48)

This is the Stokes equation. Depending on the density difference between particle and suspending liquid, the particle will either cream or settle. If we consider a single fat globule in milk with a diameter dp ¼ 3 mm the creaming rate can be calculated as 0.3 mm s1, which corresponds to about 25 mm day1. So, in a few days a fat globule would be able to reach the top of a bottle of milk. Figure 11.12 shows the effects of particle diameter and ﬂuid viscosity graphically. However, the Stokes equation is only valid for homogeneous spheres on which no other forces act than the ones shown, and the liquid should be Newtonian. Usually, these conditions are not fulﬁlled in foods, and consequently the Stokes equation will never predict the exact creaming or sedimentation rate in foods. However, it is very useful in predicting trends. For instance, the effect of particle size is quite strong because of the factor dp2 in the Stokes equation, as shown in Figure 11.12B. This knowledge is used in homogenizing milk: by reducing the particle size by about a factor of 10, creaming in milk is effectively retarded. Furthermore, if one can adjust the density difference, the rate can be inﬂuenced in a desired direction; this will however not be easy in foods. Finally, one can do something about hv and this is one of the reasons to use thickening agents: by increasing the viscosity one can effectively reduce creaming or sedimentation, as indicated in Figure 11.12A. Stokes’ equation also shows how the creaming or sedimentation rate can be enhanced by increasing acceleration in a centrifuge. Some corrections can be applied to the Stokes equation to relax the severe restrictions. One can replace the requirement for a single monodisperse droplet by replacing dp2 by a parameter derived from the particle size distribution (provided it has a reasonably narrow distribution): P ni d5 hd 2 i ¼ Pi i3 i ni di

(11:49)

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0.4

vc (µm/s)

0.3

0.2

0.1

0

0

0.1

0.2

0.3 φ

0.4

0.5

0.6

FIGURE 11.13 Effect of volume fraction w on the creaming velocity nc of a particle with d ¼ 3 mm, hv ¼ 1 mN s m2, and Dr ¼ 70 kg m3 according to Equation 11.51.

Another correction to nS can be applied for systems having moderate volume fractions (up to volume fraction f ¼ 0.05): nc ¼ nS (1 w)n

(11:50)

where the exponent n ranges from 8–9, depending on the polydispersity of the suspension. For systems with higher volume fractions, only empirical relations exist, such as nc ¼

nS (1 w) 5w 1 3 1 þ w exp 3ð1 wÞ

(11:51)

This relation is shown in Figure 11.13 and shows the large impact of volume fraction on the creaming rate. Finally, it should be mentioned that sedimentation or creaming is strongly enhanced by aggregation, which is qualitatively explained by considering that the size of the aggregates will be much larger than the size of the original particles and, in view of the large effect of particle size in Equation 11.48, this enhances the creaming rate enormously.

11.3.3

Kinetics of Coalescence

Coalescence is the phenomenon that oil droplets in an emulsion merge into bigger globules. If this process happens unhindered an emulsion would be destabilized in seconds, i.e., the result would be an oil layer on top of an aqueous phase. This is the thermodynamically stable state of the system. This is clearly unwanted and it is therefore important to know something about the speed of coalescence. Unfortunately, there is not a clear-cut theory yet to describe this. Qualitatively, however, the process is well understood. The ﬁrst thing to happen is that globules must encounter each other (which they do all the time due to Brownian motion) and stay together for a while (which they do not unless there is some

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attraction between them or when they are pushed together in a cream layer or in a sediment). When they stay together for some time, then there is a probability that the thin ﬁlm between the droplets ruptures if the repulsive energy between the droplets can be overcome. There is theory developed for this situation that we will not discuss here. For the kinetics part it sufﬁces to know that there are two stages in coalescence: ﬂocculation followed by actual coalescence. Empirically it is sometimes found that this process can be described roughly by a simple ﬁrst-order model, with N the number of emulsion droplets per unit volume: dN ¼ kN dt

(11:52a)

Nt ¼ N0 exp (kt)

(11:52b)

Or, equivalently

Experimentally, this has to be linked to changes in particle size distributions, which is not a trivial task. Great care should be taken to differentiate between ﬂocculation and coalescence. Flocculated particles can also be seen as coalesced particles by the measuring equipment. So, Equation 11.52 has to be linked to some suitable parameter characterizing particle size or particle number. Figure 11.14 gives an example of applying the logarithmic form of Equation 11.52b on the change in number of emulsion droplets as a function of time. The slope of the line represents the rate constant. It should be realized that this is purely empirical, there is no theory that predicts that it should be ﬁrst order. Figure 11.14 also shows that the relation is certainly not perfect. The stability factor W, if that can be used at all, is estimated to be in the order of magnitude of 106, but it is strongly dependent on the size of the emulsion droplets: smaller droplets are characterized by a larger stability factor.

6

5

ln [N0 /Nt]

4

3

2

1

0 0

100

200 Time (h)

300

400

FIGURE 11.14 Example of a ﬁrst-order kinetic plot for the coalescence of oil-in-water emulsions with butter oil (^) or peanut oil (&). The emulsions were stabilized by 1% whey protein and coalescence was studied at 508C. Dataset in Appendix 11.1, Table A.11.5.

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Qualitatively, the following can be said about the process of coalescence. When two emulsion droplets come close together, they are still separated by a thin ﬁlm of continuous phase. If this ﬁlm is ruptured the two droplets will merge, so processes that determine this ﬁlm rupture will determine coalescence. Basically, this rupture process is a probabilistic event. The probability of rupture will be proportional to the time that the droplets are together and to the surface area of the ﬁlm. So, ﬂattening of the droplets while they are close together will lead to a greater ﬁlm area and hence to a larger probability for coalescence. It is therefore interesting to consider factors that affect this ﬂattening phenomenon. An important characteristic for this appears to be the so-called Weber number We:

We ¼

sext dp2 8gh

(11:53)

The Weber number expresses the ratio between the local stress on a droplet pair and the Laplace pressure of the drop; g represents the interfacial tension. The local stress is the external stress sext multiplied by the stress concentration factor which is in this case dp2=h, h being the smallest distance between the drop surfaces. sext can arise due to colloidal attraction, gravitational or centrifugal stress. If We > 1, a ﬂat ﬁlm is formed, and if We 1 then a ﬁlm is not really formed and coalescence can be ruled out. What factors can now be deduced in relation to coalescence stability? If droplets are smaller, a smaller We is found and the probability for coalescence is less. If the ﬁlm between droplets is thicker, which may happen if there are strong repulsive forces between the droplets, coalescence becomes less likely. Steric repulsion is especially effective in this respect. If the interfacial tension g is higher, We becomes lower, and this lowers the probability for coalescence. The reason for this is that a higher g makes deformation of a ﬁlm more difﬁcult. Incidentally, lowering g is needed to make an emulsion which is one of the functions of a surfactant. As it happens, proteins are very suitable as emulsifying and stabilizing agents. On the one hand, they lower the interfacial tension enough for the emulsion making process, but not too much so that coalescence stability is enhanced. Moreover, proteins provide usually steric as well as electrostatic repulsion (depending on pH). An interesting complication arises when oil partly crystallizes. This phenomenon usually destabilizes an emulsion enormously. The explanation is that fat crystals partly protrude out of the emulsion droplet and rupture the thin ﬁlm between approaching droplets. The two droplets will then only partly coalesce because the crystal structures inside the droplets prohibit complete coalescence. The phenomenon is therefore called partial coalescence. Especially ﬂow has a detrimental effect on emulsion stability of partly crystallized emulsion droplets. This knowledge is exploited in churning: there is an optimal amount of solid fat content to have the highest fat yield. If the solid fat content is too high, there is too little liquid fat for coalescence, if the solid fat is too low there are too few crystals to cause destabilization. In any case, the kinetics of coalescence is strongly inﬂuenced by the presence of fat crystals in the emulsion droplets.

11.3.4

Kinetics of Ostwald Ripening

Ostwald ripening is the phenomenon that larger particles (emulsion droplets, foam bubbles, fat crystals) grow at the expense of smaller ones. It has to do with the curvature of the particles: smaller particles are more curved (have a larger surface area) than large ones. The driving force is the difference in chemical potential of the dispersed material due to the difference in radius. The resulting equation in terms of solubility is (known as the Kelvin equation) S(RP ) 2gVm ¼ exp RP RT S1

(11:54)

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The parameters are Vm the molar volume of the dispersed component S(RP) the solubility of the disperse phase in particle with radius RP S1 the solubility of the dispersed phase in the continuous phase when the interface is planar (RP !1) g the interfacial tension If the solubility of the disperse phase in the continuous phase is high enough to allow mass transfer to take place (via diffusion), the process will become noticeable. The rate of Ostwald ripening is dhRP i3 8Vm S1 gDf ¼ dt 9RT

(11:55)

where < RP > is the mean particle radius. Since the parameters Vm, g, and Df do not vary very much from one system to another, the most important factor is the parameter S1. For oil-in-water emulsions consisting of triglycerides, their solubility in water is negligible, and so Ostwald ripening does not occur readily in O=W emulsions. Water is somewhat better soluble in oil, so that Ostwald ripening does occur in W=O emulsions such as margarine and butter. In aqueous foams, the solubility of gases in water is quite high, so that Ostwald ripening is one of the main causes of foam coarsening and instability (it even has a separate name: disproportionation; the other causes of foam instability are coalescence and drainage). This knowledge can be used for beer foam stability: beer bubbles consisting of nitrogen are more stable than those of carbon dioxide because carbon dioxide is better soluble in water than nitrogen. The rate of Ostwald ripening in foams is described by the de Vries equation: R2p (t) ¼ R20

RTDf S1 g t Pd

(11:56)

This equation describes the fate of a small bubble, initially of radius R0, surrounded by much larger bubbles at a distance d; P is ambient pressure (105 Pa). This equation shows that a bubble will shrink increasingly faster when it becomes smaller. Coming back to the nitrogen and carbon dioxide solubility in beer, it can be calculated from the de Vries equation that for a bubble with radius R0 ¼ 0.1 mm and d ¼ 1 mm, a nitrogen bubble disappears in 3 min and a CO2 bubble in 4 s. Of course, these are only order of magnitude estimations, because, as usual, there are complications. For instance, when the bubble shrinks, the surface load of surfactant becomes higher which may have its bearings on the interfacial tension g and the surface dilational modules, such that the shrinkage rate becomes less. Nevertheless, the example goes to show that Ostwald ripening in foams can occur quite fast.

11.3.5

Kinetics of Gelation of Particles

When colloidal particles aggregate due to Brownian motion (perikinetic aggregation), two scenarios are possible (Figure 11.15).

v1

v2

+

FIGURE 11.15 Two scenarios of aggregation. If the rate of aggregation n1 n2, coalescence takes place according to Smoluchowski (fractal dimension df ¼ 3); if n1 >> n2, fractal aggregation takes place (fractal dimension df < 3) (see text).

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If the two particles immediately merge (coalesce) after aggregation one new particle emerges. If they do not merge so fast, fractal aggregates may develop. Fractal aggregates are not closely packed but rather they have an open, disordered structure. A fractal object like an aggregate has a characteristic fractal dimensionality df that is smaller than the dimensionality of the space in which it exists, which is 3 for a three-dimensional system, 2 for a ﬂat surface. The larger fractal aggregates are, the less dense they are. With a higher df, the structure is more dense. Fractal aggregation has been the topic of much research in the past two decades, and computer simulations have given much insight. We will not discuss this further here, but merely state some results, since our interest is in the kinetic consequences. It can make a big difference in practice whether aggregation follows conventional slow coagulation or according to fractal aggregation, as we shall see. A property of fractal aggregates is that the structure is scale invariant at length scales that are much higher than the individual particle dimensions; this is called a self-similar structure. The relation between the average number of primary particles Np in a fractal aggregate, consisting of primary particles with radius Rp, and the radius of the resulting aggregate Ragg, is Np ¼

Ragg Rp

df (11:57)

This equation deﬁnes the fractal dimension df, which is smaller than 3, except for immediate coalescence. For diffusion-limited aggregation (i.e., fast perikinetic aggregation), df ¼ 1.75–1.80. There is also reactionlimited aggregation, meaning that particles do not immediately stick upon an encounter, but rather rearrange themselves, or diffuse away again. Whether or not this happens is related to the interaction energy between the particles. In such cases, df ¼ 2.05–2.11. The end result of coagulation is either a lowdensity porous sediment that settles (or creams) as a result of gravity, or a particle gel network that ﬁlls all available space. In dilute systems, the onset of coagulation may be due to aggregation of individual particles, but at a later stage, and certainly in more concentrated systems, cluster-cluster aggregation becomes dominant. Particle gels are of importance in foods. Examples are cheese and yoghurt, which are particle gels of casein micelles. If we would have a close-packed, nonfractal, aggregate with radius Ragg, then the number of particles Nmax in such an aggregate is Nmax ¼

Ragg Rp

3 (11:58)

Combining this with Equation 11.57 shows that the average volume fraction of the fractal aggregate wagg is wagg

(Ragg =Rp )df Np ¼ ¼ ¼ Nmax (Ragg =Rp )3

Ragg Rp

df 3 (11:59)

Since df < 3 in the case of fractal aggregation, the consequence is that wagg decreases with increasing Ragg. Gelation will occur when the average volume fraction of the fractal aggregate equals the overall volume fraction of particles, hence when wagg ¼ w, in which case the system is fully packed and the system has gelled. We are interested in the kinetics of aggregation, i.e., the time after which aggregation becomes visible. The aggregation rate (reciprocal time) as depicted in Equations 11.35 and 11.40 appears in that sense to be less important than the aggregation time because at the same initial aggregation rate the aggregation time for one system may be completely different from that of another system, depending on the structures of the aggregates formed. The aggregation rate may increase abruptly after aggregates have

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reached a certain critical size (which may be well below the size of visible aggregates). How to ﬁnd the time to reach such critical aggregation size? Let us consider the relation between the concentration of primary particles and the size of an aggregate: N¼

3 f w Rdf 3 Rd agg 4p 0 p

(11:60)

The change in number of particles with change in aggregate particle size is then: dN 3df df 1 w Rdf 3 Ragg ¼ 4p 0 p dRagg

(11:61)

By combining these equations with Equation 11.35 or Equation 11.37 to ﬁnd a relation between the time for aggregation and the critical size, dt=dRagg may be integrated from the radius of the primary particle to the critical size: Rðcr

tcr ¼ Rp

dt dRagg dRagg

(11:62)

For the case of perikinetic aggregation, all this results in Rðcr

tcr ¼ Rp

phv Df 1 3df df 1 R Ragg dRagg kB T f0 p

(11:63)

When it comes to the effects of aggregation in relation to food quality, the following phenomena are important for foods, namely the formation of visible particles, the formation of a gel, or the occurrence of separation (creaming or settling). Three different critical sizes are worth considering, namely the one for gelation, the one for orthokinetic aggregation, and the one for sedimentation. The smallest value should be used for an estimate of aggregation time. The critical aggregate radius Rcr at which a gel is formed is 1 d 3

Rcr ¼ Rp w0f

(11:64)

This equation shows the large dependence of the critical aggregate size on the volume fraction. It is important to realize that the volume fraction of particles does not change if particles immediately coalesce upon an encounter; also if the particles aggregate into compact aggregates, w does hardly change. On the other hand, w does increase considerably in the case of fractal aggregation. Using the equation for the halving time (Equation 11.37), Equation 11.63 can be written as tcr ¼ t0:5

Rcr Ragg

df

! 1

phv R3p kB T

3 d 3

w0f

df d 3

¼ t0:5 f0f

(11:65)

So, this equation relates the aggregation time to the halving time, which is a rate expression. This equation is very instructive in showing the large difference between tcr and t0.5. If particles would coalesce immediately upon an encounter, df ¼ 3. Particles become visible at a few tenths of a millimeter, say, Rcr 0.2 mm. Suppose we start with particles of Rp ¼ 0.5 mm at 208C, w0 ¼ 0.01, and W ¼ 1. The halving

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time according to Equation 11.37 is then 9.7 s, so that according to Equation 11.65, tcr ¼ 6 3 108 s 20 years. In the case of fractal aggregation, say df ¼ 2, the critical value for Rcr is calculated according to Equation 11.63 and a completely different value of tcr ¼ 9.77 3 104 s 1 day is found. This shows the enormous effect of fractal aggregation. But more importantly, it also shows that prediction of aggregation time from aggregation rates can be wildly misleading because t0.5 is the same in both cases. To be sure, these numbers should be taken as orders of magnitude, not as absolute values. For orthokinetic aggregation, the critical aggregate size can be approximated as Rcr ¼

kB T 4Gs hv

1 3

(11:66)

An approximate estimate for the aggregation time due to orthokinetic aggregation is then

tcr ¼

p 4wGs

4pGs R3p

!3df 3

(11:67)

kB T

In the case of immediate coalescence upon an encounter (df ¼ 3) in orthokinetic coagulation, it can be derived that p 4hGs R3cr ln tcr ¼ kB T 4wGs

(11:68)

These equations tell us that the orthokinetic aggregation time for coalescing particles is independent of the size of the primary particles, and only slightly dependent on the critical size of the aggregate. For fractal aggregates, the aggregation time appears to be independent of Rcr but dependent on the size of the primary particles. Finally, an expression for the size at which sedimentation becomes noticeable is Rc ¼

3kB T 2pr 3df Drgg

1 df þ1

(11:69)

In this last equation, Dr is the density difference between ﬂocs and continuous phase and gg is the acceleration due to gravity. The equations stated above are derived under various assumptions. The most important ones are that it is for dilute systems, monodisperse, spherical particles and unhindered aggregation. To some extent, these assumptions can be relaxed but the reader should be aware of the fact that the equations will not be correct when directly applied to foods. However, it is reassuring that the theory (that we have touched upon only very brieﬂy) is able to predict at least some important phenomena. The following approximate equations have been derived. For perikinetic aggregation an approximate expression is tg

1

phR3p df 33 6df 3df þ w 2df þ 3 df þ 6 kB T 0

(11:70)

and an approximate expression for the gel time due to orthokinetic aggregation is tg

1 pdf w 1 4Gs (3 df ) 0

(11:71)

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TABLE 11.3 Calculated Aggregation Times as a Function of Volume Fraction w0, Fractal Dimensionality df, and Velocity Gradient Gs Gs (s1)

w0

df

tc or tg (s)

Equation Used

0.0001

3

0

6 3 1010

Equation 11.63

0.0001

1.8

0

6 3 104

Equation 11.65

0.0001

3

0.1

1 3 105

Equation 11.67

0.0001

1.8

0.1

3 3 102

Equation 11.67

0.01

1.8

0

6 3 101

Equation 11.65

0.01

1.8

0.1

6 3 101

Equation 11.65

Note: Rp ¼ 20 nm, hv ¼ 1 mPa s. For df ¼ 3, visible aggregation at Ragg > 0.2 mm, for df ¼ 1.8, gelation occurs.

As a summary, Table 11.3 shows the calculated aggregation times for various situations to show the large differences that can arise.

11.3.6

Kinetics of Crystallization

Crystallization of all kinds of materials can happen in foods. Examples are the formation of lactose crystals in ice cream, causing an unpleasant mouthfeel, crystallization and recrystallization of fat crystals in butter or margarine, crystallization of sugars in powders, retrogradation of starch (which is in fact crystallization of amylose), leading to such phenomena as staling of bread. The rate at which these changes occur is obviously of importance. However, the process of crystallization is a complex one. For crystallization from a homogeneous solution, the process can be distinguished generally in nucleation followed by crystal growth. The driving force for nucleation and subsequent crystallization is the difference in chemical potential between nucleus, or crystal, and solution. This can be expressed in supersaturation. Nucleation can be homogeneous and heterogeneous. Homogeneous nucleation occurs in the absence of a surface, while heterogeneous nucleation takes place at foreign surfaces (called catalytic impurities). The homogeneous nucleation rate depends very strongly on temperature, the heterogeneous nucleation rate much less so. If nuclei grow above a certain critical size, crystal growth can start. Crystal growth rate is basically a balance between the adsorption of molecules to a surface and the release of molecules from that surface. There are quite a few factors that inﬂuence crystal growth. First of all, the amount of supersaturation, and therefore the temperature, the shape of the crystals, the presence or absence of similar molecules that may adsorb and thus limit further adsorption, the heat of fusion that causes the temperature to rise locally. Crystal growth can be diffusion limited, or reaction limited. A general model that encompasses all the stages does not exist, though there are abundant (and competing) theoretical models available describing both the kinetics of nucleation and crystal growth. We will not discuss these here because of the many complexities involved. There is however a model that seems to be of general use, called the Avrami model, that describes crystallization for nucleation and growth if a homogeneous solution is rapidly cooled to a constant temperature. It is a semiempirical model, the equation of which is w(t) ¼ 1 expðkt nA Þ

(11:72)

w is the volume fraction of crystals in the solution. Depending on the values of the two parameters, the rate constant k and the Avrami exponent nA (usually between 0.5 and 4), the equation shows an induction period (representing the nucleation phase), a rapid increase (crystal growth) and a slowing down of the rate (when supersaturation becomes less at the end of the process). Figure 11.16 shows an example for fat crystallization.

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40 35

Solid fat (%)

30 25 20 15 10 5 0

0

50

100

Time (min )

FIGURE 11.16 Kinetics of crystallization of hardened palm oil at 408C (data points) as described by the Avrami equation (solid line). Parameter k ¼ 2.64 3 105, nA ¼ 2.67. Dataset in Appendix 11.1, Table A.11.6.

The Avrami equation has also been applied to the kinetics of retrogradation of starch. Once again, it should be realized that the Avrami equation is an empirical equation without a physical meaning.

11.4

Kinetics of Texture Changes

Many literature reports can be found describing the kinetics of texture change, for instance, in vegetables. Undoubtedly, texture changes are important for food quality. Sometimes, they are desired, as in cooking potatoes, sometimes undesirable, as in overripened fruits. Frequently, kinetics of texture changes is modeled as ﬁrst-order kinetics. First of all, it should be clear that texture changes in fruits and vegetables are the result of many molecular changes in cell wall components, such as pectin molecules. However, texture is measured by some rheological technique, not as molecular change. Consequently, it makes no sense to describe the dependence of apparent rate constants on temperature via the Arrhenius equation and to report the results as activation energy in kJ mol1. Nonetheless, one can ﬁnd many results, reporting for instance activation energy for texture degradation of cooked potatoes. It would be interesting to see a mole of potatoes! Frequently, kinetics of texture changes seems to follow a biexponential decay (Figure 11.17). It has been proposed in literature to model physical changes via a fractional conversion ﬁrst-order model rather than a normal ﬁrst-order equation. Fractional conversion would be suitable to correlate the extent of a chemical reaction with the measurement of a physical property. The fractional conversion fc is deﬁned as the ratio between what has reacted and what can potentially react. Mathematically this is translated into fc ¼ c0 is the concentration at time zero c1 is the concentration at time inﬁnity

c0 ct c0 c1

(11:73)

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140

Compression force (N)

120 100 80 60 40 20 0 0

20

40

60 t (min)

80

100

120

FIGURE 11.17 Texture (measured as compression force in N) of potato cubes as a function of time after cooking at 808C. Dataset in Appendix 11.1, Table A.11.7.

It follows that 1 fc ¼ 1

c0 ct ct c1 ¼ c0 c1 c0 c1

(11:74)

And so for a ﬁrst-order reaction this translates into ct c1 ¼ exp (kt) c0 c1

(11:75)

If c1 ! 0 we arrive again at the familiar ﬁrst-order equation. When we are not measuring concentrations but a physical property, we have F0 as the physical property at t ¼ 0, F1 the physical property at t ¼ 1, Ft the physical property at time t F0 Ft F0 F1 Ft F 1 1 fc ¼ F0 F1

(11:76)

Ft ¼ F1 þ (F0 F1 ) exp (kt)

(11:77)

fc ¼

The ﬁrst-order model is then

If this relation holds, texture change would be characterized by a single rate constant, but of course, one has to estimate the physical property at F1 at t ¼ 1, so it is a model with three parameters if F0 is estimated as well. It is not clear beforehand whether F1 is temperature dependent or not. It should be

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100 90 80

Failure stress (MPa)

70 60 50 40 T = 90C 30 T = 103C 20 10

T = 120C

0 0

2

4

6 t (min)

8

10

12

FIGURE 11.18 Kinetics of texture changes of potatoes as a function of cooking temperature (908C, 1038C, and 1208C) and time. The lines are the fractional conversion model (Equation 11.77) or the Weibull model (Equation 4.76); the ﬁts of the lines are indistinguishable. The texture parameter measured was failure stress. Dataset in Appendix 11.1, Table A.11.8.

realized that such a ﬁrst-order model is just an empirical model without a physical background, and other empirical models such as the Weibull model (Equation 4.76) may be equally applicable. Both the Weibull model (Equation 4.76) and Equation 11.77 were tested on some data describing texture change of potatoes as a function of cooking (Figure 11.18). Both models gave an equally good ﬁt and cannot be distinguished visually. Kinetics of water uptake. Water uptake by foods is important for rehydration of foods but it can also lead to quality loss if products take up water. This can be a problem, for instance, with cereals immersed in a liquid like milk: the cereals may become soggy in a short while, which is a texture problem. Or, a product can lose its crispiness by water uptake during storage. It can also be a problem when several components are mixed that have a different water activity, such as cereals mixed with raisins. Such water uptake can in principle be modeled via mass transfer models discussed earlier in this chapter, but it also possible by empirical models, of which one is the Weibull model again: aW t Mt M0 ¼ (M1 M0 ) 1 exp bW

(11:78)

Another one is the so-called Peleg model for water sorption: Mt M0 ¼ (M1 M0 )

t kþt

(11:79)

An example of both models is shown for the kinetics of uptake of water by cereals by whole milk in Figure 11.19.

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300

Mt −M0 (g H2O/100 g initial weight)

250

200

150

100

50

0

0

2000

4000 t (s)

6000

8000

FIGURE 11.19 Kinetics of water uptake by cereals immersed in whole milk at 558C modeled by the Weibull model (—, Equation 11.78) and the Peleg model (- - -, Equation 11.79). Dataset in Appendix 11.1, Table A.11.9.

Both models are seen to describe the experimental values well. Once again, these are empirical models that are nevertheless very useful, and when product composition is varied a possible relation can be found between product composition and parameters of the model. Incidentally, the data shown in Figure 11.19 are seen to be heteroscedastic (variance increases with water uptake) and this might call for a transformation to induce homoscedasticity, as discussed in Chapter 7, if one wants to ﬁnd the most probable estimates of the parameters. Here, we do not perform this exercise because the principal purpose is to show the various models.

11.5

Partitioning Phenomena

As remarked before, many foods are heterogeneous and may contain more than one phase. The possibilities are combinations of the following: . . . .

Aqueous phase Lipid phase Headspace (usually air) Solid phase

Components may divide themselves over the various phases. Although not a phase in the strict sense, micelles formed from surfactants have, to some extent, the same properties as phases, and micelles, vesicles, and microemulsions could be called a pseudophase. The fact that components may partition over various phases has an impact on kinetics because of the following reasons: . .

Concentrations of reactants and products will be affected If there is partitioning of carboxylic acids, pH changes may occur

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If there is partitioning of volatile compounds, this will effect their release from the food (upon storage but also upon cooking and eating) and will affect ﬂavor perception If there is partitioning of antioxidants, this will have an effect on oxidation kinetics

All in all, the phenomenon of partitioning in foods on kinetics deserves attention. Partitioning is, of course, a physical phenomenon, which is why it is discussed in this chapter.

11.5.1

Partition Coefﬁcients

To describe partitioning quantitatively, a partition coefﬁcient is introduced, which is linked to activities of the solute. If a solute is free to move between phases, then the activity in each phase will be the same when equilibrium is attained: fia Xia ¼ fib Xib

(11:80)

Here, Xi is the mole fraction of solute i, fi the dimensionless (rational) activity coefﬁcient, and a and b refer to two different phases (e.g., oil and water, or air and water). The thermodynamic partitioning coefﬁcient is deﬁned as x Pa=b ¼

Xia Xib

¼

fib fia

(11:81)

This equation shows that, if uneven distribution of a component over two phases occurs, this is in fact due to different activity coefﬁcients in these two phases. In actual practice, one uses the conventional partitioning coefﬁcient in terms of concentrations: Pa=b ¼

cai cbi

¼

yib yia

(11:82)

(As remarked in Chapter 3, the numerical values of activity coefﬁcients depend on the way they are deﬁned, so the value of yi may be different in Equation 11.82 than fi in Equation 11.81.) All solute–solute and solute–solvent interactions are accounted for in the activity coefﬁcients.

11.5.2

Partitioning of Volatiles

The release of components from the food matrix is important for several reasons, a very important one being the release of volatiles that act as ﬂavor compounds.* This can happen during processing and storage, and, most importantly, during eating (mastication). Many research papers deal with this topic, and models have been presented that consider mass transfer as a function of temperature and mechanical treatment, and obviously, partitioning has an important part to play. In fact the partitioning coefﬁcient is the thermodynamic measure that indicates the maximum possible extent of partitioning, and it also accounts for the inﬂuence of chemical composition of the medium. Often, the partitioning behavior of volatile compounds is characterized by the so-called log P value; this is in fact the logarithm of the n-octanol=water partitioning coefﬁcient. These log P values are not directly transferable to foods because n-octanol has different characteristics than the common fat phase in food, but the values may be converted to those for oils. log P values are frequently reported in literature * The term ﬂavor release is often used. Strictly speaking, it is better to use the term volatile release because a volatile causes only a ﬂavor when it reacts with a receptor in the nose or mouth and is perceived as a ﬂavor in the brain.

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TABLE 11.4 log P Values of Some Selected Compounds at 258C Compound

log P

Diacetyl

1.3

Acetaldehyde

0.2

Ethyl acetate

0.7

Hexanal

1.8

2-Octanone

2.4

Ethyl hexanoate

2.8

Linalool

3.5

to indicate the hydrophobicity of the compound. A positive value indicates a hydrophobic compound (the higher, the more hydrophobic) and a negative one a hydrophilic compound. Incidentally, log P values can be estimated also from the molecular structure of a molecule using appropriate software. Table 11.4 shows some log P values. If a volatile compound partitions between a liquid phase l and a vapor phase v, the partition coefﬁcient can be deﬁned as the ratio between the concentration in the vapor phase cv (g dm3) and that in the liquid phase cl (g dm3): according to Equation 11.82 Pv=l ¼

cv : cl

(11:83)

The reader is reminded that in Chapter 3 Henry’s law was introduced in Equation 3.55, describing the relation between the partial pressure Pi of a component in the vapor phase above a solution containing that compound in mole fraction Xi: Pi ¼ kH,i Xi

(11:84)

where kH,i is Henry’s constant. In relation to partitioning of volatile compounds, one uses in the literature frequently the ‘‘activity coefﬁcient at inﬁnite dilution,’’ indicated by the symbol g1 i . It should be realized that in that case one does not use Henry’s law as a reference but rather Raoult’s law. This may be a bit confusing because it was explained in Chapter 3 that the activity coefﬁcients of solutes at inﬁnite dilution approach unity by taking Henry’s law as reference. However, one is free to choose reference states, so it is equally valid to use Raoult’s law, and one uses in fact the symmetrical convention here, introduced in Chapter 3. To remind the reader of the concept, Figure 11.20 shows graphically how the activity coefﬁcient is deﬁned in the symmetrical convention (i.e., by using Raoult’s law as a reference). However, we prefer to use the symbol f to indicate activity coefﬁcients at the mole fraction concentration scale; the symbol g was chosen for the molality scale. We shall thus refer to the activity coefﬁcient at inﬁnite dilution as fi1 for Xi ! 0 as indicated in Figure 11.20. The relation between fi1 and Henry’s constant is fi1 ¼

kH,i Pi

(11:85)

Pi is the saturated vapor pressure of component i. fi1 is in fact a measure for the afﬁnity of component i for the solution in which it is dissolved. The higher this value, the less afﬁnity it has for the solution. In general, the activity coefﬁcient as a function of its mole fraction will be as indicated in Figure 11.21, if Raoult’s law is taken as reference, which implies that fi ! 1 when Xi ! 1.

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Henry ’s law P

Pi

ai =

P∗i

fi =

ai = fiXi

Pi

P iX ∗

Region of infinite dilution (Xi 0)

: Pi

=

i

aw

’s l

lt aou

Pi

P oi = P∗i

Pi

R

Pi

Xi

0

1

FIGURE 11.20 Illustration of Raoult’s and Henry’s law and the activity coefﬁcient fi according to the symmetrical convention; for Xi ! 0, fi ! fi1 .

f 1∞

f

f1 f 2∞ f2

1

X1

0

1

FIGURE 11.21 General shape of the relation between the activity coefﬁcients f and the mole fraction of compound 1 in a binary liquid mixture, showing non-ideal behaviour fi1 is found at inﬁnite dilution, i.e., when Xi 0.

Table 11.5 shows values of this activity coefﬁcient for some typical compounds. As expected, the more hydrophobic a component is, the higher its activity coefﬁcient extrapolated from inﬁnite dilution, and the higher its volatility. In fact, if the magnitude of the activity coefﬁcient at inﬁnite dilution is much larger than 1000, the following approximate relation holds: Xi

1 fi1

(11:86)

Obviously, a link can be made between partition coefﬁcients and Henry’s law because they both refer to the same phenomenon. It is assumed that the vapor behaves as an ideal gas so that the general gas law is valid and it is assumed that at inﬁnite dilution the amount of moles of the volatile can be neglected. Then, for a volatile component 1 dissolved in a solvent 2, it can be derived that: Pv=l ¼

kH,i PT =RT P1 n2 =Vl

(11:87a)

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TABLE 11.5 Some Typical Values for fi1 , the Activity Coefﬁcient Extrapolated from Inﬁnite Dilution in Aqueous Solutions, at 258C fi1

Compound Diacetyl

1.1

Acetaldehyde

4.2

Ethanol Butanol

4.8 67

Ethyl acetate

86

Benzaldehyde

559

Hexanal

1310

Linalool

14000

Note: Compiled from various sources.

PT is the total pressure, P1 the partial pressure of component 1, and n2 the amount of moles of component 2 (the solvent), Vl is the volume of the liquid phase. By combining this with Equation 11.85 an alternative equation is Pv=l ¼

fi1 Pi PT =RT P1 n2 =Vl

(11:87b)

For an aqueous solution and a vapor phase of 1 L and at 208C, this results in Pv=l ¼

kH,i 1:38 108 [Pa]

(11:88)

For a partition coefﬁcient between vapor and liquid oil, taking a molecular weight of 730 Da and a density of 910 kg m3 for oil, the relation would be Pv=l ¼

kH,i 3:1 106 [Pa]

(11:89)

The temperature dependence of partitioning of volatiles can be derived from the Clausius–Clapeyron equation, which describes actually the temperature dependence of the vapor pressure for a pure compound: Pio

¼

o Pref

DHvap 1 1 exp R T Tref

(11:90)

It seems reasonable to assume that the same relation holds for the partial pressure but with the enthalpy of hydration replacing the enthalpy of vaporization:

DHhydr 1 1 Pi ¼ Pi,ref exp R T Tref

(11:91)

The enthalpy of hydration represents the amount of energy released when 1 mol of volatile dissolves in an inﬁnite amount of water. Since the partition coefﬁcient is related to the partial pressure of component i, it may be assumed that the temperature dependence of the partition coefﬁcient is similar:

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0.016 0.014

Pair/aq, Pair/oil

0.012 0.010 0.008 0.006 0.004 0.002 0 0

20

40

60

T (C)

FIGURE 11.22 Effect of temperature on partition coefﬁcients. Methyl acetate in water (&), ethylacetate in water (*), and isopentylacetate in coffee oil (~). Dataset in Appendix 11.1, Table A.11.10.

DHhydr 1 1 Pv=l ¼ Pv=l,ref exp R T Tref

(11:92)

With increasing temperatures, partition coefﬁcients increase as well because of the increase in saturation vapor pressures. Figure 11.22 shows some examples. Figure 11.23 shows the partitioning of methanethiol between the headspace and an aqueous phase, and the headspace and a fat phase at equilibrium; methanethiol is a volatile related to cheese ﬂavor. The linear relation found is in line with Henry’s law. The activity coefﬁcient fi1 may depend also on the presence of other solutes. Figure 11.24 gives some examples for the effect of sucrose. The effects are quite large: notice the logarithmic scale. The increase in fi1 indicates less favorable interaction of the volatile with the aqueous phase. It is perhaps instructive to show the effect of the presence of a fat phase in a food on the ﬂavor composition of the headspace, because the ﬂavor proﬁle of low fat foods may be quite different from their high-fat equivalents. If we denote the total amount of a volatile present in a closed system as NT, the mass balance is NT ¼ cf Vf þ caq Vaq þ cv Vv

(11:93)

where cf, caq, and cv stand for the concentration of the volatile in the fat, aqueous and vapor phase, respectively Vf, Vaq, and Vv for the volume of the fat, aqueous, and vapor phase Using the appropriate partition coefﬁcients, we can transform Equation 11.93 as follows: NT ¼

Pf =aq Vaq Vf cv þ þ cv Vv Pv=aq Pv=aq

(11:94)

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12

Headspace (nmol dm−3)

10

8

6

4

2

0 0

5

10 15 Solution (n mol dm–3)

20

FIGURE 11.23 Partitioning of methanethiol between headspace and a solution consisting of 0.1 M sodium acetate þ37.5 g L1 NaCl, pH 5.2 (^) and between headspace and liquid milk fat (&). Dataset in Appendix 11.1, Table A.11.11.

So, if we know the volumes of the three phases and the partition coefﬁcients, we can calculate the distribution of volatiles over the various phases. Table 11.6 shows a simple calculation for several volatiles, for an imaginary full-fat cheese and a low-fat cheese, just to show the differences of the presence 6

5

log f ∞i

4

3

2

1

0 0

0.2 0.4 0.6 Mass fraction sucrose (kg kg–1)

0.8

FIGURE 11.24 Dependence of log fi1 for ethylacetate (^), butylacetate (&), and n-hexylacetate (*) on sucrose concentration in aqueous sucrose solutions. Dataset in Appendix 11.1, Table A.11.12.

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TABLE 11.6 Concentration of Volatiles in the Vapor Phase (cv) for a Full-Fat Cheese (Vf ¼ 3 mL, Vaq ¼ 4 mL) and a Low-Fat Cheese (Vf ¼ 1.5 mL, Vaq ¼ 5.5 mL) Volatile

Pf=aq

Pv=aq

Pv=f

cv in Full-Fat Cheese (nmol mL1)

cv in Low-Fat Cheese (nmol mL1)

H2S

3.5

1.31

0.41

32.2

35.5

Methanethiol

6.13

0.14

0.025

5.6

8.1

14.25

0.17

0.014

3.4

5.6

0.0006

0.0014

0.11

0.10

Acetaldehyde

0.6

0.47

2-heptanone

0.03

0.06

Dimethylsulﬁde Diacetyl

0.43

Note: Vv is taken as 20 mL and the total amount of the volatile is taken as 1000 nmol.

of a fat phase. In reality, things are more complex because there may be no equilibrium, and there may be adsorption of volatiles onto proteins, for instance. Nevertheless, Table 11.7 shows the large impact that the amount of fat can have on the partitioning of volatiles, a factor that needs to be taken into account in the design of foods. The equations given so far about partitioning are shown as being time independent, and they have been discussed to show the importance of the partitioning coefﬁcient. In the following, we will also consider the effect of time, and hence kinetics of partitioning. Release kinetics. As argued above, thermodynamic parameters give an idea about the extent of partitioning. Mass transfer on the other hand determines the rate at which partitioning occurs. Hence, both thermodynamic as well as kinetic factors determine the ultimate release or partitioning rate. Several models have been developed that describe mass transfer. We will only discuss a few of them here. References for more models are given at the end of this chapter. For situations in which there is no ﬂow of the product or the headspace, static diffusion models are used, implying only molecular diffusion. Static diffusion, i.e., without any ﬂow or movement of the material and the vapor around it, is not very realistic in practical terms, but it gives insight in the effect of key parameters. An equation describing the total amount of release of a component from a liquid l to vapor v, based on Fick’s diffusion laws, reads as follows: Mm (t) ¼

2Fc0l

rﬃﬃﬃﬃﬃ Df pt

(11:95)

Mm represents the total amount of volatiles that is released. The factor F in this equation is qﬃﬃﬃﬃﬃﬃ F¼

Pv=l 1 þ Pv=l

Df ,v Df ,l

qﬃﬃﬃﬃﬃﬃ

(11:96)

Df ,v Df ,l

TABLE 11.7 Numerical Values for Parameters Used in Equation 11.118 for the Fit of the Model to the Experimental Data in Figure 11.30 Dimethylpyrazine

Diacetyl

Acetaldehyde

Pv=l

Parameter

5.7 3 105

3.9 3 104

2.7 3 102

0.025

km

0.0191

2.7 3 103

8.98 3 104

9.97 3 105

c0

l

Dimethylsulﬁde

0.00837

0.000961

0.0011

0.000846

wv

1.17 3 106

1.17 3 106

1.17 3 106

1.17 3 106

Aa

0.001

0.001

0.001

0.001

Vv

50 3 106

50 3 106

50 3 106

50 3 106

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Relative release rate 105 (m s−1/2)

3.5 3 2.5 2 1.5 1 0.5 0

−5

−4

−3

Pv/l

−2

−1

0

FIGURE 11.25 Relative release rate (Equation 11.97) as a function of partition coefﬁcient values Pv=l. Df,l ¼ 109 m2 s1, and Dv ¼ 2 3 105 m2 s1.

This factor represents the driving force for the diffusion process. The partition coefﬁcient Pv=p appears to be the most important factor, determining differences in release rates for volatiles. Figure 11.25 gives an impression by plotting the relative release rate nr per unit area: nr ¼

Mm (t) pﬃﬃﬃﬃﬃ pt 2c0l

(11:97)

Figure 11.25 shows that the value of the partition coefﬁcient has a large effect on the release rates. Not unexpectedly, the higher the partition coefﬁcient, the higher the release rate. The differences in diffusion coefﬁcients are not so large for the various volatiles, so that parameter does not cause large differences between volatiles. Once again, the reader is advised that this situation pertains to static diffusion, which may not be very realistic for foods, but the analysis gives insight in the factors that play a role. Convective mass transfer (or eddy diffusion) models are developed for cases where a liquid phase in the product and the air phase are moving, one way or another. The transport from one phase to another is considered to take place via an interfacial layer. In general, the driving force for transport of volatiles over an interface is determined by concentration gradients and mass transfer coefﬁcients in the various phases. This transport can be described in differential equations. For solid foods that are in contact with saliva, a model is developed based on stagnant-ﬁlm theory, in which it is assumed that in the interfacial layer only molecular diffusion occurs (Figure 11.26). The equation describing the total amount Mm of volatile transported over the interface Aa reads dMm Df ,saliva ¼ Aa (t) cint saliva csaliva dt Ls

(11:98)

The differential equation describing the concentration of a volatile in the saliva (csaliva) as a function of time becomes km Aa (t) dcsaliva Df ,saliva ¼ Aa (t) cint (Psaliva=food cfood csaliva ) saliva csaliva ¼ dt Ls Vsaliva Vsaliva

(11:99)

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Concentration cfood

Solid food

Saliva

int csaliva

csaliva Ls x Stagnant interfacial layer

FIGURE 11.26 Schematic picture showing the release of a volatile from a solid food to saliva via diffusion. cfood, cint saliva , and csaliva are the concentrations of a volatile in the food, the interfacial layer, and saliva, respectively. Ls is the thickness of the stagnant interfacial layer.

where Df is the diffusion coefﬁcient of the volatile in the liquid phase Ls is the thickness of the stagnant layer Aa is the surface area of the interface Psaliva=food is the partition coefﬁcient between saliva and food Because Ls cannot be measured Df,saliva and Ls are replaced by a mass transfer coefﬁcient km. An important, saliva complication that will happen during eating is that the surface area Aa is not constant but a function of time, and some function has to be assumed to account for this. In the case of a volatile released from a liquid food to the vapor phase via convection, models can be based upon the so-called penetration theory, in which it is assumed that there is mass transport in the interfacial layer via eddy diffusion. This is supposed to happen via a volume element of the bulk phase that enters (penetrates) the interfacial layer and stays there for a short time te, during which exchange between gas and liquid phase takes place. During this period equilibrium is assumed in the interfacial layers through diffusion, so that the volatile compound can reach the gas layer before the volume element moves back to the bulk liquid. Figure 11.27 gives a schematic representation. The differential equation describing mass transfer is rﬃﬃﬃﬃﬃﬃﬃ dMm Df ,l int cv Aa cl cl ¼ km Aa cl ¼2 dt pte Pv=l

(11:100)

The parameter te cannot be measured independently, and therefore the mass transfer coefﬁcient km is introduced again: km ¼ 2

Df pte

12 (11:101)

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Concentration cl

c int l Liquid food

Headspace

c int v cv

Penetration of a bulk volume element

x Interfacial layers

FIGURE 11.27 Schematic picture showing the release of a volatile from a liquid food to a headspace via penetration int of a bulk volume element in the interfacial layer. cl, cint l , cv , and cv are the concentrations of a volatile in the liquid food, the interfacial layer at the liquid side, the interfacial layer at the vapor side, and in the vapor phase, respectively.

The solution in terms of the change in the concentration in the headspace is

! Pv=l c0,l Vl km Aa 1 exp 1 þ t cv (t) ¼ Pv=l Vv Pv=l Vv Vl þ1 Vl

(11:102)

The assumption made in this derivation is that the vapor phase is thoroughly mixed so that cint v cv. Two extreme situations can be derived from this equation. The ﬁrst is the initial release rate: cv (t ! 0) ¼

c0,l Aa km t Vl

(11:103)

The second is the situation at equilibrium: cv (t ! 1) ¼

Pv=l cg,0 t Pv=l Vv þ1 Vl

(11:104)

This last equation is in fact describing partitioning at equilibrium. Figure 11.28 shows two examples of release kinetics where this model is applied. It has also been shown that release kinetics is slowed down when the viscosity is increased by low molecular compounds but hardly so when the viscosity is decreased by high molecular weight compounds, such as pectin (provided that pectin does not bind the volatiles). It is interesting to note that the initial release is mainly determined by the mass transfer coefﬁcient, while the ﬁnal headspace concentration is determined by the partition coefﬁcient, which explains the initial differences observed in Figure 11.28 for linalool and ethylhexanoate. Effect of volatile adsorption. The situation becomes even more complex when substances are present that have an effect on the concentration of the volatile in the aqueous solution. This can happen if, for instance,

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100

Percentage of equilibrium concentration

90 80 70 60 50 40 30 20 10 0 0

2000

4000 6000 Time (s)

8000

FIGURE 11.28 Release kinetics of linalool (*) and ethylhexanoate (~) from an aqueous solution containing 0.1% NaCl. The concentrations are expressed as % of the ﬁnal equilibrium concentrations. The lines are calculated using Equation 11.102 with parameters Vv ¼ 111.63 cm3, Vl ¼ 24 cm3, Aa ¼ 17.9 cm2, Pv=l ¼ 0.0094 for linalool, and Pv=l ¼ 0.001 for ethylhexanoate, km ¼ 0.0004 cm s1 for linalool, and km ¼ 0.000092 cm s1 for ethylhexanoate. Dataset in Appendix 11.1, Table A.11.13.

volatile molecules bind with proteins or polysaccharides; we consider this binding a physical process, not a chemical one because in the latter case the volatile would not be released anymore. Another possibility is the presence of lipid phases, notably emulsions. Most volatiles are lipophilic and will tend to dissolve easily in lipid phases. As a result of all this, the partitioning of volatiles will be disturbed as compared to an aqueous phase without lipids or biopolymers. A general discussion about such phenomena is as follows. If we consider the concentration of a volatile in a product, the following mass balance is valid: ctotal ¼ cb þ cu

(11:105)

where ctotal is the total concentration of the volatile in the product cb is the concentration of the binder–volatile complex BI–F cu is the concentration of the volatile F that is unbound Suppose that the concentration of the binding substance BI is cbi, then we have the following situation at equilibrium: k1

BI þ F ! BI---F k

(11:106)

2

The equilibrium constant is then K¼

cu cbi cb

(11:107)

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Assuming that only a small fraction of the binder is complexed with the volatile, the equilibrium concentration of the binder can be put equal to its initial concentration. Rearranging and combining Equations 11.105 and 11.107 results in cu ¼

ctotal 1 þ Kcb

(11:108)

So, this equation shows clearly that the concentration of the unbound volatile in solution is reduced by a factor 1=(1 þ Kcb) and this must then also be true for the partition coefﬁcient. The above discussion is valid for the equilibrium situation in the aqueous phase. However, in the case of the presence of a headspace, the equilibrium will be disturbed and we have to consider the kinetics of these changes. The following equation applies to that situation: dcu ¼ k1 cb k2 cu cbi dt

(11:109)

The resulting model allows for binding of volatile components to food constituents such as proteins and polysaccharides: Kb is the binding constant, and cb the concentration of the ligand. The factor hD is the gas–liquid mass transfer coefﬁcient. te is the mean contact time of an element of solution with the gas phase; it is taken proportional to stirring rate. # Aa km 1 Vv 1 i 1 exp þ t cv (t) ¼ h V Vv Pv=l Vl 1 þ Kb cb 1 þ KPbv=lcb þ Vgl "

cv,0

(11:110)

Volatile release from emulsions. The equilibrium headspace above an oil-in-water emulsion (subscripts oil, aq and em, respectively) can be calculated, as was already shown above in Equation 11.94: cem ¼ (1 w)caq þ wcoil

(11:111)

cv ¼ Pv=em cem b(1 w) þ wPoil=aq c

(11:112)

The partition coefﬁcient for the gas-emulsion equilibrium is Pv,em ¼

Pv=aq 1 þ (Poil=aq 1)w

(11:113)

Equation 11.112 can be written as Pv=em c0,em

cv (t) ¼ P

v=em Vv

Vl

þ1

1 exp 1 þ

! Vem k m Aa t Pv=em Vv Vem

(11:114)

Dynamic headspace dilution. As remarked before, equilibrium situations are not easily encountered with foods. It is therefore of interest to consider situations where equilibrium is disturbed. One such situation is when the headspace above a food is disturbed by an air ﬂow, for instance when a package is opened. For the situation as displayed in Figure 11.29, the following relation can be derived. It concerns the blowing of a gas above a solution as a result of which volatiles are removed from the solution. For

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φv

11-45

civ

civ Vg

Interface gas

Area Aa

Interface liquid

cil

Vl

cil (A)

(B)

Concentration

FIGURE 11.29 Schematic representation of a cell in which a gas ﬂow blows with ﬂow wv over a solution containing a volatile in concentration cil in the bulk (A). The volume of the gas compartment is Vv, that of the solution Vl. The concentration of the volatile in the gas phase is civ. The concentration proﬁle at one moment in time is depicted (B). The concentration changes in the interface with area Aa and it is assumed that there is no concentration gradient in the bulk.

simplicity, the conditions are chosen in such a way that the concentrations of the volatile in the bulk gas phase and liquid phase can be considered constant, while there is a concentration gradient in the interface of liquid and water. A mass balance can be set up between the release of the volatile from the liquid phase and removal by the gas ﬂow: dcvi ¼ Ji Aa fv cvi dt Ji ¼ km Pi,v=l cli cvi

Vv

(11:115) (11:116)

Ji is the ﬂux. The mass transfer coefﬁcient is a function of the transfer from the bulk liquid to the interface and from the interface to the bulk gas, and since this is a process in series the following relation holds (Equation 4.86): Pi,v=l 1 1 ¼ þ km km,v km,l

(11:117)

It is assumed that there is equilibrium at the interface, and the following differential equations can be derived: dcvi km Aa km Aa þ fv v l ¼ Pv:l ci ci dt Vv Vv

(11:118)

dcli km Aa ¼ Pv=l cvi dt Vl

(11:119)

At the start of the experiment (t ¼ 0) it follows that: Pi,v=l cli ¼ cvi

(11:120)

If the time of experiment is not too large and Vl not too small, it can be assumed that c1i is constant, and a g change in ci can be predicted. Figure 11.30 shows an example of the match between experiment and

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1

0.8

c/c0

0.6

0.4

0.2

0

0

100

200 Time (s)

300

400

FIGURE 11.30 Headspace concentrations of various volatiles upon dilution of the headspace by air ﬂow. Lines are the ﬁt by the model displayed in Equation 11.118 solved numerically with the parameters shown in Table 11.7. Dimethylpyrazine (D), diacetyl (*), acetaldehyde (.), dimethylsulﬁde (&) in aqueous solutions. Dataset in Appendix 11.1, Table A.11.14.

model as displayed in Equation 11.118, which was solved numerically. The data used for the calculation are shown in Table 11.7. Overall, the model ﬁts reasonably well. To conclude this section on release kinetics of volatiles, we have discussed several models that seem to be able to ﬁt experimental data for various conditions. However, the ultimate challenge is to be able to construct models that are also capable to describe release of volatiles in the mouth where things become much more complicated. Chewing deforms the food, which has an effect on the mass transfer kinetics, there is dilution with saliva, and there is air ﬂow by inhalation and exhalation. This research area is developing quickly and models have been presented also for such cases. We will not discuss them here any further, but some references are given at the end of this chapter.

11.5.3

Partitioning of Weak Acids

Partitioning of weak acids (such as lactic acid, acetic acid, benzoic acid, sorbic acid) in fat-containing foods over aqueous and oil phases can have large effects on their potential as preservative. We can envisage the following situation for the oil and aqueous phase: þ HAoil ! HAaq ! H þA

(11:121)

A partition coefﬁcient Poil=aq can be derived for the distribution of the undissociated form over the oil and the aqueous phase: Poil=aq ¼

[HA]oil [HA]aq

(11:122)

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If we know the volume of oil (Voil) and that of the aqueous phase (Vaq), we can quantify the amount of oil as volume fraction f: f¼

Voil Voil þ Vaq

(11:123)

The mass balance for the acid reads (the subscript ‘‘T’’ indicates the total amount of acid): [HA]T Vaq ¼ [HA]oil Voil þ [HA]aq Vaq þ [A ]Vaq

(11:124)

Combining this with Equation 11.123 gives [A ] ¼ [HA]T

w [HA]oil [HA]aq 1w

(11:125)

After some algebraic rearrangement the result is [HA]aq ¼ [HA]T

1 þ Poil=aq

1 w þ 10pHpKa 1w

(11:126)

This equation is in fact a modiﬁcation of the Henderson–Hasselbalch equation, and shows the fraction of undissociated form present in the aqueous phase as a function of pH and volume fraction of oil, given the pKa and partition coefﬁcient for the acid involved. Figure 11.31 shows an example of the effect for benzoic acid, and it is obvious that a large fraction of undissociated benzoic acid is partitioning into the oil phase. This can have large consequences if benzoic acid is used as a preservative in low pH foods where it will only be effective in the aqueous phase in its undissociated (protonated) form.

1 f = 0.0

[HA]aq/[HA]T

0.8

0.6 f = 0.2 0.4 f = 0.4

0.2

f = 0.6 0 2

4

6 pH

FIGURE 11.31 Effect of volume fraction of oil (f) and pH on the fraction of undissociated benzoic acid in the aqueous phase. Calculated according to Equation 11.126 using pKa ¼ 4.19, Poil=aq ¼ 5.73.

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A solute–solute interaction that is relevant for foods in this respect is the dimerization of carboxylic acids in nonaqueous phases, which is due to hydrogen interactions. Suppose we have an aqueous phase (subscript aq) and a lipid phase (subscript oil) and a carboxylic acid HA, we can envisage the following equilibria: Partitioning: (HA)aq Ð (HA)oil

(11:127)

Dimerization: 2(HA)oil Ð (HA)2,oil

(11:128)

Hence, Poil=aq ¼

[HA]oil [HA]aq

(11:129)

A practical equilibrium constant can be given for the dimerization: Kc ¼

[(AH)2 ]oil [AH]aq

(11:130)

This leads to the following expression: [HA]oil þ 2[AH2 ]oil 2 ¼ Poil=aq þ 2Poil=aq Kc [HA]aq [HA]aq

(11:131)

This equation shows that both partitioning as well as dimerization have an effect on the ﬁnal distribution of components over the various phases. Figure 11.32 gives an example concerning the distribution of benzoic acid and shows how Equation 11.131 is obeyed experimentally.

8 7 6

coil/caq

5 4 3 2 1 0

0

2

4

caq

6

8

10

FIGURE 11.32 Distribution of benzoic acid over water and sunﬂower oil at 258C. caq is the concentration in the aqueous phase, coil is the concentration in the oil phase. Dataset in Appendix 14.1, Table A.11.15.

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In relation to kinetics, these partitioning phenomena are important because they affect the concentrations of possible reactants and products.

11.6

Concluding Remarks

To conclude this chapter on physical changes, it can be stated that it is difﬁcult to predict these in a quantitative way. This is partly due to the fact that the available theory is developed for very ideal systems, and partly to the fact that foods are so incredibly complex with respect to physical structure. Nevertheless, one can ﬁnd many studies in which kinetics of physical changes are described. A case in point is the kinetics of texture changes. Texture is an important quality aspect, related to changes in cell wall components, changes that are brought about by enzymatic activity, but also due to moisture changes, breakdown of starch and protein during the ongoing metabolism after harvest. It becomes an arduous task to derive kinetics in which all such changes are taken into account. So, the approach taken is of an empirical nature: one uses a rheological technique to describe changes, and ﬁts a kinetic equation to the data. While this is deﬁnitely useful for description of phenomena observed, one should be careful with interpretation of such results in molecular terms.

Appendix 11.1 Datasets Used for Examples in This Chapter TABLE A.11.1 MS Diffusion Coefﬁcient and Fick Diffusion Coefﬁcient for Ethanol–Water Mixtures (Figure 11.6) MS D (m2 s1)

Fick D (m2 s1)

0

1.71

1.72

0.03

1.68

1.5

0.11 0.15

1.52 1.39

1.01 0.79

0.21

1.45

0.69

0.26

1.51

0.63

0.31

1.55

0.6

0.41

1.59

0.64

0.5

1.65

0.74

0.6

1.64

0.84

0.68 0.79

1.64 1.69

1.02 1.25

0.88

1.7

1.44

0.96

1.65

1.55

1

1.65

1.65

Mole Fraction Ethanol

Source: From Wesselingh, J.A. and Krishna R., Mass Transfer in Multicomponent Mixtures, Delft University Press, Delft, the Netherlands, 2000.

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TABLE A.11.2 MS Diffusion Coefﬁcient of Lysozyme as a Function of KCl (Figure 11.7) KCl (mol m3)

2 1 D (m s )

0

5.5

1

4.14

2.5

2.89

5

2.18

50

1.39

150

1.33

Source: From Wesselingh, J.A. and Krishna R., Mass Transfer in Multicomponent Mixtures, Delft University Press, Delft, the Netherlands, 2000.

TABLE A.11.3 Mutual MS Diffusion Coefﬁcients in NaCl Solutions (Figure 11.8) log c (M)

log Dwater–Cl

log Dwater–Naþ

8.67

8.88

2

log D,

water–NaCl

1.01

8.68

8.88

10.59

0.67 0.47

8.67 8.67

8.88 8.89

10.35 10.22

0.12

8.69

8.9

9.99

0

8.69

8.93

9.89

0.21

8.71

8.94

9.77

0.33

8.75

8.98

9.68

0.44

8.78

9.03

9.64

0.52

8.82

9.09

9.59

Source: From Wesselingh, J.A. and Krishna R., Mass Transfer in Multicomponent Mixtures, Delft University Press, Delft, the Netherlands, 2000.

TABLE A.11.4 Diffusion of NaCl and Water for Cheese Immersed in a Brine (Figure 11.9) Salt data: x (m)

% Salt in Cheese Moisture

0.001 0.0025

7.76 8.12

0.0035

9.66

0.0043

10.56

0.0053

11.82

0.0069

12.64

0.0084

13.99

0.0096

14.62

0.0104 0.0118

14.98 15.7

0.0127

15.16

0.0137

16.25

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TABLE A.11.4 (continued) Diffusion of NaCl and Water for Cheese Immersed in a Brine (Figure 11.9) Salt data: x (m)

% Salt in Cheese Moisture

0.0147

16.88

0.0159

17.24

0.0169

17.24

0.0182

18.05

0.0196

18.86

0.0204 0.0224

18.59 18.32

0.0224

19.04

0.0231

19.68

0.0243

19.49

0.0249

20.31

0.0259

19.86

0.0278

20.4

0.0286 0.03

20.58 21.3

0.0312

21.48

0.0325

21.84

0.0335

21.3

0.0345

21.84

0.0363

22.29

0.0371

22.47

0.038 0.0388

22.38 22.11

0.0396

22.56

0.0412

22.56

0.0433

22.83

0.0453

22.65

0.0473

22.92

0.049

22.83

0.0518 0.0537

22.92 23.29

0.0559

22.83

0.0573

22.83

0.059

22.74

0.061

22.74

0.0625

22.83

0.0649

22.83

Water data: x (m)

% Water

0.0008

40.91

0.0037

38.58 (continued )

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TABLE A.11.4 (continued) Diffusion of NaCl and Water for Cheese Immersed in a Brine (Figure 11.9) Water data: x (m)

% Water

0.0051

37.77

0.0076

36.31

0.0098

34.85

0.0135

33.03

0.0159

32.23

0.0182 0.0224

30.69 29.67

0.0247

28.72

0.0273

28.07

0.0335

26.75

0.0369

26.17

0.0418

25.73

0.0496

25.15

0.0541 0.0643

25.07 25.15

Source: From Payne, M.R. and Morison, K.R., Int. Dairy J., 9, 887, 1999.

TABLE A.11.5 First-Order Kinetic Plot for Coalescence of Emulsions (Figure 11.14) ln N0=Nt Time (h)

Butter Oil

Peanut Oil

0

0

0.87

0.12

0

10

0.28

0.22

23 31

0.23 0.3

0.15 0.31

47

0.41

0.56

55

1.31

1.13

80

1.8

0.79

90

2

0.66

112

2.43

0.88

124

2.89

1.26

148 175

2.64 3.12

1.52 1.58

200

3.05

1.44

220

2.46

250

2.43

275

2.91

300

3.05

Source: From Das K.P. and Kinsella J.E. Droplet size and coalescence stability of whey protein stabilized milk fat peanut oil emulsions. J Food Sci 58:439–444, 1993.

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TABLE A.11.6 Crystallization Kinetics of Hardened Palm Oil at 408C (Figure 11.16) Time (min)

% Solid Fat

0

0

2.5

0

6.6

0.3

9.5

0.7

11.9

1.1

15.2

1.4

19.0

2.5

21.0 24.3

3.2 4.7

27.6

5.5

30.1

7.3

35.9

10.7

40.4

13.6

45.0

17.3

50.4

20.7

55.7 59.9

24.0 27.0

64.8

29.6

71.0

31.4

74.7

32.5

80.5

33.2

85.0

34.3

90.4

34.3

94.9

34.6

Source: From Singh, A.P., Bertolli C., Rousset P.R., and Marangoni A.G. Matching Avrami indices Achieves similar hardnesses in palm oil-based fats. J Agric Food Chem 52:1551–1557, 2004.

TABLE A.11.7 Texture of Potato Cubes as a Function of Time after Cooking at 808C (Figure 11.17) Time (min)

Compression Force (N)

0

118.59

3

100.68

5 6

89.38 79.95

10

60.16

15

52.58

20

37.48

30

33.64

40

21.32

50

18.42

60 75

21.16 10.7

90

12.47

105

12.35

Source: From Stoneham, T.R., Lund, D.B., and Tong, C.H., J. Food Sci., 65, 968, 2000.

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TABLE A.11.8 Kinetics of Texture Changes of Potatoes as a Function of Cooking Temperature and Time (Figure 11.18) Time (min)

Failure Stress (N)

908C 0

100

2 3

61.3 47.39

5

37.42

6

29.12

7

22.47

10

17.75

12

13.37

1038C 0

100

0.8

61.35

1.6

36.57

2

28.63

2.5

22.34

3.0 3.5

17.37 13.05

4

10.06

4.5

7.4

5

6.06

6.0

4.7

7

4.01

9

3.58

11 12

2.52 2.49

1208C 0

100

0.2

61.38

0.5 0.6

36.61 29.02

0.7

22.41

0.9

16.8

1.0

13.49

1.2

9.52

1.5

6.21

1.9

4.54

2.3 2.7

3.53 2.53

3.5

2.5

4.9

2.44

6.8

2.03

Source: From Rahardjo, B. and Sastry, S.K., Trans. I. Chem E., 71, 235, 1993.

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TABLE A.11.9 Kinetics of Water Uptake by Cereals Immersed in Whole Milk at 558C (Figure 11.19) Time (s)

Mt M0 g water=100 g Initial Weight

290 290

78.1 86.8

520

102.4

520

112.8

1050

133.8

1050

144.2

1580

149.5

1580

161.6

1550 2080

172 161.7

2080

173.9

2590

163.6

2590

184.4

2590

193

3140

167.2

3120

181

3120 3650

191.4 181.1

3650

191.5

4200

169.1

4180

198.6

4180

207.3

4760

191.8

4730

202.2

4730 5260

209.1 200.6

5260

209.2

5260

226.6

5790

188.5

5790

200.7

6320

207.7

6270

218.1

6270 6850

245.9 178.4

6850

195.7

6820

209.6

7380

216.6

7350

225.3

Source: From De Fatima Machado, M., Oliveira, F.A.R., and Cunha, L.M., Int. J. Food Sci. Technol., 34, 47, 1999.

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TABLE A.11.10 Effect of Temperature on Partition Coefﬁcients (Figure 11.22) Methyl Acetate in Water

Ethyl Acetate in Water

Isopentyl Acetate in Coffee Oil

25

0.00527

0.00694

0.000198

30

0.00672

0.0092

35

0.00872

T0 (8C)

0.00027 0.000359

40

0.0152

50

0.000809

Source: From Kieckbusch, T.G. and King, C.J., J. Agric. Food Chem., 27, 504, 1979.

TABLE A.11.11 c in Solution (nmol dm3)

Partitioning of Methanethiol (Figure 11.23) c in Headspace above Aqueous Solution (nmol dm3)

0

0

0.12

0.3

c in Headspace above Liquid Milk Fat (nmol dm3)

0.36

0.21

0.72 0.96

0.76 1.06

1.08

0.3

1.92

0.58

2.16

3.61

2.22

0.85

2.46

0.64

3.66

0.85

4.49 4.85

6.27

1.91

7.01

10.06

7.61

1.76

7.79

10.12

8.15

2.76

16.9

4.03

Source: From Van Boekel M.A.J.S. and Lindsay R.C. Partitioning of cheese volatiles over vapour, fat and aqueous phases. Neth Milk Dairy J 46:197–208, 1992.

TABLE A.11.12 Dependence of Activity Coefﬁcients of Volatile Components on Sucrose Concentration (Figure 11.24) Sucrose (kg kg1) 0 0.2

log fi1 Ethyl Acetate

log fi1 Butylacetate

log fi1 Hexylacetate

1.82 1.93

3.13 3.24

4.39 4.54

0.4

2.1

3.43

4.82

0.6

2.3

3.71

5.3

0.7

2.49

Source: From Bruin S. and Luyben K.C.A.M. Drying of food materials: A review of recent developments. In: Mujumdar A.S. (ed.) Advances in Drying, Vol. 1, pp. 155–215. Hemisphere Publishing Corporation, Washington, 1980.

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TABLE A.11.13 Release Kinetics of Linalool and Ethylhexanoate (Figure 11.28)

Time (s)

Ethylhexanoate % of Equilibrium Concentration

Linalool % of Equilibrium Concentration

0

0

12

0

23.68

13

6.96

25 69

32.87

80

49.03

11.42

110

17.83

180

27.3

215

72.15

260

32.6

300

74.38

400 410

82.19

42.91

1800

80

1850

85.57

4500

95.68

7250

99.94

7500

100

9000

100

Source: From Juteau A., Doublier J.L., and Guichard E. Flavor release kinetics from i-carrageenan matrices: A kinetic approach. J Agric Food Chem 52:1621–1629, 2004.

TABLE A.11.14 Headspace Concentrations of Volatiles upon Dilution of the Headspace (Figure 11.30) Time (s) 0 3.6

Dimethylpyrazine c=c0 1 0.99

5.2

Diacetyl c=c0 1

Acetaldehyde c=c0

Dimethylsulﬁde c=c0

1

1 0.96

0.88

0.84

0.84

0.74

0.95

6.8

0.97

11.7

0.96

13.4

0.84 0.79

16.5 19.6

0.95

23.1 26

0.93

0.75

31

0.8

32.7

0.73

35.9

0.93

45

0.93

0.69 0.72

0.64

49

0.48

52

0.61

58

0.59 (continued )

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TABLE A.11.14 (continued) Headspace Concentrations of Volatiles upon Dilution of the Headspace (Figure 11.30) Time (s)

Dimethylpyrazine c=c0

Diacetyl c=c0

Acetaldehyde c=c0

Dimethylsulﬁde c=c0

60

0.41

63

0.71

66

0.56

68

0.34

73

0.93

80 85

0.28 0.7

93

0.24 0.21 0.5

106

0.94

112

0.71

118

0.13

126

0.46

146.5

0.69

156 176

0.1 0.43

0.93

209

0.69

0.1

0.69

0.09

235 265

0.93

0.67

0.38

0.09

0.38

0.1

300

0.68

0.38

0.09

350

0.67

0.38

0.09

0.67

0.37

0.09

400

0.94

Source: From Marin M., Baek I., and Taylor A.J. Volatile release from aqueous solutions under dynamic headspace dilution conditions. J Agric Food Chem 47:4750–4755, 1999.

TABLE A.11.15 Partitioning of Benzoic Acid over Aqueous and Oil Phase (Figure 11.32) caq (mmol dm3)

coil=caq

1.73

4.76

1.78

4.64

3.23 3.33

5.22 5.12

4.53

5.67

4.55

5.54

5.71

6.03

5.91

5.82

6.63

6.58

6.75

6.46

7.64 7.64

6.99 6.86

9.38

7.51

9.46

7.4

Source: From Wedzicha B.L., Zeb A., and Ahmed S. Reactivity of food preservatives in dispersed systems. In: Dickinson E. (ed.) Food Polymers, Gels and Colloids, London, pp. 180–193. Royal Society of Chemistry, 1991.

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Bibliography and Suggested Further Reading General References about Physical Changes Aguilera J.M. and Lilford P. Food materials science. Principles and practice. In: Food Engineering Series, Barbosa-Cánovas G.V. (Ed.). New York: Springer, 2008. Atkins P.W. Physical Chemistry, 6th ed. Oxford: Oxford University Press, 1999. Dickinson E. An Introduction to Food Colloids. Oxford: Oxford University Press, 1992. McClements D.J. (Ed.) Understanding and Controlling the Microstructure of Complex Foods. Cambridge: Woodhead, 2007. Walstra P. Physical Chemistry of Foods. New York: Marcel Dekker Inc., 2003. Walstra, P. and Van Vliet T. Dispersed systems: Basic considerations. In: Fennema’s Food Chemistry, 4th ed. Chapter 13, Damodaran, S., Parkin, K., and Fennema, O.R. (Eds.), Boca Raton, FL: CRC Press, 2008. General Food Engineering References Bird R.B., Stewart W.E., and Lightfoot E.N. Transport Phenomena, 2nd ed. New York: John Wiley, 2002. Fryer P.J., Pyle D.L., and Rielly C.D. Chemical Engineering for the Food Industry. London: Blackie Academic Professional, 1997. Heldman D.R. and Lund D.B. Handbook of Food Engineering. New York: Marcel Dekker, 1992. Singh R.P. and Heldman D.R. Introduction to Food Engineering. 3rd ed. London: Academic Press, 2001. About Fickian Diffusion Crank J. The Mathematics of Diffusion. 2nd ed. Oxford: Clarendon Press, 1990. About Maxwell–Stefan Diffusion Payne M.R. and Morison K.R. A multi-component approach to salt and water diffusion in cheese. Int Dairy J 9:887–894, 1999. Taylor R. and Krishna R. Multicomponent Mass Transfer. New York: Wiley, 1993. Wesselingh J.A. and Krishna R. Mass Transfer in Multicomponent Mixtures. Delft, the Netherlands: Delft University Press, 2000. About Flocculation Ives K.J. The Scientiﬁc basis of ﬂocculation. Alphen aan de Rijn, the Netherlands: Sijthoff and Noordhoff, 1978. Overbeek J.T.G. Kinetics of ﬂocculation. In: Colloid Science, Vol. I. Irreversible Systems, Kruyt H.R. (Ed.). Amsterdam, the Netherlands: Elsevier, 1952. Rajagopalan R. Stability of colloidal dispersions: A thermodynamic approach. Water Sci Technol 27:117–129, 1993. About Fractal Aggregation Bremer L.G.B. Fractal aggregation in relation to formation and properties of particle gels. PhD thesis, Wageningen University, the Netherlands, 1992. Bremer L.G.B., Walstra P., and Van Vliet T. Estimations of the aggregation times of various colloidal systems. Colloids Surf A 99:121–127, 1995. Dickinson E. On gelation kinetics in a system of particles with both weak and strong interactions. J Chem Soc Faraday Trans 93:111–114, 1997. Meakin P. Fractal aggregates. Adv Colloid Interface Sci 28:249–331, 1988. Weitz D.A., Lin M.Y., and Lindsay H.M. Universality laws in coagulation. Chem Intell Lab Systems 10:133–140, 1991.

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About Emulsions McClements D.J. Food Emulsions: Principles, Practices and Techniques. Boca Raton, FL: CRC Press, Taylor & Francis, 2005. Walstra P. Emulsion stability. In Encyclopedia of Emulsion Technology, Vol. 4, Chapter 1, Becher P. (Ed.) New York: Marcel Dekker Inc., 1996. Walstra P. Emulsions. In: Fundamentals of Interface and Colloid Science, Vol. V. Soft Colloids, Chapter 8, Lyklema J. (Ed.). London: Academic Press, 2005. About Packaging Han J.H. Innovations in Food Packaging, Food Science and Technology, International Series, p. 496. Amsterdam: Elsevier, 2005. About Release of Volatiles De Roos K.B. and Graf E. Nonequilibrium partition model for predicting ﬂavor retention microwave and convection heated foods. J Agric Food Chem 2204–2211, 1995. Gosh S. and Coupland J.N. Inﬂuence of food microstructure on ﬂavor interactions. In: Understanding and Controlling the Microstructure of Complex Foods, McClements D.J. (Ed.). Cambridge: Woodhead, 2007. Harrison M. and Hills B.P. A mathematical model to describe ﬂavour release from gelatine gels. Int J Food Sci Technol 167–176, 1996. Harrison M., Hills B.P., Bakker J., and Clothier T. Mathematical models of ﬂavor release from liquid emulsions. J Food Sci 62:653–664, 1997. Harrison M. and Hills B.P. Mathematical model of ﬂavour release from liquids containing aroma-binding macromolecules. J Agric Food Chem 45:1883–1890, 1997. Hills B.P. and Harrison M. Two-ﬁlm theory of ﬂavour release from solids. Int J Food Sci Technol 30:425–436, 1995. Klooster J.R., Druaux C., and Vreeker R. Air-liquid partition coefﬁcients of aroma volatiles in frozen sugar solutions. J Agric Food Chem 53:4503–4509, 2005. Overbosch P., Afterof W.G.M., and Haring P.G.M. Flavor release in the mouth. Food Rev Int 7:137–184, 1991. Rabe S., Krings U., and Berger R.G. In vitro study of the inﬂuence of physiological parameters on dynamic in-mouth ﬂavour release from liquids. Chem Senses 29:153–162, 2004. Seuvre A.M., Philippe E., Rochard S., and Voilley A. Kinetic study of the release of aroma compounds in different model food systems. Food Res Intern 40:480–492, 2007. Van Ruth S.M. and Roozen J.P. Delivery of ﬂavours from food matrices. In: Food Flavour Technology, Chapter 6, Taylor, A.J. (Ed.). Shefﬁeld, England: Shefﬁeld Academic Press, 2002. About Kinetics of Water Sorption and Moisture Migration De Fatima Machado M., Oliveira F.A.R., and Cunha L.M. Effect of milk fat and total solids concentration on the kinetics of moisture uptake by ready-to-eat breakfast cereal. Int J Food Sci Technol 34:47–57, 1999. De Fatima Machado M., Oliveira F.A.R., Gekas V., and Singh P. Kinetics of moisture uptake and soluble-solids loss by puffed breakfast cereals immersed in water. Int J Food Sci Technol 33:225– 237, 1998. Fu Y.C., Tomg C.H., and Lund D.B. Moisture migration in solid food matrices. J Food Sci 68:2497–2503, 2003. Nisha P., Singhal R.S., and Pandit A.B. Kinetic modelling of texture development in potato cubes (Solanum tuberosum L.), green gram whole (Vigna radiate L.) and red gram splits (Cajanus cajan L.). J Food Eng 76:524–530, 2006.

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Rahardjo B. and Sastry S.K. Kinetics of softening of potato tissue during thermal treatment. Trans I ChemE 71:235, 1993. Ramesh M.N., Sathyanarayana K., and Girish A.B. Kinetic modeling of vegetable cooking at 1008C. J Texture Studies 257–264, 1996. Risbo J. The dynamics of moisture migration in packaged multi-component food systems. I. shelf life predictions for a cereal-raisin system. J Food Eng 58:239–246, 2003. Risbo J. The dynamics of moisture migration in packaged multi-component food systems II: Analytical solutions and comparison to experimental moisture transfer rate results. J Food Eng 58:247–252, 2003. Rizvi A.F. and Tong C.h. Fractional conversion for determining texture degradation kinetics of vegetables. J Food Sci 62:1–7, 1997. Stoneham T.R., Lund D.B., and Tong C.H. The use of fractional conversion technique to investigate the effects of testing parameters on texture degradation kinetics. J Food Sci 65:968–973, 2000. Verlinden B.E., Debarsy T., Debaerdemaeker J., and Deltour R. Modelling the mechanical and histological properties of carrot tissue during cooking in relation to texture and cell wall changes. J Texture Studies 15–28, 1996. About Activities Carelli A.A., Crapiste G.H., and Lozano J.E. Activity coefﬁcients of aroma compounds in model solutions simulating apple juice. J Agric Food Chem 39:1636–1640, 1991. Fanelli A. Explaining activity coefﬁcients and standard states in the undergraduate physical chemistry course. J Chem Educ 63:112–114, 1986. Kieckbusch T.G. and King C.J. Partition coefﬁcients for acetates in food systems. J Agric Food Chem 27:504–507, 1979. Letcher T.M. and Battino R. An introduction to the understanding of solubility. J Chem Educ 78:103– 111, 2001. About Partitioning of Weak Acids Wilson D.G., Wilson D.R., and Waspe C.R. Weak acids: Dissociation in complex buffering systems and partitioning into oils. J Sci Food Agric 80:471–476, 2000.

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12 Kinetics of Microbial Growth

12.1

Introduction

The microbial quality of food is of utmost importance. It is in fact the ﬁrst and foremost task of a food technologist to produce safe food. The risk of contamination and growth of pathogenic microorganisms is much larger than that of toxic or mutagenic substances. Foods are very vulnerable to microbes because the required nutrients for microbial growth are usually present in sufﬁcient amounts. In addition, foods may contain spoilage-inducing microorganisms. Thus, it is very important to be able to calculate how fast microorganisms will grow in a food. Having such knowledge is the basis for prediction of shelf life in relation to microbial activity, calculation of risks, and optimization of fermentation processes. Microbial growth is typically a kinetic problem, so it is appropriate to discuss some principles in this book. However, at the same time microbial kinetics is different from chemical kinetics discussed hitherto. In previous chapters, kinetics was related mostly to molecular events but with microbial kinetics it is about changes in numbers of cells, and, though superﬁcially the equations may look the same, the underlying principle is different: it is about population dynamics. Growth, resulting in the increase of number of cells, is the result of uncountable molecular events, and requires a different treatment. The kinetics of microbial cell growth depends on the extracellular chemical environment (pH, ionic strength, water activity, presence of nutrients, temperature) and on the intracellular environment (mainly resulting in numerous enzymatic reactions). The resulting growth is the interplay between extracellular and intracellular conditions. Predictive microbiology, as this ﬁeld of research is called, has been booming the past two decades. However, the reader is advised that this ﬁeld is rapidly expanding and too large to do justice to in this chapter. Next to some books that review the state of the art, food microbiological journals contain many articles on this topic, so the interested reader is referred to the literature if he=she is interested in more details; some selected references are given at the end of this chapter. There are quite a few growth models, some of which include the (presumed) effect of pH, water activity and temperature, mainly via empirically derived polynomials because mechanistic insight is not (yet) available. The topic is under development and rapidly evolves. At this stage it is useful to distinguish between kinetic models and stochastic models. Kinetic models are suitable for describing changes in numbers of cell, in other words, the extent of growth, whereas stochastic models are apt to describe probabilities of growth or no growth, or even death. Stochastic models are useful for quantitative microbial risk assessment (QMRA), a topic that we will not address in this book. We limit ourselves here to some typical kinetic growth models. We only give a short basic introduction in Section 12.2; as 12-1

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mentioned above, there is so much development in this area currently that it makes no sense to present the state of the art in much detail. One distinguishes between primary, secondary, and tertiary growth models in predictive microbiology. Primary models describe the growth or inactivation of microorganisms (as changes in numbers of cells), secondary models describe how kinetic parameters from the primary models depend on environmental conditions (such as pH, water activity, etc.), and tertiary models are models that attempt to describe all responses of microorganisms to their environment in experts systems or decision support systems.

12.2

Primary Growth Models

Growth models are developed with the aim to have a tool available that allows prediction of the number of microorganisms as a function of conditions as prevalent in food (notably water activity, aw, temperature, pH, and other effects, such as compounds that stimulate or inhibit growth). Incidentally, it is questionable whether aw is directly related to microbial growth; it may well be that aw is related to another mechanism, but it seems to act as a good indicator, and it is therefore modeled frequently as an independent factor. Predictive microbiology is complicated both by variability and uncertainty: we cannot yet apply kinetics based on mechanistic insight, due to lack of knowledge concerning microbial physiology in relation to the food matrix, and adaptive responses of the microbes to changing conditions. One can express microbial growth as the change in population size N (the absolute number of microorganisms per gram or per milliliter) or as the logarithm of the relative population size, ln(N=N0) or log(N=N0). The growth rate is by deﬁnition dN(t)=dt, i.e., the absolute increase in cell concentration per unit time. The speciﬁc growth rate, i.e., the increase in cell concentration per unit time per cell is: dN(t) d ln N(t) m(t) ¼ dt ¼ N(t) dt

(12:1)

Usually, a microbial growth curve is of a sigmoidal nature (Figure 12.1) and a model should then of course be cast in such a form. The ﬁrst phase, in which no growth occurs, is called the lag phase, Exponential phase

Lag phase

Stationary phase

ln N/N0

As = ln N/Nmax

m max

l

Inflection point

Time

FIGURE 12.1 Schematic depiction of a microbial growth curve. l is the lag time, mmax the maximum speciﬁc growth rate (the tangent in the inﬂection point), As the asymptotic value.

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characterized by a lag time l, in which microorganisms have to adapt to the medium in which they are present. Some damaged cells will die, some will recover, and some will start to grow. This results in a more or less stable (or sometimes slightly declining) number of microorganisms. The lag phase is difﬁcult to model, it depends strongly on the prior history of the cells. After this adaption phase, the exponential phase develops characterized by the maximum speciﬁc growth rate mmax ¼ m(t), the tangent in the inﬂection point of a ln N versus t plot, while the lag time l is deﬁned as the x-axis intercept of this tangent, see Figure 12.1. This is followed by a stationary phase (due to the exhaustion of nutrients, or the formation of toxic products such as alcohol or lactic acid) in which an asymptotic value As (expressed as logarithm of number of microorganisms) is reached. Eventually the number of microorganisms may decline in the so-called dying off phase, but this is not considered here. When the asymptotic phase is reached, foods are so much spoiled that they will not be consumed anymore. We will now consider some equations that are used to describe microbial growth curves. In fact, all these models can be related to one generic growth model, and can be considered as special cases of the generic growth model. We will not consider these relations here.

12.2.1

Differential Equations

Just as with chemical rate equations, ordinary differential equations can be set up to describe microbial growth, but as remarked in the introduction to this chapter it is about population growth, not about chemical reactions. The logistic function A logistic function is able to describe the S-shaped curve shown in Figure 12.1. An application of the logistic function is the so-called Verhulst equation (described by Verhulst in the nineteenth century) to model population growth. It is based on the two assumptions that the growth rate dN(t)=dt is proportional to (1) the momentary existing population size N(t), and to (2) the remaining resources available to the existing population. Expressed mathematically this results in dN(t) N(t) ¼ k N(t) 1 dt Nmax

(12:2)

Nmax represents the number of cells that the system can support (carrying capacity) that is to say that the population starts at N0 and asymptotically reaches Nmax. The proportionality constant k represents the growth rate constant (dimension time1). Equation 12.2 is basically a rate equation. When N(t) Nmax, the ﬁrst term in the right-hand side of Equation 12.2 dominates (i.e., growth is not limited by depletion of resources), and when N(t) ! Nmax the second term dominates (i.e., growth becomes limited by depletion of resources). Equation 12.2 can be integrated: N(t) ¼

Nmax N0 exp (kt) Nmax þ N0 ( exp (kt 1))

(12:3)

Equation 12.3 describes the characteristic shape shown in Figure 12.1 but note that the number of cells is obtained as a function of time (Figure 12.2A). However, this shape disappears when the logarithm of N is taken (Figure 12.2B).* In microbiology, it is common to work with logarithmically transformed numbers because the numbers of cells span several orders of magnitude; moreover, taking logarithms appears to stabilize the variance because of the heteroscedasticity of bacterial cell counts (Chapter 7). It is probably best to

* Some of the graphs in this chapter were produced using the freeware available on the Web as Mathematica ﬁles or as Excel ﬁles: http:==www-unix.oit.umass.edu=aew2000=

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12-4

900,000

14

800,000

13

700,000

12 11 10

500,000

ln N (t)

N (t )

600,000

400,000 300,000

8 7

200,000

6

100,000

5 4

0 0 (A)

9

10

20 30 Time (h)

40

50

0

(B)

10

20

30

40

50

Time (h)

FIGURE 12.2 Simulated growth curve using the integrated form of the Verhulst Equation 12.3 when N(t) is plotted (A) or when ln N=N0 is plotted (B). N0 ¼ 1000, k ¼ 0.5 h1, Nmax ¼ 106.

consider microbial growth data as log-normally distributed. Taking logarithms, however, has an impact on the equations. We ﬁnd for the logistic equation the following transformation: dY(t) Y(t) ¼ k Y(t) 1 dt YAs N(t) Y(t) ¼ ln N0 Nmax YAs ¼ ln N0

(12:4)

In Equation 12.4 the natural logarithm base e (ln) is used, but this can be equally well the logarithm base 10 (log). The reader is advised that although there is no principal distinction between ln and log, there is a numerical difference: ln N ¼ ln10 3 log N 2.303 3 log N. Sometimes growth is expressed as ln N and sometimes as log N and the experimental data should of course be expressed in the same way as in the model; as obvious as this may seem, mistakes are made because of this. It is important to realize that integration of Equation 12.4 leads to much longer apparent ‘‘lag times’’ than integration of Equation 12.2: see Figure 12.3, which should be compared to Figure 12.2B. As an extension of the logistic Verhulst model one can also postulate that the momentary growth rate is not proportional to N(t) and [1 N(t)=Nmax] but to N(t)a and [1 N(t)=Nmax]b: dN(t) N(t) b ¼ k N(t)a 1 dt Nmax

(12:5)

and the same can be done for the logarithmic version: dY(t) Y(t) b ¼ k Y(t)a 1 dt YAs

(12:6)

This adds more ﬂexibility to the models. If a < 1, this implies that the microorganism does not use its full capacity to grow exponentially, while a > 1 implies that it exceeds this capacity. If b > 1, this means that

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12-5

8 7

Y(t)=ln(N(t)/N0)

6 5 4 3 2 1 0

0

10

20 30 Time (h)

40

50

FIGURE 12.3 Simulated growth curve using the integrated form of Equation 12.4. N0 ¼ 1000, k ¼ 0.5 h1, YAs ¼ 7 (corresponding to Nmax ¼ 106 ).

the microorganism is more sensitive to resource depletion than for the case b ¼ 1 while the situation that b < 1 implies that it is less sensitive to this. There is, however, not an analytical solution to these last two equations, they can only be solved via numerical integration. Freeware is offered however on the web to do this, see footnote on p. 12-3. The Baranyi–Roberts model. The Baranyi–Roberts model is actually based on the logistic model discussed above. However, as shown in Figure 12.2B the logistic model is not able to ﬁt situations in which a long lag time is prominent. Therefore, a factor is introduced in the Baranyi–Roberts model to cope with longer lag times and in doing so, an attempt was made to base it on semimechanistic principles. It is semimechanistic because it uses an adjustment function that accounts for the physiological state of the cells. It is reasoned that the speciﬁc growth rate m depends on the amount of an intracellular compound q(t) . The change in this described by a function q(t). The adjustment function takes on the form q(t) þ 1 function over time is described by: dq ¼ mmax q(t) dt

(12:7)

When the function q(t) increases, implying that the amount of the critical substance per cell increases so that it becomes actually no longer rate limiting, the adjustment function thus approaches 1. In this way the transition from the lag phase to the exponential phase is characterized, as in the exponential phase the cells reach their maximal growth rate. For the transition from the exponential phase to the stationary phase the following differential equation is introduced: dN q(t) N m ¼ mmax N 1 dt 1 þ q(t) Nmax

(12:8)

The exponent m characterizes the curvature before the stationary phase is reached. For m ¼ 1 Equation 12.8 becomes similar to the logistic Equation 12.2. Note that Equation 12.8 is not completely similar to the modiﬁed logistic equation in Equation 12.5. The two differential equations (Equations 12.7 and 12.8)

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12-6

lnN

mmax = dlnN/dt

h0

l

Inflection point

t

FIGURE 12.4 Schematic picture of a growth curve and the meaning of the parameters mmax, l, and h0 in the Baranyi–Roberts model.

describe the dynamic growth of cells. Note that Equation 12.8 does not use the logarithmic ratio but the number of cells directly. A differential equation valid for the logarithm of cell concentration is dY(t) 1 ¼ m (t)ð1 exp (Y(t) YAs )Þ dt 1 þ exp (Q(t)) max Q(t) ¼ ln q(t)

(12:9)

Y and YAs are deﬁned in Equation 12.4. A transformation of q(t0) for conditions at a constant environment for the cells is 1 ¼ mmax l h0 ¼ ln 1 þ q(t0 )

(12:10)

Thus, a link is proposed between the lag time and the maximum speciﬁc growth rate (in a constant environment): the lag time l is inversely proportional to mmax with h0 the proportionality constant. Figure 12.4 illustrates the meaning of the parameters. Incidentally, the Baranyi–Roberts model as given here is a simpler version; the original one is more complicated, as it contains six parameters.

12.2.2

Algebraic Equations

Algebraic models represent static models, i.e., models that are only a function of time, not of other variables. The integrated Verhulst Equation 12.3 is an example of this. Monod model. One of the oldest (primary) growth model is the Monod model: mX ¼ where m is the speciﬁc growth rate X is the concentration of biomass

mm S X Ks þ S

(12:11)

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S is the concentration of substrate Ks is a saturation constant mm is the maximum speciﬁc growth rate The Monod equation resembles the Michaelis–Menten equation discussed in Chapter 9. The Monod equation is mainly used in fermentation kinetics. It is less to suited for food microbiology because the transition from the exponential to the stationary phase as described by Monod’s model is not so relevant for microbial growth in foods: at such microbial levels the food will be spoiled. Lag exponential. The lag exponential model is a very simple model in which it is posed that at t < l no growth takes place, while at t > l growth takes place exponentially: ln N ¼ ln N0 at t < l ln N ¼ ln N0 þ m (t l)

(12:12)

No account is given of Nmax, and so it is only applicable for growth that does not slow down. The lag exponential model can be used to estimate the order of magnitude of growth, but obviously it has its limitations. Shifted logistic function. A modiﬁed version of the logistic equation is the shifted logistic function: Y(t) ¼ YAs

1 1 1 þ expðk(tc t)Þ 1 þ expðktc Þ

(12:13)

Here, the parameters have no direct physical meaning; it is a purely empirical model. Nevertheless, the parameters can be interpreted loosely as follows. YAs corresponds more or less to the asymptotic value As (i.e., ln Nmax=N0), k measures the steepness of the growth around the inﬂection point and is therefore related to the speciﬁc growth rate, and tc locates the inﬂection point. Modiﬁed logistic equation. In the literature, more frequent use is made of the modiﬁed logistic equation, in which parameters are reparameterized in the microbial interpretable parameters As, mmax, and l: ln

N ¼ N0

As 4mmax 1 þ exp (l t) þ 2 As

(12:14)

Figure 12.5 shows the ﬁt of the shifted logistic Equation 12.13 and the reparameterized modiﬁed logistic Equation 12.14 to some experimental data. Parameter estimates are given in Table 12.1. Both models seem to give a reasonable ﬁt, but based on the Akaike criterion (introduced in Chapter 7) the shifted logistic equation performs signiﬁcantly better, at least with this dataset. Incidentally, one could also estimate the initial number N0 if so desired. One then has to ﬁt Equation 12.14 to data in the form of ln N instead of ln N=N0 and estimate ln N0 as an extra parameter. Modiﬁed Gompertz model. Another frequently used model is the Gompertz equation (developed by Gompertz in 1825 to predict human mortality as a function of age): y ¼ a exp [exp (b ct)]

(12:15)

with a, b, c parameters of the model. The modiﬁed Gompertz equation has been reparameterized into the relevant microbiological parameters: ln

N m e ¼ As exp exp max (l t) þ 1 N0 As

(12:16)

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12-8

11

11

9

9

7

7

In N/N0

13

In N/N0

13

5

5

3

3

1

1

−1

0

50

100

(A)

150 200 250 Time (h)

300

350 400

−1

0

(B)

50

100

150 200 250 Time (h)

300

350

400

FIGURE 12.5 Fit of the shifted logistic Equation 12.13 (A) and the reparameterized modiﬁed logistic Equation 12.14 (B) to growth of L. monocytogenes at 58C. Dataset in Appendix 12.1, Table A.12.1.

with the following reparameterizations for the parameters a, b, c of the original Gompertz equation: a ¼ As m e b ¼ max l þ 1 As mmax e c¼ As

(12:17)

in which As is the natural logarithm of the asymptotic value of the relative population size, As ¼ ln (Nmax=N0), e ¼ exp(1) ¼ 2.718 Also, it is worth noting that the dependent variable y has been substituted by the (natural) logarithm of the relative population, as was done with the logistic equation. This three-parameter-model has proven to be adequate in many tested cases for isothermal growth. Figure 12.6 gives an example. The original logistic and Gompertz model can be considered as mechanistic models, but using the logarithm of number of cells, rather than the number of cells directly, turns the modiﬁed logistic and Gompertz model into empirical models. When used with logarithmic ratios the term ‘‘modiﬁed’’ signiﬁes this fact. The modiﬁed Gompertz model is asymmetric about the point of inﬂection which occurs at 1=e of the distance between the lower and upper asymptote, in contrast to the modiﬁed logistic equation which is symmetric about the inﬂection point, which occurs at 1=2 of the distance between the lower and upper asymptote. The rate obtained from the modiﬁed Gompertz equation is always the maximum speciﬁc growth rate mmax and occurs at an arbitrary inﬂection point; consequently, this may lead to an

TABLE 12.1 Parameter Estimates 95% Conﬁdence Intervals and Akaike Criterion (AIC) of the Shifted Logistic Equation and the Modiﬁed Logistic Equation Applied to the Growth of L. monocytogenes at 58C Model Shifted logistic equation Modiﬁed logistic equation

AIC 50.5

0

38.0

12.5

YAs ¼ 11.9 0.6

tc ¼ 164.6 5.9 h

k ¼ 0.019 0.003 h

As ¼ 11.2 0.4

l ¼ 75.6 10.7 h

mmax ¼ 0.060 0.006 h1

Note: Same data as in Figure 12.5.

DAIC

1

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12-9

14 12

ln(N/N0)

10 8 6 4 2 0

0

10

20 30 Time (h)

40

50

FIGURE 12.6 Example of the ﬁt of the modiﬁed Gompertz equation to growth of Salmonellae in a laboratory medium. Fit parameters: As ¼ 13.14, l ¼ 4.37 h, mmax ¼ 0.70 h1. Dataset is given in Appendix 12.1, Table A.12.2.

underestimation of the growth rate. Another disadvantage of the modiﬁed Gompertz equation is that at t ¼ 0, Y ¼ ln N=N0 6¼ 0; only for t ! 1 Y ! 0. Integrated Baranyi model. Integration of the differential equations in the Baranyi model given above at constant conditions of T, pH, aw, etc., leads to 0 ln

1

B N exp (mmax AB (t)Þ 1C C ¼ mmax AB (t) lnB A @1 þ Nmax N0 exp ln N0

(12:18)

in which AB (t) ¼ t þ

1 mmax

lnðexp (mmax t) þ exp (h0 exp (mmax t h0 )Þ

(12:19)

with h0 given in Equation 12.10. Equation 12.18 thus describes isothermal growth at constant conditions, like the earlier modiﬁed logistic and modiﬁed Gompertz equations, with parameters mmax, Nmax, N0, and h0 that can be estimated from growth curves. Figure 12.7 gives an example of the ﬁt to experimental data. The above discussed models were tested on a dataset in a model discrimination study. The dataset concerned the growth of Clostridium perfringens in ground beef. Table 12.2 shows the parameter estimates, Table 12.3 the results of the model discrimination study, and Figure 12.8 the resulting ﬁts. As explained in Chapter 7, the goodness of ﬁt (which can only be done if replicates are available) indicates the sampling probability of obtaining a greater ratio than the one obtained if the null hypothesis is true that the model performs well. If this probability is high, say higher than 0.01, there is no statistical reason to reject the model because of lack of ﬁt. The model with the lowest Akaike criterion and the highest posterior probability performs the best in the statistical sense. Though slight differences can be seen in the various ﬁts in Figure 12.8, all models seem to describe the data points quite well. This is conﬁrmed by the goodness-of-ﬁt result in Table 12.3; there is no serious lack of ﬁt with any of the models: all sampling probabilities are substantially higher than 0.01. Nevertheless, it is also clear that the goodness of ﬁt differs between the models and this goes along with the two

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14 12

lnN/N0

10 8 6 4 2 0

0

5

10 15 Time (h)

20

25

FIGURE 12.7 Fit of the Baranyi model (—) to growth of C. perfringens at 308C in boneless ham with ﬁt parameters mmax ¼ 1.15 h1, As ¼ 12.93, h0 ¼ 6.19. Dataset in Appendix 12.1, Table A.12.3.

TABLE 12.2 Parameter Estimates 95% Conﬁdence Intervals for the Models Tested on the Growth of C. perfringens in Ground Beef at 308C Modiﬁed Gompertz Model

Shifted Logistic Model

mmax ¼ 2.0 0.3 h1

k ¼ 0.5 0.1 h1

As ¼ 15.3 0.6 l ¼ 3.8 0.7 h

Modiﬁed Logistic Model

Baranyi–Roberts Model

mmax ¼ 2.0 0.4 h1

mmax ¼ 1.6 0.3 h1

As ¼ 15.0 0.6

As ¼ 14.8 0.6

l ¼ 4.1 0.8 h

h0 ¼ 5.3 2.1

YAs ¼ 15.3 0.6 tc ¼ 7.7 0.4 h

l ¼ 3.2 0.8 ha a

Calculated from h0 ¼ mmax 3 l.

TABLE 12.3

Results of Model Discrimination Test (See Table 12.2 and Figure 12.8) Modiﬁed Gompertz Model

Shifted Logistic Model

Modiﬁed Logistic Model

Baranyi–Roberts Model

AIC

3.01

2.89

4.3

7.1

DAIC

0

5.9

7.31

10.11

log (PB) Goodness of ﬁt

12.6 0.526

13.3 0.241

13.5 0.193

13.8 0.121

Note: AIC, Akaike criterion; log(PB), logarithm of the posterior probability. Goodness of ﬁt: Sampling probability of greater ratio. Results are presented from best to worst performing model.

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18 16 14

ln N/ N0

12 10 8 6 4 2 0 −2

0

10

20 Time (h)

30

40

FIGURE 12.8 Fit of the shifted-logistic (), modiﬁed logistic (-.-.-), modiﬁed Gompertz (. . . . ), and Baranyi– Roberts model (- - - -) to a dataset describing growth of C. perfringens in beef at 308C. Dataset in Appendix 12.1, Table A.12.4.

model discrimination tests applied, the Akaike criterion and the posterior probability criterion, both discussed in Chapter 7. The differences are not negligible and preference is given to the modiﬁed Gompertz model, while the Baranyi–Roberts model seems to perform to a lesser extent, at least from a statistical point of view. Of course, these results cannot be generalized to a general statement about the performances of these models. The results are only valid for this particular dataset. Many more datasets need to be investigated to come to a general conclusion. This exercise is only done here to show the possibilities of model discrimination and the application of techniques discussed earlier in the book to microbial growth models. As shown in Chapter 7, it is quite instructive to study the statistical characteristics of parameter estimates via bootstrapping and Monte Carlo simulations. This is certainly true for parameter estimates for microbial growth models as these are based on variable and possibly heteroscedastic data. Since the methods are exactly the same as given in Chapter 7 we will not repeat that exercise here.

12.3

Secondary Models

It is of course desirable to be able to predict what will happen if, for instance, temperature is changed. Figure 12.9 shows an example of the effect of temperature on the growth curves of C. perfringens on cooked boneless ham. It can be seen that the upper asymptote is hardly affected by temperature, but both the maximum growth rate and the lag time are. By doing measurements at various temperatures, as shown in Figure 12.9, one can get an impression of the temperature dependence of microbial growth. However, it is of course much easier to cast this dependence in a mathematical equation, just as the temperature dependence of a chemical reaction rate constant can be cast in the form of the Arrhenius equation or the Eyring equation, discussed in Chapter 5. However, as mentioned before, the temperature dependence of microbial growth cannot be described by the Arrhenius or Eyring equation because microbial growth is not a simple elementary reaction. If an Arrhenius plot of microbial growth is made anyway, it is frequently nonlinear. Therefore, empirical models are used.

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12-12

16 14 12

ln N/N0

10 8 6 4 2 0 −2

0

5

10 15 Time (h)

20

25

FIGURE 12.9 Growth curves of C. perfringens in cooked boneless ham at 308C (&), 358C (.), and 458C (~). The ﬁts shown are the ﬁts by the Baranyi–Roberts model. Dataset in Appendix 12.1, Table A.12.5.

These secondary models, as they are called, describe the dependence of m, l, and As on environmental conditions (temperature, pH, etc.). For the effect of temperature on mmax, for instance, an empirical square root relationship is proposed (also called the Ratkowsky square root model): mmax ¼ [b1 (T Tmin )]2

(12:20)

Tmin is the minimum temperature at which growth is observed, and b1 is a ﬁt parameter (dimension 8C time1). Equation 12.20 is based on the observation that the square root of the speciﬁc growth rate varies linearly with temperature at lower temperatures. The equation only uses temperature differences so it does not matter whether the units are in K or in 8C. Figure 12.10 shows an application of this model to data obtained for growth of C. perfringens in ham and Table 12.4 shows the parameter estimates. The ﬁt as shown in Figure 12.10 and the goodness-of-ﬁt measure are excellent as well as the precision of the parameter estimates. It should be remembered though that these parameters can only be used for the range on which they are based. Especially extrapolation to higher temperatures would be very dangerous as the speciﬁc growth rate will drastically decrease at some temperature. At higher temperatures, i.e., around and above the optimum and maximum temperature, the expanded Ratkowsky model is more appropriate: mmax ¼ fb2 ðT Tmin Þð1 exp½c2 (T Tmax )Þg2

(12:21)

with Tmin, Tmax, the minimum and maximum temperature for growth (in K or 8C), respectively, and b2 and c2 ﬁt parameters. A modiﬁcation of the expanded square root model, now known as the Zwietering equation, was proposed to prevent positive values of mmax at T > Tmax: mmax ¼ [b3 (T Tmin )]2 [1 expðc3 (T Tmax )Þ]

(12:22)

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12-13

5 4.5 4

m max /h−1

3.5 3 2.5 2 1.5 1 0.5 0 20

25

30

35

40

45

50

T (C)

FIGURE 12.10 Fit of the Ratkowsky model (Equation 12.20) to a dataset describing the maximum speciﬁc growth rate of C. perfringens as a function of temperature in ham. Dataset in Appendix 12.1, Table A.12.6.

It should be noted that these equations are only valid inside the interval TminTmax, in other words, only positive growth rates can be used in the estimation procedure. Figure 12.11 gives an example of the ﬁt of the Zwietering Equation 12.22 and the Ratkowsky model to the maximum speciﬁc growth rate of Listeria monocytogenes in milk, while Table 12.5 shows the parameter estimates. The ﬁt for the Ratkowsky equation is not all that well as can be seen in Figure 12.11, and it also follows from the goodness-of-ﬁt measure. The reason is obvious: the growth rate starts to decline at the higher temperatures, which cannot be taken into account in the Ratkowsky equation. The ﬁt for the Zwietering model seems to be OK, and this is substantiated by the goodness-of-ﬁt measure. There is a problem, however, with the estimate of parameter c3 as its conﬁdence interval is larger than the estimate itself, and the conﬁdence interval of Tmax is also quite high. It is quite clear that there is not enough information in the data to estimate Tmax properly, or put differently: the experimental design was not tailored to the use of this model to estimate Tmax. Just another example is given in Figure 12.12 and Table 12.6. In this case, there is no lack of ﬁt, as in the previous example with the Zwietering model, but the parameter estimates are much more precise, showing the beneﬁt of a well-designed experiment, if the purpose was to estimate the growth rate also above the optimum temperature. For the asymptotic value As the following equation could be used: As ¼ b4 [1 expðc4 (T Tmax )Þ]

(12:23)

TABLE 12.4 Parameter Estimates 95% Conﬁdence Intervals and Goodness of Fit of the Ratkowsky Equation 12.20 to the Data Shown in Figure 12.10

Estimate

b1 (8C h1)

Tmin (8C)

Goodness of Fit

0.020 0.007

10.6 2.1

0.991

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0.4 0.35 0.3

m max/h−1

0.25 0.2 0.15 0.1 0.05 0

0

5

10

15

20 25 T (C)

30

35

40

FIGURE 12.11 Fit of the Ratkowsky model (- - -, Equation 12.20) and the Zwietering model (—, Equation 12.22) to experimental data obtained for the maximum speciﬁc growth rate of L. monocytogenes in milk as a function of temperature. Dataset in Appendix 12.1, Table A.12.7. TABLE 12.5 Parameter Estimates 95% Conﬁdence Intervals and Goodness of Fit of the Ratkowsky Equation 12.20 and the Zwietering Equation 12.22 to the Data Shown in Figure 12.11 b1=b3 (8C h1) b1 ¼ 0.060 0.004 b3 ¼ 0.020 0.007

c3 (8C1)

Tmin (8C)

—

4.9 3.2

0.1 0.2

04

Tmax (K)

Goodness of Fit

—

0.037

41 14

0.57

1.4 1.2

m max/h−1

1 0.8 0.6 0.4 0.2 0

0

10

20

30

40

50

T (C)

FIGURE 12.12 Speciﬁc growth rate mmax as a function of temperature for growth of Lactobacillus plantarum in a MRS medium modeled according to Equation 12.22. Dataset in Appendix 12.1, Table A.12.8.

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TABLE 12.6 Parameter Estimates 95% Conﬁdence Intervals and Goodness of Fit of the Zwietering Model (Equation 12.22) Applied to the Speciﬁc Growth Rate Parameter mmax as Function of Temperature for Growth of Lactobacillus plantarum in a MRS Medium

Estimate

b3

c3

Tmin (8C)

Tmax (8C)

Goodness of Fit

0.041 0.007

0.16 0.06

43

44 0.3

0.352

However, usually the parameter As is not strongly temperature dependent (as can be seen in Figure 12.9 for instance) and may be considered approximately constant in many cases. The lag time is temperature dependent and several expressions are derived as secondary models. One example is: l ¼ exp

p1 0 T q1

(12:24)

which is a hyperbola model, with T 0 in 8C. Another one is the extended hyperbola model: l¼

p2 T 0 q2

m (12:25)

As an example of a secondary model for the lag time, Equations 12.24 and 12.25 were ﬁtted to experimental data describing lag time as a function of temperature for the growth of Salmonella typhimurium on chicken meat. The results are in Figure 12.13 and Table 12.7.

50 45 40 35

l (h)

30 25 20 15 10 5 0

0

10

20

30

40

50

T (C)

FIGURE 12.13 Fit of a secondary model for the lag time (Equation 12.24) as a function of temperature for the growth of S. typhimurium on chicken meat. Dataset in Appendix 12.1, Table A.12.9.

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TABLE 12.7 Parameter Estimates 95% Conﬁdence Intervals and Goodness of Fit for a Secondary Model for the Lag Time (Equation 12.24)

Estimate

p1 (8C)

q1 (8C)

Goodness of Fit

28.9 2.1

0.43 0.56

0.336

The goodness of ﬁt seems to be in order, and the imprecision for parameter estimate p1 is acceptable, but the one for q1 is much too high. It can be seen from the ﬁt in Figure 12.13 that the residuals are not well behaved at T 0 > 208C. A further statistical analysis showed that the data were not normally distributed: see Figure 12.14 for a normal probability plot (Chapter 7). A log transformation removed this problem but the ﬁt of the model actually got worse (not shown), and the goodness-of-ﬁt parameter was well below 0.0001. As a next attempt the extended hyperbola model in Equation 12.25 was modeled. Again, a log transformation was needed to stabilize the variance, and the resulting ﬁt and probability plot is shown in Figures 12.15 and 12.16, respectively, while the parameter estimates are shown in Table 12.8. The normal probability plot of the transformed data looks much better though there is still a hint of nonnormality (right skewed, see Figure 7.16). The ﬁt and the residuals are well behaved and the parameter estimates shown in Table 12.8 are also satisfactory, even though the extended hyperbola lag time model has one parameter more than the hyperbola equation (as a reminder: more parameters give a better ﬁt but less precise parameter estimates). This chapter shows that it is worthwhile to use residuals and normal probability plots as diagnostics, and to take action if the diagnosis is that something is not right. Schoolﬁeld model. An attempt has been made to model the temperature dependence of mmax based on mechanistic reasoning, the Schoolﬁeld model. The reasoning is that only one enzyme reaction is rate

3

2

Residuals

1

0

−1

−2

−3 −3

−2

−1

0 z-score

1

2

3

FIGURE 12.14 Normal probability plot for the data and the model shown in Figure 12.13.

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4.5 4 3.5 3

ln l

2.5 2 1.5 1 0.5 0 −0.5

0

10

20

T (C)

30

40

50

FIGURE 12.15 Fit of the extended hyperbola model (Equation 12.25) to the logarithm of lag time for growth of Salmonella on chicken. Same data as in Figure 12.13 but logarithmically transformed.

0.6 0.5 0.4

Residuals

0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −2.5

−1.5

0.5 −0.5 z-score

1.5

2.5

FIGURE 12.16 Normal probability plot for the data and the model shown in Figure 12.15.

TABLE 12.8 Parameter Estimates 95% Conﬁdence Intervals and Goodness of Fit for a Secondary Model for the Lag Time (Equation 12.23)

Estimate

p2

q2

m

Goodness of Fit

28.9 2.1

5.6 0.8

1.3 0.1

0.336

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controlling, following a zero-order reaction, and that this enzyme is reversibly denatured at very high and very low temperatures. The total amount of rate-controlling enzyme per cell is considered constant. Furthermore, the temperature dependence of the enzyme reaction as well as the inactivation reaction is supposed to follow an Arrhenius dependency (Equation 5.12). This results in the so-called Schoolﬁeld equation:

mmax

Ea A exp RT ¼ Eal Eah 1 þ Al exp þ Ah exp RT RT

(12:26)

A is the pre-exponential factor Ea the activation energy for the enzyme reaction, Al, Ah Eal, Eah, the pre-exponential factor and activation energy for the inactivation at low and high temperature, respectively As discussed in Chapter 7, this equation should be reparameterized because of the strong correlation between the parameters. Although this is not a problem as such, the problem with the Schoolﬁeld equation is that it contains many (six) parameters, and that the mechanistic basis is doubtful: it is not very likely that only one enzyme is rate controlling. This is reﬂected in the fact that the values derived for the parameters do in quite some cases not reﬂect biologically meaningful values. The Ratkowsky and Zwietering equations, although of an empirical nature, seem to be sounder from a modeling point of view. Secondary models for empirical parameters. As introduced above, some empirical models do not use the established microbiological parameters m, l, and As. For instance, in the shifted logistic Equation 12.16, the temperature dependence of the parameters YAs, k, and tc can be expressed as YAs (T) ¼ constant

(12:27)

k(T) ¼ k0 exp (ck T 0 )

(12:28)

tc (T) ¼ tc0 expðctc T0 Þ

(12:29)

These are purely empirical models that nevertheless can give a very good ﬁt, but they should, as with all empirical models, not be used outside the range for which they were derived. Figure 12.17 gives an example, obtained from ﬁtting the shifted logistic Equation 12.13 to growth curves of Pseudomonas species in ﬁsh. The resulting equations are just convenient ﬁts to the data obtained, without mechanistic meaning. Gamma concept and cardinal models. The so-called gamma concept relates so-called cardinal parameters (pH, T, aw) to each other on the assumption that they are not a function of the other parameters, so that they can be multiplied: gtotal ¼ g(T) g(aw ) g(pH)

(12:30)

The speciﬁc growth rate is related to temperature, pH, and water activity as follows: m ¼ mopt g(T)g(pH)g(aw )

(12:31)

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7

0.05 0.045

6

0.04 5

0.035 0.03

YA

s

k (h−1)

4 3

0.025 0.02 0.015

2

0.01 1

0.005

0 0 (A)

2

4 6 T (C)

8

0

10

0

2

4 6 T (C)

6

8

10

(B)

8

10

180 160 140 120 t c (h)

100 80 60 40 20 0

0

2

4

T (C)

(C)

FIGURE 12.17 Fits of empirical models to express temperature dependence of parameters of the shifted logistic Equation 12.13 to growth curves of Pseudomonas spp. on ﬁsh. YAs ¼ constant ¼ 5.86 (A), k ¼ 0.017 exp(0.125T 0 ) (B), tc ¼ 159.2exp(0.17T 0 ) (C). Dataset in Appendix 12.1, Table A.12.10.

in which g(T) ¼ g(pH) ¼

T Tmin Topt Tmin

2 (12:32)

(pH pHmin )(pHmax pH) (pHopt pHmin )(pHmax pHopt )

(12:33)

aw aw, min 1 aw, min

(12:34)

g(aw ) ¼

In this way, a correction is applied to the optimal growth rate if the pH, aw, or temperature are not optimal.

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Cardinal parameter models (CPMs) are also empirical secondary models and also assume that the inhibitory effect of factors such as temperature, pH, and water activity are multiplicative. CPM consist of a term (CMn) for each factor, while each term has a numerical value between 0 and 1 and acts as correction term to the optimal speciﬁc growth rate. Thus, when a term is unity, the growth rate is optimal for that factor. For optimal growth, all terms have to be unity. For CMn terms describing the effect of temperature, pH, water activity, a cardinal model would read: mmax ¼ mopt CM2 (T) CM1 (pH) CM2 (aw )

(12:35)

This equation can be expanded, if so desired, for any term CMn(X) that has an effect on the growth rate. Each term of CMn(X) is expressed as for X Xmin : CMn (X) ¼ 0 for Xmin < X < Xmax : CMn (X) ¼

(X Xmax )(X Xmin )n (Xopt Xmin )n1 (Xopt Xmin )(X Xopt ) (Xopt Xmax )((n 1)Xopt þ Xmin nX)

for X Xmax : CMn (X) ¼ 0

(12:36)

CPMs perform much like square root models discussed above. There is an extended literature on CPMs that will not be discussed here. Some references are given at the end of this chapter.

12.4

Nonisothermal Growth Modeling

The integrated modiﬁed logistic models, the modiﬁed Gompertz model, and the integrated Baranyi model are (semi)empirical models that are so-called time-explicit models because growth is an explicit function of time. This means that there are potential problems in describing time-varying changes in environmental conditions (notably temperature). Also, the modiﬁed gomperty model does not give fully correct values at t < l, as can be seen in Figure 12.6, as well as can be analytically derived from Equation 12.16. The problem becomes apparent when there is a sudden jump in temperature: the model also predicts a sudden jump in number of microorganisms (Figure 12.18). This is of course impossible: although growth rate will change, it cannot affect the number of microorganisms immediately. For this reason, there is a need for so-called dynamic growth models. Models that consist of differential equations such as the modiﬁed logistic model (Equation 12.4) and the Baranyi model, Equations 12.7 and 12.8 (which is actually based upon the logistic model) are capable to predict growth under dynamic conditions, such as varying temperatures. What is needed for this? First, the temperature proﬁle needs to be known, so that the temperature can be expressed as a function of time. Second, the parameters in the dynamic model must be made a function of temperature, and since temperature has been made a function of time, the parameters can be made time dependent also. As a next step, the differential equations can be integrated where the parameters are made time dependent as indicated. Using the Baranyi model for instance, the parameter mmax can be made temperature dependent using the square root model or the Zwietering model and using the known temperature–time dependency, mmax can be calculated at any time. Another way to describe population growth under varying temperature is due to Peleg and coworkers; they propose a method to estimate nonisothermal growth from isothermal growth curves. This is basically the same approach as discussed in Chapter 5 for the empirical modeling of chemical changes with variable temperature kinetics. Let us take the shifted logistic Equation 12.13 as an example. The momentary growth rate at constant temperature can be found by differentiating Equation 12.13 with respect to time:

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Kinetics of Microbial Growth

ln (N/N0)

12-21

T1 T0

Sudden temperature jump Time

FIGURE 12.18 Simulation showing the effect of a temperature jump (1.7 increase in mmax) on the Gompertz model (—) as compared to the growth curve at constant temperature (---).

dY(t) kYAs expðk(tc t)Þ ¼ dt T ð1 þ expðk(tc t ÞÞ2

(12:37)

This equation is valid at any temperature as long as this temperature is constant, but the values of the parameters change with temperature, so to make Equation 12.37 valid for every temperature we have to write: dY(t) k(T)YAs (T) expðk(T)(tc (T) t)Þ ¼ dt ð1 þ expðk(T)(tc (T) t ÞÞ2

(12:38)

As before, we need to make temperature a function of time from the known time–temperature relationship, and we need an expression that describes the dependence of the parameters as a function of temperature; for this we could use Equations 12.27 through 12.29. To relate time to the corresponding number of cells, we can reverse the original model (that describes the number of cells as a function of time) and calculate what Y(t) is at a certain time t* for any constant temperature. For the shifted logistic model this results in 1 expðk(T)tc (T)ÞðYAs (T) þ Y(t)ð1 þ expðk(T)tc (T)ÞÞÞ ln t* ¼ k(T) YAs (T) expðk(T)tc (T)Þ Y(t)ð1 þ expðk(T)tc (T)ÞÞ

(12:39)

So, we can now calculate the time that corresponds to the number of cells at a certain temperature. By making temperature a function of time and by combining Equations 12.38 and 12.39 we obtain the nonisothermal rate equation: dY(t) kðT(t)ÞYAs ðT(t)Þ expðkðT(t)Þðtc (T(t)) t*ÞÞ ¼ dt ð1 þ expðk(T(t))(tc (T(t)) t*)ÞÞ2

(12:40)

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Even though Equation 12.40 looks very awkward, it is a straightforward ordinary differential equation that can be solved numerically using appropriate software, but this can be done even in spreadsheets: see footnote on p. 12–3. It is basically the same approach as was discussed in Chapter 5 for variable temperature kinetics. The difference between Equations 12.40 and 12.37 is that Equation 12.37 is valid for any constant temperature, while Equation 12.40 is valid for a temperature that changes with time, i.e., nonisothermal conditions. Figure 12.19 and 12.20 gives an example using this approach. It concerns the growth of C. perfringens in ground meat during cooling after a heating process. Growth starts at temperatures below 458C. From isothermal experiments, the temperature dependence of the parameters of the shifted logistic equation was determined. The parameter YAs was virtually independent of temperature and was therefore held constant at a value of 5.8. The parameters k and tc were modeled with ad hoc empirical models (Figure 12.19). The results shown in Figure 12.20 are seen to give an excellent prediction of the nonisothermal dynamic growth; note that these are not model ﬁts but real predictions (i.e., not based on the same experiments for which the parameters were derived). Incidentally, this exercise can be done with any other equation than the shifted logistic equation, of course. The ﬁeld of predictive microbiology is rapidly developing, and it is not possible, also not very useful, to list all proposed equations. The ones that were given are reasonably well established, and most newly presented models are extensions or modiﬁcations of these.

12.5

Bayesian Modeling

1.6

16

1.4

14

1.2

12

1

10

0.8

8

0.6

6

0.4

4

0.2

2

0 20 (A)

tc (T ) (h)

k (T ) (h−1)

The Bayesian approach introduced in Chapter 7 can, of course, also be applied to microbial growth. As a ﬁrst example, we show a growth curve modeled using Markov Chain Monte Carlo (MCMC) integration as is possible with the program WINBUGS (introduced in Chapter 7). Figure 12.21 shows the code with the priors used. Figures 12.22, 12.23 and 12.24 show the results for ﬁtting the Gompertz model to growth data of Salmonellae while Table 12.9 shows the estimates for the parameters; the same data were used in Figure 12.6. Using the Bayesian approach just for simple growth models would not be a big advantage. However, the real advantage of this approach is that it allows tackling uncertainty and variability, which are

25

30

35 40 T (C)

45

0 20

50

(B)

25

30

35 T (C)

40

45

50

FIGURE 12.19 Empirical models describing temperature dependence of the shifted logistic equation parameters k(T) ¼ 0.27 þ 0.000855T2 (A) and tc ¼ 158 exp(0.09T) (B) for the case of isothermal growth of C. perfringens in ground ham. Dataset in Appendix 12.1, Table A.12.11.

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50

6

50

45

45 1

40 35

25 20

0.4

T (C)

30 0.6

35 30

3

25 20

2

15 10

0.2

40

4 log N/N0

0.8 log N/N0

5

15 10

1

5

5

0

0 0 (A)

1

2 3 Time (h)

4

T (C)

1.2

0 0

5 (B)

5

10

0 15

Time (h)

FIGURE 12.20 Fit of the shifted logistic model (Equation 12.40) to growth of C. perfringens in ground ham during cooling for two different cooling patterns. Dataset in Appendix 12.1, Table A.12.12.

important in predictive microbiology. Uncertainty refers to lack of knowledge on the values of model parameters (such as mmax), variability is due to biological variation in the behavior of the microorganism under study as well as in the food. The models discussed hitherto do not take into account uncertainty and variability explicitly. Bayesian hierarchical models on the other hand can take uncertainty and variability into account. The Bayesian approach was explained in Chapter 7, as well as a small introduction to hierarchical Bayesian models. Parameters in the Bayesian framework are considered as random

#WINBUGSmodel for bacterial growth model{ for (j in 1:N){ #likelihood Y[j]~dnorm(mu[j],tau); mu[j] 0, and the cumulative form is bW ! t F(t) ¼ exp aW

(13:7)

with aW and bW the two parameters of the distribution; aW is a scale parameter (a characteristic time) and bW is the so-called shape parameter for reasons that will become obvious below. In terms of a survival ratio S(t) ¼ N(t)=N0, the cumulative function is then

t S(t) ¼ exp aW

bW ! (13:8)

In microbiology, the logarithm of the survival ratio is commonly used: log S(t) ¼

bW 1 t 2:303 aW

(13:9)

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1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

S (t)

Frequency

13-6

0.5

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0 0 (A)

0.5

0.4

1 2 3 Time (arbitrary units)

4

0

1 2 3 Time (arbitrary units)

(B)

4

0 −0.5 −1

log S

−1.5 −2

−2.5 −3 −3.5 −4 (C)

0

1 2 Time (arbitrary units)

3

FIGURE 13.4 Simulations describing microbial inactivation using the Weibull distribution: density function (A), cumulative function (B), and a semilog plot (C) for aW ¼ 1 and bW ¼ 2.

The Weibull distribution corresponds to a concave upward survival curve if bW < 1 and concave downward if bW > 1. Figure 13.4 shows an example of the Weibull distribution. Interestingly, the Weibull distribution reduces to an exponential distribution for bW ¼ 1 and the cumulative distribution looks the same as the familiar ﬁrst-order equation: t S(t) ¼ exp aW

(13:10)

Although this exponential distribution has the same form as the ﬁrst-order Equation 13.3, the meaning of the parameters D and aW is different. D is the reciprocal of a ﬁrst-order rate constant, whereas aW represents the mean of the distribution describing the death times tc of the microbial population. It is worthwhile to explore the parameter aW a little bit further. At t ¼ aW, it represents a characteristic time at which the survival function log S(t) ¼ 0.434 (corresponding to S(t) ¼ exp(1)), regardless of the value of parameter bW. This becomes clear from Figure 13.5, where several theoretical cumulative functions are depicted. The parameter aW is called the hazard rate in reliability engineering. It is constant in the case of bW ¼ 1, increasing with time if bW < 1 and decreasing when bW > 1.

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13-7

0 −0.2 −0.4 −0.6 −0.8 log S

b W = 0.5

Characteristic time a W

b W = 1.0

−1 −1.2

b W = 2.0

−1.4 −1.6 −1.8 −2

0

0.5

1 1.5 2 Time ⫻ aW (arbitrary units)

2.5

3

FIGURE 13.5 Simulation describing microbial inactivation using the Weibull distribution for the shape parameter bW < 1, bW ¼ 1 and bW >1 and aW ¼ 1.

For the time needed to reduce the number of microorganisms by a factor 10 (analogous to the D-value) one could use the concept of reliable life tR used in reliability engineering, which is in fact the 90% percentile of the failure time distribution (Equation 13.11): tR ¼ aW (ln 0:1)1=bW ¼ aW (2:303)1=bW

(13:11)

These times are indicated by dashed lines in Figure 13.6 for various values of bW, and it shows its large dependence on bW. tR could play the same role as the parameter D, even though it has a different meaning. One could generalize Equation 13.11 into td ¼ aW ln (10d )1=bW

(13:12)

in which d is the number of decimal reductions. Thus for a 12D reduction, as is commonly used in sterilization td ¼ aW (25:328)1=bW . This equation shows that we need values for parameters aW and bW in order to do process calculations. These values need to be derived from experimental measurements, so this is what we have to discuss next. The Weibullian model. If we now move to practical applications, there is a complication with the numerical values of the Weibull parameters aW and bW if they are obtained by regression from experimental results. The problem lies in the scatter in experimental data and the log transformation commonly applied that changes the error distribution (Chapter 7). The parameters obtained experimentally from a plot of log S(t) depicted by Equation 13.10 may differ from the original parameters derived in Equation 13.9 when the untransformed survival ratio S(t) is used for estimation. This effect depends on the experimental errors (to be sure, in the absence of experimental errors there would be no difference). Therefore, it is better to introduce a ‘‘Weibullian’’ alternative, basically a power-law relation:

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13-8 tR

0

tR

tR

−0.2 −0.4 −0.6

b W = 0.5

log S

−0.8 −1 −1.2

b W = 1.0

b W = 2.0

−1.4 −1.6 −1.8 −2

FIGURE 13.6

0

1

2 3 4 Time ⫻ aW (arbitrary units)

5

6

Simulated survival curves showing the effect of the Weibull parameters on the reliable life tR.

log S(t) ¼ b t n

(13:13)

Mathematically, this equation is similar to Equation 13.9 with b ¼ 1=2.303 3 (1=aW)b and n ¼ bW, but it serves to make a distinction in the parameters. The Weibullian model is quite ﬂexible and is capable to ﬁt microbial survival curves quite well. Figures 13.7 through 13.9 give some examples. It should be understood that the Weibull model is an empirical one in the sense that no attempts are made to link to mechanistic theories about microbial death. The only thing that is done is to give a statistical account of a failure time distribution. However, according to the author this is rather an 0 −0.5 −1

log S

−1.5 −2 −2.5 −3 −3.5 −4 −4.5

0

1

2

3

4 5 Time (min)

6

7

8

FIGURE 13.7 Fit of the Weibullian model to inactivation of Salmonella typhimurium with parameters b ¼ 0.037 and n ¼ 2.4. Dataset in Appendix 13.1, Table A.13.2.

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Kinetics of Inactivation of Microorganisms

13-9

0 −0.5 −1

log S

−1.5 −2

−2.5 −3 −3.5 −4

0

1

2 Time (min)

3

4

FIGURE 13.8 Fit of the Weibullian model to inactivation of Salmonella enteriditis in egg yolk with parameters b ¼ 2.6 and n ¼ 0.3. Dataset in Appendix 13.1, Table A.13.3. 0 −0.2 −0.4 −0.6 log S

−0.8 −1 −1.2 −1.4 −1.6 −1.8

0

10

20 Time (min)

30

40

FIGURE 13.9 Fit of the Weibullian model to inactivation of Saccharomyces cerevisiae with parameters b ¼ 0.046 and n ¼ 1.0. Dataset in Appendix 13.1, Table A.13.4.

advantage, because most likely there is not a single cause of death. Nevertheless, a link can be made with microbial inactivation as follows. If n < 1, it means that the remaining cells have less probability of dying, indicating that the remaining cells are the sturdy ones, or perhaps adapting to the stress. With n > 1 it means that the remaining cells become increasingly susceptible to heat; in other words this indicates that there is cumulative damage occurring making it increasingly difﬁcult for the cells to survive. With n ¼ 1 it means that the probability of dying does not depend on time, in other words each cell is equally susceptible no matter how long the treatment lasts. The implicit assumption of ﬁrst-order kinetics, when interpreted in a stochastic sense, is thus that there is no biological variation of the cells in response to stress, in other words, k or D do not depend on time. This seems unrealistic and it would explain why so few linear survival curves are actually found.

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In the language of reliability engineering, regression models take into account how the two parameters aW and bW depend on other conditions (covariates). Relevant for microorganisms are pH, temperature, water activity, pressure, ionic strength, etc. Experiments can be devised to determine the dependency of the Weibullian parameters on such conditions. It is usually assumed in reliability engineering that the shape parameter bW is independent of external conditions. We will discuss the effect of temperature below, but ﬁrst we will take a look at the behavior of spores. The conclusion is that the Weibullian model is very suitable to model nonlinear survival curves, and may be helpful to pinpoint relevant physiological effects caused by heating. As remarked before, there are certainly other distributions applicable than the Weibull model. The choice made here for the Weibull model is because it is a simple as well as a ﬂexible model that has proven to be useful in many applications.

13.3

Kinetics of Inactivation of Spores

Section 13.2 dealt with vegetative cells. It is also necessary to consider the inactivation of spores because they are known to be quite resistant (which is the very reason for the organism to form them), and because of that they can be a real nuisance if they are present in raw materials and foods. First-order inactivation kinetics. Like with vegetative cells, inactivation of spores is usually analyzed as if it were a ﬁrst-order reaction, even though more often than not a nonlinear survival curve is observed (see for instance Figure 13.10). One usually neglects the nonlinear part and takes the slope of the linear part as a measure for the rate constant (mostly expressed as D-values). Sometimes even an initial increase in numbers of spores is found when heating spore suspensions or foods containing spores, for instance, as shown in Figure 13.11. There is a perfectly logical explanation for this phenomenon, namely the presence of dormant spores that are not detected with plate counts. However, they may become recoverable by a heat shock and, if this happens, more spores are counted than in the case where no heat shock was given (i.e., at t ¼ 0). One cause for recovery is heat, other causes are high pressure, radiation, oxidizing and reducing agents, sonication. This recovery is usually called activation in the literature. It should be understood that dormant spores do not grow during the 0

0

−0.5

−1

−1 −2 log S (t)

log S (t)

−1.5 −2

−2.5

−3 −4

−3 −5

−3.5 −4 (A)

0

−6 100 200 300 400 500 600 700 0 (B) t (s)

1

2

3 t (min)

4

5

6

FIGURE 13.10 Inactivation of spores of B. licheniformis in concentrated milk ﬁtted with the Weibullian model, n ¼ 1.5, b ¼ 0.00021 (A) and of spores of C. botulinum ﬁtted with the Weibullian model, n ¼ 0.3, b ¼ 3.3 (B). Dataset in Appendix 13.1, Table A.13.5.

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Kinetics of Inactivation of Microorganisms

13-11

22 20

ln Sa/Sa0

18 16 14 12 10 8

0

100

200 300 Time (min)

400

500

FIGURE 13.11 Semilogarithmic inactivation plot of a suspension of spores of B. stearothermophilus heated at 1058C. The different symbols refer to replicate experiments. Dataset in Appendix 13.1, Table A.13.6.

experiment, they are present but not detected at t ¼ 0. With Sd the number of dormant spores, Sa the number of active spores, and Si the number of active spores, the simplest kinetic scheme is then in the framework of ﬁrst-order kinetics, ﬁrst proposed by Shull: Sd

kact

Sa

kinact

Si

SCHEME 13.1 Shull model: First-order kinetics scheme to describe inactivation of spores from dormant spores via activated to inactivated spores. Sd: number of dormant spores, Sa: number of active spores, Si: number of inactivated spores.

This is actually in the form of the consecutive reaction scheme A ! B ! C, as discussed in Chapter 4. The rate constant kact accounts for the rate of activation of dormant spores and kinact for the rate of inactivation of activated spores. This sequential kinetic model assumes that dormant spores are not inactivated; they ﬁrst have to be activated. It is not easy to determine the number of dormant spores because they do not grow on nutrient-agar plates. The difference between the total number of spores, determined by other methods of counting (microscopy, for instance), and active spores (those that grow on a nutrient-agar plate) gives the number of dormant spores, on the assumption that all counted particles are spores, that all spores are viable (i.e., no dead spores), and that there are no vegetative cells present (they would be counted as spores). In normal practice one therefore only determines the number of spores that grow on a plate, i.e., the number of activated spores. The fact that heat causes recovery (activation) complicates the subsequent inactivation process. For that reason, in laboratory situations, spores are sometimes ﬁrst activated by giving them a sublethal heat shock and then, after a while, they are subjected to a further heat treatment to determine inactivation. There are two objections to such a procedure. First, this does not correspond to the practice of heat treatment of food (no heat shock is given, and a considerable number of spores is supposed to be in a dormant state), and second, the time allowed between the heat shock and the further heat treatment may be crucial: it is not really known how long it takes for a spore suspension to completely transform from the dormant state into an activated

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state. So, it would be best to study inactivation kinetics using the spores as they would be present in the food, i.e., a mixture of dormant and active spores. Let us take Scheme 13.1 as a starting point of kinetic analysis, assuming that we can only determine the number of active spores Sa. The resulting equation is (Equation 4.48) Sa ¼ Sa0 exp (kinact t) þ Sd0

kact ðexp (kact t) exp (kinact t)Þ kinact kact

(13:14)

in which Sd0 is the initial number of dormant spores and Sa0 the initial number of active spores. Normalizing the number of spores we ﬁnd: Sa Sd0 kact ¼ exp (kinact t) þ ðexp (kact t) exp (kinact t)Þ Sa0 Sa0 kinact kact

(13:15)

and it is seen that the decrease in number of active spores is determined by the initial ratio of dormant to active spores and the two rate constants. Some simulations using this equation give interesting results (see Figure 13.12 in which the inactivation rate constant is varied and Figure 13.13, in which the ratio of

0.5

0.2

0.4

0.1 0 −0.1 ln Sa/Sa0

ln Sa/Sa0

0.3 0.2

−0.2

−0.3

0.1

−0.4

0

−0.5

−0.1 −0.2

−0.6 0

20

0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1

(C)

−0.7

100 (B)

0

20 40 60 80 Time (arbitrary values)

100

0 −0.5 ln Sa/Sa0

ln Sa/Sa0

(A)

40 60 80 Time (arbitrary values)

−1

−1.5 −2

0

20

40 60 80 Time (arbitrary values)

−2.5 100 0 (D)

20

40 60 80 Time (arbitrary values)

100

FIGURE 13.12 Simulated semilog plots for inactivation of spores according to Equation 13.16 with Sd0=Sa0 ¼ 10 and kact ¼ 0.01, kinact ¼ 0.05 (A), kinact ¼ 0.08 (B), kinact ¼ 0.098 (C), kinact ¼ 0.3 (D).

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Kinetics of Inactivation of Microorganisms

13-13

0.5

0

0.4

− 0.1 − 0.2 − 0.3

0.2

ln Sa/Sa0

ln Sa/Sa0

0.3

− 0.4 − 0.5

0.1

− 0.6

0

− 0.7

− 0.1 − 0.2

(A)

− 0.8

0

20 40 60 80 Time (arbitrary values)

100

−0.9 0 (B)

20 40 60 80 Time (arbitrary values)

100

0

0

−0.5 −1 −1.5

−1

ln Sa/Sa0

ln Sa/Sa0

−0.5

−2

−2.5

−1.5

−3

−2

−3.5

−2.5

−4.5

−4

(C)

0

20 40 60 80 Time (arbitrary values)

100

0

(D)

20 40 60 80 Time (arbitrary values)

100

FIGURE 13.13 Simulated semilog plots for inactivation of spores according to Equation 13.16 with kact ¼ 0.01, kinact ¼ 0.05, and Sd0=Sa0 ¼ 10 (A), Sd0=Sa0 ¼ 5 (B), Sd0=Sa0 ¼ 1 (C), Sd0=Sa0 ¼ 0.1 (D).

dormant to active spores is varied). This ﬁrst-order model does actually result in nonlinear plots, depending on the values of the rate constants involved. In most cases reported in the literature, plots as in Figure 13.12B and C and Figure 13.13B are found. The important point is that the rate constant (or D-value) determined from the linear part of a logarithmic plot is actually the resultant of both rate constants kact and kinact and in some cases the slope could be signiﬁcantly inﬂuenced by the rate of activation rather than the rate of inactivation. The ﬁrst-order model given in Scheme 13.1 has been extended by Sapru to include also inactivation of dormant spores (Scheme 13.2): Si

kinact2

Sd

kact

Sa

kinact1

Si

SCHEME 13.2 Sapru model: First-order kinetics scheme to describe inactivation of spores, including the inactivation of dormant spores. Sd: number of dormant spores, Sa: number of active spores, Si: number of inactivated spores.

A possible problem with the Sapru model is that too many parameters are needed to describe inactivation while only Si can be measured; as a result they may be either inestimable or too imprecise to be useful.

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13-14

A further aspect to discuss is the applicability of ﬁrst-order kinetics. If the conclusion from Section 13.2 on inactivation of vegetative cells is plausible that ﬁrst-order kinetics is not really applicable, this same conclusion should hold for spore inactivation. We therefore need to consider other models as well. Alternative models for nonlinear spore survival curves. A logical extension of Section 13.2 on inactivation of vegetative cells would be to use the Weibull model again, but now also allowing for the phenomenon of ‘‘activation’’: log S(t) ¼ b1 t n1 b2 t n2

(13:16)

The ﬁrst part of the right-hand side refers to the activation part of the survival curve, and the second part to the inactivation part of the survival curve. If there is no activation shoulder, which implies that b1 ¼ 0, the model reduces to the Weibullian model (Equation 13.13). Another empirical model to handle activation shoulders was proposed by Peleg: log S(t) ¼

t f1 lnð1 þ exp (b3 t Þn3 g k1 þ k2 t

(13:17)

These empirical models are certainly not exclusive, others may perform equally well. It is perhaps instructive to compare the performance of the models discussed hitherto. A dataset was used that contained replicates and included activation shoulders and concerned the inactivation of Bacillus stearothermophilus spores. Four models were tested, the Shull and Sapru models displayed in Schemes 13.1 and 13.2, respectively, and the double Weibullian model and the Peleg model in Equations 13.16 and 13.17, respectively. Figure 13.14 shows the ﬁts obtained. Although the ﬁts do not differ that much by visual inspection, and they all seem to describe the experimental data quite well, there are big differences. The Shull and Sapru model suffer from very imprecise and sometimes indeterminate estimates, which is the reason that they coincide. This problem did not exist with the double Weibullian and the Peleg model. A model discrimination study with

2

1.5 1

1

0.5 0

0

−1

log S(t)

log S(t)

− 0.5

−1.5

−1 −2

−2 −3

−2.5 −3

−4

−3.5 −4 (A)

0

100

200

300 400 t (min)

500

−5

600 (B)

0

50

100 t (min)

150

200

FIGURE 13.14 Fits of four models to heat inactivation of spores of B. stearothermophilus at four temperatures. Peleg model (—), Shull model (- - -), Sapru model (. . . . ), double Weibullian model (— —) at 1058C (A), 1108C (B) Dataset in Appendix 13.1, Table A.13.7.

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13-15

2

2

1

1

0

0

−1

−1

−2

−2 log S(t)

log S(t)

Kinetics of Inactivation of Microorganisms

−3 −4

−3 −4

−5

−5

−6

−6

−7

−7

−8

0

10

20

(C)

FIGURE 13.14 (continued)

30 40 t (min)

50

60

70

−8

0

5

10

15

t (min)

(D)

1158C, (C), 1208C (D).

these data and models is shown in Tables 13.1 through 13.4. The criteria used have been discussed in Chapter 7; the goodness of ﬁt indicates the sampling probability of obtaining a higher ratio if the model is correct, the model having the lowest Akaike criterion number and the highest posterior probability performs best. The results clearly indicate that the double Weibullian and the Peleg model perform much better. The goodness of ﬁt is much less for the Sapru and Shull model at 1108C, while there is lack of ﬁt for all models at 1208C. As already mentioned, some parameter estimates were indeterminate with the Shull and Sapru model. All in all, it seems that the empirical models perform much better than the ones based on

TABLE 13.1 Goodness of Fit, Akaike Criterion (AIC, DAIC), and Logarithm of the Posterior Probability (log(PB)) for Four Models and Inactivation Data of Spores of B. stearothermophilus at 1058C DAIC

log(PB)

118.5

8.1

8.1

116.5

10.1

8.1

0.923

122.0

4.6

7.1

0.995

126.6

0

7.7

Model

Goodness of Fit

AIC

Shull

0.783

Sapru

0.783

Double Weibullian Peleg

TABLE 13.2 Goodness of Fit, Akaike Criterion (AIC, DAIC), and Logarithm of the Posterior Probability (log(PB)) for Four Models and Inactivation Data of Spores of B. stearothermophilus at 1108C Model

Goodness of Fit

AIC

DAIC

log(PB)

Shull

0.005

110.6

31.7

8.4

Sapru

0.005

108.6

33.7

8.4

Double Weibullian

0.941

139.5

2.9

4.4

Peleg

0.814

142.3

0

4.0

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13-16

TABLE 13.3 Goodness of Fit, Akaike Criterion (AIC, DAIC), and Logarithm of the Posterior Probability (log(PB)) for Four Models and Inactivation Data of Spores of B. stearothermophilus at 1158C Model

Goodness of Fit

AIC

DAIC

log(PB)

Shull

0.866

70.3

7.5

6.4

Sapru

0.866

68.3

9.5

6.4

Double Weibullian Peleg

1.000 0.999

77.3 77.8

0 0.5

5.6 5.6

TABLE 13.4 Goodness of Fit, Akaike Criterion (AIC, DAIC), and Logarithm of the Posterior Probability (log(PB)) for Four Models and Inactivation Data of Spores of B. stearothermophilus at 1208C Model

Goodness of Fit

AIC

DAIC

log(PB)

Shull

0.001

110.0

4.3

5.5

Sapru

0.001

108.0

6.3

5.5

Double Weibullian

0.005

114.3

0

4.8

Peleg

0.002

113.3

1.0

5.1

ﬁrst-order kinetics, at least for this dataset. However, if these models are to be used more datasets should be tested, and of course, also the temperature dependence should be studied. Temperature dependence is the topic of Section 13.4.

13.4

Temperature Dependence of Microbial Inactivation

Thermal death time (TDT) curves. The temperature dependence of microbial inactivation according to the classical ﬁrst-order model was already discussed in Chapter 5 when D- and Z-values were introduced. The analysis is based upon the assumption that the logarithm of the D-value varies linearly in a so-called TDT plot (see Equation 5.38 and Figure 5.14) and the Z-value can then be derived from the slope of the TDT plot. An example is given for the heat inactivation of Listeria monocytogenes suspended in a buffer (Figure 13.15A). As shown, the survival curve was more or less linear (one of the few examples to be found in the literature), so a D-value could be derived, and the D-values derived at four temperatures were used for a TDT plot (Figure 13.15B). The TDT curve could be regarded as linear, though there may also be a hint on nonlinearity but there are too few data points to check that. In any case, it is dangerous to extrapolate beyond the range (which is rather narrow: only 68C) for which the relation is derived. Most TDT curves published in the literature are based upon only three, sometimes four temperatures that are close together and then it is hard to discover nonlinearity. Moreover, the resulting parameter estimates are very imprecise due to the limited number of data, leading to very inaccurate predictions. A similar analysis for nonlinear survival curves by calculating a D-value makes no sense. However, one could investigate the behavior of the Weibullian parameters as a function of temperature; this is done below. D- and Z-values are the commonly used parameters in the literature to characterize the inactivation of microbes as a function of temperature. The inactivation of microbes in a process (lethality of a process) is commonly expressed in a so-called F0 value. This is an equivalent time at a reference temperature Tref that gives the same inactivation as the actual process:

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Kinetics of Inactivation of Microorganisms

13-17

0

0.6 0.4

−1

0.2 0 log D

log S(t)

−2 −3

−0.2 −0.4

−4

−0.6 −5 −6 (A)

−0.8 −1 0

1

2 3 t (min)

4

5

6

55

56

57

(B)

58 59 T ⬘ (⬚C)

60

61

62

63

FIGURE 13.15 Survival curve of L. monocytogenes at 588C in a buffer, pH 7.4 with a log-linear regression line (A) and a TDT curve for the same microorganism using the D-value calculated for four temperatures (B). Dataset in Appendix 13.1, Table A.13.8.

ðt T(t)Tref F0 ¼ 10 Z dt

(13:18)

0

The log reduction is related to F0 as F0 1 ¼ log S(t) ¼ Dref Dref

ðt 10

T(t)Tref Z

dt

(13:19)

0

where Dref is the D-value at the reference temperature. For vegetative cells the reference temperature is chosen usually around 708C (pasteurization range) and for spores 1218C (sterilization range). Sometimes, one uses ﬁrst-order rate constants and plots them according to the logarithmic form of the Arrhenius equation (Chapter 5). This author is of the opinion that the Arrhenius equation is not applicable at all to microbial inactivation. Arrhenius’ law was derived for simple elementary reactions, not for complicated events such as microbial death. Interpretation of activation energy in terms of J mol1 is impossible because one cannot refer to a speciﬁc elementary reaction, and it is hard to imagine a mole of bacteria anyway. Also, when the Arrhenius equation is applicable, it indicates that a reaction rate constant has a ﬁnite value both at low and high temperature and this makes sense for a chemical reaction: it will take place also at a low temperature, be it at a very low rate. With microbial inactivation, however, it becomes difﬁcult to imagine that inactivation takes place at a slow rate according to the same mechanism as the one at the higher temperature. On the contrary, at moderate temperature microorganisms grow and at temperatures below their minimum growth temperature they do not grow but they are usually not inactivated. Furthermore, inactivation of microbes takes place over a very narrow temperature interval and then it makes no sense to use the inverse of the absolute temperature, as is commonly done with the Arrhenius equation, nor does it makes sense to make a logarithmic transformation. So, we will not discuss the analysis of Arrhenius in microbial inactivation.

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One could argue that a Z-value does make sense, as long as it is not connected to activation energy (as in Equation 5.35). In other words, the Z-value could be used just as an empirical parameter to express temperature dependence of a rate parameter. However, the problem remains that if the derivation of the Z-value is based upon D-values that are derived from nonlinear plots, this Z-value has actually little value and, as stated before, there are not that many linear inactivation plots to be found. If we accept the Weibullian model (Equation 13.13) as the alternative model for microbial inactivation kinetics, the dependence of its two parameters b and n on temperature has to be studied. Literature analysis of microbial inactivation showed that the logarithm of the scale parameter b depended linearly on temperature, analogous to the classical D-value. However, the temperature dependence of the shape parameter n was not so clear. In only a few cases the shape parameter seemed to depend on temperature, in a linear way. In all other cases, no statistically signiﬁcant (linear) relation with temperature could be found. In most cases, the shape parameter n was larger than 1, and in fewer cases smaller than 1.

0

0

−0.5

−1

−1 −2

log S(t)

−1.5

−3

log S(t)

−2 −2.5

−4

−3 −5

−3.5 −4

0

100

200

(A)

300 400 t (min)

500

−6

600

0

500

−1.5 −2 log S(t)

log S(t)

400

−1

−3 −4

−2.5 −3 −3.5

−5

−4

−6

FIGURE 13.16 54.68C (D),

200 300 t (min)

−0.5

−2

(C)

100

0

−1

−7 0

0

(B)

−4.5 20

40

60 80 t (min)

100

120

−5

140 (D)

0

20

40 t (min)

60

80

Inactivation of E. coli K12 MG1655 suspended in growth medium at 49.98C (A), 528C (B), 548C (C), (continued )

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Kinetics of Inactivation of Microorganisms

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0

0

−0.5

−1

−1

−2

−2

log S(t)

log S(t)

−1.5

−2.5 −3 −3.5

−4 −5

−4

−6

−4.5 −5

−3

0

10

20

30 t (min)

(E)

40

50

−7

60

0

5

10

15 t (min)

(F)

0

20

25

30

0 −1

−1

−2 −3 log S(t)

log S(t)

−2 −3

−4 −5

−4

−6 −5 −6 (G)

−7 0

2

4

t (min)

6

8

−8

10 (H)

0

1

2

t (min)

3

4

5

FIGURE 13.16 (continued) 558C (E), 56.68C (F), 58.68C (G), and 60.68C (H). The drawn lines are the ﬁts of the Weibullian model for a variable n and b, the broken line is for a ﬁxed n ¼ 1.6 (the average value for n), ﬁxed for all temperatures. Dataset in Appendix 13.1, Table A.13.9.

Only very rarely is the shape parameter n ¼ 1, indicating that the classical ﬁrst-order kinetics approach is indeed the exception rather than the rule. Most experimental results obtained so far point out that the n parameter is not, or only weakly, dependent on temperature. Figure 13.16 gives an example of nonlinear survival curves for an E. coli strain. The data were ﬁtted ﬁrst by allowing parameters n and b to vary. The variation of n with temperature is shown in Figure 13.17. It can be seen that parameter n does not have a constant value for each dataset, but on the other hand there does not seem to be a systematic variation with temperature; in this particular case the estimate for the data at 58.68C seems to be an outlier. Similar analyses from other datasets indicate the same behavior of the parameter n as a function of temperature, so to a ﬁrst approximation n can be assumed constant with varying temperature. For the data shown the average value of parameter n was 1.6. However, because of the variation, this adds uncertainty to subsequent ﬁts. To show the effect, the ﬁts with the ﬁxed value of n ¼ 1.6 are also shown in Figure 13.16.

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13-20

3

2.5

n (T⬘)

2

1.5

1

0.5

0 45

50

55 T⬘ (⬚C)

60

65

FIGURE 13.17 Dependence of the Weibullian parameter n on temperature for the ﬁts shown in Figure 13.16.

A ﬁxed value of n is needed if we want to study the effect of temperature on parameter b because the parameters b and n are correlated. By ﬁxing n at its average value 1.6 for this particular dataset, the dependence of b on temperature can be studied. A suitable model to capture the dependence of b on temperature T 0 is the log-logistic model: b(T 0 ) ¼ ln½1 þ expðk(T 0 Tc Þ

(13:20)

with k (8C1) and Tc (8C) as parameters. When T 0 > Tc, b(T 0 ) k(T 0 Tc). Another relation could be a power-law relation as depicted in the next equation: 0

b(T 0 ) ¼ 10aT þc

(13:21)

The ﬁt of these two models to the parameter estimates b for the datasets shown in Figure 13.16 is shown in Figure 13.18. For the data shown in Figure 13.18, the ﬁts by the two models are almost indistinguishable. This approach is an alternative for the TDT approach discussed above. To deal with nonisothermal situations the following analysis has been developed by Peleg and coworkers (see also Chapter 5 and Chapter 12 where this approach was applied). Nonisothermal conditions. If we take the Weibullian model (Equation 13.13) as our model, and we ﬁx its parameter n at a constant value (i.e., temperature independent) for the moment, the momentary rate at constant temperature can be calculated from this model as d logðS(t)Þ ¼ b(T) n t n1 dt T

(13:22)

It shows that the momentary rate varies with time. We need to know also how temperature varies with time as T(t). This information should be available from the experiment and the relation can be cast in any algebraic equation that ﬁts the experimental conditions. Then, we can make parameter b time dependent

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Kinetics of Inactivation of Microorganisms

13-21

0.7 0.6

b (T⬘)

0.5 0.4 0.3 0.2 0.1 0

45

50

55 T⬘ (⬚C)

60

65

FIGURE 13.18 Fit of the loglogistic model (Equation 13.20, solid line) and the power-law model (Equation 13.21, hyphenated line) to the Weibullian parameter b as a function of temperature for the ﬁts found to the dataset shown in Figure 13.16 while keeping the parameter n constant at its average value of n ¼ 1.6.

so that it can be written as b(T(t)). To know the survival ratio at any time t* the Weibullian Equation 13.13 can be inverted to ﬁnd: t* ¼

logðS(t)Þ bðT(t)Þ

1=n (13:23)

By combining these last two equations we ﬁnd d logðS(t)Þ logðS(t)Þ n1=n ¼ bðT(t)Þ n dt bðT(t)Þ

(13:24)

This equation can be numerically integrated if T(t) is known and the relation for b(T) has been established, for instance via Equations 13.20 or 13.21, or any other equation that would do the job. Thus, it allows calculating the survival ratio as a function of varying temperature in a relatively simple way. As an example, Figure 13.19 shows the inactivation of L. monocytogenes in a buffer at pH 4 at four temperatures, modeled via the Weibullian model. The shape parameter n was found to be relatively constant over this temperature range; the average value was n ¼ 1.265 and the results shown in Figure 13.19 are for this average value. From such data the temperature dependence of b can be derived. The temperature dependence of b(T) is shown in Figure 13.20 for two models, one is the log-logistic model of Equation 13.20, and the other the power-law relation shown in Equation 13.21. In this case, the log-logistic model appears not to be appropriate and the power-law model is performing well. So, the temperature dependence of parameter b(T) via the power-law model was used in Equation 13.24 to predict nonisothermal behavior of this bacterium. the results are shown in Figure 13.21 for a linear increasing temperature proﬁle.

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Kinetic Modeling of Reactions in Foods 0

0

−1

−1

−2

−2 log S (t)

log S (t)

13-22

−3

−3

−4

−4

−5

0

1

2

(A)

3 4 Time (min)

5

−5

6

0

0.5

1

(B)

0

1.5 2 Time (min)

2.5

3

0

−1

−1

−2 log S(t)

log S(t)

−2

−3

−3 −4

−4

−5 (C)

−5

0

0.5

1 Time (min)

1.5

−6 0 (D)

0.2

0.4 Time (min)

0.6

0.8

FIGURE 13.19 Survival curves of L. monocytogenes in a buffer of pH 4 at 548C (A), 568C (B), 588C (C), 608C (D). The curves are the ﬁts by the Weibullian model (Equation 13.13) for a ﬁxed shape parameter n ¼ 1.265 and varying parameter b. Dataset in Appendix 13.1, Table A.13.10.

Note that the predictions given in Figure 13.21 are real predictions: the data shown were obtained in independent experiments. The predictions look reasonable, though certainly not perfect. One possible reason for the discrepancy is the imprecision, but it is also not unthinkable that microorganisms respond differently to a variable temperature treatment than to a constant temperature treatment. As mentioned, this analysis can be done with any other model; one could also include temperature dependence of the parameter n if so desired. It is thus not necessary to work with activation energies to be able to do calculations for nonisothermal treatments.

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Kinetics of Inactivation of Microorganisms

13-23

18 16 14 12

b

10 8 6 4 2 0 40

45

50

55

60

65

T ⬘(⬚C)

FIGURE 13.20 Temperature dependence of the Weibullian parameter b for the inactivation of L. monocytogenes as shown in Figure 13.19 for the log-logistic model (hyphenated line, k ¼ 3.1, Tc ¼ 57.08C) and the power-law model (solid line, a ¼ 0.23, c ¼ 12.8).

0

−1

log S(t)

−2

−3

−4

−5

−6

0

10

20

30 40 Time (min)

50

60

FIGURE 13.21 Nonisothermal inactivation of L. monocytogenes in buffer at pH 4 for a linearly increasing temperature proﬁle from 308C with a rate of 0.58C min1 (&), 18C min1 (*), 28C min1 (~), and 58C min1 (*). The solid lines are the predictions by Equation 13.24, using the temperature dependence of b via the power-law model as indicated in Figure 13.20. Dataset in Appendix 13.1, Table A.13.11.

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Kinetic Modeling of Reactions in Foods

13-24

13.5

Food Matrix Effects

0

0

−1

−1

−2

−2

−3

−3 log S

log S

The kinetics of inactivation of microorganisms can be inﬂuenced strongly by the surroundings in which the microorganisms are. It is not really possible to predict the effects, sometimes the matrix appears to protect the microorganisms, and sometimes it is the other way around. Just to show the magnitudes of possible effects two examples are shown here. The ﬁrst example concerns the protective effect of the presence of fat on inactivation of a Salmonella species in meat (Figure 13.22). Table 13.5 shows the Weibullian parameter estimates for the data shown in Figure 13.22. Both the graphs and the parameter estimates show that the higher the fat content the longer the time needed for inactivation. In other words, fat appears to protect the microorganism. It is interesting to see that the shape parameter n increases with fat content, suggesting that the microorganism is better able to withstand the thermal stress when more fat is present. It is hard to think of a mechanism that could explain this, however. Another example is shown in Figure 13.23. This concerns the inactivation of spores of Bacillus subtilis in various media: 0.05 M phosphate solutions containing 3% and 5% NaCl, whole milk, soy sauce, and kayu (Japanese porridge). Table 13.6 shows the parameter estimates for Weibullian ﬁts to the data shown in Figure 13.23. It can be seen that in this case inactivation is faster in the food matrices shown as compared to a buffer system. Moreover, the shape factor is seen to vary as well, which may indicate that the inactivation mechanism depends on the food matrix. However, these few data do not allow ﬁrm conclusions to be drawn. The only thing that can be concluded is that matrix effects cannot be neglected, but unfortunately it is not possible to predict the magnitude or the direction of these effects. In any case, it shows that it can be dangerous to extrapolate from simple aqueous systems to real foods.

−4

−4

−5

−5

−6

−6

−7 (A)

0

1

2

3

Time (min)

4

−7

5 (B)

0

1

2

3

4

5

Time (min)

FIGURE 13.22 Effect of fat content on heat inactivation of Salmonellae species at 658C in beef (A) with 24% fat (^), 18% fat (~), 12% fat (*), 7% fat (&), and pork meat (B) with 28% fat (^), 24% fat (~), 10% fat (*), 4% fat (&). The drawn lines are ﬁts of the Weibullian model. Dataset in Appendix 13.1.

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Kinetics of Inactivation of Microorganisms

13-25

TABLE 13.5 Weibullian Parameter Estimates for the Inactivation of Salmonellae Species in Meat with Varying Fat Content (Figure 13.22) Matrix Beef 24% fat

b (min)

n

0.37

1.91

Beef 18% fat

0.60

1.57

Beef 12% fat Beef 7% fat

0.79 1.02

1.45 1.40

Pork 28% fat

0.21

2.26

Pork 24% fat

0.55

1.62

Pork 10% fat

0.67

1.51

Pork 4% fat

0.86

1.43

9 8 7

log N

6 5 4 3 2 1 0 0

10

20 30 Time (min)

40

50

FIGURE 13.23 Heat inactivation of B. subtilis spores in phosphate buffer with 3% NaCl (&) and 5% NaCl (^), whole milk (~), soy sauce (&), kayu (*). Dataset in Appendix 13.1, Table A.13.13.

TABLE 13.6 Parameter Estimates for the Weibullian Fits Shown in Figure 13.23 Matrix

b (min)

n

Phosphate buffer, 3% NaCl

0.02

1.37

Phosphate buffer, 5% NaCl

0.02

1.28

Whole milk

0.18

1.07

Soy sauce

0.06

1.62

Kayu

0.36

0.83

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13-26

13.6

Concluding Remarks

Kinetics of microbial inactivation has been investigated for almost a century now. For most of that century, ﬁrst-order kinetics has been applied, and not without success, because processed foods are actually quite safe. It is all the more remarkable that survival curves are not straight lines but rather are curved. The evidence is overwhelming that ﬁrst-order kinetics does not apply. This does not mean that it cannot be used in practice, but alternative models are widely available now and they seem to perform better in many cases from a statistical point of view. They also do more justice to the probabilistic character of inactivation phenomena. In the author’s view they deserve to be further investigated for practical applications. As a ﬁnal remark, it can be mentioned that the suggested models are applicable to other treatments than heat. Weibull models have already been tested quite extensively for inactivation due to pressure, pulsed electric ﬁelds, and chemical agents. We did not discuss this here, but the methods given can be applied directly. Some references are given in the Bibliography section.

Appendix 13.1

Datasets Used for Examples in This Chapter

TABLE A.13.1 Time 0

Survival Data for E. coli in Apple Cider (Figure 13.1) log N at 528C

log N at 558C

log N at 588C

9.65

9.65

9.65

1

9.10

2

8.75

3

8.35

4

7.12

5 6

5.45 3.77

10

9.2

8.05

15

8.9

6.78

20

8.4

6.25

25

7.95

4.75

30

7.5

35

6.8

45

5.8

Source: From Splittstoesser D.F., McLellan M.R., and Churey J.J. Heat resistance of Escherichia coli O157:H7 in apple juice. J Food Prot 59:226–229, 2000.

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Kinetics of Inactivation of Microorganisms TABLE A.13.2 (Figure 13.7) Time (min)

13-27

Inactivation of Salmonella typhimurium log S

0

0

0.47

0.041

0.984

0.124

1.476

0.145

1.968

0.373

2.527

0.373

2.997

0.539

3.556 4.048

0.622 1.119

4.607

1.513

5.099

1.575

6.128

2.632

6.597

3.565

7.089

3.876

Source: From Mackey B.M. and Derrick C.M. Elevation of the heat resistance of Salmonella typhimurium by sublethal heat shock. J Appl Bacteriol 61:389–393, 1986.

TABLE A.13.3 Inactivation of Salmonella enteriditis in Egg Yolk (Figure 13.8) Time (min) 0

log S 0

0.057

1

0.113 0.192

1.478 1.523

0.17

1.682

0.34

1.978

0.509

2.159

0.668

2.182

0.736

2.318

1.483

2.841

2.208 2.943

3.296 3.728

Source: From Michalski C.R., Brackett R.E., Hung Y.-C., and Ezeike G.O.I. Use of capillary tubes and plate heat exchanger to validate U.S. Department of Agriculture pasteurization protocols for elimination of Salmonella enteritidis from liquid egg products. J Food Prot 62:112– 117, 1999.

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13-28

TABLE A.13.4 Inactivation of Saccharomyces cerevisiae (Figure 13.9) Time (min) 0

log S 0

2.3

0.051

4.2

0.103

5.9

0.276

8.1

0.396

10.0

0.417

12.2

0.665

14.1 16.2

0.762 0.761

17.8

0.896

20.2

0.947

22.0

1.06

24.0

1.111

26.0

1.163

28.0

1.29

29.4 31.8

1.456 1.537

Source: From Van Uden N. and Vidal-Leiria M.M. Inactivation of S. cerevisiae in an aqueous solution of yeast nitrogen base and glucose. Arch Microbiol 108:293, 1976.

TABLE A.13.5 Inactivation of Spores of B. licheniformis in Concentrated Milk (Figure 13.10A) Time (s) 0

log N=N0 0

70

0.039

80

0.135

110

0.135

150 160

0.328 0.309

220

0.772

280

1.004

320

1.274

380

1.641

440

1.911

600

3.069

650

3.436

Source: From Behringer R. and Kessler H.G. Heat resistance of spores in skimmilk and skimmilk concentrates and determination of reaction kinetic parameters. Milchwissenschaft 46:488–492, 1991.

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TABLE A.13.6 Inactivation of Spores of C. botulinum (Figure 13.10B) Time (min)

log S 0.069

0 0

0.069

0.5

2.552

0.5

2.368

0.75

3.127

0.75

2.989

1

3.104

1 1.7

3.495 4

1.7

3.862

2.7

4.529

2.7

4.322

5

5.564

5

5.426

Source: From Anderson W.A., McClure P.J., Baird-Parker A.C., and Cole M.B. The application of a log-logistic model to describe the thermal inactivation of Clostridium botulinum 213B at temperatures below 121.18C. J Appl Bacteriol 80: 283–290, 1996.

TABLE A.13.7 Inactivation of Spores of B. stearothermophilus (Figures 13.11 and 13.14) Data for 1058C Time (min)

log S

0

0.145

0

0.140

0

0.055

1

0.282

1

0.112

1

0.191

2.5 2.5

0.404 0.191

2.5

0.305

5

0.594

5

0.421

5

0.446

10

0.594

10

0.305

10 15

0.506 0.700

15

0.552

15

0.565

25

0.828

25

0.627 (continued )

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13-30

TABLE A.13.7 (continued) Inactivation of Spores of B. stearothermophilus (Figures 13.11 and 13.14) Data for 1058C Time (min)

log S

30

0.571

30

0.648

35

0.755

35 45

0.663 0.860

45

0.520

45

0.627

60

0.714

60

0.594

60

0.673

90

0.367

90 120

0.546 0.673

120

0.367

120

0.316

150

0.033

150

0.161

170

0.056

180

0.056

180 300

0.112 0.772

300

0.578

300

0.480

390

1.140

390

1.372

480

3.332

480

1.523

480

3.166

Data for 1108C Time (min)

log S

0

0.101

0

0.017

0

0.096

0.5

0.415

0.5 0.5

0.376 0.343

1

0.443

1

0.460

1

0.406

1.5

0.585

2

0.538

2

0.415

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13-31

TABLE A.13.7 (continued) Inactivation of Spores of B. stearothermophilus (Figures 13.11 and 13.14) Data for 1108C Time (min)

log S

2

0.682

4

0.832

4

0.545

4 6

0.832 0.735

6

0.592

6

0.832

10

0.832

10

0.667

10

0.794

15

0.832

15 15

0.656 0.095

20

0.832

20

0.740

20

0.832

30

0.702

30

0.509

30

0.573

40 40

0.434 0.151

40

0.309

60

0.151

60

0.071

60

0.032

90

0.402

90

0.885

90 120

0.566 1.308

120

2.247

120

1.471

180

3.991

180

4.690

180

4.512

Data for 1158C Time (min)

log S

0

0.101

0

0.017

0

0.095

0.25

0.565

0.25

0.460

0.25

0.406 (continued )

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13-32

TABLE A.13.7 (continued) Inactivation of Spores of B. stearothermophilus (Figures 13.11 and 13.14) Data for 1158C Time (min)

log S

0.5

0.702

0.5

0.616

0.5

0.692

0.75 1

0.634 0.794

1

0.753

1

0.650

1.5

0.899

2

0.832

2

0.867

2

0.794

2.5 3

0.753 0.794

3

0.867

3

0.983

5

0.899

5

0.716

10

0.452

10

0.271

15 15

0.017 0.206

15

0.832

20

0.366

20

0.816

30

1.414

30

2.323

45

4.101

45 45

3.168 5.017

60

6.946

Data for 1208C Time (min)

log S

0

0.101

0

0.017

0 0.08

0.095 0.386

0.08

0.524

0.08

0.284

0.17

0.753

0.17

0.592

0.17

0.434

0.33

0.744

0.33

0.832

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Kinetics of Inactivation of Microorganisms

13-33

TABLE A.13.7 (continued) Inactivation of Spores of B. stearothermophilus (Figures 13.11 and 13.14) Data for 1208C Time (min)

log S

0.33

0.672

0.67

0.622

0.67

0.867

0.67 1

0.899 0.566

1

0.794

1

0.794

1.33

0.365

1.33

0.794

1.33

0.697

1.67

0.321

1.67 1.67

0.604 0.559

2

0.133

2

0.610

2

0.538

3

0.167

4

0.366

4

0.548

4 6

0.584 1.361

6

1.414

6

1.384

8

2.703

8

2.483

8

3.168

10

4.688

10 10

5.001 4.645

12

5.832

14

6.645

Source: From Sapru V. Mathematical modeling of bacterial spore population dynamics for design and validation of ultra high temperature sterilization processes. PhD thesis, University of Florida, Florida (1991).

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Kinetic Modeling of Reactions in Foods

13-34 TABLE A.13.8 (Figure 13.15)

Survival Curve of L. monocytogenes

Data for Figure 13.15A Time (min)

log S

0

0 0.33 0.96

0.5 0.98 1.49

1.52

1.99

1.8

2.5

2.73

2.97

3.19

3.55

3.54

4.06

3.97

4.51 5.02

4.35 4.76

5.5

5.32

Data for Figure 13.15B T (8C)

log D

56

0.35

58

0.01

60 62

0.31 0.79

Source: From Hassani M., Alvarez I., Raso J., Condón S., and Pagán R. Comparing predicting models for heat inactivation of Listeria monocytogenes and Pseudomonas aeruginosa at different pH. Int J Food Microbiol 100: 213–222, 2005.

TABLE A.13.9 Inactivation of E. coli K12 MG1655 (Figure 13.16) 49.88C Time (min)

log S

0 50

0 0.02

66

0.07

85

0.07

100

0.17

125

0.25

150

0.3

165

0.32

185 200

0.55 0.54

220

0.64

235

0.79

250

0.74

265

0.77

270

1.02

290

1.12

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13-35

TABLE A.13.9 (continued) Inactivation of E. coli K12 MG1655 (Figure 13.16) 49.88C Time (min)

log S

300

1.24

325

1.42

350

1.59

370

1.84

390

1.79

410 430

2.17 2.47

450

2.29

475

2.54

490

2.52

510

2.99

570

3.69

528C Time (min)

log S

0

0

15

0.07

40

0.22

55

0.4

75

0.5

85 105

0.62 0.62

120

0.72

130

1.05

150

1.3

200

1.75

225

2.68

250

3.13

280 300

3.13 3.3

340

3.73

375

4.35

415

4.28

430

4.78

548C Time (min)

log S

0

0

3

0.15

10

0.33

25

0.53

32

0.75

44

1.5

45 50

1.83 2.36

60

2.51 (continued )

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13-36

TABLE A.13.9 (continued) Inactivation of E. coli K12 MG1655 (Figure 13.16) 548C Time (min)

log S

62

2.83

74

3.01

75

3.71

87

3.79

92

3.91

100 120

5.62 5.72

54.68C Time (min)

log S

0

0

9

0.13

11 18

0.05 0.35

25

0.73

30

1.46

35

1.74

40

2.39

45

2.49

50

3.07

55 60

3.15 3.35

Kinetic Modeling of Reactions in Foods

Boekel/Kinetic Modelling of Reactions in Foods DK3903_C000 Final Proof page ii 12.11.2008 10:19pm Compositor Name: JGanesan

FOOD SCIENCE AND TECHNOLOGY Editorial Advisory Board Gustavo V. Barbosa-Cánovas Washington State University–Pullman P. Michael Davidson University of Tennessee–Knoxville Mark Dreher McNeil Nutritionals, New Brunswick, New Jersey Richard W. Hartel University of Wisconsin–Madison Lekh R. Juneja Taiyo Kagaku Company, Japan Marcus Karel Massachusetts Institute of Technology Ronald G. Labbe University of Massachusetts–Amherst Daryl B. Lund University of Wisconsin–Madison David B. Min The Ohio State University Leo M. L. Nollet Hogeschool Gent, Belgium Seppo Salminen University of Turku, Finland John H. Thorngate III Allied Domecq Technical Services, Napa, California Pieter Walstra Wageningen University, The Netherlands John R. Whitaker University of California–Davis Rickey Y. Yada University of Guelph, Canada

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Kinetic Modeling of Reactions in Foods Martinus A. J. S. van Boekel

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-57444-614-2 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Van Boekel, Martinus A. J. S. Kinetic modeling of reactions in foods / Martinus A. J. S. van Boekel. p. cm. -- (Food science and technology) “A CRC title.” Includes bibliographical references and index. ISBN 978-1-57444-614-2 (alk. paper) 1. Food--Analysis. 2. Chemical kinetics--Mathematical models. 3. Food adulteration and inspection. I. Title. II. Series. TX545.V36 2008 664.001’54--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

2008044086

Boekel/Kinetic Modelling of Reactions in Foods DK3903_C000 Final Proof page v 12.11.2008 10:19pm Compositor Name: JGanesan

Dedication To my wife Corrie For her patience and understanding

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Contents Preface ................................................................................................................................................. xiii Author ................................................................................................................................................ xvii

1

Kinetic View on Food Quality ............................................................................................ 1-1 1.1 Introduction .......................................................................................................................................... 1-1 1.2 Food Quality ......................................................................................................................................... 1-1 1.3 Foods as Complex Reaction Media ................................................................................................. 1-6 1.4 Outline of the Book ............................................................................................................................ 1-7 Bibliography and Suggested Further Reading ......................................................................................... 1-8

SECTION I The Basics

2

Models and Modeling ........................................................................................................... 2-1

3

Chemical Thermodynamics in a Nutshell ....................................................................... 3-1

2.1 Introduction .......................................................................................................................................... 2-1 2.2 Models and Modeling ......................................................................................................................... 2-1 2.3 Concluding Remarks ....................................................................................................................... 2-12 Bibliography and Suggested Further Reading ...................................................................................... 2-12

3.1 Introduction .......................................................................................................................................... 3-1 3.2 Quantiﬁcation of Reactants and Products ..................................................................................... 3-1 3.3 Thermodynamics of Reactions ......................................................................................................... 3-6 3.3.1 Heat and Work ..................................................................................................................... 3-6 3.3.2 Energy ..................................................................................................................................... 3-8 3.3.3 Enthalpy ................................................................................................................................. 3-9 3.3.4 Entropy ................................................................................................................................. 3-11 3.3.5 Free Energy .......................................................................................................................... 3-15 3.3.6 Chemical Potential ............................................................................................................. 3-18 3.3.7 Ideal Solutions .................................................................................................................... 3-20 3.3.8 Ideal Dilute Solutions ........................................................................................................ 3-21 3.3.9 Real, Nonideal Solutions: Activity Concept ................................................................. 3-22 3.3.10 Standard States ................................................................................................................... 3-27 3.3.11 Solvent Activity and Water Activity .............................................................................. 3-29 3.3.12 Chemical Potential and Equilibrium ............................................................................. 3-33 vii

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Contents

viii

3.3.13 Equilibrium Constants .................................................................................................... 3-36 3.3.14 Thermodynamic Potentials and Conjugate Variables ............................................. 3-42 3.3.15 Nonequilibrium or Irreversible Thermodynamics ................................................... 3-48 3.4 Concluding Remarks ..................................................................................................................... 3-52 Appendix 3.1 Datasets Used for Examples in This Chapter ......................................................... 3-53 Bibliography and Suggested Further Reading ...................................................................................... 3-57

4

Chemical Reaction Kinetics ................................................................................................ 4-1

Introduction ....................................................................................................................................... 4-1 Foods as Chemical Reactors? ......................................................................................................... 4-2 Rate and Extent of Reactions in Closed Systems ....................................................................... 4-4 4.3.1 Kinetics of Elementary Reactions .................................................................................. 4-9 4.3.2 Kinetics of Experimentally Observed Reactions ....................................................... 4-16 4.3.3 Steady-State Approximation and Rate-Controlling Steps ...................................... 4-28 4.4 Catalysis ............................................................................................................................................ 4-33 4.4.1 General Catalysis ............................................................................................................. 4-33 4.4.2 Acid–Base Catalysis ......................................................................................................... 4-34 4.5 Kinetics of Radical Reactions ...................................................................................................... 4-37 4.6 Kinetics of Photochemical Reactions ......................................................................................... 4-41 4.7 Diffusion-Limited Reactions in Aqueous Solutions ............................................................... 4-42 4.8 Kinetics in Open Systems ............................................................................................................. 4-46 4.9 Concluding Remarks ..................................................................................................................... 4-53 Appendix 4.1 Datasets Used for Examples in This Chapter ......................................................... 4-54 Bibliography and Suggested Further Reading ...................................................................................... 4-61 4.1 4.2 4.3

5

Temperature and Pressure Effects ..................................................................................... 5-1

6

Charge Effects ......................................................................................................................... 6-1

5.1 Introduction ....................................................................................................................................... 5-1 5.2 van’t Hoff Equation .......................................................................................................................... 5-1 5.3 Transition State Theory ................................................................................................................... 5-3 5.4 Arrhenius’ Law .................................................................................................................................. 5-8 5.5 Empirical Relations to Describe Temperature Dependence ................................................ 5-15 5.6 Activation Energy and Catalysis ................................................................................................. 5-16 5.7 Parameters Used in Food Science .............................................................................................. 5-18 5.8 Enthalpy=Entropy Compensation .............................................................................................. 5-21 5.9 Variable Temperature Kinetics ................................................................................................... 5-23 5.10 Effect of Pressure ............................................................................................................................ 5-32 5.11 Concluding Remarks ..................................................................................................................... 5-36 Appendix 5.1 Datasets Used for Examples in This Chapter ......................................................... 5-36 Bibliography and Suggested Further Reading ...................................................................................... 5-41

6.1 6.2

Introduction ....................................................................................................................................... 6-1 Models for Ion Activities ................................................................................................................. 6-1 6.2.1 Debye–Hückel Type Models ........................................................................................... 6-4 6.2.2 Mean Spherical Approximation Theory ....................................................................... 6-8 6.2.3 Pitzer Equations ............................................................................................................... 6-11

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Ion Pairing Models ........................................................................................................................ 6-12 6.3.1 Mass Action Law ............................................................................................................. 6-15 6.3.2 Pytkowicz Model .............................................................................................................. 6-18 6.3.3 Binding MSA Model ....................................................................................................... 6-23 6.4 Kinetics of Reactions between Ions ........................................................................................... 6-25 6.4.1 Primary Salt Effect ........................................................................................................... 6-25 6.4.2 Secondary Salt Effect ....................................................................................................... 6-29 6.4.3 Examples Showing the Primary Salt Effect on Kinetics ......................................... 6-31 6.5 Concluding Remarks ..................................................................................................................... 6-40 Appendix 6.1 Datasets Used for Examples in This Chapter ......................................................... 6-40 Bibliography and Suggested Further Reading ...................................................................................... 6-46 6.3

7

Kinetics and Statistics ........................................................................................................... 7-1

Introduction ....................................................................................................................................... 7-1 Some Background on Statistical Approaches ............................................................................. 7-2 7.2.1 Classical Sampling Theory ............................................................................................... 7-3 7.2.2 Maximum Likelihood ....................................................................................................... 7-3 7.2.3 Bayesian Statistics .............................................................................................................. 7-4 7.2.4 Resampling Methods ........................................................................................................ 7-7 7.3 Experimental Design: Statement of the Problem ....................................................................... 7-9 7.4 On Errors and Residuals .............................................................................................................. 7-13 7.4.1 Deterministic and Stochastic Models .......................................................................... 7-13 7.4.2 Least Squares Regression ............................................................................................... 7-14 7.4.3 Sums of Squares and ANOVA ..................................................................................... 7-15 7.4.4 Error Structure of Data: A Variance Model .............................................................. 7-16 7.5 Linear and Nonlinear Models ..................................................................................................... 7-20 7.6 A Closer Look at Assumptions for Parameter Estimation ................................................... 7-21 7.7 Normal Probability Plots and Lag Plots ................................................................................... 7-25 7.8 Goodness of Fit and Model Discrimination ............................................................................ 7-29 7.9 Precision of Regression Lines and Parameter Estimates ....................................................... 7-40 7.9.1 Jackknife Method ............................................................................................................. 7-50 7.9.2 Bootstrap Method ............................................................................................................ 7-50 7.9.3 Grid Search Method ........................................................................................................ 7-53 7.9.4 Monte Carlo Method ...................................................................................................... 7-57 7.9.5 Bayesian Analysis Using Markov Chain Monte Carlo Methods .................................................................................................................. 7-57 7.10 Variability and Uncertainty ......................................................................................................... 7-64 7.11 Transformation of Parameters: Reparameterization .............................................................. 7-68 7.12 Propagation of Errors .................................................................................................................... 7-72 7.13 Sensitivity Analysis ........................................................................................................................ 7-74 7.14 Experimental Design ..................................................................................................................... 7-76 7.14.1 Systematic and Random Errors: Accuracy and Precision .................................................................................................................... 7-77 7.14.2 Experimental Design for Kinetic Models ................................................................... 7-78 7.15 Concluding Remarks ..................................................................................................................... 7-87 Appendix 7.1 Datasets Used for Examples in This Chapter ......................................................... 7-87 Bibliography and Suggested Further Reading ...................................................................................... 7-94 7.1 7.2

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SECTION II

8

Application of the Basics to Chemical, Biochemical, Physical, and Microbial Changes in the Food Matrix

Multiresponse Kinetic Modeling of Chemical Reactions ............................................. 8-1

Introduction ....................................................................................................................................... 8-1 What Is Multiresponse Modeling? ................................................................................................ 8-1 Determinant Criterion ..................................................................................................................... 8-3 Model Discrimination and Goodness of Fit for Multiresponse Models .............................. 8-5 Examples of Multiresponse Modeling of Reactions in Foods ................................................. 8-7 8.5.1 Heat-Induced Acid Hydrolysis of Sucrose ................................................................... 8-7 8.5.2 Degradation of Chlorophyll ............................................................................................ 8-8 8.5.3 Aspartame Degradation ................................................................................................. 8-16 8.5.4 Maillard Reaction ............................................................................................................ 8-19 8.6 Concluding Remarks ..................................................................................................................... 8-26 Appendix 8.1 Datasets Used for Examples in This Chapter ......................................................... 8-27 Bibliography and Suggested Further Reading ...................................................................................... 8-29 8.1 8.2 8.3 8.4 8.5

9

Enzyme Kinetics .................................................................................................................... 9-1

Introduction ....................................................................................................................................... 9-1 Michaelis–Menten Kinetics ............................................................................................................. 9-4 9.2.1 Linearized Plots ................................................................................................................ 9-13 9.3 Enzyme Inhibition ......................................................................................................................... 9-16 9.4 Progress Curves .............................................................................................................................. 9-20 9.5 Kinetics of Two-Substrate Reactions ......................................................................................... 9-29 9.6 Other Types of Enzyme Kinetics ................................................................................................ 9-32 9.7 Temperature Effects ....................................................................................................................... 9-36 9.8 pH Effects ........................................................................................................................................ 9-39 9.9 Experimental Design for Enzyme Kinetics .............................................................................. 9-42 9.10 Enzyme Kinetics in Foods ............................................................................................................ 9-43 9.11 Concluding Remarks ..................................................................................................................... 9-47 Appendix 9.1 Datasets Used for Examples in This Chapter ......................................................... 9-48 Bibliography and Suggested Further Reading ...................................................................................... 9-58 9.1 9.2

10

Kinetics of Protein and Enzyme Denaturation ............................................................ 10-1

11

Kinetics of Physical Changes ............................................................................................ 11-1

10.1 Introduction ..................................................................................................................................... 10-1 10.2 Protein Stability .............................................................................................................................. 10-1 10.3 General Kinetic Schemes Describing Enzyme Inactivation ................................................ 10-13 10.4 Food Matrix Effects ..................................................................................................................... 10-26 10.5 Concluding Remarks ................................................................................................................... 10-29 Appendix 10.1 Datasets Used for Examples in This Chapter .................................................... 10-29 Bibliography and Suggested Further Reading .................................................................................... 10-36

11.1 Introduction ..................................................................................................................................... 11-1 11.2 Kinetics of Diffusion ...................................................................................................................... 11-2 11.2.1 Fick’s Laws ....................................................................................................................... 11-2 11.2.2 Maxwell–Stefan Approach ............................................................................................ 11-8

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11.3 Kinetics of Changes in Dispersity ............................................................................................ 11-14 11.3.1 Kinetics of Aggregation of Colloids ......................................................................... 11-14 11.3.2 Kinetics of Creaming or Settling ............................................................................... 11-19 11.3.3 Kinetics of Coalescence ............................................................................................... 11-21 11.3.4 Kinetics of Ostwald Ripening .................................................................................... 11-23 11.3.5 Kinetics of Gelation of Particles ................................................................................ 11-24 11.3.6 Kinetics of Crystallization ........................................................................................... 11-28 11.4 Kinetics of Texture Changes ...................................................................................................... 11-29 11.5 Partitioning Phenomena ............................................................................................................. 11-32 11.5.1 Partition Coefﬁcients ................................................................................................... 11-33 11.5.2 Partitioning of Volatiles .............................................................................................. 11-33 11.5.3 Partitioning of Weak Acids ........................................................................................ 11-46 11.6 Concluding Remarks ................................................................................................................... 11-49 Appendix 11.1 Datasets Used for Examples in This Chapter .................................................... 11-49 Bibliography and Suggested Further Reading .................................................................................... 11-59

12

Kinetics of Microbial Growth ........................................................................................... 12-1

13

Kinetics of Inactivation of Microorganisms .................................................................. 13-1

14

Modeling the Food Matrix ................................................................................................ 14-1

12.1 Introduction ..................................................................................................................................... 12-1 12.2 Primary Growth Models ............................................................................................................... 12-2 12.2.1 Differential Equations .................................................................................................... 12-3 12.2.2 Algebraic Equations ....................................................................................................... 12-6 12.3 Secondary Models ........................................................................................................................ 12-11 12.4 Nonisothermal Growth Modeling ............................................................................................ 12-20 12.5 Bayesian Modeling ....................................................................................................................... 12-22 12.6 Experimental Design ................................................................................................................... 12-28 12.7 Effects of the Food Matrix ......................................................................................................... 12-28 12.8 Concluding Remarks ................................................................................................................... 12-29 Appendix 12.1 Datasets Used for Examples in This Chapter .................................................... 12-30 Bibliography and Suggested Further Reading .................................................................................... 12-40

13.1 Introduction ..................................................................................................................................... 13-1 13.2 Kinetics of Inactivation of Vegetative Cells ............................................................................. 13-1 13.3 Kinetics of Inactivation of Spores ............................................................................................ 13-10 13.4 Temperature Dependence of Microbial Inactivation ........................................................... 13-16 13.5 Food Matrix Effects ..................................................................................................................... 13-24 13.6 Concluding Remarks ................................................................................................................... 13-26 Appendix 13.1 Datasets Used for Examples in This Chapter .................................................... 13-26 Bibliography and Suggested Further Reading .................................................................................... 13-42

14.1 Introduction ..................................................................................................................................... 14-1 14.2 Speciﬁc Effects in Aqueous Solutions ........................................................................................ 14-3 14.2.1 Water Activity and the Effect of Cosolutes .............................................................. 14-4 14.2.2 Water Activity and Food Stability ............................................................................ 14-11 14.2.3 Ionic and Nonionic Solute Interactions .................................................................. 14-14 14.2.4 Signiﬁcance of pH in Food ......................................................................................... 14-18 14.3 Transport Phenomena and Molecular Mobility in the Food Matrix ............................... 14-23 14.4 Micellar Effects ............................................................................................................................. 14-32

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14.5 Effect of Molecular Crowding in the Food Matrix .............................................................. 14-34 14.6 Concluding Remarks ................................................................................................................... 14-36 Appendix 14.1 Datasets Used for Examples in This Chapter .................................................... 14-37 Bibliography and Suggested Further Reading .................................................................................... 14-53

15

Retrospective and Outlook ................................................................................................ 15-1

15.1 Introduction ..................................................................................................................................... 15-1 15.2 Shelf Life Modeling as an Integrative Approach .................................................................... 15-1 15.2.1 Shelf Life from the Product Point of View ............................................................... 15-2 15.2.2 Shelf Life from the Consumer Point of View .......................................................... 15-4 15.3 Some Developments .................................................................................................................... 15-12 15.4 Concluding Remarks ................................................................................................................... 15-14 Appendix 15.1 Datasets Used for Examples in This Chapter .................................................... 15-15 Bibliography and Suggested Further Reading .................................................................................... 15-19

Appendix A

Some Calculus Rules .......................................................................................... A-1

Appendix B

Ways to Express Amounts of Reactants and Products .............................. B-1

Appendix C

Interconversion of Activity Coefﬁcients Based on Mole Fractions, Molalities, and Molarities ............................................. C-1

Appendix D

Differential and Integrated Rate Equations for Kinetic Models of Complex Reactions ........................................................................................ D-1

Appendix E

McMillan–Mayer and Lewis–Randall Framework and Equations for the Mean Spherical Approximation Theory ........................................... E-1

Appendix F

Probability Laws and Probability Models ...................................................... F-1

Appendix G Use of Matrix Notation in Model Representation and Regression Analysis .................................................................................... G-1 Appendix H Some Thermodynamic Activity Coefﬁcient Models ................................... H-1 Appendix I

Reliability Engineering and the Weibull Model ............................................ I-1

List of Symbols and Units ........................................................................................................... 16-1 Index ................................................................................................................................................... I-1

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Preface The topic of food quality is receiving ever-increasing attention. Consumers are concerned about the quality of their food and have high demands. At the same time, consumer demands are rapidly changing, and the food supply chain needs to match these changing demands in order to be able to deliver food of a desired quality at the end of the chain. However, the quality of a food changes continuously along its way through the food chain. It is therefore important to have tools to control and predict food quality (including food safety) and to be able to quickly change food design according to changing consumer expectations. This is useful for consumers because it helps to ensure that their needs are fulﬁlled and that they obtain safe food. Obviously, it is helpful for the food industry because it provides a suitable tool to connect physical product properties with consumer wishes. I am convinced that the use of mathematical models for modeling of quality attributes of foods is going to be of great help in these matters. This book is about how to model changes taking place in foods, for which the scientiﬁc term is kinetics. The aim of this book is to introduce appropriate kinetic models and modeling techniques that can be applied in food science and technology. It is fair to say that mathematical modeling is already used to some extent in the food science and technology world, but in the author’s opinion there are many more opportunities than those currently applied. This book aims to indicate directions for the use of modeling techniques in food science. It will be argued that modeling of food quality changes is in fact kinetic modeling. However, this is not just another book on kinetics. Rather, it integrates food science knowledge, kinetics, and statistics, so as to open the possibility to predict and control food quality attributes using computer models. Moreover, much more information can be extracted from experiments when quantitative models are used. I hope to show with this book that the quality of modeling can be improved considerably with proper mathematical and, especially, statistical techniques. The choice of topics reﬂects my research interests. Obviously, this choice is subjective and reﬂects my ideas about how modeling of food quality should be done. Quality changes in foods are related to the chemical, biochemical, physical, and microbiological changes taking place in the food, in relation to processing conditions. I have attempted to apply kinetic models using general chemical, physical, and biochemical principles, but allowing for typical food-related problems. The general principles mentioned are usually derived for only very simple, dilute, and ideal systems. Foods are all but simple, ideal, and dilute. Another important point in my view is that allowance should be made for variability and uncertainty, and therefore I consider the use of statistics as indispensable. A substantial part of this book is devoted to the use of statistical techniques in kinetic modeling, which is another reason it is not a typical kinetics book. I introduce the concept of Bayesian statistics, which is hardly known in the food science world. I feel it has great potential, and I intend to show that in this book. The book is ﬁrst of all meant for food scientists who want to learn more about modeling. It was written with two objectives in mind. The ﬁrst was to introduce the topic of kinetics and its application to foods to students and graduates in food science and technology. I teach kinetics to food science students in an advanced MSc course called ‘‘Predicting Food Quality’’ and in an advanced PhD course called ‘‘Reaction Kinetics in Food Science’’ at Wageningen University. The response of the students is encouraging. xiii

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Preface

The book could therefore be helpful as a textbook in advanced MSc and PhD courses at other universities. The second objective was to write a reference book to be used by professional researchers active in food-related work. It should be useful, therefore, for graduates working in the food industry who have a keen interest in modeling, and who are willing to apply modeling concepts in food product design. It could even be useful for nonfood disciplines such as biotechnology, pharmacy, nutrition, and general biology and chemistry. It is on an advanced level in the sense that it builds upon basic food science and technology knowledge, as well as basic mathematical knowledge of calculus and matrix algebra. Also, basic statistical knowledge is assumed, although some introduction is given to Bayesian statistics because this will be new to most food scientists. As a reminder for the reader, appendices consisting of the basic background on all these matters are provided. The ultimate aim is to guide students, graduates, and postgraduates in such a way that they can understand and critically read articles in the literature concerning this topic, and can apply the principles in their own research, be it fundamental or applied. It is, of course, unavoidable that there are many equations in this book since it deals with mathematical models. Fortunately, mathematical complexity can be kept to a minimum using appropriate software such as Mathematica, MathCad, Maple, and even well-known spreadsheets such as Microsoft Excel. I used MathCad and Excel quite extensively for this book, as well as some specialized software where indicated. The reader should try to look beyond the equations and math involved and it will be very helpful to work out the examples given. Wherever possible, I will express in words also what is expressed in an equation. Nevertheless, I do realize that the many mathematical and statistical equations are not easy to digest. Therefore, I have strived to illustrate the concepts introduced with many real-life examples rather than using hypothetical data, or examples that are less relevant for food science problems. The data for the examples were either read directly from tables published in papers, or digitally scanned by computer from graphs. Occasionally, authors supplied me with data, for which I am very grateful, and I also used my own data. All datasets used are supplied in appendices to the chapters, including their sources, so that the interested reader can work with these examples by himself or herself. I would like to stress that the examples chosen are not meant to criticize results; they are chosen because they illustrate the points I want to make. I am actually quite grateful that authors made it possible to extract data from the publications; this is actually as it should be. I have used many references from literature in compiling my own text, by going well beyond the food science and technology literature. However, I decided not to indicate literary references in the text itself to improve readability. Rather, whenever substantial use was made of a particular reference that reference was mentioned at the end of the chapter. I do acknowledge all the excellent articles that are available and which substantially helped me to formulate my own text. Finally, I would like to acknowledge several persons who have been instrumental in helping me realize this book. First of all, I would like to acknowledge Professor Dr. Bronek Wedzicha from the Procter Department of Food Science, University of Leeds, United Kingdom. Thanks to his hospitality, I have been able to spend two sabbatical periods of three months at the University of Leeds in the summers of 1999 and 2004 and during these periods we had very intensive discussions over the topics covered in this book. Moreover, he and his wife Glenis have been very generous to me on a personal level by inviting me to many lovely dinners at their house, and for entertaining walks in beautiful Yorkshire. I do regret not having Professor Wedzicha as a coauthor; the book would have been much better had this been the case. However, his critical spirit has been essential for my writing and many of his thoughts are reﬂected in this book. This is especially true for Chapter 14, which has been inspired strongly by his ideas and lectures on this topic. Furthermore, I would like to thank Professor Pieter Walstra from Wageningen University for stimulating me to take this path in my academic career, and for critically reading several drafts of the chapters. I would also like to thank Professor Willem Norde from Wageningen University for very useful comments on the chapter on thermodynamics. Having acknowledged Bronek Wedzicha, Pieter Walstra, and Willem Norde for their invaluable contributions, I am of course fully responsible for the text,

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including all errors and mistakes. I would very much appreciate remarks, criticism, and corrections from readers. Last but not least, I would like to take this opportunity to thank my wife Corrie for being very patient with me, for not complaining about my physical absence of two periods of three months abroad, not to mention the countless evening and weekend hours, just so that I could do my writing. It is well appreciated and I dedicate this book to her.

M.A.J.S. (Tiny) van Boekel Wageningen

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Author Martinus van Boekel received his BSc, MSc, and PhD in food science and technology from Wageningen University, Wageningen, the Netherlands. Immediately after, from 1980 to 1982, he worked at the Food Inspection Service at Rotterdam, the Netherlands, as a food chemist. He then returned to Wageningen University to work as an assistant professor from 1982 to 1994, as an associate professor from 1994 to 2001, and as a full professor from 2001 onward in the ﬁeld of food science and technology. In 2006, he became the scientiﬁc director of the graduate school VLAG (Food, Nutrition, Agrotechnology, and Health) for 4 years. His research and teaching encompass modeling of food quality attributes in an integrative way, that is, integrating the various food science disciplines but also nutrition, marketing, economics, and quality management. He has been a visiting professor at the University of Madison, Wisconsin, and the Procter Department of Food Science, University of Leeds, United Kingdom. He is the author and coauthor of about 160 refereed scientiﬁc papers and author= editor of six books.

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1 Kinetic View on Food Quality

1.1

Introduction

The aim of this book is to discuss kinetics of reactions in foods in relation to food quality. By reactions we mean all type of change taking place in the food whether they be chemical, enzymatic, physical, or microbial. Kinetics is about change. For the moment it sufﬁces to describe kinetics as the translation of knowledge (theoretical as well observational) on a time-dependent chemical, physical, microbial, reaction into an equation describing such changes in mathematical language. The mathematical relations result in models that we can use to design, optimize, and predict the quality of foods. It should also be helpful in choosing the technology to produce them. We thus need chemical, physical, microbial knowledge to build mathematical models as well as knowledge on composition and structure of foods, i.e., food science; it is assumed that the reader is familiar with basic principles of food science and technology. The major part of the book is concerned with modeling the kinetics of relevant reactions in foods and deals with questions such as: what is kinetics, what are models, how do we apply kinetics to practical problems in foods, what are pitfalls and opportunities, how to deal with uncertainty, and how to interpret results. A key question to be answered is why the kinetics of reactions in foods is often different from, say, that of chemical engineering processes. In this chapter, we discuss some important determinants of food quality. While the subject of quality deserves a book in its own right, the purpose here is to put the relationship between kinetic modeling and food quality in perspective, to be developed in subsequent chapters.

1.2

Food Quality

What then is food quality? There are many deﬁnitions and descriptions of quality. One useful but very general description is ‘‘to satisfy the expectations of the consumer.’’ Although the idea of quality seems to be somewhat elusive, it is important to understand the concept because, as food technologists, we need to be able to control and predict food quality attributes. Food quality attributes are all those product attributes that are relevant in determining quality. The ultimate test for quality is acceptance or rejection by the consumer. When a consumer evaluates a product, a ﬁrst impression arises from so-called quality cues: attributes that can be perceived prior to consumption and that are believed to be indicative of quality. Examples are red color of meat, or information concerning the origin of the product. This leads to certain quality expectations. When the consumer starts eating, he is confronted with the physical 1-1

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Kinetic Modeling of Reactions in Foods

1-2

Quality attributes (intrinsic and extrinsic)

Quality cues

Physical product properties

Quality expectations

Quality experience

Quality assignment If higher than quality limit

If less than quality limit

Acceptance

FIGURE 1.1

Rejection

Schematic picture of aspects involved in quality evaluation by a consumer.

product properties (e.g., texture, taste, ﬂavor) and this leads to a quality experience. If the quality expectation and the quality experience, integrated with each other, exceed a certain quality level, the consumer will accept the product, if not he will reject it. Figure 1.1 shows this process schematically, but the reader is advised that this scheme is an oversimpliﬁcation. Quality is multidimensional, it contains both subjective and objective elements, it is situation speciﬁc and dynamic in time. A consumer however does not analyze all elements of food quality consciously but gives an integrated response based on complex judgments made in the mind. In order to make quality more tangible for the food scientist, it is suggested to make a division into intrinsic quality attributes, i.e., inherent to the product itself, and extrinsic attributes, linked to the product but not a property of the food itself. Extrinsic factors are, for instance, whether or not a food is acceptable for cultural=religious or emotional reasons, or whether the way it is produced is acceptable (with or without fertilizer, pesticides, growth hormones, genetically modiﬁed, etc.) and its price. Extrinsic factors are therefore not part of the food itself but are deﬁnitely related to it (as experienced by the consumer). On the other hand, the chemical composition of the food, its physical structure, the biochemical changes it undergoes, the microbial and chemical condition (hazards from pathogens, microbial spoilage, presence of mycotoxins, heavy metals, pesticides, etc.), its nutritional value and shelf life, the way packaging interacts with the food, are intrinsic factors. We can propose a hypothetical quality function Q: Q ¼ f (Qint , Qext )

(1:1)

In words, this equation states that quality can be decomposed in intrinsic and extrinsic quality attributes. The nature of this function remains as yet obscure. We do not know, for instance, whether we are allowed to sum intrinsic and extrinsic quality attributes, or that we need to multiply them, or do yet something else. In terms of modeling, quality assignment is usually done either from the consumer perspective or from the product perspective. It would be better if the two approaches were integrated. Techniques like quality function deployment (QFD) try to do this. We will not discuss this further in this book.

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Kinetic View on Food Quality

1-3

Raw materials and ingredients

Product formulation

Processing

Number, type of bacteria

Bacterial growth

Interactions in the food matrix

Ii =

Taste

Flavor

Shape, color

Nutritional Convenience value

Texture

Shelf life

Food safety

Packaging Intrinsic quality Qint

FIGURE 1.2

Schematic presentation of intrinsic quality attributes Ii.

It helps however to disentangle intrinsic and extrinsic quality attributes to make clear which factors are controllable by a technologist. Figure 1.2 shows a further decomposition of Qint into intrinsic quality attributes Ii: Qint ¼ f (I1 , I2 , . . . , In )

(1:2)

Figure 1.3 does the same for extrinsic quality attributes Ei: Qext ¼ f (E1 , E2 , . . . , En )

Ei = Values, norms

Regulations

Availability

Brand name

Extrinsic quality Qext

FIGURE 1.3

Schematic depiction of extrinsic quality attributes Ei.

(1:3)

Price

After sales service

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1-4

As with the overall quality function Q, we do not know the nature of the functions Qint and Qext. In other words we do not know how the quality attributes interact and are integrated by the consumer into one ﬁnal quality judgment; moreover it will differ from consumer to consumer. Much more can be said about quality, but that is beyond the scope of this book. We focus now on intrinsic quality attributes. To be sure, we will not attempt to ﬁnd a relation for Qint in this book; rather we focus on how to characterize the listed quality attributes from a technological point of view. Even though the ﬁnal quality judgment is not based on intrinsic factors alone, measurable objective quality attributes such as food safety, nutritional value, and color are of utmost importance. In food science literature, intrinsic factors such as those mentioned in Figure 1.2 are usually called quality attributes, though this is not strictly correct as shown in Figure 1.1. To satisfy the (dynamic) expectation of consumers, with diversity in needs and markets, a producer must be prepared to be very ﬂexible with respect to intrinsic quality attributes. Insight in these quality attributes is thus a prerequisite to survive in a competitive market. We propose that with the kinetic tools presented in this book these intrinsic quality attributes can be controlled and predicted. Intrinsic food quality attributes can be studied at several levels as shown in Figure 1.4. With reference to Figure 1.4, this book will deal mainly with modeling activities at levels 1 and 2, with some attention to level 3 concerning the design of experiments for food product design. Kinetic modeling of food quality attributes can be a powerful tool as part of the steps to be taken in food product development. Also, it can be the basis for the development of expert systems and management systems, especially with reference to risk analysis and food safety issues. Certain chemical reactions may serve as indicators for speciﬁc quality attributes. For instance in milk, the concentration of lactulose (an isomerization product of lactose) is an indicator of the heat treatment to which the milk has been exposed but it is not a quality attribute by itself. Clearly, the kinetics of the chemical reaction that serves as a quality indicator need to be closely related to the kinetics of the chemical or physical changes that determine the relevant quality attributes it represents. The way quality is monitored and safeguarded

Product market Level 4

Increasing complexity

Food design

Level 3

Functional properties, model system, processing

Properties of real foods, product development

Component interactions, kinetics

Level 2

Chemical, physical, microbial behavior of compounds Level 1

FIGURE 1.4

Consumer + sensory research

Several levels at which food quality can be studied.

Basic (bio)chemical, physical research

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Kinetic View on Food Quality

1-5

is a particular aspect of food quality. This involves quality management, the introduction of systems such as hazard analysis and critical control points (HACCP), ISO systems, and good manufacturing practices (GMP). The statement of quality is made with reference to speciﬁc technical speciﬁcations, in other words such an approach requires integration of technological and management knowledge (technomanagerial approach). Basically, this comes down to realizing the fact that food quality is not only determined by the product itself or the technology applied but also by the people that handle the product. We will not discuss these aspects here; some references are given at the end of this chapter. The basic message is thus that quality is not a property of the food but is determined by the consumer who translates his perception into quality attributes. Some of these attributes can be related to measurable properties of the food though this is not always possible for user-related factors. The crispness of potato crisps, for instance, relates to mechanical properties of the (fried) potato cell wall; the sweetness of pastry is related to its sugar content as well as the sweetening intensity of the sugar used; the color of a food is for the most part attributable to components that absorb light at a particular wavelength and=or scatter light. There are also intrinsic factors that cannot be perceived directly by the consumer, such as the presence of toxic components or pathogenic bacteria. Such ‘‘hidden’’ quality attributes can, however, in most cases be measured. This book is concerned only with intrinsic factors, and particularly how we can ‘‘capture’’ these quality attributes within mathematical models. The advantage of using such models is that they can be linked to other models describing for instance stimulus–response relationships and consumer preference. The following intrinsic quality attributes are the most important ones for the food scientist: . . . . . . . .

Safety (microbial, toxic, mutagenic) Wholesomeness, nutritional value Usage (handling) properties Storage stability=shelf life Texture Color Appearance Flavor, taste compounds

Some of these attributes are the result of the interaction of stimuli picked up by the senses and are called sensory properties. Sensory properties can be estimated using sensory panels (though this is a different type of measurement process than using laboratory instruments). It is, however, important to make a distinction between product properties and the perception of these properties. Sensory measurements are, therefore, the result of product properties (causing stimuli) and the processing of these stimuli by the consumer. Some quality attributes are the resultant of several phenomena. For instance, the color of a food may be the result of the presence of several components absorbing or reﬂecting light of a certain wavelength. Even though color can be measured instrumentally, it is not immediately obvious which compound is responsible for the color observed. Another example is the quality attribute nutritional value, which is determined not only by vitamin content but also by the type and amounts of amino acids, type and amounts of fatty acids, etc. That is why we propose to decompose quality attributes further into quality performance indicators. In the above examples, a quality performance indicator for color may be the concentration of a carotenoid, and the content of the amino acid lysine may be one of the quality indicators for nutritional value. Many quality indicators can be measured directly using physical or chemical measurements. Examples include the presence=absence of pathogenic microorganisms, the protein content and the biological value of the protein, vitamin content, bioavailability, etc. These indicators clearly cannot be determined via sensory panels; they are hidden to the consumer, although they may have a subliminal effect on food choice. Kinetic approach. When we speak of food quality in this book we address these physical, chemical, biochemical, and microbial quality indicators. We accept that this is only a part of the quality perceived

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by the consumer. However, we limit ourselves deliberately to the indicators mentioned because we consider them the principal domain of the food technologist. An important consideration is that these indicators tend to change with time, and therefore they have to be characterized by a kinetic approach, the subject of this book. Food technology is, in short, concerned with the transformation of raw materials into foods and their stabilization (preservation), taking into account all boundary conditions of food safety and quality mentioned above. Raw materials and foods are subject to change because of their thermodynamic instability: reactions take place driving the system toward thermodynamic equilibrium (as will be discussed in Chapter 3). Foods may deteriorate soon after harvesting (sometimes even during harvesting), and deterioration should be read as loss of quality. Prevention and control of this thermodynamic instability is the main task of food technologists. It is the characterization of the changes taking place that is important because this provides us with possibilities to control quality. This is then the domain of kinetics. Kinetics plays thus an important part in the modeling of food quality. The purpose of this book is to explain how kinetics and kinetic models can be used in a meaningful way, thus to supply valuable tools to describe changes in quality performance indicators and attributes, and most importantly to supply tools to control and predict these quality indicators and attributes. Still, foods are so incredibly complex from a chemical and physical point of view that we need to resort frequently to systems mimicking foods. Otherwise, there will be so many interfering factors that the predictive capabilities of mathematical models will be very limited. Model systems mimicking foods are by their very nature simpliﬁcations but, on the other hand, they need to approach real foods in some sense. Ignoring speciﬁc properties of foods when designing model systems may lead to serious mistakes when one extrapolates from the model systems to real foods. Since this is not straightforward, a special chapter (Chapter 14) discusses this in detail for some relevant food aspects. Overall, the philosophy presented in this book is that it is essential to understand what is happening at the molecular level (occasionally the colloidal level) and for this reason the material presented is at the fundamental level of thermodynamics and chemical kinetics. It is the author’s view that such understanding is needed in order to come to models that will be able to control and predict food quality. In addition, kinetic modeling as such is a tool in understanding what is going on because proposed mechanisms need to be confronted with experiments, and if the two do not match something was apparently wrong with the proposed mechanism. Having said that, it is also appreciated that we sometimes have to resort to empirical models due to the complexity of foods. This statement may seem contradictory to the philosophy that fundamental insight is needed but it is not. It is merely a recognition of the fact that our understanding of what is going on in foods is far from complete, and it would be foolish to stick to models that are derived from situations in very simple and ideal systems while they are not capable to grasp the real situation. Especially if we want to be able to predict real-life situations in a realistic way, empirical models may actually perform better than mechanistic models in some situations. That is why the reader will also be introduced to empirical models. Admittedly, empirical models will not directly provide molecular insight. It is therefore important to have attention for both approaches.

1.3

Foods as Complex Reaction Media

When considering reactions in foods, the medium in which these reactions take place is obviously of importance. We may have solid, liquid, and vapor phases in and around foods. Most of the relevant reactions in foods will take place in the liquid phase. In many cases this will be an aqueous phase but also lipid phases are possible, or ethanol may be present which gives different properties to the reaction medium. There may be partitioning between phases. Solid phases may become of importance because they may result from exceeding solubility products; an important solid phase is, of course, ice, but also salts and sugars may be present as crystalline material, or sometimes as amorphous materials. Moreover, solid phases may induce adsorption of reactants and products and catalyze or inhibit reactions. Then we have the presence of amorphous phases, like in glasses. The vapor phase is of importance when a

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Food Food composition composition and and structure structure

Chemicalpotential, potential, Chemical Activities activities

Ionic Ionic

Compartments within a continuous phase

Nonionic Nonionic

Foambubbles bubbles Foam

FIGURE 1.5

Cells Cells

Molecular mobility mobility Molecular

Viscosity Glassy Glassystate state Viscosity

Emulsiondroplets droplets Emulsion

Networks Networks

Crystals Crystals

Overview of the complexity of foods as reaction media.

headspace is present, or in the case of foams, and the partitioning of volatiles is a very relevant phenomenon in relation to sensorial aspects. When we want to study kinetics in foods, we have to take all these various aspects into account, but there are few, if any, theoretical frameworks available with which to do this. Most theories have been developed for ideal, dilute, and homogeneous systems. Foods are multicomponent, concentrated systems with various phases present at the same time, and consequently foods behave all but ideal. Complications with foods arise because of deviations from simple diffusion laws, complications with molecular mobility, partitioning phenomena, and volume exclusion effects. These questions will be addressed in the book. The food matrix is usually very complex, consisting of water-insoluble material (e.g., cell membranes), a complicated aqueous solution of ionic and nonionic compounds of high and low molar mass, amorphous materials, various phases (fat globules, foam bubbles, crystals), and to complicate matters further, foods can also be in a glassy state. All this has a large impact on kinetics. Figure 1.5 summarizes the aspects involved, and serves as a guideline for the topics to be discussed in this book.

1.4

Outline of the Book

After this introduction, the book is divided in two parts. The ﬁrst part is called ‘‘The basics’’ and attempts to describe the ﬁrst principles of modeling (Chapter 2), thermodynamics (Chapter 3), chemical kinetics (Chapter 4), temperature and pressure effects (Chapter 5), and charge effects (Chapter 6). Chapter 7 introduces the use of statistics in kinetics. In the author’s view this is a crucial topic that deserves a great deal of attention and the topic is therefore treated at some length. Though the treatment in Part I is basic and general, food examples are used wherever possible. Part II is called ‘‘Application of the basics to chemical, biochemical, physical, and microbial changes in the food matrix,’’ in which we direct our attention subsequently to chemical, physical, biochemical, and microbiological aspects relevant for foods. Thus, Chapter 8 discusses the possibilities and advantages of multiresponse modeling, a topic that lends itself very well for food science problems, especially when they are of a chemical nature. Surprisingly, the concept is hardly used in food science literature, and accordingly we describe the principles, applications, and potential problems in detail and apply it to some chemical changes. As indicated above, there is more to food quality than chemical changes. The chapters to follow are devoted to enzyme kinetics and kinetics of protein and enzyme inactivation (Chapters 9 and 10), kinetics of physical processes (Chapter 11), kinetics of microbial growth as well as inactivation (Chapters 12 and 13, respectively). Chapter 14 attempts to address speciﬁc problems arising in the food matrix when dealing with kinetics. This concerns discussions as to why kinetics in foods can be quite different from reactions in simple model

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systems in test tubes, how we can identify such problems and take them into account when using model systems (to get around the problem of variability and complexity). Finally, we give a retrospective and an outlook by discussing some trends and developments in modeling in general and with some attention for shelf life modeling in particular because that requires integration of several aspects (Chapter 15).

Bibliography and Suggested Further Reading About Quality Aguilera J.M. and Lilford P. Food Materials Science. Principles and Practice. New York: Springer, 2008. Bruin S. and Jongen T.R.G. Food process engineering: The last 25 years and challenges ahead. Compr Rev Food Sci Food Saf 2:42–81, 2003. Damodaran S., Parkin K.L., and Fennema O.R. Fennema’s Food Chemistry. 4th ed. Food Science and Technology, p. 1144. Boca Raton: CRC Taylor & Francis, 2008. Jongen W.M.F. and Meulenberg M.T.G. Innovation in Agri-Food Systems. Product Quality and Consumer Acceptance, p. 399. Wageningen: Wageningen Academic Publishers, 2005. Fito P., LeMaguer M., Betoret N., and Fito P.J. Advanced food process engineering to model real foods and processes: The ‘‘SAFES’’ methodology. J Food Eng 83:173–185, 2007. Gaonkar A.G. and McPherson A. Ingredient Interactions. Effects on Food Quality. 2nd ed., p. 554. Boca Raton: CRC Taylor & Francis, 2006. Kind M. Product engineering. Chem Eng Process 38:405–410, 1999. Labuza T.P. and Riboh D. Theory and application of Arrhenius kinetics to the prediction of nutrient losses in foods. Food Technol 36:66–74, 1982. Labuza T.P. Application of chemical kinetics to deterioration of foods. J Chem Educ 61:348–358, 1984. Linnemann A.R. and Van Boekel M.A.J.S. Food Product Design. An Integrated Approach, p. 236. Wageningen: Wageningen Academic Publishers, 2007. Lund D. Predicting the impact of food processing on food constituents. J Food Eng 56:113–117, 2003. Luning P.E., Marcelis W., and Jongen W.M.F. Food Quality Management. Wageningen, the Netherlands: Wageningen Pers, 2002. Luning P., Devlieghere F., and Verhé R. Safety in the Agri-Food Chain, p. 688. Wageningen: Wageningen Academic Publishers, 2006. Martens H. and Martens M. Multivariate Analysis of Quality. Chichester: John Wiley & Sons, 2001. Molnar P.J. A model for overall description of food quality. Food Qual Pref 6:185–190, 1995. Niranjan K. Chemical engineering principles and food processing. Trends Food Sci Technol 5:20–23, 1994. Norton I., Fryer P., and Moore S. Product=process integration in food manufacture: Engineering sustained health. AIChe J 52:1632–1640, 2006. Saguy I. and Karel M. Modeling of quality deterioration during food processing and storage. Food Technol 34:78–85, 1980. Sloof M., Tijskens L.M.M., and Wilkinson E.C. Concepts for modelling the quality of perishable products. Trends Food Sci Technol 7:165–171, 1996. Tijskens, L.M.M. Discovering the Future: Modelling Quality Matters. PhD thesis, Wageningen University, the Netherlands, 2004. Van Boekel M.A.J.S. Kinetic modeling of food quality. Compr Rev Food Sci Food Saf 7:144–157, 2008. Van Trijp H.C.M. and Steenkamp J.B.E.M. In: Innovations in Food Production Systems. Jongen W.M.F. and Meulenberg M.T.G. (Eds.). Wageningen: Wageningen Press, 1998.

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The Basics

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2 Models and Modeling

2.1

Introduction

Since this book is about kinetic modeling, it is appropriate to explain the philosophy about models. Models are certainly not a panacea for all problems. They offer opportunities but also have limitations. It is essential that the reader be aware of this and it is the intention of this chapter to provide this basic awareness.

2.2

Models and Modeling

So, what are models and what is modeling? The answer is not straightforward because it depends on the goals of modeling and the type of model used. Generally speaking, models attempt to formulate the behavior of systems from knowledge of the properties of their component parts. Invariably, models are simpliﬁcations of the real world, designed to facilitate predictions and calculations. They are a tool to help us handle complex situations. Thus, the modeler should be under no illusion with regard to the physical reality of models. Models exist in the mind of the scientist, not in nature. Modeling is an attempt to approximate the real world (the truth), but the truth (whatever that is) will never be reached (if we would know the truth it would not be necessary to use models). This does not detract at all from the usefulness of models but an awareness of the nature of models will help us to see the opportunities as well as the limitations. Thus, models can be seen as a way of communicating a view of the world and they are open to scientiﬁc debate. This applies, of course, equally well to kinetic modeling of reactions in foods. Let us try to picture the various ways in which we can use models to describe a system. Suppose that an input is given to a system that will respond with an output: see Figure 2.1. If we know the input I and we can measure the response R, we can use a model to learn about the system S. For instance, if we heat a food (heat is the input) and we measure the effect on protein denaturation (the response) we could learn something about the behavior of proteins in that particular food matrix (the system). If we know the input I as well as the system S, we can use a model to predict the response. For instance, if we know how much heat we put into a system and we know how the proteins in the system respond to this, we can predict the level of denaturation. If we know the system S as well as the response R it produces upon a certain input we can use a model to control, or to design, which input we need to produce a desired output. For instance, if we want to achieve a certain level of protein denaturation in a food, then we can calculate how much heat is needed to achieve this. These simple examples show that models can be used for various goals. In relation to food quality, all three goals are important. Our system is the food, inputs can be processing conditions, and responses can be changes in food quality attributes. We can use models to learn about the ‘‘physical’’ processes taking place in the food that govern food quality attributes, to 2-1

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Input I

FIGURE 2.1

System S

Response R

System S responding to an input I by a response R.

predict food quality attributes as a function of inputs for a given food, and to control food quality for a given food by managing the input. It is perhaps meaningful to spend a few more words on the concept of prediction because there seems to be some confusion on this in literature. A distinction should be made between ‘‘model ﬁt’’ and ‘‘prediction.’’ A model ﬁt is obtained when comparing the performance of a model with the experimental data. Prediction means that it is possible, via models, to predict events or situations that were not in any way used in setting up the model. This can be future events, or events that were obtained independently in other studies. To be clear, we use the words model ﬁt and prediction in this sense throughout the book. Schemes such as in Figure 2.1 are sometimes referred to as conceptual models, i.e., a hypothesis about how a system works and responds to changes in inputs. In other words, it is a set of qualitative assumptions. If we are able to turn somehow these qualitative assumptions into quantitative ones, and if we can describe this with mathematical equations, a conceptual model changes into a mathematical one. Throughout this book, we will conﬁne ourselves to models that describe ‘‘physical’’ phenomena in a mathematical way, i.e., chemical, physical, or microbial events are translated into mathematical equations. Examples include the nonenzymatic browning of foods, the growth of bacteria in a food, and the sedimentation of cocoa particles in a chocolate drink. Mathematical models relate responses to variables via parameters in one or more equations. Say that we are interested in the fate of a vitamin in a food during processing and storage, as a measure for a change in nutritional quality. What is useful to know then is the change in concentration of such a vitamin over time at a certain temperature, or possibly at ﬂuctuating temperature. As we will see in later chapters, a possible mathematical relation that describes the vitamin concentration (denoted as [vitamin]) as a response to the variables time t and temperature T could be: Ea t [vitamin] ¼ [vitamin]0 A exp RT

(2:1)

[vitamin]0 represents the initial concentration. A and Ea represent parameters characteristic for the degradation of the vitamin in the food under study (these are the pre-exponential factor A and the activation energy Ea in the Arrhenius equation, Chapter 5). Such parameters can be estimated in controlled experiments. The parameter R is a fundamental physical constant (the gas constant), which is known and need not be estimated. Preferably, the parameters should be physically interpretable. Time and temperature are controllable variables that can be manipulated by the experimenter or operator. If an equation such as Equation 2.1 is established and the parameters estimated from experimental results, one could then, in principle at least, control and predict the change in vitamin concentration at any relevant time and temperature because we can control time and temperature, and thereby control this food quality indicator. Quality change modeling. We can generalize this further by following the food along its way in the food production chain. Generally, there will be changes in food quality when the food moves from the producer to the consumer; quality is not static. There may be losses, for instance of vitamins during storage of fruits and vegetables. On the other hand, the nutritional value of a processed food may increase as compared to the raw material, for instance, bioavailability or digestibility may be enhanced. To illustrate this change in quality, suppose that a quality performance indicator (for instance the concentration of a vitamin) is built up during the growth of a vegetable or a fruit. Loss of quality usually starts immediately after harvesting, so postharvest storage may already result in some losses. Processing may perhaps result in

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Processing

Concentration

Postharvest Primary storage production

Storage Distribution

Time

FIGURE 2.2 Schematic depiction of change in a quality performance indicator (e.g., vitamin concentration) along the food processing chain.

much higher losses, while storage and distribution may give a further gradual decrease of quality. Figure 2.2 gives a schematic example of such a quality loss. For modeling purposes, it is convenient to identify and quantify the various factors leading to quality changes in each chain element. The output of one chain element is the input for the next (Figure 2.3); a production chain can be seen in this way as a cascade of unit operations. We propose to describe this as quality change modeling. In doing so, various models need to be connected to each other with proper use of mass and energy balances. However, if we want to maintain a high quality at the end of the chain, i.e., when the food arrives at the consumer, the trick is to optimize quality all over the chain, rather than locally in one of the chain elements. By analyzing quality in this way, it becomes possible to optimize quality from an analysis of what happens in the various elements in the food chain. Analogous to the term HACCP, we propose to describe this as Quality Analysis Critical Control Points (QACCP). In the case of a situation as in Chain element 1

Quality attribute in

Chain element 2

Chain element 3

Processes affecting quality indicators comprising the quality attribute

Chain element 4

Chain element 5

Quality attribute out

Chain element 6

FIGURE 2.3

Schematic presentation of quality models in elements of the food chain.

Factors affecting quality indicators

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Figure 2.2, it is clear that much could be improved in the processing step, so that attempts to increase quality should focus on processing conditions. In other cases, losses during storage or distribution may actually be larger than in processing, and then storage conditions may have to be optimized (for instance, by changing temperature, or relative humidity). In order to be able to derive graphs such as Figure 2.2, one must understand what is happening to that particular quality attribute. The main theme of this book is to apply methods correctly to describe such changes quantitatively in every element of the food chain. Mass and energy balances may be helpful in this respect because terms in such balance equations involve kinetics. To be sure, changes in quality attributes as depicted in Figure 2.3 apply, of course, also to microbial growth as a (negative) quality factor, which is the subject of much research nowadays, sometimes referred to as predictive microbiology, or quantitative microbiology. Of course, much work has already been done and published in the past. Unfortunately, many of these studies published cannot be used to develop predictive models because external conditions as well as essential information on the food were not reported. It is also essential that quantitative data are reported in full rather than as averages for developing and validating models. Deterministic empirical and mechanistic models. If we now generalize equations such as Equation 2.1 in a more abstract way, a description of a mathematical model can be given as: h ¼ f (u, jv )

(2:2)

where h represents measurable response(s) (such as vitamin concentration in Equation 2.1) u symbolizes the parameters of the model (A and Ea in Equation 2.1) jv represents the controllable variables (t and T in Equation 2.1) The notation f(..) should be read as: ‘‘is a function of.’’* In kinetic models, h would thus represent concentrations or rates; u rate constants, activation energies, diffusion constants; and jv reaction time, temperature, pressure, or initial concentrations. The main purpose of kinetic modeling is to cast the relevant quality attribute in some mathematical equation and to ﬁnd the actual form of Equation 2.2, followed by estimation of the characteristic parameters. There may be two different objectives for setting up a mathematical model in the form of Equation 2.2: 1. To obtain an estimate of responses over a range of variables that are of interest, either by interpolation between experimental measurements or in a predictive way. 2. To determine the underlying physical mechanism of the process under study, i.e., to ﬁnd the nature and signiﬁcance of the function h ¼ f(u,jv). For objective 1, a theoretical model is not really needed, although it could be useful if one is to stray outside the boundaries of experimental measurements. All one needs is a suitable mathematical function (such as a polynomial function) that accurately describes the experimental results. This is often referred to as empirical modeling or response surface methodology (RSM). It can be very useful for situations where an underlying mechanism is not readily available. The approach obviously has its limitations. It cannot be used to build a mechanistic model because the parameters have no physical signiﬁcance. It is also very dangerous to extrapolate outside the region of variables for which the function was derived (and sometimes even interpolation is tricky). The situation is different for objective 2. Here a scientiﬁc theory is required on which to base a mathematical function. While a model can never represent the complete real world, an adequate mechanistic model should be based nevertheless on a scientiﬁc theory and the model should be able to predict experimental results or commonly observed phenomena accurately. The parameters in the model * The notation in Equation 2.2 using Greek symbols is commonly used in the statistical literature. We adopt this here because we will apply statistics frequently in this book.

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Models and Modeling

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should have physical signiﬁcance and in the ﬁeld of kinetics they include, for instance, rate constants and diffusion coefﬁcients. It is also of importance to state the conditions clearly under which the parameters have been deﬁned. It should be less dangerous in this case to extrapolate outside the experimentally tested regions. The two types of models, empirical and mechanistic, represent extremes; in reality the situation is somewhere in between, certainly for foods with all their complexity. Thus, even with empirical models, one may have some idea of the underlying mechanism. For instance, some microbiological growth models are, strictly speaking, empirical because the manner in which water activity or pH affect microbial growth is not (fully) understood. On the other hand, the functional behavior of the response, e.g., whether linear or logarithmic with respect to pH, may provide clues as to the underlying mechanism. Conversely, a model that is claimed to be mechanistic may still contain unexplained aspects; a rate constant, for instance, can be apparent, i.e., reﬂect more than one reaction step, as discussed in Chapter 4. Stochastic models. At this point it is essential to introduce yet another element in the discussion about models. Mathematical models as such are deterministic, i.e., they produce a certain outcome, usually expressed in a number (e.g., a vitamin concentration). The model displayed in Equation 2.1 produces a so-called point estimate (when the parameters are known and the controllable variables time and temperature are set). However, we do not live in a deterministic but rather in a stochastic world (from the Greek word ‘‘stoxastikos,’’ meaning guessing, surmising) and a number as such can be misleading because it suggests certainty. In other words, deterministic models provide an answer that is in a sense not realistic because it ignores (random) variability. When we use models to predict something, we have to accept that there will be an element of uncertainty in our prediction. Suppose, for instance, that we are able to predict the content of a vitamin as predicted by Equation 2.1 as a function of time and temperature. We want to use this to predict the shelf life of a product; when the concentration falls below a certain level the product is not deemed acceptable anymore. This could result in a graph as depicted in Figure 2.4A: A critical time tc can be estimated from this. At a time longer than tc the product is not acceptable anymore. However, because there is uncertainty in the value of the parameters A and Ea, there will be uncertainty in the outcome as well and this results in variation in the prediction and consequently the estimation of critical times tc is also variable (Figure 2.4B). If we are somehow able to estimate this variation, it will be possible to predict the uncertainty, and this will usually be in the form of a probability distribution, in this case of critical times tc (Figure 2.4C). Incidentally, this probability distribution need not be a normal distribution. Variability and uncertainty. Uncertainty, in other words, can and should be modeled! In this respect, it is useful to subdivide the total uncertainty in its two constituents variability and uncertainty. Variability comprises the natural variation in the real world. For foods, this comes down to the biological variation in the composition of raw materials and in the behavior of living materials, especially microorganisms. It can also relate to such things as a slightly varying temperature in a supply chain: even though the temperature may be ﬁxed at a certain value, it will show some stochastic variation that will have an effect on the outcome of our prediction. This variation is inherent in the nature of our physical world. We can measure this variation via statistical methods, but we cannot reduce it (at least not without changing the system). Incidentally, this is often the very purpose of using controlled model systems that simulate behavior of foods, for instance by using a solution of an amino acid and a reducing sugar to simulate the Maillard reaction occurring in foods. In this way we can control or even eliminate biological variation and direct our attention to the reaction of interest, which is very useful to understand the mechanism at hand, but when we translate the results back to real foods we should not forget the biological variation. In fact, variability does give important and essential information about the system under study and should be studied accordingly. The other element is uncertainty. This reﬂects the state of our knowledge (or ignorance) about the system. For instance, a parameter (such as an activation energy) in a mathematical model can be estimated from data but there will be an error involved in this estimate because the data are obtained by using an error-prone method. By doing more experiments (and perhaps better designed when we get

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80

60 Minimum acceptable level

40

20

0 (A)

100 Concentration (arbitrary units)

Concentration (arbitrary units)

100

0

tc Time (arbitrary units)

80

60

20

0 25 (B)

Minimum acceptable level

40

0

tc tc tc Time (arbitrary units)

tc

25

Frequency

100

0 (C)

10

tc

20

FIGURE 2.4 Hypothetical example showing the prediction of the change in vitamin content as a function of time at constant temperature. tc is the time at which the minimum acceptable level is reached. Deterministic result (A), variable result because of uncertainty in the parameters (B), frequency distribution for the critical times tc (C).

to know the system better) we can reduce this uncertainty. This is the very reason why it is useful to split total uncertainty up into variation and uncertainty. If total uncertainty is determined mainly by variation, it makes no sense to try to reduce the uncertainty by doing more measurements. If it is, however, determined by uncertainty we can reduce total uncertainty by doing more and better measurements. Such considerations are very important for risk–beneﬁt analyses in a broad sense, i.e., not only microbial risk assessment but also optimization of concentration or bioavailability of certain food components (beneﬁt assessment). A very good impression of total uncertainty can nowadays be obtained via Monte Carlo simulation (for which we need probability models, to be discussed later). It is the author’s opinion that this way of thinking will become increasingly important for food design problems. It means that we should be prepared to introduce elements of stochastic modeling into our mathematical models, i.e., to introduce probability distributions rather than point estimates in our model. So, instead of a ﬁxed value for the activation energy in Equation 2.1 we could insert the probability distribution of the activation energy in the equation (reﬂecting our state of ignorance) and simulate stochastic variation in the prediction by drawing random numbers by computer. This is done typically thousands of times

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(i.e., Monte Carlo simulation) and results in a probability distribution of the outcome, i.e., a description of the range of values that the outcome may take together with the probability that the variable will take any speciﬁc value. A probability can be seen as a numerical measurement of the likelihood of an outcome. This stochastic nature of modeling is the reason why we spend considerable attention to statistics in this book. Not every scientist seems convinced of the usefulness of statistics, sometimes expressed in the phrase ‘‘how to lie with statistics.’’ This is unfortunate because statistics should be seen as an important tool in the scientiﬁc learning process, to cope with the phenomena of variability and uncertainty, and to be able to draw general conclusions from a limited amount of data. A very useful branch of statistics is the so-called Bayesian statistics, especially in relation to modeling. Bayesian statistics treats probability as plausibility of a hypothesis in view of data obtained and expresses this as a so-called posterior probability, whereas ‘‘classical’’ statistics interprets probability in terms of frequency (a proportion in a large number of repetitions of the random process), and uses signiﬁcance tests to see if a hypothesis can be conﬁrmed. There are fundamental differences in the two approaches and few food technologists appreciate Bayesian statistics, as they have been trained, most likely, in classical statistics. We consider Bayesian statistics important enough to introduce it and discuss some of its elements in Chapter 7. It is also important for risk–beneﬁt analysis and decision analysis concerning food safety. Model uncertainty. We now come to a very important philosophical point in modeling. What we actually are trying to do is to approximate truth or reality with our models. However, it is important to realize that we will never be able to capture reality fully (if we could we would not need a model). The only thing we can do is to infer something from the data that we have obtained (either by observational studies or by doing planned experiments, but we will not consider observational studies in this book). So, in other words, we try to capture the truth behind the data, i.e., the processes or mechanisms that cause the data to be as they are; we do not model the data themselves. It is the information contained within the data in which we are interested and that is expressed in mathematical models. How do we know that we select models that come as close to the truth as possible? Information theory is quite helpful in this respect, providing tools to aid in model selection. We will discuss this in some detail in Chapter 7. For the moment it is important to realize that more than one model may come close to the truth (even though we will never know what truth is). We stress this point because this is the essence of modeling: we will never reach truth (and we do not need to!) as models are just approximations. The important consequence is however that this aspect adds to uncertainty, namely the uncertainty as to how far the model is away from the truth. Figure 2.5 gives some hypothetical situations. With reference to Figure 2.5 it is obvious that model 1 comes reasonably close to reality, model 2 follows the trend to some extent but with considerable bias, and model 3 is completely off. Obviously, we

Model 2

Model 1

Reality

Model 3

FIGURE 2.5

Hypothetical examples of ‘‘reality’’ and three models that approximate reality.

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would like to reduce this model uncertainty but, because we do not know the truth we cannot measure this in an absolute sense, only in a relative sense. One such measure is the so-called Akaike criterion, discussed in Chapter 7. The Akaike criterion focuses on predictive accuracy and provides a methodology to see which model performs the best in predictive accuracy. The topic of model discrimination, i.e., differentiation between a good model and one that is less good, or a bad model, is thus another essential topic. A good model is able to extract the relevant structural information from the data and separate this essential information from noise. It may be that more than one model applies, and the choice for a particular model introduces again uncertainty in our endeavor to approximate to the truth. In some cases it may be better to do some form of model averaging rather than choosing just one model. Some methods of model discrimination are discussed in Chapter 7. By comparing models we actually evaluate the amount of information in the data relative to the information capacity of the model (the more complex the model, the more information capacity it has). So, it should be clear by now that models are always wrong, but some of them may be useful (to paraphrase the famous statistician George Box). Box and coworkers (see bibliography at the end of the chapter) suggest that one should ‘‘tentatively entertain a model’’ rather than assume it to be correct. This implies that one should always be prepared to put models in jeopardy, and subsequently revise them in the light of new evidence. This is the very basis of the scientiﬁc method, where hypotheses and theories are subject to peer review and amended (or indeed rejected). The process of modeling is, therefore, iterative in nature (Figure 2.6). All the elements in this iterative cycle are essential for modeling. Although a cycle is depicted in Figure 2.6 with no apparent starting point, we suggest that, whenever possible, the cycle is started with the box called conjecture. The reason for this is that this is the point where science comes in (for food science basically (bio)chemistry, physics, and microbiology). In the case of a chemical reaction, for instance the Maillard reaction, it helps enormously if the researcher is aware of the possible basic mechanisms because that will give structure to the planning of subsequent experiments. It also means that the researcher should propose some possible models already at this stage, which will be tested and compared later on. This conjecture can be a very simple idea based on literature, observation of the phenomena, basic chemical knowledge, or even intuition. In any case, it is important to think hard before doing experiments and to ask the right questions and apply the appropriate science. Admittedly, there may be situations in which it is impossible to pose models beforehand, and that one needs to do some starting experiments in order to get a feel for the problem at hand. However, in most cases, there will be some idea of an approximating model. Experiments are designed to test the original idea.

Conjecture, hypothesis

Analysis of experimental results

Design of experiments

Experiments

FIGURE 2.6

Scheme showing the iterative nature of the various stages in modeling.

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Experimental design is an essential part of modeling, and its importance is often overlooked. Statistical methods are available to support this stage of experimental design. The design determines and limits the information that can be obtained from a data set. The goal of experimental design is to optimize the information content of a data set with the least possible effort. Experimental design is likely to depend on the purpose of the investigation, whether it be model discrimination or parameter estimation. Chapter 7 pays attention to this aspect of kinetic modeling. Doing the actual experiment can be relatively straightforward in principle, but may be complicated in practice. For instance, with an experiment designed at one temperature the heating-up time may be considerable and cannot be neglected. For the analysis of the data, the use of statistics is indispensable because experiments always contain noise (i.e., noninformation caused by unexplainable variation) and we need to be able to differentiate between this noise and essential information contained in the data. Once again we stress that we are trying to model the information contained within the data, not the data themselves. With properly analyzed results, the original idea can be tested for its validity (or if more than one model had been proposed, model discrimination is accomplished). This may well lead to adjustment of the original idea (the objective is not to accept or to reject a model but to improve it). Experimental data only become meaningful in the framework of a model, as data in isolation do not provide this type of information. However, a model is never deﬁnitive and we must accept the iterative nature of modeling. The already mentioned Bayesian statistical approach ﬁts very well into this philosophy because it describes this learning process in a mathematical way: prior knowledge and data are combined in posterior knowledge, as discussed in Chapter 7. In any case, the combined use of statistics and mechanistic understanding is needed here because it will be necessary to differentiate between noise in the data and the information contained within the data. Model parameters. Model parameters constitute the core of a model. One should always strive for the lowest number of parameters possible in a given model because, as it happens, any model will ﬁt a data set if the number of parameters is made high enough. The penalty for this so-called overparameterization is that the model will be indiscriminate and often worthless: the variance of the parameters will increase too much for proper use of the model (e.g., making predictions). Fortunately, proper use of statistics could signal this and appropriate measures can be taken, as discussed in Chapter 7. On the other hand, if the number of parameters is lowered, the bias between the model and the data increases. Modeling is thus a delicate balance between over- and underparameterization. Figure 2.7 illustrates this. It is in fact a depiction of a famous quote from Einstein: ‘‘Models should be as simple as possible but no simpler than

Bias between model and data

Uncertainty in parameter estimates

Number of parameters p ( p ≥ 1)

FIGURE 2.7 Schematic picture showing the bias between a model and experimental data and the uncertainty in parameter estimates as a function of the number of parameters in a model.

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that.’’ As simple as possible refers to the idea that models should simplify reality taking into account the details that matter and neglecting the details that are not so important. No simpler than that means it should be possible to do calculations with the model that tell the essential things about reality. An interesting situation may arise when mechanistic insight requires a certain parameter, whereas the statistical analysis tells us that that parameter is redundant. It may be that the data set does not contain enough information to estimate the parameter, or it may be that the mechanistic insight needs revision. In any case, a sensible interplay between statistics and mechanistic knowledge (chemical, physical, microbiological, and biochemical in the case of foods) is required. The following guidelines are of importance, and are discussed in more detail in following chapters. 1. The art of keeping the number of parameters in a model at a minimum is called the principle of parsimony, or Ockham’s Razor (after the fourteenth century English philosopher William of Ockham) stating that ‘‘things should not be multiplied beyond necessity,’’ or ‘‘shave away all things unnecessary.’’* In other words, a simple model is better than a complex one, but as indicated above a right balance needs to be found between over- and underﬁtting. A proper procedure for model discrimination will contain a penalty function for increase in the number of parameters. This is discussed in Chapter 7. Also of importance in this respect is that the greater the number of parameters, the greater the extent of nonlinear behavior (in the case of nonlinear models), also discussed further in Chapter 7. 2. Parameterization. The extent to which parameters in nonlinear models behave nonlinearly varies greatly. It may be that some parameters need to be reparameterized in order to ﬁnd the best estimation properties. This is discussed further in Chapter 7. 3. Range of applicability. The data should cover the full range over which the model is applied. This is further explained in subsequent chapters. 4. Stochastic speciﬁcation. It is very important to model not only the underlying mechanism but also the error terms involved (i.e., uncertainty). This is discussed extensively in Chapter 7. 5. Interpretability. Preferably, the parameters should have a physical meaning and not just be ﬁt parameters. This is in fact one of the main themes of the book. In relation to kinetics it is discussed in depth in the following chapters. A complicating factor may be that a conﬂict arises between interpretability of parameters and their statistical estimation properties. To summarize, the aims of modeling in the food science area are as follows: 1. Models can be very helpful to control and predict food quality and provide a tool to optimize quality and costs. 2. Critical points determining quality along the various elements of the food production chain can be identiﬁed. 3. Research results in different domains of the food chain can be combined. 4. Models provide tools to identify biological variability as well as uncertainty of parameters and thus provide a basis for structured data acquisition. It is very helpful to state a goal as well as a purpose when building a model, e.g., to develop a microbial growth model (the goal) to predict microbial shelf life (the purpose). The goal can be different for different purposes. It is quite common to make several assumptions when applying models, for instance, that a constant pH exists, etc. It is necessary to state these assumptions explicitly and to consider them again after a model is used: were the assumptions reasonable for the problem at hand, or are perhaps one or more of the assumptions violated? This may help greatly in evaluating the usefulness of a model. * The relevant statements that can be found in William of Ockham’s writings are ‘‘Pluralitas non est ponenda sine necessita’’ (one should not pose more things than is necessary) and ‘‘Frustra ﬁt per plura quod potest ﬁeri per pauciora’’ (it is vain to do with more what can be done with less). The statement ‘‘Entia non sunt multiplicanda praeter necessitatem’’ (entities should not be multiplied unnecessarily) was probably made by a later scholar. See: www.weburbia.com=physics=occam.html

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Models and Modeling

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Modeling and mathematical terminology. Mathematics should be seen as a language to express relations in a concise, logical, and straightforward way. It may be helpful for the remainder of the book to explain brieﬂy some commonly used terms. Mathematical models in relation to kinetics can appear in several forms. When systems are not changing in time, the state of that system remains constant (static). Variables describing the state of the system (e.g., temperature, concentration) do not change in time in such cases. Models describing such a condition are named static models or steady-state models. Incidentally, steady state is not synonymous with equilibrium, and equilibrium is a special case of a steady-state situation. Equilibrium is a thermodynamic concept. A system is in equilibrium when its free energy cannot be decreased any further under the conditions applied (Chapter 3). A mass balance is typically a static model. A chemical reaction in equilibrium is described also by a static model. Static models can be described with algebraic equations. When a model describes a system that changes in the course of time, as will be the case with most (if not all) reactions in foods, we speak of a dynamic model. Dynamic models are typically described by ordinary differential equations (ODEs), relating the state of a system (such as a concentration) to the rate of change of that state (change in concentration). Another classiﬁcation is that of spatial models, when things are not only changing in time but also as a function of space. These can be described by partial differential equations (PDEs). It is important to realize that much information can be obtained from systems that are changing. A system in steady state that is disturbed at a certain moment will respond to this disturbance and ﬁnd its TABLE 2.1

Overview of Terminology Used with Mathematical Models

Term

Description

Mechanistic model

Mathematical model based on a mechanism; a translation of a physical, chemical, and biological theory

Empirical model

Mathematical model that is optimized to give the best ﬁt to the observed data without an underlying chemical, physical, or biological theory

Deterministic model

Gives outcomes as exact numerical values (point estimates); it produces always the same output with the same input

Probabilistic model, stochastic model Static=steady state=stationary model

Gives an outcome with associated total uncertainty (as a probability distribution); it does not produce exactly the same output with the same input A solution of the state equations when the time derivatives of the state variables are all set to zero, i.e., described by algebraic equations. No description of future states

Dynamic model

Results depend on time and space, described by differential equations. Representation of future system states or conditions

Linear model

A model that is linear in the parameters (not necessarily a linear relationship between x and y)

Nonlinear model

A model that is nonlinear in the parameters

Spatial models

Objects have a position in space (or a ﬁnite region in space)

State variable

Quantity describing the state of the system (e.g., concentration at a certain time)

Independent variable

Variable that determines the change in the state of the system (usually time and space) and is controlled by the modeler=experimenter Equations that specify the particular solution (initial values, for instance) for state variables as a function of the independent variables

State equations Transient

Temporal proﬁle of the state variables after a perturbation on the boundary conditions

Boundary conditions

Constraints that apply to the solution of state equations (e.g., initial values, mass balances)

Parameters Sensitivity coefﬁcient

Constants in the state equations (constant in a particular case, but may vary in different cases) Partial derivative of a state variable with respect to variations in a parameter

Black box model

A model that is not based on any supposed mechanism

White box model

A model that is based on a supposed mechanism

Gray box model

A model that contains both empirical and mechanistic elements

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way to a new steady state. If we were to look only at the two steady states without looking at how the system changes in between, we would lose much information. Relations that describe these changes are very informative. Physical and biological models often arise as solutions of differential equations. Therefore, some knowledge of calculus is needed (more details can be found in Appendix A). Depending on the complexity of the problem at hand, analytical solutions can or cannot be found. If not, one has to resort to numerical solutions, but this is relatively easily achieved with appropriate software packages. Another important factor is that regression models are frequently nonlinear in the parameters, the implications of which are discussed in Chapter 7. To conclude this chapter, Table 2.1 gives an overview of terminology used with mathematical models.

2.3

Concluding Remarks

This chapter has attempted to put the opportunities and limitations of models in perspective of the scientiﬁc method. The most important ‘‘take home message’’ is that models are tools to get a grip on reality, but they are deﬁnitely not the truth or reality. Obvious as this may seem, one sometimes gets the impression that researchers prefer models over reality, and this is, of course, a capital sin in modeling. In that sense, data should never be ﬁtted to models as is sometimes stated in literature; it should always be the other way around!

Bibliography and Suggested Further Reading About Modeling Box G.E.P. and Hunter W.G. A useful method for model building. Technometrics 4:301–318, 1962. Box G.E.P. and Hunter W.G. The experimental study of physical mechanisms. Technometrics 7:23–42, 1965. Box G.E.P. Science and statistics. J Am Stat Assoc 71:791–799, 1976. Burnham K.P. and Anderson D.R. Model selection and inference. A Practical Information-Theoretic Approach. New York: Springer Verlag, 1998. Chatﬁeld C. Problem Solving. London: Chapman and Hall, 1995. Cullen A.C. and Frey H.C. Probabilistic Techniques in Exposure Assessment: A Handbook for Dealing with Variability and Uncertainty in Models and Inputs. New York: Plenum Press, 1999. Haefner J.W. Modeling biological systems. Principles and Applications. 2nd edn. New York: Springer, 2005. Kohn M.C. Use of sensitivity analysis to assess reliability of metabolic and physiological models. Risk Anal 22:623–631, 2002. Phair R.D. Integrative bioinformatics. Practical kinetic modelling of large scale biological systems. www. bioinformaticsservices.com=bis=resources=cybertext=Ibcont.html Ratkowsky D.A. Handbook of Nonlinear Regression Models. New York: Marcel Dekker, 1990. Tijskens L.M.M., Hertog M.L.A.T.M., and Nicolaï B.M. (Eds.). Food Process Modelling. Cambridge, United Kingdom: Woodhead Publishing Limited, 2001. Vose D. Risk Analysis: A Quantitative Guide. Chichester, United Kingdom: Wiley & Sons, 2000. Zwietering M.H. and Hastings A.P.M. Modelling the hygienic processing of foods—a global process overview. TransIChemE 75(C3):159–167, 1997. About Ockham’s Razor Jefferys W.H. and Berger J.O. Ockham’s razor and Bayesian analysis. Am Scientist 80:64–72, 1992.

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3 Chemical Thermodynamics in a Nutshell

3.1

Introduction

Although the topic of this book is on kinetics, we consider it appropriate to include a short chapter on chemical thermodynamics, which is the branch of thermodynamics that studies how chemical reactions come to equilibrium. The reason to do this is that thermodynamics and kinetics are complementary to the study of chemical reactions. Thermodynamic parameters are also used in the formulation of chemical kinetics, as we shall see later on in this book. Thermodynamics is, in short, the science of conversion of energy and matter. Reversible thermodynamics makes statements about systems in equilibrium, and gives an answer to the question: what drives chemical reactions toward equilibrium or to completion? Reversible thermodynamics is reasonably established by now. In contrast, irreversible thermodynamics, which makes statements on processes and systems that are not in equilibrium, is still debated. Most of this chapter will be on reversible thermodynamics, with some remarks about irreversible thermodynamics at the end of the chapter. Irreversible thermodynamics makes a nice link to kinetics. Kinetics gives an answer to the question, how does it happen and at what rate, and that will be addressed in subsequent chapters. The chapter is organized as follows. Both for thermodynamics and kinetics we need to be able to express quantitatively the progress of a reaction, so we start with a section on how to quantify reactants and products. This is not only needed for this chapter but also for the rest of the book. Then we move to chemical thermodynamics, by ﬁrst highlighting the concepts of energy, enthalpy, entropy, and free energy. Then we discuss the difference between ideal systems and real, nonideal systems and how we cope with nonideal systems via the activity concept. We continue by considering how these concepts can be used to state something about direction of processes and equilibrium positions of reactions. Finally, we conclude the chapter by making the move from reversible to irreversible thermodynamics, and from there we will make a connection to kinetics.

3.2

Quantiﬁcation of Reactants and Products

Ways to express amounts and concentrations. The amount of chemicals present in a solution can be expressed in many ways. The chemical present in the highest amount is called the solvent, and the chemical present in lower amount the solute (obviously there can be more than one solute in a solvent), 3-1

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and the whole of solutes and solvent is called a solution. We will mostly refer to thermodynamics and kinetics in solution, and the most common way to express amounts is then via molarity, i.e., the number of moles of solute per liter of solution. This is indicated by the symbol M (molar). Another way is via molality, which is the number of moles of solute per kilogram of solvent, indicated by the symbol m (molal). Yet another way is molinity, i.e., the number of moles per kilogram solution, but this unit is hardly ever used. In most cases relevant for foods, the solvent is water. Since the density of water is about 1 kg dm3, there will not be too much difference between molarity and molality for diluted aqueous solutions. However, it should be realized that it depends on the density of the solution in how far molality and molarity differ, and for foods this can make quite a difference. Also, molarities do depend on temperature because the volume changes with temperature while molalities do not. Furthermore, for some foods, such as cheese, or tomato paste, it is hard to envisage what a liter of solution actually means. So, it would not be a bad idea to use molalities instead of molarities for foods, but it is not very common. Another possible way of expressing amounts is via the mole fraction, i.e., the number of moles of solute divided by the total number of moles present in the system. Mole fractions are mostly used for more concentrated systems, but this unit only makes sense for well deﬁned, simple systems, not for foods. Molarities=molalities are more used for dilute systems. Then, it is of course also possible to express amounts in mass fractions, indicated by the symbol w=w (e.g., g solute=kg solution), mass–volume fractions w=v (e.g., g solute=L solution), and volume–volume fractions v=v (e.g., mL solute=L solution); these are expressed often as percentages, and sometimes also as ppt (parts per thousand, e.g., g solute=kg solution) or ppm (parts per million, e.g., mg solute=kg solution). In this book, we use mostly concentrations expressed as molarity or molality, but occasionally other measures will also be used. In any case, it will always be indicated how amounts are expressed as this is a prerequisite for thermodynamics and kinetics. Of course, the various ways to express amounts can be converted into each other, but extra information may be needed. For instance, to convert molar concentration into molal concentration, the density of the solution is needed. Appendix B gives an overview of ways to express amounts and some formulas for conversions. To show the intricacies involved for foods, the example of milk is a nice one. A liter of milk is not a pure solution; it also contains fat globules and casein micelles, i.e., these components are not dissolved but dispersed. Lactose is really dissolved and is part of the milk serum (the part of milk that can be considered as aqueous solution). So, the concentration of a solute like lactose can be expressed in several ways, as shown in Table 3.1, and it does make a difference which unit is used to express the amount of lactose present in milk. Sometimes, components are quantiﬁed in solutions that are ﬁrst separated from a food before the analysis. It may be that the ratio of a component to water in such a separated solution is larger than that ratio in the food from which the solution is obtained. This is due to the phenomenon of steric exclusion, which implies that the volume occupied by one particle is not available for another particle. Solute molecules need to keep a distance from the surface of large particles, and this distance depends on the size of the solute. In other words, a part of the volume is not accessible to solute molecules but is accessible to water molecules: this is the nonsolvent water. Obviously, the effect is larger for larger particles. Figure 3.1 illustrates this effect graphically. Corrections should be applied when concentrations

TABLE 3.1

Various Ways to Express the Amount of Lactose in Full-Fat Milk

Expression

Unit

Result

Mass percentage (%)

g=100 g milk (%)

Concentration (molinity)

mol kg1 milk

0.134

Concentration (molarity)

mol L1 milk (M)

0.138

Concentration (molality)

mol kg1 water (m)

0.154

3

Note: 4% fat, 12.8% dry matter, density 1030 kg m .

4.6

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No water

Water molecules Protein

No solute

FIGURE 3.1

Solute molecules

Graphical illustration of the steric exclusion effect resulting in the presence of nonsolvent water.

determined in a solution obtained from a food are recalculated to the corresponding concentration in the food itself. Stoichiometry of reactions. In a chemical reaction, molecules interact (react) with each other and the result is the formation of products in which atoms are arranged differently than in the original reactant molecules. When a chemical reaction occurs, the proportions of the amounts of participating molecules change. Suppose we have a reaction represented symbolically as nA A þ nB B ! nP P þ n Q Q

(3:1)

where nA, nB, nP, nQ represent the number of molecules or moles of reactants A and B, and products P and Q, respectively (it is just a convention to call the components on the left-hand side reactants and the ones on the right-hand side products). The equation is a standard notation for a chemical transformation and is called a stoichiometric equation (from the Greek words stoixeoy [stoicheon], meaning element, and metroy [metron], meaning measure). The importance of a stoichiometric equation is that it deﬁnes the exact change from initial number of molecules to the ﬁnal composition; in other words, such a change is not arbitrary. The bidirectional arrows indicate that reactants can also be formed from the products, and when there is a balance between formation of products and reactants a steady state is reached, which may or may not be an equilibrium; a more formal deﬁnition for equilibrium will be given later on. nA,B,P,Q are called the stoichiometric constants, which can be seen as physical quantities representing the change in the number of molecules of a component per chemical transformation as indicated by the reaction equation. They indicate ﬁxed proportions of the number of moles upon chemical transformation. We can generalize Equation 3.1 for more reactions in the following way. Suppose there are n species Ai (i ¼ 1, . . . , n) and r independent reactions ( j ¼ 1, . . . , r). An example is the decarboxylation of amino acids, an important reaction in ripening cheese with respect to ﬂavor development. The following reactions are possible: Reaction 1:

R-CH-NH2 ! R-CH2 -NH2 þ CO2

Reaction 2:

j COOH 2R-CH-NH2 þ O2 ! 2R-CO-COOH þ 2NH3 j COOH

Reaction 3:

R-CO-COOH ! R-CHO þ CO2

(3:2)

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3-4 TABLE 3.2

Stoichiometric Table for the Reactions in Equation 3.2

Species i Reaction j

R-CH(COOH)NH2 (i ¼ 1)

O2 (i ¼ 2)

R-CH2NH2(i ¼ 3)

CO2 (i ¼ 4)

R-COCOOH (i ¼ 5)

R-CHO (i ¼ 6)

NH3 (i ¼ 7)

j¼1

1

0

þ1

þ1

0

0

0

j¼2

2

1

0

0

þ2

0

þ2

j¼3

0

0

0

þ1

1

þ1

0

So, we have three reactions (r ¼ 3) and seven species (n ¼ 7). We can set up a table for the stoichiometric constants (Table 3.2). The convention is that stoichiometric constants are indicated with a negative sign for reactants (because the number of these molecules decreases) and a positive sign for products (because the number of product molecules increases as the reactions starts with reactants A and B). Such a table can be summarized in the so-called stoichiometric matrix 2

1 4 2 0

0 1 0

1 1 0 0 0 2 0 1 1

3 0 0 0 25 1 0

(3:3)

Matrices are just convenient ways to represent a bunch of numbers. A concise way to represent balanced reaction equations such as Equation 3.2 is n X

nij Ai ¼ 0

(3:4)

i¼1

In the case of just one reaction, r ¼ 1, Equation 3.4 reduces to n X

n j Ai ¼ 0

(3:5)

i¼1

A balanced reaction equation such as in Equation 3.1 does not necessarily represent the actual mechanism of a reaction. What it does represent is the proportion in which changes occur in amounts of reactants and products. It is important to be aware of this distinction. It may well be that at the molecular level (i.e., elementary reactions) the reaction depicted in Equation 3.1 is as follows: nA A ! nA A1 nA A1 þ nB B ! nB B1 nB B1 ! nP P þ nQ Q

(3:6)

A1 and B1 represent transient intermediates that do not appear in the overall stoichiometric equation. An example of such a reactive species is a radical, for instance formed in several oxidation processes in foods. The danger in using an overall reaction equation such as Equations 3.1 and 3.2 is that it represents the ﬁnal outcome of a reaction. One is, however, not allowed to add up reaction equations (such as Equation 3.6) as if they were algebraic equations. This is especially so for a kinetic treatment: sometimes the intermediates are very important from a kinetic point of view. It can cause much confusion if reaction (Equation 3.1) is taken as the actual mechanism when it only reﬂects the balance in amounts of reactants and products. When we want to study how a reaction progresses we are in need of a parameter that describes this progress. Such a parameter is j, the extent of the reaction.

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Extent of reaction. With the number of moles expressed as ni, and ni,0 the initial number of moles for component i, the extent of reaction is deﬁned as j¼

ni ni,0 Dni ¼ ni ni

(3:7)

or expressed in terms of the number of moles ni ¼ ni,0 þ ni j

(3:8)

The stoichiometric constant ni indicates the amount of component i in the reaction. j is a timedependent variable describing the advancement of a reaction and is proportional to the net number of transformations in going from left to right in Equation 3.1. j could be referred to as the amount of chemical transformations (or number of events) expressed in number of moles. j ¼ 0 means that the amount of (at least one) product is zero. A maximum value jmax is reached when at least one reactant is exhausted; this will happen for a reaction going to completion. The exhausted reactant is called the stoichiometrically limiting reagent. The amount of a limiting component consumed in the reaction must be equal to the amount initially present. The limiting reagent is therefore the component that has the ni,0 and jmax is numerically equal to this lowest value (as follows from Equation 3.7). An lowest value of jni j example may clarify this a bit more. Suppose we have a reaction depicted by: 2A þ B þ 3C ! P þ Q and the reaction starts with initial amounts as shown in Table 3.3. The limiting reagent for the initial conditions indicated is thus component C. In the case of a reaction not going to completion, an equilibrium jeq will be reached as discussed below in the section on equilibrium. j speciﬁes the composition of a reaction mixture during the course of the reaction; it is a useful parameter because it describes the extent of reaction regardless of which compound is considered. It should be realized that j can be larger than 1. For more than one reaction ( j ¼ 1, . . . , r) with n components (i ¼ 1, . . . , n), Equation 3.8 becomes ni ¼ ni,0 þ

r X

nij ji

(3:9)

j¼1

It is possible to turn the parameter j into a dimensionless parameter, for instance by using jmax ar ¼

j jmax

(3:10)

ar is called the degree of reaction, with a numerical value between 0 (only reactants present) and 1 (only products present). TABLE 3.3 Component ni,0 (mol) ni jmax (mol)

Example of Calculation of jmax for the Reaction 2A þ B þ 3C ! P þ Q A

B

C

P

Q

1

1

1

0

0

2 1=2

1 1=1

3 1=3

1

1

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Another frequently used measure for the progress of a reaction is the fractional conversion parameter, fc fc ¼

ni,0 ni ni,0

(3:11)

In contrast to j, fc depends on the species i and is related to the extent of reaction as follows (cf. Equations 3.7 and 3.11): j¼

fc ni,0 ni

(3:12)

These parameters are needed to calculate the amounts of reactants and products as a reaction progresses. We will use them in the sections and chapters to follow. Now that we have learned about the parameters needed to express quantities we can turn to thermodynamics and, subsequently, kinetics.

3.3

Thermodynamics of Reactions

Thermodynamics is the science of energy; it includes all aspects of energy and energy transformation and relationships among the properties of matter. For instance, a cup of hot coffee will cool down but it will never reach a higher temperature by itself, and thermodynamics offers an explanation for that. Thermo is the Greek word for heat (uermh), and dynamics for power (dynamis). Chemical thermodynamics is the study of the interrelation of heat with chemical reactions or with a physical change of state. It can provide information about the equilibrium position in a chemical process, i.e., whether or not the reaction can take place when one starts with the reactants A and B, to what extent, and in which direction. The possible exchanges of work, heat, or matter between a system and its surroundings take place across a boundary proceeding from an initial state to a ﬁnal state. Then, one may want to know how this equilibrium can be inﬂuenced by, for instance, temperature and pressure changes. It is for these reasons that we would like to remind the reader of some relevant thermodynamic principles for (chemical) reactions.

3.3.1

Heat and Work

As mentioned, the basic considerations for thermodynamics are heat effects (thermo) and work effects (dynamics). Heat is the reﬂection of random molecular motion, work is organized motion (force 3 distance). Work can appear in various forms: gravitational, expansion, tension, electrical, dissipative force of friction, and also chemical. Heat and work are processes by which energy is transferred between a reactive system and its surroundings. Work is the energy needed to do something. Thermodynamics allows one to account quantitatively for how much energy goes where. In principle, thermodynamics is model independent and does not refer necessarily to molecular events; this model independency is the case of classical thermodynamics. Thermodynamics can be built also upon quantum mechanics and this is called statistical thermodynamics. Statistical thermodynamics determines the distribution of a given amount of energy E of n identical systems. The goal of statistical thermodynamics is to understand and to interpret the measurable macroscopic properties of materials in terms of the properties of their constituent particles and the interactions between them. This is done by connecting thermodynamics functions to quantum mechanic equations. Since it does help in understanding to consider molecules, the discussion given below contains elements of both classical and statistical thermodynamics. It is important to realize that thermodynamics only applies to macroscopic amounts, i.e., large numbers of molecules. Before we discuss some aspects of thermodynamics it may be useful to recall some basic terminology (Table 3.4). The reader is advised that the following thermodynamic considerations are not as rigorous as they perhaps should be. It is not the intention of this book to discuss thermodynamics in great detail; it merely aims to make the link with kinetics. There are many excellent textbooks on the topic and some suggestions for further reading are given at the end of this chapter.

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Descriptions of Some Terms Used in Thermodynamics

Term System

Meaning Part of the universe that is studied

Surroundings

Part of the universe outside the system

Open system

System can exchange mass and energy with surroundings

Closed system

System can exchange energy with surroundings but cannot exchange mass

Isolated system

System can neither exchange mass nor energy with surroundings

Adiabatic system Isothermal system

System that does not exchange heat with surroundings Uniform temperature, i.e., no temperature gradients in the system

Isobaric system

Uniform pressure, i.e., no pressure gradients in the system

Isochoric system

System at constant volume

Diathermic system

System that permits heat transfer

Intensive variable

Does not depend on the amount of the quantity it refers to

Extensive variable

Does depend on the amount of the quantity it refers to

Equilibrium state

No net production or consumption of components (chemical reactions), no unbalanced forces (mechanical), no temperature gradients (thermal)

Spontaneous process

A process that occurs in a system without any work being done on the system

Reversible process

The system as well as the surroundings return into the same state when a process is reversed by inﬁnitesimal changes of a variable

Irreversible process

Net changes in the state of a system and surroundings have occurred as a result of the process that cannot be reversed

System, surroundings, and universe. First of all, we divide the universe into system and surroundings (Figure 3.2). A system can be any part of the universe that we want to study, so it can be a single reaction, an aggregate, an emulsion droplet, a whole food, a food factory, etc. The system and its surroundings are distinguishable because one can imagine some type of boundary between them, and this boundary does or does not permit exchange of matter and energy (see Table 3.4 for the possibilities and the terminologies). For instance, a can containing food can exchange heat with the surroundings but not matter, and canned foods are therefore closed systems. Hot coffee in a closed Dewar ﬂask forms an isolated system (in principle at least, though in practice not completely because eventually the coffee temperature will assume the temperature of the surroundings). Freshly baked bread left on the table is an open system: there will be exchange of heat (the bread cools down, the surroundings warm up) and of matter (water will System evaporate from the bread). Incidentally, these simple boundary examples demonstrate why packaging of foods is so important, because one can inﬂuence then the interaction of a food with its surroundings through manipulating the boundaries (gas permeation, solubility of System Universe components in the packaging material, etc.), and thereby one can control reactions. The conditions that describe a system are called collectively the state of the system. A change in conditions implies thus a change of state. Conditions that must be speciﬁed to establish the state of a system are Surroundings called state variables (such as pressure, temperature, FIGURE 3.2 Schematic representation of the volume, number of molecules). An equation of state universe and its constituents: a system and its describes mathematically how state variables are related. A change from one state to another is called a surroundings, separated by a boundary.

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process. A reversible process in thermodynamic terms is brought about by inﬁnitesimal changes of a variable and it means that going along the same path in reverse restores the system as well as the surroundings to its original state. With irreversible processes, permanent changes have occurred in the system and=or surroundings. This terminology may cause some confusion because a reversible reaction in chemical terms refers to a reaction in which products are formed from reactants but reactants can also be formed from products; an irreversible reaction means essentially that reactants are completely converted into products. The difference is thus in the words ‘‘process’’ and ‘‘reaction.’’ A reversible process does not really occur in nature, it is rather an idealized process, a thought experiment, which is nevertheless useful in practice because it helps in determining the limits of real processes. Real, natural processes are irreversible, though some real processes may approximate reversible processes, such as the melting of ice around 08C. An important point is, however, that some thermodynamic functions can only be evaluated when considering reversible processes. We describe ﬁrst classical thermodynamics for reversible processes; at the end of this chapter we will spend some words on irreversible thermodynamics because of its practical importance. Another important consideration is that of ideality; ideal gases and solutions are characterized by the absence of interactions between molecules. They represent limiting cases of real behavior. For instance, ideal solutions do not show a heat or volume effect upon mixing, whereas real solutions do. This is particularly of importance when trying to describe properties of foods in thermodynamic terms. Foods behave by no means as ideal solutions and gases; they are usually inhomogeneous, concentrated, and show many interactions between components. Ideal solutions and gases could be the starting point to derive trends but one should expect complications when applying such concepts to foods. In this book we consider reactions in foods mainly as if they take place in solution, at least as the reference point, acknowledging however, that solid (e.g., crystals) and gaseous phases (e.g., ﬂavor volatiles in headspaces, bubbles in foam) are also important. We will come back to some speciﬁc food complications in Chapter 14. A physical change of state (such as melting, or expansion) implies that the atoms or molecules involved do not change. A chemical change of state implies that the amounts and identities of reactants and products change, in addition to a possible physical change of state. A system is characterized by intensive and extensive parameters. An intensive parameter (or quantity, variable) such as temperature is independent of the amount of substance in the sample and an extensive parameter such as mass, or volume, or energy does depend on the amount of substance. However, the ratio of two extensive parameters, such as density calculated from the ratio of mass and volume, yields an intensive parameter. When chemical reactions occur it is useful to distinguish between species and components. A component can be the sum of several species. For instance, a salt (the component) can consist of several ions (the species). An important concept in thermodynamics is about state and path functions. State functions describe variables that only depend on the difference between one state or another but not on how the change occurs. Path functions on the other hand describe variables that depend on how the changes have occurred. Heat and work are path functions. Energy is a state function. This brings us to the topic of energy.

3.3.2

Energy

The basic thermodynamic property is the internal energy E. We cannot measure this internal energy in absolute terms, but we can measure energy changes. This is an important point to note because it explains the need for a reference point, and that is why standard states are introduced in thermodynamics. We will come back to this shortly. Energy comes in several forms: kinetic energy (i.e., translational, vibrational, or rotational energy of motion of molecules), potential energy (gravitational, chemical, electrical), thermal energy (characterized by incoherent motion of molecules), radiant energy, and even mass is related to energy according to the Einstein equation E ¼ mc2. In terms of chemical reactions, energy mainly refers to potential and kinetic energy in molecules, namely the bond energies of molecules, and translational, rotational, and vibrational molecular motions, respectively. The forms of energy mentioned are all

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interchangeable, and the ﬁrst law of thermodynamics states that energy is conserved throughout. This means that energy cannot be lost. However, the quality of energy, when expressed as its ability to do work, can diminish, as we will discuss shortly. Exchange of energy between system and surroundings can happen via heat (q), or via work (w); work ¼ force 3 distance. There are different kinds of work possible: chemical, mechanical, electrical, and magnetic. A ﬂow of thermal energy is known as heat; thermal energy itself is an energy content. It is important to understand that heat q and work w are modes of energy transfer; they depend on the path the process takes in going from the initial to the ﬁnal state. This path dependency means mathematically that the integral of the differential of work or heat depends not only on the initial and ﬁnal states but also on the path connecting them; therefore the differentials are so-called inexact differentials (indicated by the operator D). In contrast, the integral of an exact differential (indicated by the operator d) is independent of the path between initial and ﬁnal states, and therefore depends only on the initial and ﬁnal states. The result is a state function. However, the sum of these two inexact differentials is an exact differential. So we should write dE ¼ Dq þ Dw

(3:13)

But when the ﬁnal state is reached we can write for the change in energy DE ¼ q þ w

(3:14)

Energy of a system depends only on the current state, not on how it reached that state, in other words, the sum of q and w is always independent of the path between a given initial and a given ﬁnal state. Heat and work relate to processes; the energy content of a system cannot be divided into a heat part and a work part after the process. According to the ﬁrst law of thermodynamics, the following relation should hold: DEsystem ¼ qsystem þ wsystem ¼ DEsurroundings

(3:15)

Expressed in words, if the system gains energy the surroundings will lose the same amount of energy and vice versa, and these energy changes are brought about by heat and work. Once again, this refers to changes in energy content (symbolized by the symbol D), not to the energy content itself. Once the transfer of energy is completed, the contributions of heat and work are no longer distinguishable. One cannot say that a system contains so much heat or so much work; it only contains energy. Temperature is not energy but a measure for the average kinetic energy of atoms or molecules. There is a relation between temperature change (DT), the amount of heat transferred (q), the amount of material that is involved (N), and the type of material (expressed as molar heat capacity at constant pressure CP) DT ¼ Tfinal Tinitial ¼

3.3.3

q NCP

(3:16)

Enthalpy

Most chemical reactions occur at constant pressure, which means that volume changes may occur, so that work is done against the external pressure. For convenience, a new thermodynamic property called enthalpy is deﬁned H E þ PV

(3:17)

The equation that accounts for changes in enthalpy at constant pressure is DH ¼ DE þ PDV

(3:18)

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If the only work done is work against pressure (wsystem ¼ PDV, which is negative by convention because the work is done by the system on the surroundings), the combination of Equation 3.18 with Equation 3.15 shows that DH ¼ qsystem þ wsystem þ PDV ¼ qsystem PDV þ PDV ¼ qsystem

(3:19)

In words, the enthalpy change accounts for the heat ﬂow at constant pressure. Enthalpy is an extensive quantity like energy. Incidentally, volume changes for solids and liquids are usually negligible, so that for these systems DE DH. One can calculate enthalpy changes for chemical reactions using tabulated values for known compounds. Since we are interested in differences rather than absolute values, so-called standard states are used as reference points. Standard states refer to the state of an element, of a gas, a solid or a liquid, or a solvent, or a solute, at a pressure of 1 bar.* It is the convention in thermodynamics to assign zero enthalpy to all chemical elements (not components!) in their most stable state at 1 bar pressure. It should be realized that this is entirely arbitrary and it is just a matter of agreement. What may be confusing is that standard states for components can be deﬁned in various ways, as we shall see later when discussing chemical potentials. For the moment, let us denote a standard state by the superscript ‘‘ 8,’’ so that H8 indicates the value of the enthalpy at the standard state, for instance of one mole of gas at P ¼ 1 bar. Later on we will make a distinction between the various standard states and indicate this also by a different superscript. Next to the standard state, temperature has to be speciﬁed, and usually the temperature chosen is 258C (298.15 K), but again, this is arbitrary (a standard state refers to a concentration or pressure, not to a temperature). The standard reaction enthalpy change DrH8 is thus the change in enthalpy when reactants in their standard states change to products in their standard states (the subscript ‘‘r’’ indicates reaction). Besides reaction enthalpies there are also enthalpies of physical change, for instance, enthalpies of fusion or evaporation. A standard reaction enthalpy covers the overall process from pure unmixed reactants in their standard states to pure separated products in their standard states. A thermochemical equation is a combination of a chemical equation and a standard reaction enthalpy change. Referring to Equation 3.1, a general formula to calculate the standard reaction enthalpy is Dr H ¼ nP HP þ nQ HQ nA HA nB HB

(3:20)

where HA , HB , HP , HQ are the standard enthalpies of A, B, P, Q per mole, respectively, with the unit of J mol1. For instance, the balanced chemical equation for the oxidation of glucose is C6 H12 O6 þ 6O2 ! 6CO2 þ 6H2 O

(3:21)

The difference in enthalpies of reactants and products in their standard state is: 1 Dr H ¼ (6 Hcarbon dioxide þ 6 Hwater ) (1 Hglucose þ 6 Hoxygen ) ¼ 2820 kJ mol

The negative sign indicates that the system releases energy when this reaction takes place. This energy was stored as chemical energy in the glucose molecules (it is in fact an exothermic reaction; we will come back to this shortly). Incidentally, this energy was originally captured by plants from sunlight via photosynthesis, and since energy is conserved it can be regained back from glucose. As indicated, the reference state of an element is its most stable state at the speciﬁed temperature and at 1 bar and is deﬁned as zero by convention. The standard enthalpy of formation of a substance is the standard reaction

* Many authors use a pressure of 1 atm. According to International Union of Pure and Applied Chemistry (IUPAC) recommendations it should be 1 bar.

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enthalpy for the formation of the compound from its elements in their reference states, expressed as enthalpies per mole of compound. Enthalpy changes can be measured as heat exchanges at constant pressure via thermochemical experiments. Also, reaction enthalpies can be calculated from enthalpies of formation DfH8 of reactants and products, which are tabulated for many components Df H ¼

X products

nDf H

X

nDf H

(3:22)

reactants

These tabulated values are based on many accurate thermochemical experiments done in the past. Thus, it is now very easy and convenient to calculate enthalpy changes for almost every reaction. The ﬁrst law of thermodynamics allows one to calculate energy changes when reactions take place, but there are limitations on the ability to convert heat energy into work. Also, the ﬁrst law does not indicate the direction of processes: it applies to any process in which energy is conserved, but there is a direction for processes in real life, as is everyone’s experience. For example, sugar will dissolve easily in hot coffee, but the reverse process that sugar crystallizes all of a sudden in a hot cup of coffee will not happen (or put more correctly, the probability that this will happen is so extremely low that we can safely state that it will never happen). This is where the second law of thermodynamics comes in: there is a certain direction in which reactions go. In other words, there is a reason why sugar does not crystallize in hot coffee and this reason is called entropy S.

3.3.4

Entropy

Entropy is a rather abstract concept. It is based on the Greek words ey and troph (meaning: turning into). It is sometimes referred to as disorder, but this has lead to much confusion, and serious misinterpretation. Even though in some limited cases disorder could be a right term, it is preferable and unambiguous to refer to entropy of a system at constant energy as the number of ways energy can be stored in that system; increase of entropy reﬂects dispersal of energy at some temperature T. So, the entropy of a substance at a temperature T is a measure of the total quantity of energy that had to be dispersed within that substance from T ¼ 0 K to T to exist as a solid, or liquid, or gas at that temperature T. With such a notion of entropy, phase changes and colligative properties (such as freezing point depression, boiling point elevation, and osmosis) can be explained. In addition, it also allows us to understand why some reactions occur and others do not, as will be discussed shortly. The consequence of real-world processes, which are irreversible, is that in every energy transformation the entropy of the universe increases and because of that the potential of the energy available to do work diminishes. So, its ‘‘quality’’ is reduced. Hence, there is a good reason to be careful with the use of energy in society, even though energy cannot be lost according to the ﬁrst law. A suitable measure for the quality of energy is called exergy, and exergy can be lost. We will not discuss this any further here, except to state that exergy calculations are a useful tool for the food industry to design processes as efﬁcient as possible with regard to energy conversion. The second law of thermodynamics states that there is only one direction for irreversible processes, namely those resulting in an increase in entropy S in an isolated system and the entropy is maximal at the equilibrium state. If changes occur spontaneously (i.e., without any work being done on the system) the total entropy of the system and its surroundings (i.e., of the universe) must increase. This does not mean that the entropy of a system cannot decrease; this actually happens during crystallization or condensation, or in a refrigerator, but the requirement remains that the total entropy of the universe must increase. However, the entropy of an isolated system cannot decrease, as will be made clear shortly. The change in entropy determines whether or not a reaction will occur; the energy changes in a reaction have an effect in as much as they increase or decrease the entropy of the system and its surroundings. The dispersal of energy is the ‘‘driving force’’ for physical as well as

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chemical reactions but it is of a probabilistic nature. That means that it is possible for energy to be concentrated temporarily in a system, for instance because there is some barrier that prevent things from happening (otherwise life would not be possible!). This does not conﬂict with the second law of thermodynamics. We will come back to this when discussing kinetics. A frequently used term in relation to entropy is spontaneous process (Table 3.4). A spontaneous process is something that occurs without the input of additional energy from outside the system, but there is a direction indicated by entropy, namely the only direction is the one in which the entropy of the universe increases. Although it is not wrong per se to talk about a spontaneous reaction, the adjective spontaneous does not add much. A nonspontaneous reaction does not occur, so the word is actually superﬂuous. Another possible confusing notion of the word spontaneous is that it should occur immediately, and this is deﬁnitely not the case. A spontaneous reaction such as the oxidation of glucose can take a very long time. In equilibrium thermodynamics, time is not a variable. So far, entropy has been discussed as a qualitative measure, and to make it a quantitative measure, two possibilities exist, in principle at least. The ﬁrst is to count the number of possibilities for energy dispersal via statistical thermodynamics. From molecular thermodynamics and statistical mechanics, it follows that energies of particles and the probability of position in space are coupled inseparably. Such combinations of energy and space are called microstates. A macrostate is the result of many different microstates of individual particles and a macroscopically observable quantity is an average of these microstates (which depend on the conditions of the system); variables such as temperature, volume, number of molecules, measure a macroscopic state. Microstates are thus quantum mechanical descriptions of the ways that molecules can differ in their energy distribution and probable location. Energy dispersion relates to the way in which the energy of particles (i.e., atoms, ions, molecules) are distributed over vibrational, rotational, and translational energy levels. When particles have the ability to access a higher number of energy levels, they can spread out their energy and as a result the entropy is increased. Hence, the entropy of a macrostate (such as a solution) is a measure of the number of ways in which a system can be different in the energetic distribution of the constituting molecules. The number of possible microstates, V, that correspond to a given macrostate is linked to entropy quantitatively via Boltzmann’s relationship S ¼ kB ln V

(3:23)

in which kB is Boltzmann’s constant (1.381 3 1023 J K1), and V the number of possible microstates. V represents the different ways of distributing particle energies over levels leading to the same macrostate. To make this more comprehensible, the following explanation may help. Dispersion of total energy stored in particles is highest when the number of occupied energy levels is as large as possible, while at the same time the distance between the energy levels is as low as possible. Entropy is a measure of the number of occupied energy levels, in other words a measure of the dispersion of energy among accessible microstates. V, the number of microstates, is actually the number of possible permutations of the particles while keeping the number of particles the same in the different energy levels P ni ! V ¼ Qi i ni !

(3:24)

Q ni is the number of particles occupying an energy level i. The operator i represents the continued Q product: i ni ¼ n1 n2 n3 n3 . The lowest energy level is most populated, the highest accessible energy level the least. This is quantitatively expressed by the Boltzmann distribution

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ei e 0 ni ¼ n0 exp kB T

(3:25)

where e0 is the lowest energy level n0 the number of particles occupying energy level e0 An important effect is the distance between energy levels. If this distance increases, quantum leaps will be larger, energy levels less accessible, and consequently entropy will decrease. The decreased accessibility of the higher energy levels limits the number of microstates. An increase in temperature makes higher energy levels accessible to more particles, which explains why entropy increases with T. It should be clear that there is only one kind of entropy change in a system, namely the spreading of energy among a changed number of accessible microstates regardless whether this is due to a change in volume, composition, or temperature. When processes such as phase transitions and chemical reactions take place, this may result in a change in the number of particles occupying a certain energy level, in a changed accessibility of energy levels, and in a change in distance between energy levels. Figure 3.3 illustrates the number of microstates for 10 particles distributed over 3 and 4 energy levels, respectively, while the distance between the energy levels is kept the same. (Incidentally, Figure 3.3 is inaccurate by depicting only energy levels; quantum levels involve energy as well as position in space and the latter are not shown in the ﬁgure.) Using Equation 3.24 it follows that for the situation in Figure 3.3A the number of microstates V ¼ 12,600 while for the situation in Figure 3.3B V ¼ 3,150, which implies that the situation depicted in Figure 3.3A is much more probable than the one in Figure 3.3B, in other words the entropy of the system A is much higher than in B. To be sure, the situation depicted in Figure 3.3B is not impossible, but the probability that it occurs is very low. The example of Figure 3.3 is of course purely hypothetical. For real systems, the number of microstates becomes incredibly high and difﬁcult to comprehend, even when expressed logarithmically as in Equation 3.23. This statistical interpretation of the second law leads to the description of an irreversible process as that process that moves the system toward a state of greater probability in the absence of external forces. In reversible processes, the system remains in a state of maximum probability (or very close to it). A change in entropy between an initial and ﬁnal state is expressed as Vfinal DS ¼ kB ln Vinitial

(3:26)

Entropy change is thus expressed as the spreading of energy among a changed number of accessible microstates. This can happen, for instance, due to a change in volume (increased density of microstates),

e3

n3 = 1

e2

n2 = 2

n2 = 2

e1

n1 = 3

n1 = 4

e0

n0 = 4

n0 = 4

(A)

(B)

FIGURE 3.3 Schematic depiction of 10 particles occupying energy levels. Higher energy levels (>e0) are less accessible than lower energy levels. The situation depicted in Figure 3.3A is more likely than the one in Figure 3.3B (see text).

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composition, or temperature (more energy levels accessible at higher temperature). Boltzmann’s formula applies strictly speaking only to isolated systems, and there are some other restrictions as well that we will not discuss here. It sufﬁces here to think of the statistical interpretation of entropy in terms of microstates that belong to the macrostate the system is in. The second way to look at entropy change is a phenomenological view according to classical thermodynamics, namely by linking the change in energy dispersal to the heat ﬂow between a system and its surroundings. Clausius proposed that for a reversible process a change in entropy is proportional to the heat absorbed and the proportionality constant is 1=T. As remarked before, heat ﬂow is path dependent and we have to work with inexact differentials (indicated by the operator D) dS ¼

Dq T

(3:27)

1=T acts as a so-called integrating factor turning the inexact differential Dq into an exact one for dS. Entropy is thus a state function and we can consider the entropy change DS by looking at the difference in initial and ﬁnal state. Hence, for a heat ﬂow (q) at constant temperature, it follows that the entropy change corresponds to the heat transferred divided by absolute temperature DS

q T

(3:28)

This is the so-called Clausius inequality. The equality sign ( ¼ ) refers to reversible reactions, the ‘‘higher than’’ sign (>) to irreversible reactions. q=T is the entropy change induced in the surroundings by transferred heat. Changes in the entropy function can only be measured along reversible paths. Hence, the entropy change of a reversible process at constant T can be determined from measurements of the heat transferred and the temperature at which this occurs. The entropy change of a system is thus related to the heat transfer by q=T DSsys. For a reversible process q=T ¼ DSsys. For an irreversible process q=T < DSsys and the heat transfer is less than TDSsys. So, the phenomenological point of view for entropy (as opposed to the statistical mechanical interpretation) is that entropy is a measure of energy degraded during irreversible processes. For the universe the following relation holds: DSuniv ¼ DSsys þ DSsur 0

(3:29)

If one sums up all of the entropy changes in an isolated system, the entropy change DS 0. It is zero if there is equilibrium between all of the subsystems and it is positive when an irreversible change has taken place. Thermodynamic processes which develop so slowly as to allow each intermediate step to be an equilibrium state are said to be reversible processes. In a reversible process the system changes such that the system and surroundings can be put back in their original states by exactly reversing the process. These changes are inﬁnitesimally small. If an irreversible process has occurred the entropy change of the universe must have increased. Statistical mechanics deals with elementary steps which are perfectly reversible. Elementary reactions are reversible. But for a large system, irreversibility becomes important. This is nicely illustrated by looking at the dissolution of a sugar cube in a hot drink. The sugar molecules can in principle move in and out of the crystal lattice in a reversible manner, but the dissolution of the sugar cube as a whole is an irreversible process, because that is the most probable distribution for a large system. We conclude this section on entropy by summarizing the second law in Table 3.5.

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Chemical Thermodynamics in a Nutshell TABLE 3.5

3-15

Overview of Entropy Changes dS as a Function of Conditions Reversible Process

3.3.5

Irreversible Process

Isolated system

dS ¼ 0

dS > 0

Nonisolated system

dS ¼ Dq=T

dS > Dq=T

Free Energy

Since the second law reveals the direction of the reaction, namely that which increases the entropy of the universe, we need to be able to calculate this. However, this is an impossible task in terms of the universe. One can get around this problem by deﬁning two other derived thermodynamic quantities called free energy. Free energy can be deﬁned in two ways. At constant volume and temperature we have the so-called Helmholtz free energy F F E TS

(3:30)

At constant pressure and temperature, the so-called Gibbs free energy G is deﬁned as G H TS

(3:31)

We will concentrate here on the Gibbs free energy, but what will follow is valid for the Helmholtz free energy as well. Following Equation 3.31 one can consider the change in Gibbs free energy of a system as follows at constant pressure and temperature DGsystem ¼ DHsystem D(TS)system

(3:32)

Let us now consider a process at constant T so that D(TS)system ¼ (TS)final (TS)initial ¼ TDSsystem

(3:33)

So, at constant T it follows that DGsystem ¼ DHsystem TDSsystem

(3:34)

Now, when also the pressure is constant DHsystem ¼ qsystem (cf. Equation 3.19) and qsystem ¼ qsurroundings (Equation 3.15), it follows that DHsystem ¼ qsurroundings

(3:35)

From Equation 3.28 it follows that for a reversible process qsurroundings ¼ TDSsurroundings. Then it also follows that DHsystem ¼ TDSsurroundings

(3:36)

and now Equation 3.34 can be rewritten, at constant T and P, as DGsystem ¼ TDSsurroundings TDSsystem ¼ T(DSsurroundings þ DSsystem ) ¼ TDSuniverse

(3:37)

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So, the result of this algebraic exercise is that the change of the Gibbs energy at constant temperature and pressure is the change in free energy of the system and it equals the entropy change of the universe multiplied by the temperature. T and DSuniverse can only be positive, hence DGsystem always has to be negative for a process to occur (once again: at constant temperature and pressure). The second law of thermodynamics has been rephrased now in the sense that there is only one direction for processes, namely the direction of decrease in free energy of the system. This is the reason why entropy can never decrease in an isolated system: the entropy of a system can only decrease if energy is put into it and this is not possible for an isolated system by deﬁnition (Table 3.4). Upon increase of entropy, energy is still conserved but something is lost (namely part of the capacity to do work; in other words, useful energy is converted into less useful energy) and this is associated with the increase in entropy (namely TDS). Thus, entropy change is a convenient measure of the loss of capacity of the system to do work. This all means that if we can measure this free energy change, we can state something about the direction of a process. Equation 3.34 is a very important equation because we now have a tool to calculate whether or not a reaction will take place, provided of course that we can calculate free energy changes. DG is a state function. We are interested in changes, and standard states are also deﬁned for free energies, so the molar standard free energy change DGo can be calculated from the standard values of enthalpy and entropy for a particular change DGo ¼ DH o TDSo

(3:38)

Reactions in the nonstandard state. Of course, reactions do not always occur under standard conditions, and if a reaction cannot occur under standard conditions, it may take place under nonstandard conditions (e.g., by changing concentrations and=or temperature). Therefore, one must determine how DG depends on temperature and concentration under nonstandard conditions. This can be done as follows. At higher temperature, energy can be dispersed over more levels, so entropy is clearly temperature dependent. However, temperature affects entropies of reactants and products roughly in the same way, so the effect of temperature on the entropy change DrS due to a reaction is usually small. This means that DrS can be considered independent of temperature. For the same reason, DrH can be considered also independent of temperature. Thus, to account for a temperature different from the one speciﬁed for the standard condition while keeping the concentration at standard condition, this can be calculated as o o Dr GoT Dr HT¼298 K TDr ST¼298 K

(3:39)

The next step is to account for a concentration different from standard conditions. As it happens, DHo depends hardly on concentration, but entropy does: a more dilute sample has a larger entropy because of more accessible translational energy levels. This can be expressed as a function of pressure as S ¼ So R ln

P Po

(3:40a)

in which Po ¼ 1 bar (standard pressure). A similar equation holds for the dependence of entropy on concentration S ¼ So R ln

c co

where c is the molar concentration co is the unit standard concentration (chosen as 1.0 whichever unit is chosen) R is the gas constant

(3:40b)

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Since Po and co have by deﬁnition a numerical value of 1, they are usually omitted; however, it should be realized that c must be expressed in the same units as the chosen standard concentration in order to cancel units. So, for the free energy change under nonstandard conditions Equation 3.34 changes to Dr G ¼ DH o TDr S ¼ DH o T(DSo R ln c) ¼ DH o TDSo þ RT ln c ¼ Dr GoT þ RT ln c

(3:41)

To summarize, if concentrations do not refer to the standard state at the speciﬁed temperature, free energy calculations must be done in two stages. First, one has to correct for the temperature being different from the speciﬁed one (usually 298 K), using Equation 3.39 to obtain Dr GoT , and then that result can be used in Equation 3.41. These equations show clearly that DrG depends on concentration as well as on temperature, and this implies that the direction of a reaction can be manipulated by manipulating temperature and concentration. In this respect, it is instructive to consider both enthalpy and entropy changes. If the reaction enthalpy is negative, implying heat transfer from the system to the surroundings, this is called an exothermic reaction. If the reaction enthalpy is positive, this implies a heat ﬂow from the surroundings to the system, and then it is called an endothermic reaction. An endothermic reaction is only possible if the change in reaction entropy is positive. So, for a reaction as depicted in Equation 3.1 the possibilities are as shown in Table 3.6. An irreversible adiabatic process (no heat exchange between system and surroundings, see Table 3.4) necessarily leads to an increase in entropy of the system in which the process takes place. The tendency of increasing entropy in a given system can be counteracted by putting energy into that system. The total energy of the system plus surroundings remains, of course, constant (ﬁrst law) and the total entropy increases (second law), but the entropy of the system receiving the energy may increase, decrease, or remain constant. If the entropy change of the system is negative, the system must lose heat such that the entropy of the surroundings increases by at least the same amount; in that case the process is necessarily exothermic. If the entropy change of the system is positive the system can absorb heat such that the decrease in entropy of the surroundings matches the increase in the system; the chemical process can then be either exothermic or endothermic. It is thus possible that an endothermic reaction proceeds, if the gain in entropy (dispersal of energy) is sufﬁcient. This gain in entropy in the system is able to overcome the loss of entropy in the surroundings brought about by the inﬂux of heat (or other forms of energy, such as electric energy) from the surroundings into the system. For the sake of completeness we also mention the third law of thermodynamics which states that the entropy of a perfect crystal would be zero at 0 K, where all motion would cease. This gives a reference point to calculate absolute entropy values. Both the Gibbs and Helmholtz energy are deﬁned in terms of the entropy function, implying that they must be evaluated along reversible paths via reversible processes. For most reactions in foods, the TABLE 3.6

Direction of a Reaction A þ B > P þ Q as Dictated by Values of DH, DS, and the Resulting DGa

DH

DS

DG

Negative (exothermic)

Positive

Always negative: reaction from left to right at all temperatures

Negative (exothermic)

Negative

Negative at low temperature: reaction from left to right Positive at high temperature: reaction from right to left

Positive (endothermic)

Positive

Positive at low temperature: reaction from right to left

Positive (endothermic)

Negative

Negative at high temperature: reaction from left to right a

Referring to the reaction going from left to right.

Positive at all temperatures: reaction only from right to left

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differences between using the Gibbs or Helmholtz free energy will be small. We will use mainly Gibbs energy in this book. The Gibbs free energy change in a reversible process at constant temperature and pressure equals the work done exclusive of PdV work. It thus represents the maximum quantity of useful energy from a chemical reaction at constant temperature and pressure in a reversible process, and the Helmholtz free energy that at constant volume; useful energy means interconvertible energy (electrical, chemical, mechanical). If there is excess of free energy, or if the free energy cannot be harnessed as useful energy, the free energy represents the maximum extent of energy dispersal to the universe. Gibbs energy and dissolution. Before considering reactions in more detail, it is also of interest to consider what happens when a compound dissolves in a solvent. For foods, the solvent water is the most important one. The dissolution can be in principle endothermic, exothermic, or a-thermic. It all depends on changes in interactions between solute–solute molecules, solute–solvent molecules, and solvent– solvent molecules. For instance, the dissolution of a crystal depends on the decrease in enthalpy when solvation occurs and the increase in enthalpy needed to disrupt the solvent structure and possibly enthalpy effects of dissociation in the case of electrolytes. Table 3.7 gives a few examples. Table 3.7 shows that the dissolution of NaCl in water is endothermic but with a positive entropy change. The dissolution of CaCl2 on the other hand is exothermic but it leads to a negative entropy change. In both cases the Gibbs energy is negative, indicating that dissolution is possible under these conditions. At temperatures higher than 258C, NaCl would have a greater tendency to dissolve and CaCl2 less, whereas at lower temperatures the opposite is the case. To conclude this section, two remarks must be made. 1. Being state functions, changes in DE, or DH, or DS, or DG, or DF, can be added. However, this can only be done for processes that are independent. This is a requirement that may not always be easy to realize in practice. 2. It should be realized that variables such as enthalpy, entropy, Gibbs, and Helmholtz energies describe huge numbers of particles; these variables do not have any meaning at the level of the individual particles.

3.3.6

Chemical Potential

Every substance has the tendency to change, that is to say, it can . . .

React with other substances Transform into another state of aggregation Migrate to another place

This tendency is described by a single physical quantity, the chemical potential m. Its value depends on a speciﬁc substance; it also depends on T, P, and in solution also on concentration and the kind of solvent, and the state of aggregation. So, when we want to evaluate chemical reactions in thermodynamic terms, the chemical potential of components is a very useful concept. In general, a potential indicates the ability

TABLE 3.7 Examples of Standard Molar Enthalpies, Entropies, and Gibbs Energy of Solution in Water at 258C Compound NaCl CaCl2 CH3OH (methanol) C2H5OH (ethanol)

DsolH8 (kJ mol1) 3.89 81.3 7.28 10.6

DsolS8 (J mol1 K1)

DsolG8 (kJ mol1)

43.1

8.9

44.8

67.9

6.3

9.2

12.1

7.0

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to do something, and a chemical potential indicates the ability for a component to react, or to move to another phase, etc. The chemical potential m of a pure substance is deﬁned at constant temperature and pressure as m¼

@G @n T, P

(3:42)

This shows the chemical potential to be the molar Gibbs energy, i.e., it shows how the Gibbs free energy changes as a function of the change in number of moles at constant temperature and pressure. Using Equations 3.40a and 3.37, it can be shown that the following relation holds P G ¼ Go þ RT ln o P

(3:43)

As remarked before, thermodynamic quantities can be evaluated only relative to different states, so we need a reference, which is called the standard state, at which Go in Equation 3.43 is evaluated. For a perfect gas, this is the state at which Po is the standard pressure of 1 bar. In a perfect gas there are no intermolecular interactions; the pressure is purely the result of kinetic energy of the molecules. It follows then that for a perfect gas P m ¼ m þ RT ln o P o

(3:44)

mo is the chemical potential of the component in its standard state. Moving on now to liquids, imagine a pure liquid compound i that is in equilibrium with its vapor having a vapor pressure P*. The chemical potential of i in the vapor phase is then (with (g) indicating the gaseous state) P* o ¼ m (g) þ RT ln m*(g) i i Po

(3:45)

The superscript ‘‘*’’ indicates that we are dealing with the pure compound. At equilibrium the chemical potential of component i in the vapor phase (g) must be equal to that of the component in the liquid ¼ m*(g), hence Equation 3.45 can be written also as phase (l): m*(l) i i ¼ m*(l) i

moi (g)

P* þ RT ln o P

(3:46)

Imagine now that component i is not present as a pure compound but is one of the components in a binary liquid mixture causing a partial vapor pressure Pi. Equation 3.45 can then be written as Pi mi (g) ¼ moi (g) þ RT ln o P

(3:47)

Again, at equilibrium mi(l) ¼ mi(g), hence Pi mi (l) ¼ moi (g) þ RT ln o P

(3:48)

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By eliminating moi (g) from Equations 3.46 and 3.48 it follows that Pi þ RT ln mi (l) ¼ m*(l) i Pi*

(3:49)

We now have an equation that describes the chemical potential of a component in a liquid mixture as a function of vapor pressures. The next step is to connect this to the composition of liquid mixtures. We can do this via the concept of ideal solutions.

3.3.7

Ideal Solutions

As shown, thermodynamic relations are frequently based upon ideal gases. When discussing equilibria in liquid mixtures, it is necessary to know how the chemical potential of a compound in a solution varies with composition. We can make a link between solutions and gases via Raoult’s law, which is formulated as follows for each component i: Pi ¼ Xi Pi*

(3:50)

Pi is the partial vapor pressure of compound i present in the solution at mole fraction Xi and Pi* is the vapor pressure of pure compound i. If the total vapor pressure of a solution is the sum of the partial pressures related to mole fractions in solution as expressed by Raoult’s law, we speak of an ideal solution. Figure 3.4 shows Raoult’s law graphically. Substituting Equation 3.50 in Equation 3.49 results in the following equation: þ RT ln Xi mi (l) ¼ m*(l) i

(3:51)

This equation describes the dependence of the chemical potential of component i in an ideal solution on its composition expressed as mole fraction, which is what we were looking for.

Pvap P ∗2 P ∗1 P1 =

∗

X1 P ∗ 1

0

P

P2 =X2 2

X2

1

FIGURE 3.4 Illustration of the vapor pressure Pvap above an ideal solution consisting of components 1 and 2 as a function of mole fraction X2. The solid line represents the vapor pressure of the vapor in equilibrium with a liquid mixture of components 1 and 2. The broken lines indicate the partial vapor pressures of components 1 and 2 according to Raoult’s law.

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Even though only very few mixtures behave as ideal solutions, they are the starting points to describe the behavior of real mixtures. An ideal solution does not have the same properties as a perfect gas. In a perfect gas, there are no interactions between molecules. In an ideal solution, there is interaction between molecules, but on average the interaction between the solute–solvent, solvent–solvent, and solute–solute molecules are the same as in the pure liquids. In an ideal solution there are no volume and heat effects upon mixing, i.e., DmixV ¼ 0 and DmixH ¼ 0. For the entropy change DmixS of mixing in an ideal solution the following relation holds: Dmix S ¼ R

X

Xi ln Xi

(3:52)

This shows that the mixing entropy is always positive for an ideal solution. If a solute at mole fraction X2 is dissolved in a liquid with mole fraction X1, then in an ideal solution the following relation holds (known as the Hildebrand equation): R ln X2 ¼ Dfus H2

1 1 T Tfus,2

(3:53)

The subscript ‘‘fus’’ indicates fusion. Likewise, for an ideal solution of a gas in a liquid the following holds: R ln X2 ¼ Dvap H2

1 Tvap,2

1 T

(3:54)

The subscript ‘‘vap’’ indicates evaporation. As shown, these relations are independent of the solvent in an ideal solution.

3.3.8

Ideal Dilute Solutions

There are solutions in which at very low concentrations of the solute (i.e., X1 1, X2 1, whereas 0 < ar < 1, which is more convenient for the present discussion), the reaction Gibbs energy under standard conditions can be expressed as ð Þ ¼ G G ¼ 1 a þ G G þ G þ G þ a G þ G G þ G þ a G r r r r A B P Q A B P Q A B

(3:89)

This can be rewritten as G r ¼ GA þ GB þ ar Dr G

(3:90)

So, when G r is plotted as a function of ar, a straight line with slope Dr G is obtained (Figure 3.15). Of course, it is a hypothetical situation for reactants and products to remain both in their standard state during the reaction. Reactants and products mix due to the very reaction: at ﬁrst there are only reactants but products start to be formed as the reaction progresses. Therefore, reactants and products cannot remain in the standard state. So, we have to study what will happen under nonstandard conditions. An extra contribution to the free energy comes from the mixing effect so that Equation 3.90 becomes G ¼ G A þ GB þ ar Dr G þ Dmix G

(3:91)

As discussed above, for a real mixture the relation holds mi ¼ m*i þ RT ln fi Xi

(3:92)

and for the mixing term Dmix G ¼

X

Xi (mi m*) i ¼

X

Xi RT ln Xi þ

X

Xi RT ln fi

(3:93)

GA + GB

G

∆rG

0

GP + GQ

ar

1

FIGURE 3.15 Free energy of a reaction is depicted as a function of the degree of reaction ar with both reactants and products in the standard state. A, B represent reactants, P, Q represent products.

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As before, the deviation from ideality is accounted for in the activity coefﬁcient. The ﬁrst term in the right-hand side of Equation 3.93 reﬂects ideal mixing and is always negative while the second term reﬂects nonideal mixing. This second term is called the excess Gibbs energy of mixing. Recall that we are looking for an equation that expresses how G changes with ar. For ideal solutions, there is no volume change when the composition changes and DmixH ¼ 0, so in that case the extra contribution in DmixG is due to entropy effects, as was shown already in Equations 3.40. Equation 3.40 shows that DS contains both a concentration-independent term and a concentration-dependent term; see also Equation 3.52. The concentration-dependent term is called the entropy of mixing. It reﬂects the change in number of accessible microstates upon mixing of two or more chemical species. It is the dispersal of energy of each species within a larger volume due to mixing that affects entropy. Hence, the entropy of mixing is the difference between the entropy of the mixed state and the entropy of the corresponding unmixed state. Expressed in mole fractions Xi the expression for a reaction as depicted in Equation 3.1 taking place in an ideal solution would be (cf. Equation 3.52) Dmix S ¼ NR(XA ln XA þ XB ln XB þ XP ln XP þ XQ ln XQ ) ¼ NR

X

Xi ln Xi

(3:94)

i

N is the total number of moles in the system, and R is the gas constant, as usual. When a reaction starts with reactants A and B, the reaction mixture becomes more mixed when products are formed, and consequently, the entropy of mixing increases. When the reaction would go to completion, i.e., no reactants left anymore, some demixing occurs, leading to a decrease in mixing entropy. The change in entropy of mixing with ar can be derived from Equation 3.94 and when activities are used instead of mole fractions it results in nP nQ aP a d(Dmix S) ¼ R ln nA QnB aA aB dar

(3:95)

DmixS ¼ 0 in the unmixed state and 0 when the reaction progresses. It reaches its maximum value when reactants and products become maximally mixed. Using the relation DG ¼ DH TDS, and remembering that DmixH ¼ 0 for an ideal solution, it follows that nP n Q aP a d(Dmix G) d(TDS) ¼ ¼ RT ln nA QnB ¼ RT ln Qr aA aB dar dar

(3:96)

Thus, the reaction quotient Qr is deﬁned as n

Qr ¼

anPP aQQ anAA anBB

(3:97)

The expression of how G changes with ar can now be found by evaluating Equation 3.90 with respect to ar and substituting Equations 3.96 and 3.97 dG d(ar Dr G ) d(Dmix G) ¼ þ ¼ Dr G þ RT ln Qr dar dar dar

(3:98)

This equation shows the contribution of the entropy of mixing to the change in free energy but now expressed as RT ln Qr. One can thus calculate the change in free energy as a function of the extent of reaction

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Gar Gar ¼0 ¼ ar Dr G þ RT ln Qr

(3:99)

If dG=dar < 0 the reaction will proceed from left to right because that is the direction of free energy decrease, and if dG=dar > 0 for such a reaction it will move in the opposite direction from right to left, again because that decreases free energy. Equilibrium is attained when dG=dar ¼ 0 and then the reaction quotient Qr has become equal to the equilibrium constant Keq n

anPP aQQ anAA anBB

(3:100)

n i mi ¼ 0

(3:101)

Keq ¼ Another expression that describes equilibrium is X

3.3.13

Equilibrium Constants

Recalling Equation 3.1 nA A þ nB B ! nP P þ nQ Q Equations 3.97 and 3.100 are an expression of the famous law of mass action; we will come back to this when discussing chemical kinetics in Chapter 4. Equation 3.100 shows how activities are related to each other when equilibrium is attained. Note that Keq is dimensionless because activities are dimensionless. This is called a thermodynamic equilibrium constant and it is this constant that is reported in tables. In practice, however, it may be more convenient to use constants based on pressures, concentrations, or molalities but it should be realized that these are actually not true constants (except at very low concentrations) because they depend on activity coefﬁcients, and these vary with composition. The following analysis shows how practical equilibrium constants are related to the true thermodynamic equilibrium constant. It is easiest to start with considering the reaction in Equation 3.1 in the gas phase, and consider it as an ideal mixture of perfect gases so that we can use pressures instead of fugacities, and Equation 3.100 can be written as nP nQ PP PQ Dn (PP )nP (PQ )nQ 1 nP þ nQ nA nB 1 Po Po ¼ K Keq ¼ nA nB ¼ P nA nB o PA PB (PA ) (PB ) P Po o o P P

(3:102)

KP is the equilibrium constant in terms of pressure. Since Po ¼ 1 bar, this equation shows that Keq ¼ KP, as it should be, because we have assumed an ideal gas. To change from pressures to concentrations, we can use the ideal gas law that reads P¼

N RT V

(3:103)

The concentration is in fact N=V so that Equation 3.100 changes to Keq ¼

Dn [P]nP [Q]nQ RT Dn RT ¼ Kc o o nA nB [A] [B] P P

(3:104)

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This shows that the equilibrium constant in terms of concentration (with a hypothetical standard state of 1 mol dm3) is different from Keq unless Dn ¼ 0. The relation between Kc and KP follows from Equations 3.102 and 3.104 Kc ¼ KP (RT)Dn

(3:105)

Similarly, the relation between KX, the equilibrium constant in terms of mole fraction and KP is KX ¼ KP PDn

(3:106)

(P is the total pressure). Unless Dn ¼ 0, these equations show that the mole fractions or the concentrations in an equilibrium mixture will depend on the pressure P, even though KP does not. As mentioned, thermodynamic equilibrium constants are tabulated for activities, not for concentrations. The expression for Keq as in Equation 3.100 based on activities is universally valid and Keq is a true constant (though depending on temperature) but the difﬁculty is of course in relating activities to measurable quantities. To convert activities to concentrations we need activity coefﬁcients, and as we have seen these depend on the choice of standard state. Practical equilibrium constants such as Kc depend on concentration because activity coefﬁcients depend on concentration. This becomes clear from the following analysis: n

Keq ¼

n

n

anPP aQQ [P]nP [Q]nQ gnPP gQQ yPnP yQQ nA nB ¼ n A n B ¼ Kc n A n B nA nB aA aB [A] [B] gA g B yA yB

(3:107)

Activity coefﬁcients depend on concentration which implies that Kc must also depend on concentration to keep Keq constant. The same is of course true for Km based on molalities. So, Km and Kc are not really true equilibrium constants unless the activity coefﬁcients are equal to one; such a situation may occur in (very) diluted solutions. The standard states need not be the same for all components that are involved in the reaction. However, it is essential that the same standard state is used for the standard chemical potential and for the activity coefﬁcient because these two determine the chemical potential (as is shown in several equations above, e.g., in Table 3.8). The reader is reminded that activities in Equation 3.100 are by deﬁnition equal to 1.0 for pure solids and pure liquids while for gases it is equal to pressure (at a standard pressure Po ¼ 1 bar). Since the activities of pure liquids and solids hardly depend on pressure, we do not need to include activities of pure liquids and solids in calculation of equilibria because they are approximately equal to 1.0 at moderate pressures. For instance, the equilibrium constant for dissociation of water þ H2 O(l) ! H (aq) þ OH (aq)

Keq ¼

aHþ aOH ¼ aHþ aOH a H2 O

(3:108)

with aH2O ¼ 1 and this constant is commonly called the water dissociation constant Kw. For the dissolution of a salt in water, assuming a saturated solution 2þ CaCl2 (s) ! Ca (aq) þ 2Cl (aq)

the equilibrium constant is Keq ¼

aCa2þ a2Cl ¼ aCa2þ a2Cl aCaCl2

(3:109)

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The latter equilibrium constant is usually referred to as the solubility product. For the formation of water from oxygen and hydrogen: 2H2 (g) þ O2 (g) ! H2 O(l) The equilibrium constant is Keq ¼

aH2 O 1 ¼ 2 2 P PH P H2 PO2 2 O2

(3:110)

Perhaps it is useful to come back here brieﬂy to the biochemists’ standard state. To recall, the hydrogen ion concentration for the standard state is deﬁned as 107 M, and as a result the change in standard Gibbs free energy will be different for reactions in which Hþ ions are produced or consumed. So, if we have a reaction: A þ B ! P þ nHþ Hþ Equation 3.98 becomes [P]

dG 0 c ¼ Dr G þ RT ln dar

nHþ

[Hþ ] 107 M [A][B]

c c

(3:111)

Comparing this with the ‘‘normal’’ standard state of c ¼ 1 mol dm3, it follows that Dr G ¼ Dr G þ nHþ RT ln

1 107

(3:112)

0

If, for instance, nHþ ¼ 1, the difference between DrG and DrG is 40 kJ mol1. Similarly, if Hþ ions are taken up in the reaction, the difference is 40.0 kJ mol1 for nHþ ¼ 1. For reactions in which no Hþ is involved, there is no difference in standard free energies. Equilibrium constants and free energy changes. When equilibrium is attained so that dG=dar ¼ 0 and Qr ¼ Keq, it follows from Equation 3.98, that the equilibrium constant of a reaction is linked to the standard Gibbs energy change Dr G ¼ RT ln Keq

(3:113)

This is a very important relationship because it shows that equilibrium constants can be predicted from standard Gibbs energy changes, which are tabulated for many reactions, or can be calculated from standard Gibbs energies of formation (by analogy of Equation 3.20). Table 3.10 shows some numerical examples and gives a feel what the values imply for the relative amount of reactants and products. Incidentally, the fact that DrG appears in Equation 3.113 is due to the relation between the chemical potential and molar Gibbs energy (Equation 3.42). In the case of constant temperature and volume, we should use the Helmholtz energy F, but DrG then still appears in the equations. For instance, Equation 3.98 in the case of constant T, V becomes dF ¼ Dr G þ RT ln Qr dar

(3:114)

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Chemical Thermodynamics in a Nutshell TABLE 3.10

3-39

Relationship between DrG and Keq at 208C

DrG (kJ mol1)

Keq

Composition of the Reaction Mixture

75

2.4 3 10

Only products, amount of reactants negligible

15

472

Mainly products

5

7.8

More products than reactants

0

1

Equal amounts of products and reactants

5

0.13

More reactants than products

15 75

2.1 3 103 4.3 3 1014

Mainly reactants Only reactants, amount of products negligible

13

In other words, Equation 3.113 describes the equilibrium condition for both constant T, P and constant T, V. Distinction between DrG , DrG, and dG=dar. It is essential to appreciate the difference between DrG , DrG, and dG=dar. DrG indicates the difference in free energy between reactants and products in their standard state when they are unmixed. If DrG is positive, it does not mean that a reaction cannot take place. A positive DrG implies an equilibrium constant smaller than 1 (cf. Equation 3.113, and Table 3.10), so the concentration of products may be small but products can be formed nevertheless, despite a positive DrG . It all depends on concentrations of reactants and products whether or not a reaction will take place in a speciﬁed direction, in other words on the value of the reaction quotient Qr. There is a difference between Qr and Keq. Qr can have any value for a reaction, whereas Keq has a speciﬁed, constant value. Qr is a useful parameter to predict the direction of change at a certain composition of the reaction mixture, as indicated in Table 3.11, which shows the connection between Qr, Keq, and dG=dar. Table 3.11 shows the meaning of dG=dar in terms of the direction of a reaction. dG=dar indicates how much the Gibbs free energy changes with the advancement of the reaction. DrG indicates the difference in free energy of ﬁnal and initial states at equilibrium at nonstandard state conditions. As for DrG, if it is positive, the reaction cannot take place, in contrast to when DrG is positive. There is much confusion about this in papers and textbooks, especially when it concerns so-called coupled reactions in biochemistry. Frequently, it is claimed that a reaction under conditions for which DrG < 0 can ‘‘drive’’ a coupled reaction for which DrG > 0. This is simply not true. A reaction showing a positive free energy change just does not occur. However, it is possible that a reaction has a positive standard free energy DrG and if the concentrations of reactants and products are such that dG=dar is negative, then the reaction will occur. What can be said about coupling of reactions is that a reaction having DrG < 0 can drive a reaction having DrG > 0. Another way of making the difference clear between DrG and DrG is that DrG is a constant, while DrG is constantly changing depending on the concentrations, until the reaction composition reaches the point that DrG ¼ 0. A case in point is the formation of glucose-6-phosphate in the body from phosphate and glucose. The standard free energy change DrG for this reaction is þ12.5 kJ mol1. The cellular concentration of glucose is about 0.0001 M, that of phosphate 0.01 M and that of glucose-6phosphate 0.0001 M. The DrG can then be calculated as þ24 kJ mol1. This reaction is therefore not possible in the way that it is written. As mentioned, it is frequently suggested that this positive free energy can be overcome by thermodynamic coupling to the hydrolysis of ATP, which has a negative DrG ¼ 50.4 kJ mol1 at cellular conditions. The overall reaction is depicted as:

TABLE 3.11

Relation between Qr, Keq, and dG=dar for a Reaction A þ B > P þ Q

Reaction Goes to the Right When

There Is Equilibrium When

Reaction Goes to the Left When

Qr< Keq

Qr ¼ Keq

Qr > Keq

dG=dar < 0

dG=dar ¼ 0

dG=dar > 0

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3-40 TABLE 3.12

Data for Mutarotation of Glucose in the Standard State at 258C

Reactant

DHu kJ mol1

a-glucose

1.15

b-glucose

1.15

DSu J mol1 K1

DGu kJ mol1

Keq

0.92

1.425

1.77

0.92

1.425

1.77

Dr G ¼ þ24 kJ mol1 Glucose þ phosphate ! glucose-6-phosphate ! Dr G ¼ 50:4 kJ mol1 ATP ADP þ Pi 1 ! Glucose þ ATP glucose-6-phosphate þ ADP Dr G ¼ 26:4 kJ mol Such a presentation may lead to a wrong conclusion because the ﬁrst reaction cannot take place under any condition! The error made here is that the reactions shown are overall reactions, but they do not take place at the molecular level as they are written down, in other words they may not be added up like this. What really happens is that glucose is phosphorylated via an indirect reaction path in several steps, and for each step the free energy change DrG is necessarily negative; incidentally, these steps take place enzymatically in living organisms but that is not essential for this discussion because also enzymes cannot overcome positive free energy changes! A further example may be helpful here. It concerns the mutarotation of glucose at pH 7 and 258C a-glucose ! b-glucose

(3:115)

Table 3.12 gives some data needed for the calculation. With these data we can calculate the change in G as a function of ar. It is instructive to plot the contribution of the mixing and the nonmixing contribution to the free energy plot as a function of the degree of reaction (Figure 3.16).

0 Nonmixing contribution

Gαr −Gαr=0/kJ mol–1

−0.5

−1

∆rG

Mixing contribution −1.5

∆rG −2 Total

−2.5

Equilibrium position −3 0

0.2

0.4

0.6

0.8

1

ar

FIGURE 3.16 Change in free energy for the mutarotation of glucose at 258C as a function of the degree of reaction ar.

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The equilibrium is at ar ¼ 0.64 in this example, where a minimum in free energy is reached. The minimum in the mixing contribution, at ar ¼ 0.5 in this example, does not coincide with the equilibrium position, which makes sense because the equilibrium not only depends on DmixS but also on enthalpy and temperature. The minimum in the mixing contribution is always reached at Qr ¼ 1, as follows from Equation 3.96. At this position the afﬁnity of the reaction, dG=dar, which is the slope of the graph of Gar Gar ¼ 0 versus ar, equals DrG , see Equation 3.98. Figure 3.16 also shows what would happen if the standard free energy change is positive. This would be the case if the reaction is started with b-glucose as reactant: then DrG ¼ þ1.425 kJ mol1 (cf. Table 3.12). Even though this is a positive standard free energy change, the reaction will nevertheless take place and a-glucose will be formed from b-glucose up to ar ¼ 0.36 because DrG is negative in going from pure b-glucose to this equilibrium. One may now wonder whether reactions can go ever to completion to the point that ar ¼ 1, when it takes place in solution and when reactants or products are not removed. The answer is no, not completely, but the equilibrium position can lie very far to the right or to the left. In that case, the nonmixing contribution to the free energy is largely determining the free energy change. It is perhaps instructive to show this in a graph (Figure 3.17). This is for the same reaction as in Figure 3.16 but now the standard free energy has been changed, for the sake of argument, from 1.425 to 14.25 kJ mol1, leaving DrS ¼ 0.92 J mol1 K1 so that DrH ¼ 14,249.08 J mol1. The equilibrium would then be found at ar ¼ 0.997 mol, hence the reaction would have gone almost to completion but there is still a tiny contribution from the mixing entropy that prevents it from reaching total completion. Calculation of equilibrium concentrations. With knowledge of standard Gibbs energies, or equivalently, equilibrium constants, equilibrium concentrations can be calculated. This is most easily done using the degree of reaction, ar. An example will show this. It concerns the dissociation of benzoic acid, a food preservative, in pure water. The equilibrium constant at 258C is 6.4 3 105. We can set up the following table (Table 3.13). The following relation holds at equilibrium: Kc ¼

ar c0 ar c0 a2r c0 ¼ 6:4 105 ¼ (1 ar )c0 (1 ar )

(3:116)

0 −2

Mixing contribution

Gαr - Gαr= 0/kJ mol–1

−4

Nonmixing contribution

−6

∆rG

−8 ∆rG

−10 Total

−12 −14

Equilibrium position

−16 0

0.2

0.4

ar

0.6

0.8

1

FIGURE 3.17 Change in free energy when the nonmixing contribution would be 14.25 kJ mol1 rather than 1.425 kJ mol1 as in Figure 3.15.

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3-42

TABLE 3.13 Calculation of Concentration as a Function of ar for the Dissociation of Benzoic Acid in Water at 258C Degree of Reaction

C6H5COOH

C6H5COO

Hþ

ar ¼ 0

c0

0

0 (10

ar ¼ 1

0

c0

c0

0 < ar < 1

(1 ar)c0

arc0

arc0

7

in water)

This can be rearranged to a2r c0 þ 6:4 105 ar 6:4 105 ¼ 0

(3:117)

TABLE 3.14 Degree of Reaction at Equilibrium for Various Initial Concentrations for the Dissociation of Benzoic Acid c0 (mol L1)

ar

0.01

0.077

0.1

0.025

1

0.008

Such an equation can be solved following the algebraic rules to derive the roots for a square root equation ar ¼

6:4 105

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (6:4 105 )2 4c0 6:4 105 2c0

(3:118)

This shows that the degree of reaction at equilibrium depends on the initial concentration in this case. Table 3.14 shows this dependence of ar for some values of c0. It is interesting to note that the degree of reaction at equilibrium does not depend on the initial concentration if the number of moles of reactants and products is the same (Sni ¼ 0); this would be the case for mutarotation of glucose, for instance. Perhaps, it is helpful to list some more terms that are used in addition to Table 3.4. Especially in biochemistry textbooks, the terms exergonic and endergonic are used. This means a negative and a positive standard free energy change DrG (but not a negative or positive DrG as is frequently suggested). The terms exergonic and endergonic should be distinguished from exothermic and endothermic; the latter two terms pertain to enthalpy changes at constant pressure (and no work other than by volume changes). Table 3.15 lists the possibilities, including the meaning of enthalpy- and entropy-driven reactions. In conclusion, many reactions do not go to completion in the sense that ar ¼ 1 because the entropy of mixtures of reactants and products is greater than that of separate products or reactants. However, the magnitude of this effect strongly depends on how exothermic the reaction is. A highly exothermic reaction causes a much greater entropic effect to the surroundings than the mixing entropy effect. It is stressed once again that the equations used are for ideal solutions. For real solutions, one also has to take enthalpy and volume changes into account. The trends will be the same but the magnitude of the effect may be different.

3.3.14

Thermodynamic Potentials and Conjugate Variables

Here, we brieﬂy recapitulate the meaning of thermodynamic potentials, which are the quantitative measures of the energy contained within a system. This section forms the bridge between sections on

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Chemical Thermodynamics in a Nutshell TABLE 3.15

3-43

List of Terms Used in Thermodynamics of Chemical Reactions

Term

Symbol

Meaning

Exothermic

DH < 0

Release of energy by the system to the surroundings

Endothermic

DH > 0

Uptake of energy by the system from the surroundings

Exergonic

DrG < 0

Negative free energy difference between reactants and products in their standard state; Keq > 1

Endergonic

DrG > 0

Positive free energy difference between reactants and products in their standard state; Keq < 1

Enthalpy-driven reaction

If jDHj >> jTDSj

Enthalpic effect is much stronger than entropic effect in determining the magnitude of DG

Entropy-driven reaction

If jDHj 0 and dS 0 it follows that Af dj 0. This is called the de Donder inequality. The equation for Gibbs free energy for a system with changing composition is (already shown in Equation 3.67) dG ¼ SdT þ VdP þ

X i

mi dni

(3:130)

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For a system in which chemical reactions occur, it can be changed into, analogous to Equation 3.128 dG ¼ SdT þ VdP þ

X

mi dni 00 Af dj

(3:131)

i

At constant temperature and pressure it follows that for a closed system (dG)T, P ¼ Af dj

(3:132)

Or with k multiple reactions (dG)T, P ¼

X

Af ,k djk

(3:133)

k

Similarly for the Helmholtz free energy dF ¼ SdT PdV þ

X

mi dni 00 Af dj

(3:134)

i

(dF)T,V ¼ Af dj

(3:135)

The afﬁnity is a state variable and the advantage in using it is that the conditions under which a reaction is taking place do not need to be speciﬁed like with Gibbs free energy (constant T,P) or Helmholtz free energy (constant T,V). A positive afﬁnity indicates that the reaction takes place in the direction of j (forward reaction), a negative afﬁnity points at the reverse reaction. If Af ¼ 0, the system is at equilibrium. This is shown schematically in Figure 3.18; note that the slope is negative, i.e., dAf=dj is always negative, regardless of which variables are held constant. Some useful relations for the chemical afﬁnity are @G @F ¼ Af ¼ @j T,P,{ni 00 } @j T,V,{ni 00 } @E @H ¼ ¼ @j S,V,{ni 00 } @j S,P,{ni 00 } @S ¼ @j E,n,{ni 00 }

Af +

0

−

xeq

x

FIGURE 3.18 Relation between afﬁnity Af and extent of reaction j. The equilibrium position is indicated at jeq and Af ¼ 0.

(3:136)

Conjugate pairs. Energy is the ability to do work. In mechanics, energy transfer results from a force Fi that causes a displacement ui, with Fi 3 ui being the amount of energy involved. Similarly, changes in thermodynamic systems can be thought of as the result of a generalized force causing a generalized displacement, with the product of the two being the amount of energy transferred. These thermodynamic force–displacement pairs are known as conjugate pairs or variables. The thermodynamic force is always an intensive variable, and the

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Kinetic Modeling of Reactions in Foods

3-46 TABLE 3.16 Conjugate Pair

Overview of Some Common Conjugate Pairs Force F (Intensive Variable)

Displacement u (Extensive Variable) Volume V

PdV

Pressure P

TdS

Temperature T

Entropy S

mdn

Chemical potential m

Number of particles n

Afdj

Afﬁnity

Advancement of reaction j

MdH

Magnetization M

Magnetic ﬁeld H

gdA Efde

Surface tension g Electromotive force Ef

Surface area A Amount of charge e

displacement is always an extensive variable, and the energy transfer is always extensive. The above equation for energy (as well as those for enthalpy, Gibbs, and Helmholtz energy) can thus in general be displayed as a product of force and displacement dE ¼

X

Fi dui

(3:137)

i

The product Fidui represents conjugate variables or pairs, e.g., TdS, PdV, midni, etc. The above equations show that, next to TdS and PdV, mdn as well as Afdj can be considered as conjugate pairs. These conjugate pairs all represent the ﬂow of energy across the boundary of the system. Table 3.16 gives an overview of conjugate pairs discussed so far, along with some other ones, less important perhaps for the topic of this book, but shown anyway to illustrate the general application of this concept. The magnitude of an intensive variable determines the magnitude of the resulting energy change. For instance, a change in entropy has a much higher effect on the energy change at a high temperature than at a low temperature. The sum Fidui in the left-hand side of Equation 3.137 indicates that there are several ways in which work can be done on or by the system. Change of any of the conjugate variables Fi or ui may affect equilibrium activities of reactants and products and thereby equilibria will shift. One can choose a forcing parameter Fi for perturbation to which the equilibrium is sensitive. The relaxation time is generally shorter if during equilibration the extensive variables ui instead of the intensive variables are held constant. However, experimentally it is easier to vary the intensive variables Fi. A remark should be made about the meaning of the words ‘‘force and displacement’’ in relation to chemical reactions and diffusion. Of course, molecules do not feel a force because of a difference in chemical potential; they just move about randomly and when there are spatial concentration differences this will eventually disappear because of this random motion, not because of a force. Similarly, a chemical reaction is basically a stochastic process and molecules do not feel a force that pulls them toward a chemical reaction in the mechanical sense. So, it should be understood that thermodynamic forces and displacements are just convenient ways to state quantitatively what will happen; they should not be taken literally in the case of diffusion and chemical reaction. Analogies have their limitations! In the case where gravity is important, particles do ‘‘feel’’ a real force (for instance, cocoa particles in a chocolate drink) and in the case of electric potentials charged particles also experience a force (ions, proteins, and charged colloidal particles such as casein micelles). We consider some of these cases in Chapter 11. Le Châtelier’s principle. The well-known Le Châtelier’s principle can be formulated in various ways. A popular one is if a chemical system in equilibrium is subjected to a perturbation, the system will shift to a new equilibrium such that the perturbation is partially undone. This principle is often used as a rule of thumb to make qualitative predictions. Unfortunately, it is not a very rigorous principle and is in principle only valid for a perfect gas. It is actually much better to work with de Donder’s afﬁnity Af. If a system is in equilibrium, Af ¼ 0. It implies that one of the thermodynamic functions is at an extremum (i.e., at a maximum or a minimum). In the case of an isolated system the entropy is maximal at

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Chemical Thermodynamics in a Nutshell

3-47

Af +

0 T2 < T0 T0 T1 > T0 x1

FIGURE 3.19

x0

x2

x

Af versus j for different temperatures in the case of an exothermic reaction. See text for explanation.

equilibrium, and the Gibbs free energy is minimal at constant temperature and pressure, etc. A useful rule that replaces Le Châtelier’s principle is the following. If in a system at equilibrium a particular extensive variable is changed, the equilibrium will shift in the direction that tends to reduce the change in the corresponding intensive variable. Similarly, if in a system at equilibrium a particular intensive variable is changed, the equilibrium will shift in the direction that tends to increase the change in the corresponding extensive variable. It is possible to make quantitative predictions on how an equilibrium will shift upon perturbation by deriving equations of jeq as a function of the perturbed variable and of Af versus j for various values of the perturbing parameter. In that way, it can be derived, for instance, how a reaction at equilibrium will respond to a temperature change: see Figure 3.19 where the situation is depicted for an exothermic reaction. When an exothermic reaction originally at equilibrium at T0 and j0 is perturbed by a sudden temperature increase (T1 > T0), Af becomes negative and the reaction will move in the reverse direction toward a new equilibrium (j1 < j0). The opposite will happen when the temperature is reduced (T2 < T0): j2 > j0. Similarly, it can be derived quantitatively how equilibria will respond to pressure changes and to addition or removal of species. For details on this approach, we refer the interested reader to literature references given at the end of this chapter. Limits of equilibrium thermodynamics. What are the limits to which equilibrium thermodynamics may be used to treat nonequilibrium transformations in real life? In the equilibrium state there are no unbalanced potentials=driving forces. The focus is on changes that begin and end at equilibrium states and the system is considered uniform throughout, deﬁned macroscopically by quantities such as T, P, V. To consider transformations between different equilibrium states, additional constraints are imposed, or removed. We can, for instance, change the constraints temperature and pressure, or add a chemical to a system. There are speciﬁc expressions that relate the effect of temperature and pressure to the equilibrium constant. This is discussed in more detail in Chapter 5. When a constraint is removed from or added to a system that is at equilibrium, the system will then relax to a new equilibrium state. Reversing such a process would require performing work on the system to reimpose the removed constraint. Work done on a system irreversibly leaves the universe in a higher entropy state. The usefulness of reversible thermodynamics is that it gives information for open systems whether or not a process can occur. Diagrams such as Figures 3.15 and 3.16 allow one to predict the value of j or ar at a particular composition for the reaction, and hence to predict the direction of the reaction, but not the rate at which this occurs. In other words, we do not know anything about dj=dt. Time is not a variable in

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thermodynamics of reversible processes as equilibrium is a time-independent state. Equilibrium positions will change as a function of temperature and pressure. Reversible thermodynamics is about equilibria; the equations are invariant with respect to time. However, entropy has a time element in it: for real processes, entropy increases, so it is in fact a function of time. It is impossible to bring a system subject to an irreversible process backwards in time to its original state without changing anything. The time dependence of entropy generation is made explicit in irreversible thermodynamics.

3.3.15

Nonequilibrium or Irreversible Thermodynamics

A stated repeatedly, real processes are irreversible, proceeding toward an equilibrium state, dissipating the driving force. As a result, such systems are nonhomogeneous: at least some of the intensive parameters are functions of time and position. As mentioned in Section 3.3.14, entropy production is an essential feature of irreversible processes, so we need to take a look at this phenomenon in somewhat more detail. Entropy generation. It is, of course, always possible to bring a system back to its initial state after a process, whether it is reversible or irreversible, but in the case of a reversible process no trace is left whereas for irreversible processes the surroundings do usually some work on the system and will not return to their original state. Obviously, reversible processes do not occur in reality, rather they are idealizations of the real processes. Reversible processes can be seen as theoretical limits for the corresponding irreversible processes. If reversible processes would be possible, they would deliver the most work in the case of work-producing devices (think of engines and turbines) and consume the least work for work-consuming devices (such as pumps, fans, compressors). What makes real processes irreversible? This is due to such factors as friction, mixing of ﬂuids, and heat transfer over ﬁnite temperature differences, electrical resistance, inelastic deformation, and also chemical reactions; these factors are sometimes called irreversibilities. The magnitude of generated entropy depends on the process, it is not a property of the system. As a reminder, the ﬁrst law makes no difference between heat transfer and work; they are considered as equals. The second law, however, does make a distinction: an energy interaction accompanied by entropy transfer is heat transfer, and an energy interaction not accompanied by entropy transfer is work. Mass contains entropy as well as energy and so, energy and entropy contents of a system are proportional to mass. These considerations are very general, and they do of course also apply to the speciﬁc branch of chemical thermodynamics that deals with systems whose chemical composition changes during a process, which is why we are interested in it in the framework of this book. If in the course of a reaction free energy is released, this is either lost as entropy or used as work. In living organisms, for instance, this free energy is used to perform work, e.g., by transport of ions, muscle contraction, etc. However, irreversibilities do occur and not all free energy will be harnessed into useful work. The entropy of a system can change because of two reasons: an internal entropy change diS from within the system (e.g., due to a chemical reaction, or a phase change, etc.) and an external entropy change across the boundary from the surroundings to the system (or vice versa) deS due to heat transfer: deS ¼ Dq=T, hence dSsys ¼ di S þ de S

(3:138)

We can also write for the entropy change of the system dSsys ¼ dSuniv dSsur

(3:139)

By combining the previous two equations we ﬁnd di S þ de S ¼ dSuniv dSsur

(3:140)

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The external entropy change for the system must mirror that of the surroundings de S ¼ dSsur

(3:141)

di S dSsur ¼ dSuniv dSsur

(3:142)

di S ¼ dSuniv

(3:143)

Hence, Equation 3.140 can be written as

We arrive at the ﬁnding that

In other words, the internal entropy change in the system equals the entropy change of the universe. For a reversible process, of course, diS ¼ 0 but for an irreversible process diS > 0. In other words, entropy generation is a measure of the magnitudes of the irreversibilities present during that process. Entropy transfer from surroundings to system and vice versa can happen by two mechanisms: heat transfer and mass ﬂow. In contrast, there is no entropy transfer associated with energy transfer due to work. Figure 3.20 shows all this schematically; it is in fact a more detailed picture of Figure 3.2. Entropy production is thus as a measure for dissipation taking place in a process: how much of the energy is used for work, how much is dissipated to the environment. An irreversible process occurs in response to a thermodynamic force (driving force) which results from the system’s nonequilibrium state. Hence, in a nonequilibrium situation something must ﬂow, i.e., change its spatial distribution. Fluxes tell something about the speed of the process. As we have seen in Section 3.3.14, generalized forces could be differences in chemical potential, difference in temperature, difference in electrical potential, etc., and these are all very relevant for irreversible processes. Examples of generalized ﬂuxes are chemical reactions moving toward equilibrium, heat ﬂow, diffusion, electrical current, etc. The rate of entropy production < (J K1 s1) is deﬁned as > k2, Equation 4.94 reduces effectively to a consecutive irreversible ﬁrst- and second-order reaction and it follows from Equation 4.94 that kobs k1

(4:95)

When however k3[C] k2, it follows from Equation 4.93 that d[P] k1 k3 [A] [C] dt k2

(4:96)

k1 k3 ¼ Kc k3 k2

(4:97)

and kobs

with Kc being the equilibrium constant for the reversible reaction A ! B. If, again, [C] is made large, and thereby effectively constant, the reaction becomes ﬁrst order in [A]. The observed rate constant is in this case seen to be determined by all three elementary rate constants, even though the actual ratelimiting step is the second reaction to form P. Only when the ﬁrst of several sequential steps is rate-controlling, is the overall rate determined by the smallest elementary rate constant. Otherwise, rate constants of all steps prior to and including the rate-limiting one are included in the observed rate constant. The dependence of the overall rate on [C] also shows that the rate-controlling step may depend on the concentration of reactants. As stated above, steady-state approximations are not necessary anymore. It is also possible to ﬁnd out whether or not a particular step is rate-controlling via computer simulation using the parametric sensitivity function as a quantitative measure. The parametric sensitivity function is the partial derivative of the overall rate r with respect to a rate constant ð@[email protected] Þkj . The parameter that has the highest sensitivity can be identiﬁed as the rate-controlling step. In a simple, irreversible consecutive reaction, the rate-controlling step is, as intuitively expected, the one with the smallest rate constant: this step will determine the overall rate. So, for instance for the following consecutive reaction, where the overall rate refers to the formation of end product D:

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Chemical Reaction Kinetics

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18

Parametric sensitivity for [D]

Concentration (arbitrary units)

1 [A]

0.8

[B] 0.6

0.4

[D]

0.2 [C] 60

20 40 Time (arbitrary units)

(A)

d[D]/dk2

14 12 10 8 6 4

d[D]/dk3

d[D]/dk1

2 0 0.00

0 0

16

20.00

(B)

40.00

60.00

Time (arbitrary units)

FIGURE 4.20 Graph of the reaction shown in Equation 4.98 for k1 ¼ 0.1, k2 ¼ 0.02, k3 ¼ 0.5. [A]0 ¼ 1, [B]0 ¼ [C]0 ¼ [D]0 ¼ 0 (arbitrary units) (A) and the respective parametric sensitivity functions (B). k1

k3

k2

A ! B ! C ! D

(4:98)

the course of the reaction is simulated in Figure 4.20A for k1 ¼ 0.1, k2 ¼ 0.02, k3 ¼ 0.5, [A]0 ¼ 1, [B]0 ¼ [C]0 ¼ 0 (arbitrary units). The sensitivity functions are shown in Figure 4.20B and indeed k2, having the lowest value, shows the highest parametric sensitivity for the formation of end product D. However, when the rate constants are of comparable magnitude, a single rate-controlling step cannot be identiﬁed as shown in Figure 4.21A and B, for k1 ¼ 0.05, k2 ¼ 0.06, k3 ¼ 0.07. For slightly more complex reactions, the observed rate is not always solely determined by the smallest elementary rate constant. For instance, if a reversible step is introduced in a consecutive reaction: k3

k1

A ! B ! C k

(4:99)

1

6

0.8

5 [A]

Parametric sensitivity

Concentration (arbitrary units)

2

[D] 0.6 0.4

[B] [C]

0.2

d[A]/dk2

4 3

d[A]/d k3

2 1

0

0 0

(A)

d[A]/d k1

20

40

60

Time (arbitrary units)

80

100

0 (B)

20

40 60 Time (arbitrary units)

80

100

FIGURE 4.21 Graph of the reaction shown in Equation 4.98 for k1 ¼ 0.05, k2 ¼ 0.06, k3 ¼ 0.07, [A]0 ¼ 1, [B]0 ¼ [C]0 ¼ [D]0 ¼ 0 (arbitrary units) (A) and the respective parametric sensitivity functions (B).

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1

8

0.8

6

Parametric sensitivity for (C)

Concentration (arbitrary units)

d[C]/d k1

[A]

0.6

[C]

0.4

0.2 [B]

(A)

20

d[C]/dk3

2 0

0

20

40

60

80

100

−2 d[C]/dk2 −4

0 0

4

40

60

80

100

Time (arbitrary units)

(B)

Time (arbitrary units)

FIGURE 4.22 Graph of the reaction shown in Equation 4.88 for k1 ¼ 0.05, k2 ¼ 0.01, k3 ¼ 0.1, [A]0 ¼ 1, [B]0 ¼ [C]0 ¼ 0 (arbitrary units) (A) and the respective parametric sensitivity functions (B).

it now depends on the relative magnitude of each rate constant which step becomes rate determining. For the case of k1 ¼ 0.05, k2 ¼ 0.01, k3 ¼ 0.1 for instance, the parametric sensitivity is highest for parameter k1, even though k2 is smaller, and the formation of C is also sensitive to the other parameters (see Figure 4.22). In other words, there does not appear to be a clear rate-controlling step. Just another example is the case of reversible parallel reactions, discussed in Equation 4.29, and Figure 4.4B. Here it can be seen that the rate-controlling step can change during the course of a reaction (Figure 4.23): at ﬁrst the decrease in [A] depends strongly on k2, later on much more on k4. In conclusion, this analysis shows that the concept of a rate-controlling step is not an easy one, and, most importantly, cannot be predicted beforehand. The availability of modern simulation software and the possibility to do parametric sensitivity analysis seems to be helpful.

5

d[A]/d k4

Parametric sensitivity

4

3

2 d[A]/dk1,3 1 d[A]/dk2 0

0

1000

2000 3000 Time (arbitrary units)

4000

FIGURE 4.23 Parametric sensitivity for the reaction depicted in Equation 4.29 with k1 ¼ 1, k2 ¼ 0.01, k3 ¼ 0.1, and k4 ¼ 0.0005, [A]0 ¼ 1, [B]0 ¼ [C]0 ¼ 0 (arbitrary units).

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Catalysis

The IUPAC description of the process of catalysis is as follows: ‘‘A catalyst is a substance that increases the rate of a reaction without modifying the overall standard Gibbs energy change of the reaction. The process is called catalysis and a reaction in which a catalyst is involved is known as a catalyzed reaction.’’ Hence, the position of the equilibrium is not affected by the catalyst. The mechanism by which a catalyst operates is to effectively reduce the activation free energy for a reaction (Chapter 5) by providing an alternative pathway to the ‘‘normal,’’ uncatalyzed one. Consequently, the reaction proceeds much faster at the same temperature. A catalyst does not undergo a chemical change itself, or if it does, it can reversibly return to its original state, and, as already mentioned, it does not change the position of the equilibrium (DG remains the same). One can distinguish between homogeneous catalysis (in which the reactants and the catalyst are in the same phase, it concerns protons, ions, atoms, molecules) and heterogeneous catalysis (in which the reactants and the catalyst are in different phases, it concerns solid surfaces, interfaces of emulsion droplets, membranes). Both types of catalysis can occur in foods. Very important catalysts are of course enzymes. The behavior of enzymes can be so different from ‘‘normal’’ catalysts that they are discussed in a separate chapter (Chapter 9); their action is very important for food quality.

4.4.1

General Catalysis

Suppose we have a monomolecular uncatalyzed reaction k1

A ! B

(4:100)

which can also be catalyzed by component C to give the same product via an alternative, catalyzed route: kc

A þ C ! B þ C

(4:101)

in which kc is the second-order catalytic constant. The rate of disappearance of A is now: d[A] ¼ k1 [A] kc [A] [C] ¼ (k1 kc [C]) [A] ¼ kobs [A] dt

(4:102)

with kobs the observed rate constant, which is constant if [C] is constant. It thus means that the catalyzed and uncatalyzed reactions take place in the proportion kc[C]=k1 and that the relative contributions of both routes can be changed by changing [C]. A completely different type is heterogeneous catalysis where the transformation takes place at surfaces. This is very common in the chemical industry, but for foods this may be less relevant because catalysts are not normally added (except of course enzymes). One exception is the hardening of fats in which hydrogen is added to unsaturated fatty acids with the aid of a solid catalyst. It can also happen with reactions at, for instance, precipitated milk salts in heat exchangers, in which case, for instance, the isomerization of lactose is promoted. With heterogeneous catalysis several processes take place: transport of reactant molecules to the surface, adsorption of reactant molecules at the surface, reaction at the surface, desorption of products from the surface, and transport of the products away from the surface. Models that describe such catalysis are mostly of an empirical nature, such as the Langmuir–Hinshelwood–Hougen–Watson (LHHW) relation for a reaction in the gas phase A þ B ! C catalyzed at the surface of a catalyst: r¼

kKA PA KB PB 1 þ KA PA þ KB PB þ KC PC

(4:103)

with PA,B,C the partial pressures of components A, B, C, and KA,B,C their adsorption coefﬁcients.

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4.4.2

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Acid–Base Catalysis

Many reactions are catalyzed by acids and=or bases. The classical example is the hydrolysis of esters. Acid and base catalysis can take place in two ways, namely speciﬁc acid (base) catalysis and general acid (base) catalysis. The acid (proton donor) and bases (proton acceptor) are characterized as: þ HA þ H2 O ! H3 O þ A

(4:104)

A þ H2 O ! HA þ OH

(4:105)

The practical equilibrium constants for ionization are respectively: KHA ¼

[H3 Oþ ] [A ] [HA] [H2 O]

(4:106)

KA ¼

[HA] [OH ] [A ] [H2 O]

(4:107)

Taking the concentration of water as essentially constant (in fact, we may take the water activity to be unity), we can deﬁne an apparent equilibrium constant as follows: Ka ¼ KHA [H2 O] ¼

[H3 Oþ ] [A ] [HA]

(4:108)

Kb ¼ KA [H2 O] ¼

[HA] [OH ] [A ]

(4:109)

These two equilibrium constants are linked via the ionization constant of water Kw: Kw ¼ Ka Kb ¼ [H3 Oþ ] [OH ]

(4:110)

Now, suppose we have a very simple reaction of a reactant turning into a product (R ! P) and we want to consider what happens if this reaction is catalyzed by acids and bases. The uncatalyzed reaction is k0

R ! P

(4:111)

The acid-catalyzed reaction is kHþ

R þ H3 Oþ ! P þ H3 Oþ

(4:112)

The base-catalyzed reaction is kOH

R þ OH ! P þ OH

(4:113)

If the reaction is also catalyzed by the undissociated acid, we have: kHA

R þ HA ! P þ HA

(4:114)

and likewise for the reaction with the conjugate base: kA

R þ A ! P þ A

(4:115)

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Based on the above equations, the following rate equations can be derived: r0 ¼ k0 [R]

(4:116)

rHþ ¼ kHþ [H3 Oþ ] [R]

(4:117)

rOH ¼ kOH [OH ] [R]

(4:118)

rHA ¼ kHA [HA] [R]

(4:119)

rA ¼ kA [A ] [R]

(4:120)

The overall rate of the reaction R ! P is then the sum of all these contributions because these reactions run in parallel, cf. Equation 4.87: r ¼ r0 þ rHþ þ rOH þ rHA þ rA ¼ (k0 þ kHþ [H3 Oþ ] þ kOH [OH ] þ kHA [HA] þ kA [A ]) [R] ¼ kc [R]

(4:121)

The catalytic rate constant kc is deﬁned in the above equation as the sum of all separate rate constant contributions. Now we can consider several possibilities concerning general acid–base and speciﬁc acid– base catalysis. First of all, we can consider the case of general acid–base catalysis. This means that the rate is not affected by Hþ or OH ions so that kHþ and kOH can be neglected, as well as k0. Hence, the main contribution to the catalytic rate constant is kc ¼ kHA [HA] þ kA [A ] ¼ kHA [HA] þ kA

Ka [HA] ¼ [Hþ ]

kHA þ kA

Ka [HA] [Hþ ]

(4:122)

So, the result is that the rate constant is proportional to the undissociated acid concentration [HA]. Ka is known, and [Hþ] can be set so that kHA and kA can be calculated from various [HA] and pH experiments. For the case of speciﬁc acid catalysis, the rate is proportional to [Hþ], and now kHA, kA, k0, and kOH can be neglected. Consequently, the catalytic rate constant becomes: kc ¼ kHþ [Hþ ]

(4:123)

Similarly for speciﬁc base catalysis: kc ¼ kOH [OH ] ¼ kOH

Kw [Hþ ]

(4:124)

So, a plot of log kc versus pH yields a straight line with slope 1 and intercept log kþ H for speciﬁc acid catalysis and a slope of þ1 and intercept log(Kw k OH ) for speciﬁc base catalysis. Such a plot is called a pH–rate proﬁle. A slope of about 0 shows that the rate is independent of pH. Figure 4.24 gives a schematic example. Many variations are possible, i.e., speciﬁc acid catalysis at low pH, no effect of pH at higher pH and vice versa, and of course, also slopes that are different from 1 are found. In any case, plots such as the ones in Figure 4.24 can be the starting point for further mechanistic research. A food-related example of speciﬁc acid catalysis is the hydrolysis of sucrose, and an example of speciﬁc base catalysis the isomerization of sugars at alkaline pH or near neutral pH as occurs concurrently in the Maillard reaction. Figure 4.25 gives yet another example of speciﬁc acid catalysis for the

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0 −1

log k

−2 −3 −4 −5 −6 −7 0

FIGURE 4.24

2

4

6 pH

8

10

12

Schematic example of a pH–rate proﬁle.

hydrolysis of fructo-oligomers, which is straightforward. Figure 4.26 gives a rather intricate example of a pH–rate plot of the demethylation of aspartame, an artiﬁcial sweetener. Aspartame can lose methanol via hydrolysis to yield the dipeptide L-a-aspartyl-L-phenylalanine, or it can yield 3-carboxymethyl-6benzyl-2,5-diketopiperazine via intramolecular aminolysis. At low pH, the decomposition is typically speciﬁc acid catalysis leading to L-a-aspartyl-L-phenylalanine; the species that reacts is positively charged because of the protonated amino group. At high pH (which is actually unrealistic for foods) it is a typical speciﬁc base-catalyzed reaction, leading again to L-a-aspartyl-L-phenylalanine; in this case the reactive species is negatively charged due to the dissociated carboxyl group. At intermediate pH 6–10, the main reaction product is diketopiperazine and the effect of increasing [OH] is to increase the concentration of the negatively charged species of aspartame. When this has reached its maximum concentration at about pH 8–9, there is no longer an effect of pH on this reaction and the

0

log (rate)

−1

−2

Slope = −1

−3

−4

−5 1

2

3 pH

4

5

FIGURE 4.25 Example of a reaction showing speciﬁc acid catalysis: acid hydrolysis of a fructo-oligomer. Dataset in Appendix 4.1, Table A.4.9.

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0 −1 Specific base-catalysis: slope = 1

−2

log k

−3 −4 −5

Specific acid-catalysis: slope = −1

−6 −7 −8 0

5

10 pH

FIGURE 4.26 pH–rate proﬁle of the demethylation of aspartame as a function of pH. Speciﬁc acid catalysis of demethylation of aspartame leading to L-a-aspartyl-L-phenylalanine (^), speciﬁc base catalysis of formation of L-a-aspartyl-L-phenylalanine (&), formation of 3-carboxymethyl-6-benzyl-2,5-diketopiperazine (~). Dataset in Appendix 4.1, Table A.4.10.

speciﬁc base-catalyzed reaction takes over. The aspartame example shows that one has to be careful with the interpretation of pH–rate proﬁles. It requires mechanistic insight in what is actually happening before one can draw conclusions about catalytic effects.

4.5

Kinetics of Radical Reactions

Radical reactions in foods are of importance, especially in relation to autoxidation of fats. Such reactions lead to quality loss, ﬁrst of all because of the formation of undesired ﬂavor compounds (oxidative rancidity), and second because of possible formation of hazardous compounds. Radical reactions are also of importance in the body because reactive oxygen species that are formed in all kinds of biochemical reactions may cause damage to DNA, proteins, and cell membranes. Antioxidants in foods could be considered as health-promoting compounds, because they may slow down radical reactions in the body if and when they are absorbed in the body. However, there is currently much debate in the literature whether or not antioxidants in foods can be considered as health promoting. Antioxidants are added also to foods to prevent autoxidation. In any case, radical reactions are of importance and so we spend some attention to the kinetics of radical reactions. However, we limit ourselves to the basics because radical reactions are very complex and it would take too much space to discuss all intricacies. Readers who are interested in more details are referred to some selected references suggested at the end of this chapter. A radical is an atom or a group of atoms possessing one or more unpaired electrons, sometimes also called a free radical (the word free seems to be superﬂuous). A radical is usually indicated with a dot, e.g., H or OH. Radicals can be formed from so-called homolytic scissions of covalent bonds: A ! B þC

(4:125)

as opposed to heterolytic scissions that result in ions: A ! Bþ þ C

(4:126)

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Homolytic scissions are more likely to occur in the absence of solvent. Radicals are unstable and they recombine with each other or attack other groups. In general, the kinetics of radical reactions are quite intricate, and it is not well possible to give a general scheme. In most cases, derivations are made based on steady-state assumptions. This could result in a scheme such as ki

A! B þC

Initiation: Propagation:

kp

A þ B ! B þ D k0p

D ! B þE kt

k0t

C þC ! C C Overall reaction:

(4:128) (4:129)

B þB ! B B

Termination:

(4:127)

k

A! B þ E þ B B þ C C

(4:130) (4:131) (4:132)

Suppose we are interested in the kinetics of formation of compound B: d[B] ¼ kp [A] [B] dt

(4:133)

It is necessary to ﬁnd an alternative equation for this because we cannot easily determine the concentrations of radicals. If we apply the steady-state approximation, which implies that we assume that d[radicals]=dt ¼ 0: d[B] ¼ ki [A] kp [A] [B] k0p [D] 2kt [B]2 0 dt

(4:134)

d[D] ¼ kp [B] [A] k0p [D] 0 dt

(4:135)

It then follows that k0p [D] ¼ kp [B] [A] and substituting this in Equation 4.134 results in: ki [A] kp [A] [B] þ kp [B] [A] 2kt [B]2 ¼ ki [A] 2kt [B]2 ¼ 0

(4:136)

Hence: [B] ¼

ki 2kt

1=2 [A]1=2

(4:137)

We now have an expression for [B] which we can substitute in Equation 4.133: 1=2 d[B] ki [A]3=2 ¼ k[A]3=2 ¼ kp 2kt dt

(4:138)

A kinetic equation having a fractional order is a typical result for radical reactions. Instead of using the steady-state approximation, one can also resort to numerical solutions of the differential equations that describe each step in the proposed reaction mechanism.

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Lipid peroxidation. For foods, lipid peroxidation is most important; it is an oxidative degradation of lipids such as unsaturated fatty acids, sterols, carotenoids, phospholipids, etc. It is called autoxidation because the reaction takes place with molecular oxidation via a self-catalytic mechanism. The classical scheme for lipid peroxidation is divided in three phases, namely initiation, propagation, and termination. A special role is played by oxygen. Normal oxygen is in the so-called triplet state (3 O2 ), with two unpaired electrons in the 2pp orbital and this is not a very reactive oxygen species. However, oxygen can also exist in the singlet state (1 O2 ), which has two paired electrons in the 2pp orbital at one atom and none at the other atom, and this is a very reactive, electrophilic reagent. Singlet oxygen can be formed in various reactions, such as electromagnetic radiation, photodecomposition (see Section 4.6), or irradiation with g-rays, enzymatic reactions involving such enzymes as lipoxygenase, peroxidase, xanthine-oxidase, and due to catalytically acting metals (especially iron and copper). If RH represents a lipid, and X a radical, this results in the following scheme: Initiation:

3

O2 ! 1O2

(formation of singlet oxygen)

RH þ 1 O2 ! ROOH (formation of hydroperoxides) ROOH ! ROO þH Propagation:

(formation of peroxyl radicals)

ROO þRH ! ROOH þ R

R þ 3 O2 ! ROO

Termination:

2ROO ! ROOR þ O2

R þROO ! ROOR R þR ! RR

(4:140) (4:141) (4:142) (4:143)

RH þ 3 O2 ! ROOH

Overall reaction:

(4:139)

(formation of peroxides)

(formation of peroxides)

(formation of peroxides)

(4:144) (4:145) (4:146) (4:147)

The products formed in the termination step are nonradicals. Once radicals are formed, also triplet oxygen may react in the propagation step (Equation 4.143). The hydroperoxides formed (ROOH) are very unstable and decompose into aldehydes and ketones, which are the end products and the cause of rancid ﬂavor. Figure 4.27 shows a general, schematic proﬁle for the course of lipid oxidation. There are many empirical kinetic expressions available in literature describing the course of oxidation as a function of oxygen concentration, the type of lipids (the amount of unsaturated bonds and their location are particularly important), presence of metals, temperature, and notably water activity.

Concentration

Oxygen consumption

Peroxides formation End products formation

Time

FIGURE 4.27 Schematic picture showing the course of lipid autoxidation.

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Lipid oxidation is much faster at low water activity. The empirical expressions differ in the assumptions made to come to a kinetic expression. It is not well possible to list all derived expressions here. The interested reader is referred to references cited at the end of this chapter. Antioxidants. Antioxidants are very important in foods as they can protect fat autoxidation to some extent. Antioxidants are present naturally in foods, such as tocopherols, vitamin C and vitamin E, polyphenols, carotenoids; also synthetic antioxidants are sometimes added, such as butylated hydroxytoluene (BHT). Primary antioxidants interfere directly in the radical reactions as radical scavengers, secondary antioxidants sequester trace metals or quench singlet oxygen, so that the initiation phase is inhibited. Antioxidants are obviously reducing agents, i.e., they donate electrons, and they act in combination with an oxidizing agent. Denoting an antioxidant as AH, a simple scheme for a primary antioxidant would be: R þAH ! RH þ A

(4:148)

A þR ! AR

(4:149)

The radical A that is formed out of the antioxidant is considered less unstable because the unpaired electron is stabilized because of electron resonance in the resulting molecule structure. Figure 4.28 shows the effect of an antioxidant (a-tocopherol) on b-carotene oxidation. Once the antioxidant is consumed, autoxidation can still proceed, which is the reason that the stability of carotene shown in Figure 4.28 depends on the concentration of a-tocopherol. To indicate the complexity of the reactions taking place, 17 differential equations formed the basis for a kinetic model that was able to ﬁt the data shown in Figure 4.28 (the model is not shown here). In fact, this type of kinetic analyses is well suited for the multiresponse technique (Chapter 8).

0.8

[β-carotene] (mol L–1)

0.6

0.4

0.2

0 0

20

40

60

80

100

Time (h)

FIGURE 4.28 Oxidation of b-carotene at an oxygen composition of 40 mol% at 608C, without a-tocopherol (~), with 3.8 mmol L1 (&) and 7.5 mmol L1 a-tocopherol (). Dataset in Appendix 4.1, Table A.4.11.

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Besides in lipid oxidation, radical reactions occur also in the Maillard reaction, during high temperature treatments such as roasting, and during photochemical reactions. Also reactions associated with the heme group in the blood protein hemoglobin are radical reactions, of importance for meat products.

4.6

Kinetics of Photochemical Reactions

As such, photochemical reactions are of utmost importance because that is the way sunlight is captured in photosynthesis of plants. In relation to foods, photochemical reactions are less desired because some components in foods are light sensitive. For instance, beer develops a stale ﬂavor when it is exposed to light, which is the reason why beer is bottled in dark brown or green bottles. Riboﬂavin (vitamin B2) in milk is degraded when exposed to light, so it is not wise to put a bottle of milk in the sunlight. The riboﬂavin decomposes into two biologically inactive products (loss of nutritional value) but in addition it leads to ﬂavor defects. These reactions can be classiﬁed as photochemical. Photochemical reactions can also lead to formation of radicals, which can be the initiation of lipid peroxidation. A photochemical reaction starts with the absorption of a photon hn by a reactant molecule; this is then called an excited molecule. The excited molecule can loose its excitation by collision with other molecules or the wall of a system, or it may turn into a product. This is summarized in the scheme: k1

A þ hv ! A* k2

A* ! A k3

A* ! P

(4:150) (4:151) (4:152)

In photochemistry one uses the concept of primary quantum yield fq, which is the number of reactant molecules producing speciﬁed primary products, and the concept of overall quantum yield Fq, which is the number of reactant molecules that react for each photon absorbed. We can write the following rate law for the rate of disappearance of A:

d[A] ¼ Iabs dt

(4:153)

Iabs is the rate at which photons are absorbed divided by the volume in which absorption occurs. A quantitative expression for the amount of moles of photons absorbed is n¼

Pw Dt (hP cl =l)NAV

(4:154)

In this equation, Pw is the power of the light source in watt, Dt the time over which absorption takes place, hP Planck’s constant, cl the speed of light, l the wavelength, and NAV Avogadro’s number. Using Lambert’s Beer law and denoting the molar absorption coefﬁcient as «M, l the length over which absorption occurs, and the incoming light intensity as I0, Equation 4.153 can be rearranged into:

d[A] ¼ I0 [1 exp («M [A]l)] dt

(4:155)

When the light absorption is high, the rate is proportional to I0 and does not depend on [A], hence it is zero order. If the light absorption is low, the rate is proportional to [A] and the equation turns into a ﬁrstorder reaction. If we include the reactions depicted in Equations 4.150 through 4.152 and assume that [A*] is in steady state, it follows that:

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2

2

1.5

1.5

log (c/c0)

log (c/c0)

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1

1

0.5

0.5

0

0 0

(A)

FIGURE 4.29 Table A.4.12.

0.5

1 Time (days)

1.5

2

0 (B)

1

2 Time (days)

3

4

Photodecomposition of riboﬂavin in water (A) and in milk serum (B). Dataset in Appendix 4.1,

d[A] k2 ¼ Iabs ¼ Fq Iabs k2 þ k3 dt

(4:156)

An example of photochemical degradation is that of riboﬂavin in milk. Figure 4.29 shows some results, which conform to a ﬁrst-order reaction. A typical phenomenon is that in milk serum a two-phase decomposition is observed while there is a one-phase decomposition of riboﬂavin in water. This appears to be linked to the generation of reactive oxygen species in the case of milk serum, which initiates another decomposition reaction of riboﬂavin but at another rate. Another example is the photodecomposition of aspartame, as shown in Figure 4.30. In this case, a zero-order reaction is observed. Photosensitization is the phenomenon that a molecule that does not absorb photons itself can be stimulated by another absorbing molecule. The energy is then transferred by the excited molecule. An example of this is the photosensitization of ascorbic acid by riboﬂavin, in which probably also singlet oxygen is involved. Figure 4.31 shows an example of the magnitude of the effect. It is clear that such phenomena are not desirable for food quality and should therefore be minimized.

4.7

Diffusion-Limited Reactions in Aqueous Solutions

As remarked before, up until now we have assumed that solutions are ideally mixed and that the rate with which reacting species meet each other is not limiting. We will now investigate the conditions when this is not the case. If the observed rate depends on the rate at which solutions mix, this is called macroscopic diffusion control. If the rate is inﬂuenced by the rate of molecular diffusion this is called microscopic diffusion control. We consider this latter condition in more detail here: the rate is then limited by the encounter frequency of reactant molecules. If two species are going to react, they will have to come together ﬁrst. The situation may arise that if the reaction itself is very fast that the actual rate is determined by the encounter frequency and not by the chemical or physical reaction itself. We take reactions in solution as the starting point for discussion. Therefore, solvent effects are important to consider. Solvent effects are classiﬁed as ‘‘physical’’ when the solvent molecules cause the elementary reactions between reactants to be different from what they would have been in the gas phase. The most

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1.3

[Aspartame] (1⫻10−3 M)

1.2

1.1

1

0.9

0.8

0.7

0.6 0

2

4 6 Time (days)

8

10

FIGURE 4.30 Photodecomposition of aspartame in a phosphate buffer, pH 7, 258C, as a function of different light intensities. No light (^), 1100 lx (&), 3300 lx (~), 5500 lx (). Dataset in Appendix 4.1, Table A.4.13.

1.2

[Ascorbic acid] (M ⫻ 10−4)

1

0.8

0.6

0.4

0.2

0 0

5

10

15

Time (min)

FIGURE 4.31 Photosensitization of 1.2 3 104 M ascorbic acid as a function of riboﬂavin content: 0 ppm (^), 1.2 ppm (&), 2.4 ppm (~), 3.6 ppm (3), 6.0 ppm (). Dataset in Appendix 4.1, Table A.4.14.

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important physical effect of solvents is that they can induce ionization (if the solvent is polar enough, certainly true for water), and another effect is the rapid energy transfer of the abundant collisions between reactant and solvent molecules. The solvent molecules are of course also subject to diffusion. A third physical effect is due to the dielectric character of the solvent, which has a large inﬂuence on interactions between ions (Chapter 6). Furthermore, if reactants and products are surrounded by large numbers of solvent molecules, they are more or less trapped; this is called the ‘‘cage effect.’’ Solvent effects are classiﬁed as ‘‘chemical’’ if the solvent molecules participate in the mechanism. These can be catalytic effects, or the solvent molecules can act as reactant or product, which means that solvent molecules can be permanently consumed or generated. An example of this is the hydrolysis of a disaccharide or an ester, in which the solvent water acts also as a reagent. Bimolecular reactions require that molecules will have to come very close before they can interact. Encounters may result because of diffusion and ﬂow (though ﬂow becomes only important in this respect when particles are of colloidal size [Chapter 11]). Here we consider mainly molecular mobility. There are three types of molecular motions: vibrational, rotational, and translational. Vibration concerns changes in size and shape of the molecule by stretching, bending, and rotation of bonds; these are intramolecular effects. Rotational motion is about spinning of the molecule about its axes in three-dimensional space. Translational motion is about the change in location in three-dimensional space. We are mainly concerned with translational motion, which is a random walk characterized by a mean square distance x2, where x is the net distance traveled by the molecule in a time span Dt in a given direction. Einstein derived the following relation: x2 ¼ 2Df Dt

(4:157)

Df is the diffusion coefﬁcient, for which the following relation holds: Df ¼

kB T fs

(4:158)

fs is the friction factor. For spheres, fs was derived by Stokes, and combination of this friction factor with Equation 4.158 results in the Stokes–Einstein equation: Df ¼

kB T 6phv RP

(4:159)

RP is the hydrodynamic radius of the molecule, hv the viscosity. Df is called also the self-diffusion coefﬁcient. For the sake of completion, we also mention the expression for rotational diffusion, the Debye–Stokes– Einstein relation: Df ,r ¼

kB T 8phv R3P

(4:160)

Note that Df,r has units of s1, not m2 s1. Translational motion exists in the gas phase and liquid phase but hardly at all in the solid phase. Based on the Smoluchowski theory for Brownian motion in a viscous liquid, the encounter rate per unit volume and time due to diffusion in solution, rdif, is rdif ¼ 4p(DA þ DB )(RA þ RB )NA NB

(4:161)

This is derived for the case that the particles react immediately upon the encounter. In this equation DA, DB represent the diffusion coefﬁcients, RA and RB the particle radii, and NA and NB the number of

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particles=molecules A, B per unit volume, respectively. For two spherical particles with RA ¼ RB ¼ RP, DA ¼ DB ¼ Df, and assuming that the hydrodynamic radius and the collision radius are the same, it follows that the rate constant for bimolecular encounters, kdif is kdif ¼ 4p(DA þ DB )(RA þ RB )

(4:162a)

After some rearrangement to convert the units of kdif to that of the usual dm3 mol1 s1 the equation becomes: kdif ¼ 16pDf RP 103 NAV (dm3 mol1 s1 )

(4:162b)

in which NAV is the Avogadro number. Combination of this equation with the Stokes–Einstein relation yields an expression for kdif: kdif ¼

8 103 RT (dm3 mol1 s1 ) 3hv

(4:163)

hv is the viscosity of the solution as ‘‘sensed’’ by the diffusing particles (i.e., not necessarily equal to the bulk viscosity). The size of the molecules appears to have no effect as shown in Equation 4.163, because larger molecules move more slowly but are a larger target for collision and as it happens, these two effects compensate each other. If particles would react immediately upon an encounter, it means that the rate of reaction is controlled by diffusion; such reactions are called diffusion controlled. If we take Equation 4.163 as the measure for the fastest bimolecular reaction possible, it is found that for hv ¼ 1 mPa s (viscosity of water at 208C) kdif ¼ 6.6 3 109 dm3 mol1 s1 and at 1008C kdif ¼ 3 3 1010 dm3 mol1 s1. These should be roughly the upper limits for bimolecular reaction rate constants in aqueous solutions at the temperature indicated. The effect of temperature on the encounter rate is incorporated via the effect of temperature on the viscosity of the solvent. A drawback of Equation 4.163 is that it is valid for initially well-mixed systems, and it is quite likely that highly reactive systems are not well mixed initially. If, in addition, the actual chemical reaction takes time, a rate constant with a lower value will be found. For monomolecular dissociation in solution, the rate is determined by the rate at which the products can diffuse away. For uncharged reactants it follows that kdif ¼

kB T phv R3P

(4:164)

Therefore, the upper limit for monomolecular reaction rate constants (uncharged species) would be roughly 1012 s1. When a chemical reaction occurs it is obvious that ﬁrst the reactants must meet before they can react. The overall reaction rate is thus the resultant of a reaction in series and we can apply Equation 4.86. It is of interest to compare the chemical reaction rate to the diffusion rate to see when the chemical reaction is rate limiting and when diffusion becomes rate limiting. Following Equation 4.86: 1 1 1 ¼ þ koverall kchem kdif

(4:165)

If we take a value for kdif, e.g., kdif ¼ 7 3 109 dm3 mol1 s1, we can plot koverall as a function of kchem. This plot is shown in Figure 4.32. It shows clearly that when kchem is 1012 the reaction rate is completely dominated by diffusion.

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7.3E + 09

kdif = 7 ⫻ 109

6.4E + 09

koverall

5.5E + 09 4.6E + 09 3.7E + 09 2.8E + 09 1.9E + 09 1.0E + 09 1.0E + 08 8.00

9.00

10.00 log kchem

11.00

12.00

FIGURE 4.32 Overall reaction rate constant as a function of chemical reaction rate constant, assuming a diffusion rate constant kdif ¼ 7 3 109 dm3 mol1 s1.

When 109 < kchem < 1012, the overall rate constant is determined by both processes. This is then called partial microscopic diffusion control. Since in most reactions in solutions kchem < 109 dm3 mol1 s1, it appears that the rate of encounters is then not limiting the overall rate. Notable exceptions are radical reactions and photochemical reactions where diffusion may become rate-limiting indeed. When diffusion is hindered, however, the picture may become different. This can be the case particularly for foods. We will discuss this further in Chapter 14. When molecules encounter each other they need not react immediately. This can be due to the activation energy barrier (Chapter 5), but also to a geometrical constraint. It may be that the molecules did not approach each other in the right way for a reaction to be possible, and they may need to realign. This is where rotational diffusion can be important. In order to compensate for this delay, it is sometimes proposed to introduce a factor that takes this into account. Although this effect will no doubt play a role at the molecular level, it is currently not clear how to apply this concept to reactions in foods where many other complications also lead to a reduction in rate. Therefore, we will not discuss this in further detail. The rates of elementary reactions are seldom measured directly, since the experimental rate measurements, and the effects of concentration changes etc., on it, reﬂect usually the overall reaction. This will certainly be true for foods with its many components, reaction cascades and complicated structure. We will pay more attention to these phenomena in Chapter 14.

4.8

Kinetics in Open Systems

So far, we have assumed that reactions take place in closed systems, that is to say, without mass transfer to and from the surroundings. While this may be valid for many cases, there are also examples where this is not the case. It is especially important for processing when foods are treated in continuous processes such as in heat exchangers. The topic is vast and we will only show the basic principles as to how the kinetics change when mass transfer comes into play. The reader who is interested in more details is referred to some literature references given at the end of this chapter.

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A reactor allowing mass transfer was already introduced in Figure 4.1, in which the average residence time t is t¼

V w

(4:166)

Suppose next that we are able to introduce a certain amount of a component in the inlet of the CSTR as a pulse. Since the system is considered ideally mixed, the concentration is instantaneously equal in the system at t ¼ 0. We can consider then how the concentration changes at the outlet of the system, supposing that the component is not subject to a reaction, so rA ¼ 0 in Equation 4.1. The following mass balance should hold: win cin wout cout V

dcreactor ¼0 dt

(4:167)

Because the reactor is ideally mixed, the exit stream has the same composition as that in the reactor, i.e., creactor ¼ cout, hence: dcout ¼0 dt

(4:168)

dcout w 1 ¼ dt ¼ dt cin cout V t

(4:169)

win cin wout cout V This can be rearranged to:

The boundary conditions are that at t ¼ 0 cout ¼ c0 (because the reactor is considered ideally mixed) and at t > 0: cin ¼ 0. This leads to the following expression: t cout ¼ c0 exp t

(4:170)

Figure 4.33 shows the change in concentration in this situation. It basically shows how a compound is washed out of a reactor due to mass transfer in the case that there is no further chemical change occurring. The next question is what happens if the compound is subject to a chemical reaction. We need to know, of course, the kinetics of the reaction in order to calculate the change quantitatively. If we suppose a ﬁrst-order decay reaction in A, assuming certain values for the parameters of interest, the equation becomes: t 1 ¼ cA,in exp exp (kt) ¼ cA,in exp t þ k t t

cA,out

(4:171)

The change in the amount of component A is shown in Figure 4.34 for the conditions assumed, as well as the concentration of component B formed out of A. Both component A and B are washed out of the reactor. This result should be compared to the rate of change of A in a closed system as depicted in Figure 4.5, in which component B accumulates. Equation 4.171 shows that the chemical reaction will dominate the disappearance of the compound if k >> 1=t, while the physical removal due to mass transport will dominate if k 1=t. That gives the opportunity to direct the desired change via the ﬂow rate w and the volume of the reactor V.

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1 0.9

Introduction of a certain amount of component A at t = 0

0.8

f in

fout

cout (mol dm−3)

0.7 rA = 0

[A]

0.6 0.5 0.4 0.3

Amount of component A at time t

0.2 0.1 0

0

5

10 t (s)

15

20

FIGURE 4.33 Change in concentration at the outlet of a CSTR when a compound A is added as a pulse at t ¼ 0 and when the compound is not subject to reaction. V ¼ 10 dm3, w ¼ 2 dm3 s1, [A]0 ¼ 1 mol dm3.

Now, consider the situation that an amount of A is constantly supplied at the entrance of the CSTR. Then the concentration of A at the outlet will be building up and eventually reaches the concentration at the inlet if no reaction in the reactor takes place: this is the point where a steady state is reached. The equation describing this situation is t cout ¼ cin cin exp t

(4:172)

1 0.9 Introduction of a certain amount of component A at t = 0

0.7 A→B rA = − d[A] = k[A] dt

fout

cout (mol dm−3)

fin

0.8

0.6 0.5

[A]

0.4 0.3 0.2

Amount of component A and B at time t

[B]

0.1 0 0

5

10 t (s)

15

20

FIGURE 4.34 Schematic representation of a CSTR into which an amount of component A is introduced and for which a ﬁrst-order decomposition reaction is assumed. V ¼ 10 dm3, w ¼ 2 dm3 s1, [A]0 ¼ 1 mol dm3, k ¼ 0.1 s1.

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1 0.9 0.8

cout (mol dm−3)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10 t (s)

15

20

FIGURE 4.35 Pre-steady-state concentration change at the outlet of a CSTR reactor into which a continuous input of A is given from t ¼ 0 onwards, without a chemical reaction taking place. V ¼ 10 dm3, w ¼ 2 dm3 s1, [A]0 ¼ 1 mol dm3.

A graphical representation is given in Figure 4.35. When a compound is also subject to a reaction in the reactor, the concentration changes because of mass transfer and the reaction. For a ﬁrst-order decay reaction in a CSTR, the situation is described by the following equation: t k exp cout ¼ cin cin exp t t

(4:173)

Assuming certain values for the parameters, this equation is depicted in Figure 4.36. The result displayed in Figure 4.36 shows that eventually a steady-state situation is reached (which is not the same as equilibrium!). This situation is distinctly different from that in Figure 4.33 in which the concentration of A and B becomes zero eventually. A different type of open system is a plug ﬂow reactor. It is then assumed that material ﬂows through, say, a pipe such that the residence time is the same for each element in the material and it is assumed that there is no mixing, i.e., no axial dispersion (see Figure 4.37). If a reaction takes place inside the reactor, a concentration gradient will develop in the direction of the ﬂow. We can set up a mass balance over an inﬁnite small slice perpendicular to the direction of the ﬂow having a volume dV. At the beginning of the slice the concentration of a component A is cA, at the end it will have changed to cA þ dcA. At steady state, when there is no change in the amount of component A, the following equation should hold (see Equation 4.1): 0 ¼ wcA w(cA þ dcA ) þ rA dV

(4:174)

This equation can be transformed to: dcA dV ¼ rA w

(4:175)

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1 0.9 0.8

cout (mol dm−3)

0.7 0.6

[A]

0.5 0.4 [B]

0.3 0.2 0.1 0 0

5

10 t (s)

15

20

FIGURE 4.36 Pre-steady-state concentration changes at the outlet of a CSTR reactor into which a continuous input of A is given from t ¼ 0 onwards and in which a ﬁrst-order reaction A ! B takes place. V ¼ 10 dm3, w ¼ 2 dm3 s1, [A]0 ¼ 1 mol dm3, k ¼ 0.1 s1.

Integration over the whole reactor leads to: cA,out ð

cA,in

dcA ¼ rA

ðV 0

dV w

(4:176)

dcA rA

(4:177)

Or alternatively: V t¼ ¼ w

cA

cA,out ð

cA,in

cA + dcA

f , cA,in

f , cA,out

x=0 t=0

FIGURE 4.37

Schematic drawing of a plug ﬂow reactor.

x=L t=τ

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The kinetics in a plug ﬂow reactor are actually the same as in a batch reactor, except that the reaction time is ﬁxed at the residence time t. This becomes apparent if we integrate the expression in Equation 4.175 for, e.g., a zero-order reaction, for which rA ¼ k: cA,out ð

t¼ cA,in

dcA 1 ¼ (cA,out cA,in ) k k

(4:178)

! cA,out ¼ cA,in kt The same exercise for a ﬁrst-order reaction (rA ¼ kcA) yields: cA,out ð

t¼ cA,in

dcA 1 ¼ ln (cA,out cA,in ) kcA k

(4:179)

! ln cA,out ¼ ln cA,in kt These equations are thus exactly the same as for a batch reactor, given earlier in this chapter. Figure 4.38 shows the change in a compound subject to a ﬁrst-order reaction in a plug ﬂow reactor. If one knows the kinetics of a reaction and the initial concentration, one can calculate the outcoming concentration straightforwardly using the equations given in this chapter. Of course, one can vary the residence time by changing the volume V and the ﬂow rate w. The residence time for a plug ﬂow reactor is also given by Equation 4.166. Although this is also an idealized situation, it can be approximated quite closely if the ﬂow inside the reactor is turbulent. In real life, however, there is a distribution of residence times that,

cA + dcA

cA

f , cA,out

f, cA,in 1 0.9

cout (mol dm-3)

0.8 0.7

[A]

0.6 0.5 0.4 0.3 0.2 0.1 0

0

x=0 t=0

5

10 t/s

15

20 x=L t=τ

FIGURE 4.38 Change in the concentration of a compound A in a plug ﬂow reactor subject to a ﬁrst-order reaction.

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incidentally, can be determined experimentally by introducing a tracer at the entrance of the reactor and following its concentration at the exit. It is instructive to study the performance of the various reactor types after reaching steady-state conditions. The general kinetic equation for a CSTR at stationary conditions is t CSTR ¼

cA,out cA,in rA

(4:180)

The general kinetic equation for plug ﬂow is cA,out ð

t PF ¼ cA,in

dcA rA

(4:181)

This last equation is also valid for a batch reactor, as shown above. For a zero-order reaction (rA ¼ k) in a CSTR, it follows that under the condition that tCSTR < cA,in=k: cA,out kt CSTR ¼1 cA,in cA,in

(4:182)

(for the condition that tCSTR cA,in=k, cA,out=cA,in ¼ 0). For a ﬁrst-order reaction (r ¼ kcA) in a CSTR, it follows from Equation 4.177 that: cA,out 1 ¼ cA,in 1 þ kt CSTR

(4:183)

For a zero-order reaction in a plug ﬂow reactor the equation is cA,out kt PF ¼1 cA,in cA,in

(4:184)

Note that this equation is similar to the one for CSTR. And for a ﬁrst-order reaction in a plug ﬂow reactor it follows: cA,out ¼ exp (kt PF ) cA,in

(4:185)

Since Equations 4.182 and 4.184 are the same it follows that the type of reactor does not matter for a zero-order reaction. This is not so for a ﬁrst-order reaction, as shown in Figure 4.39. A plug ﬂow reactor gives a higher degree of conversion than a CSTR for equivalent residence times t, provided that we deal with ﬁrst-order kinetics. One can of course exploit this phenomenon. It makes immediately clear that if one wants to reach a high conversion rate, that a plug ﬂow reactor is much more efﬁcient than a CSTR. This is especially relevant for the killing of microorganisms (assuming for the moment that this can be described by a ﬁrst-order reaction), which would be very inefﬁcient in a CSTR. Furthermore, one can design a reactor and its operating conditions (thereby effectively setting the residence time t) in such a

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1

cA,out/cA,in

0.8

0.6 CSTR 0.4

0.2

Plug flow reactor 0 0

FIGURE 4.39 reactor.

5

10

15 t (s)

20

25

30

Comparison of a ﬁrst-order reaction having the same rate constant in a CSTR and a plug ﬂow

way that the desired effect is maximum. It is perhaps worth mentioning that a series of CSTRs is going to approach the performance of a plug ﬂow reactor (for the same equivalent residence time). There is much more to be said about kinetics in open systems. Here we only showed some basic principles to show the differences. The reader who is interested in more details is referred to some selected literature references at the end of this chapter. It is especially important to be aware of the phenomenon of residence time distribution. As a ﬁnal remark we would like to point out that a very important difference between open and closed systems is that in closed systems thermodynamic equilibrium is the time-invariant condition, while for open, continuous systems the steady state is the time-invariant condition, and the reader should appreciate the differences between these two conditions as they are very relevant for foods and food processing.

4.9

Concluding Remarks

In this chapter kinetic models have been introduced and a connection has been made with thermodynamics. The treatment has been kept very general, but at the same time limited to cases that are of interest for foods. Therefore, we have illustrated the material as much as possible with relevant examples from food science. It has been discussed how rates can be expressed and it has been shown when reaction rates may become diffusion controlled. For reactions in solutions this does not seem to be the case very often, except perhaps for radical and photochemical reactions. Furthermore, the basis has been shown how to tackle kinetics in the case of transport of material in addition to a chemical reaction. This chapter forms the basis for further discussion of kinetic models throughout the book. In the next chapter, we will continue the discussion by focusing on the effects of temperature and pressure.

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Appendix 4.1

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Datasets Used for Examples in This Chapter

TABLE A.4.1 Degradation of 1-Methyladenosine in Heated Milk (Figure 4.11) Concentration (mmol L1)

Heating Time (s) at 1358C 0 2

0.41 0.38

12

0.36

31.7

0.35

45

0.29

60.1

0.32

70

0.26

128

0.24

156

0.15 Initial Rate (mmol L1 s1) at 1358C

Initial Concentration (mmol L1) 0.41

0.011041

0.53

0.014196

0.64

0.01735

0.71

0.019243

1.18

0.031861

1.19

0.032177

2.35 5.92

0.063407 0.159306

Source: From Schlimme E., Ott F.G., and Kiesner C. Reaction kinetics of the heat-induced formation of N6-methyladenosine in milk. Int Dairy J 4:617–627, 1994. Note: The author would like to thank Prof. Schlimme for providing the data.

TABLE A.4.2 Nonenzymatic Browning of Whey Powder during Storage (Figure 4.12) Optical Density Exp. 1, 258C

Optical Density Exp. 2, 258C

0

1.8

1.9

30

4.3

4.1

60

6.3

6.1

90 120

7.4 9.6

7.6 9.8

150

12.0

11.8

180

12.5

12.7

210

14.5

14.8

Time (days)

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TABLE A.4.2 (continued) Nonenzymatic Browning of Whey Powder during Storage (Figure 4.12) Optical Density Exp. 1, 358C

Optical Density Exp. 2, 358C

0

1.7

1.6

10

5.2

5.0

20

7.9

7.9

30

10.6

10.5

40

13.7

13.8

50 60

16.3 20.2

16.5 20.1

70

23.2

23.4

95

27.8

27.7

Optical Density Exp. 1, 458C

Optical Density Exp. 2, 458C

0

1.7

1.7

2

5.2

5.2

4

7.1

7.0

7 11

22.4 25.2

22.4 25.3

Time (days)

Time (days)

18

31.7

31.7

28

44.4

44.2

35

50.7

50.9

Source: From Labuza Th.P. Reaction kinetics and accelerated tests. Simulation as a function of temperature. In: Computer Aided Techniques in Food Technology, Vol. I, Saguy (ed.). New York: Marcel Dekker, 1983, pp. 71–115.

TABLE A.4.3 Degradation of Betanin in Aqueous Solution at 758C (Figure 4.13) Time (min)

Betanin mg L1

0 10

4.29 3.65

20

3.04

25

2.83

35

2.30

40

2.20

50

1.83

60

1.49

90 100

0.96 0.70

Source: From Saguy I., Kopelman I.J., and Mizrahi S. Thermal kinetic degradation of betanin and betalamic acid. J Agric Food Chem 26:360–362, 1978.

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TABLE A.4.4 Lysine Loss in Milk Heated at 1608C (Figure 4.14) Time (s)

Lysine (mg L1 in Milk)

0

2.93

50

2.62

100

2.40

200

2.12

350

1.98

600

1.54

1000

1.01

1500 2000

0.84 0.62

Source: From Horak F.P. Über die Reaktionskinetik der Sporenabtötung und chemischer Veränderungen bei der thermischen Haltbarmachung von Milch zur Optimierung von Erhitzungsverfahren, PhD thesis. Technical University of Munich, Germany, 1980.

TABLE A.4.5 Reduction of Hexacyanoferrate (III) ([B]) by Ascorbic Acid ([A]) at Ionic Strength of 0.0384 M (Figure 4.15) Time

ln(A=B)

0 1.02

3.26967

1.87

3.33875

3.05

3.43085

3.96 4.98

3.54598 3.61506

5.99

3.73019

7.12

3.82229

8.03

3.89137

9.1

3.98347

10.06

4.05255

11.19

4.14465

12.15 13.22

4.25978 4.35189

14.08

4.39794

15.2

4.51307

Source: From Watkins K.W. and Olson J.A. Ionic strength effect on the rate of reduction of hexacyanoferrate (III) by ascorbic acid. J Chem Ed 57:158–159, 1980.

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TABLE A.4.6 Heat-Induced Denaturation= Aggregation of b-Lactoglobulin at 658C (Figure 4.16) Initial Concentration (g L1)

Initial Rate

1.485

9.096e-6

4.0771

3.094e-5

7.902

8.113e-5

1.62

3.262e-4

2.407

5.0e-4

3.554

8.954e-4

4.483

1.221e-3

5.505 7.128

1.64e-3 2.342e-3

8.373

2.646e-3

Time (s)

c=c0

0

1

5,000

0.9

10,000

0.85

20,000

0.78

20,000

0.78

30,000

0.72

40,000

0.7

50,000 90,000

0.55 0.4

125,000

0.35

180,000

0.25

Source: From Roefs S.P.F.M. and de Kruif C.G. A model for the denaturation and aggregation of b-lactoglobulin. Eur J Biochem 226:883–889, 1994. Note: The author would like to thank Dr. Roefs for supplying the data.

TABLE A.4.7 Degradation of Chlorophyll A at 1158C (Figure 4.17) Time (min)

c=c0

0

1

0.5 1 1.5 2 2.5

0.71

4

0.55

5

0.27

6

0.21

7 8 9

0.11

10

0.08

Source: From Canjura F.L., Schwarz S.J., and Nunes R.V. Degradation kinetics of chlorophylls and chlorophyllides. J Food Sci 56:1639–1643, 1991.

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TABLE A.4.8 Sucrose Hydrolysis at 708C and pH 2.5 (Figure 4.18) Sucrose in g L1, exp. 1

Sucrose in g L1, exp. 2

0

0.76

0.76

20

0.72

0.71

30

0.68

0.66

50

0.61

0.59

70

0.55

0.54

85

0.49

0.50

115

0.48

0.44

140 175

0.37 0.5

0.37 0.32

215

0.27

0.26

260

0.22

0.19

320

0.14

0.12

400

0.08

0.07

535

0.03

0.04

Time (min)

Source: From Pinheiro Torres A., Oliveira R., Silva C.L.M., and Fortuna S.P. The inﬂuence of pH on the kinetics of acid hydrolysis of sucrose. J Food Process Eng 17:191–208, 1994.

TABLE A.4.9 Effect of pH on Acid Hydrolysis of a Fructo-Oligomer (Figure 4.25) pH

Log (rate) for Raftilose P95

2 2.6

1.6 2

3

2.7

3.5

3

4

3.7

4.2

4

Source: From Blecker C., Fougnies C., van Herck J.C., Chevalier J.P., and Paquot M. Kinetic study of the acid hydrolysis of various oligofructose samples. J Agric Food Chem 50:1602–1607, 2002.

TABLE A.4.10 Demethylation Kinetics of Aspartame in aqueous solution at 258C (Figure 4.26) pH

Log k1

0.28

4.96

0.59

5.31

0.91

5.67

1.33

6.15

1.82 2.25

6.60 7.0

5.82

Log k2

Log k3

5.96

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4-59 TABLE A.4.10 (continued) Demethylation Kinetics of Aspartame in aqueous solution at 258C (Figure 4.26) pH

Log k1

Log k2

Log k3

6.19

5.62

6.78

5.12

7.16 7.38

4.74 4.40 4.30

7.66 8.92

4.66

3.69

9.58

4.19

3.63

10

3.84

3.59

10.41

3.38

3.70

10.87

3.00

3.65

11.21 11.43

2.59 2.40

3.57 3.65

Source: From Skwierczynski R.D. and Connors K.A. Demethylation kinetics of aspartame and L-phenylalanine methyl ester in aqueous solution. Pharm Res 10:1174–1180, 1993.

TABLE A.4.11 Tocopherol Time (h) 0

Effect of a-Tocopherol on b-Carotene (Figure 4.28) 7.5 mmol dm3

3.8 mmol dm3

0 mmol dm3

b-Carotene (mmol dm3)

b-Carotene (mmol dm3)

b-Carotene (mmol dm3)

0.76

1

0.71

1.5

0.66

2.5

0.6

3.75

0.53

6

0.72

8 10

0.39 0.26 0.15

12

0.09

13

0.7

14 20

0.65

24

0.66

39 44 47

0.63 0.63 0.62

52

0.56

55 56

0.47 0.62

60

0.18

63 81

0.09 0.57 (continued )

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4-60 TABLE A.4.11 (continued) Tocopherol Time (h)

Effect of a-Tocopherol on b-Carotene (Figure 4.28)

7.5 mmol dm3

3.8 mmol dm3

0 mmol dm3

b-Carotene (mmol dm3)

b-Carotene (mmol dm3)

b-Carotene (mmol dm3)

90

0.39

92

0.29

94

0.18

99

0.06

Source: From Takahashi, A., Shibasaki-Kitakawa, N., and Yonemoto, T., J. Am. Oil Chem. Soc., 80, 1241, 2003.

TABLE A.4.12 Photodecomposition of Riboﬂavin in Water and Milk Serum (Figure 4.29) Time (days)

Log c=c0 in Water

0

2

0.1

1.92

0.15

1.83

0.17

1.77

0.42

1.67

0.44

1.57

0.59 0.78

1.48 1.41

0.79

1.34

0.95

1.3

0.94

1.22

1.19

1.17

1.13

1.11

1.38

1.05

1.3 1.57

0.99 0.95

1.55

0.89

Time (days)

Log c=c0 in Milk Serum

0

2

0.06

1.97

0.13

1.9

0.34

1.79

0.42

1.72

0.59

1.66

0.63 0.95

1.58 1.56

0.8

1.5

1.1

1.48

1.39

1.4

1.58

1.34

2.06

1.28

2.34

1.22

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TABLE A.4.12 (continued) Photodecomposition of Riboﬂavin in Water and Milk Serum (Figure 4.29) Time (days)

Log c=c0 in Water

2.68

1.15

3.18

1.12

3.39

1.07

3.81

1.05

3.9

0.98

Source: From Toyosaki T., Yamamoto A., and Mineshita T. Kinetics of photolysis of milk riboﬂavin. Milchwiss 43: 143–146, 1988.

TABLE A.4.13

Photodecomposition of Aspartame (Figure 4.30) 0 lx

1100 lx

Time (days)

3300 lx

5500 lx

(30.001 M)

0

1.2

1.2

1.2

1.2

2 4

1.18 1.16

1.18 1.14

1.16 1.09

1.12 1.03

6

1.15

1.13

1.02

0.9

8

1.12

1.1

0.96

0.81

10

1.09

1.04

0.92

0.73

Source: From Kim, S.K., Jung, M.Y., and Kim, S.Y., Food Chem., 59, 273, 1997.

TABLE A.4.14 Riboﬂavin

Photosensitization of Ascorbic Acid by Riboﬂavin (Figure 4.31) 0 ppm

1.2 ppm

0

1.20

1.20

1.21

1.21

1.20

3 6

1.18 1.18

1.00 0.76

0.87 0.50

0.74 0.31

0.60 0.13

Time (min)

2.4 ppm

3.6 ppm

6.0 ppm

(30.0001 M)

9

1.18

0.54

0.23

0.032

0.02

12

1.17

0.35

0.04

0.003

0

15

1.17

0.22

0.004

0

0

Source: From Jung, M.Y., Kim, S.K., and Kim, S.Y., Food Chem., 53, 397, 1995.

Bibliography and Suggested Further Reading About Kinetics Andraos J. A streamlined approach to solving simple and complex kinetic systems analytically. J Chem Ed 76:1578–1583, 1999. Baird J.K. A generalized statement of the law of mass action. J Chem Ed 76:1146–1150, 1999. Clegg R.M. Derivation of diffusion controlled chemical rate constants with the help of Einstein’s original derivation of the diffusion constant. J Chem Ed 63:571–574, 1986. Corradini M.G. and Peleg M. A model of non-isothermal degradation of nutrients, pigments and enzymes. J Sci Food Agric 84:217–226, 2004. Croce A.E. The application of the concept of extent of reaction. J Chem Ed 79:506–509, 2002.

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Cvitaš T. A new look at reaction rates. J Chem Ed 76:1574–1577, 1999. Cvitaš T. and Kallay N. A mole of chemical transformations. Ed Chem 17:166–168, 1980. Davis J.S. Practical aspects of kinetic analysis. Methods Enzymol 210:374–390, 1992. De Levie R. Stochastics, the basis of chemical dynamics. J Chem Ed 77:771–774, 2000. Diemente D. A closer look at the addition of equations and reactions. J Chem Ed 75:319–321, 1998. Dyson R., Maeder M., Puxty G., and Neuhold Y.M. Simulation of complex chemical kinetics. Inorganic Reaction Mechanisms 5:1–8, 2003. Eberhart J.G. and Levin E. A simpliﬁed integration technique for reaction rate laws of integral order in several substances. J Chem Ed 72:193–195, 1995. Gellene G.I. Application of kinetic approximations to the A Ð B ! C reaction system. J Chem Ed 72:196–199, 1995. Laidler K.J. Rate-controlling step: A necessary or useful concept? J Chem Ed 65:250–254, 1988. Le Vent S. Rate of reaction and rate equations. J Chem Ed 80:89–91, 386, 2003. Lee J.Y. The relationship between stoichiometry and kinetics revisited. J Chem Ed 78:1283–1284, 2001. Lin K.C. Understanding product optimization: Kinetic versus thermodynamic control. J Chem Ed 65:857–860, 1988. Loudon G.M. Mechanistic interpretation of pH-rate proﬁles. J Chem Ed 68:973–984, 1991. Puxty G., Maeder M., and Hungerbühler K. Tutorial on the ﬁtting of kinetics models to multivariate spectroscopic measurements with non-linear least-squares regression. Chemometr Intell Lab Sys 81:149–164, 2006. Pyun C.W. Steady-state and equilibrium approximations in chemical kinetics. J Chem Ed 48:194–196, 1971. Quisenberry K.T. and Tellinghuisen J. Textbook deﬁciencies: Ambiguities in chemical kinetics rates and rate constants. J Chem Ed 83:510–512, 2006. Ramachandran B.R. and Halpern A.M. A novel experiment in kinetics: The A Ð B ! C reaction system. J Chem Ed 74:975, 1997. Schmitz G. What is a reaction rate? J Chem Ed 82:1091–1093, 2005. Snadden B. A new perspective on kinetic and thermodynamic control of reactions. J Chem Ed 62:653–655, 1985. Stumm W. (Ed.) Aquatic Chemical Kinetics. Reaction Rates of Processes in Natural Waters. New York: Wiley-Interscience, 1990. Toby S. The relationship between stoichiometry and kinetics. J Chem Ed 77:188–190, 2000. Villota R. and Hawkes J.G. Reaction kinetics in food systems. Handbook of Food Engineering, Heldman D.R. and Lund D.B. (Eds.), New York: Marcel Dekker, 1992. Viossat V. and Ben-Aim R.I. Validity of the quasi-stationary-state approximation in the case of two successive reversible ﬁrst-order reactions. J Chem Ed 75:1165–1169, 1998. About Radical Reactions Antunes F., Salvador A., Marinho H.S., Alves R., and Pinto R.E. Lipid peroxidation in mitochondrial inner membranes. 1. An integrative kinetic model. Free Radic Biol Med 21:917–943, 1996. Brimberg U.I. and Kamal-Eldin A. On the kinetics of the autoxidation of fats: Inﬂuence of pro-oxidants, antioxidants and synergists. Eur J Lipid Sci Technol 105:83–91, 2003. Brimberg U.I. and Kamal-Eldin A. On the kinetics of the autoxidation of fats: Substrates with conjugated double bonds. Eur J Lipid Sci Technol 105:17–22, 2003. Chen B.H., Chen T.M., and Chien J.T. Kinetic model for studying the isomerization of alpha- and betacarotene during heating and illumination. J Agric Food Chem, 42:2391–2397, 1994. Ingold K.U. Kinetic and mechanistic studies of free radical reactions in the 21st century. Pure Appl Chem 69:241–243, 1997. Karel M. Kinetics of lipid oxidation. In: Physical Chemistry of Foods, Schwartzberg H.G. and Hartel R.W. (Eds.), pp. 651–668, New York: Marcel Dekker, 1992.

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Labuza T.P. Kinetics of lipid oxidation. Crit Rev Food Sci Nutr 2:355–404, 1971. Minn D.B. and Boff J.M. Chemistry and reaction of singlet oxygen in foods. Comp Rev Food Sci Food Safety 1:58–72, 2002 Takahashi A., Shibasaki-Kitakawa N., and Yonemoto T. A rigorous kinetic model for b-carotene oxidation in the presence of an antioxidant, a-tocopherol. J Am Oil Chem Soc 80:1241–1247, 2003. Yanishlieva N.V., Kamal-Eldin A., Marinova E.M., and Toneva A.G. Kinetics of antioxidant action of aand g-tocopherols in sunﬂower and soybean triacylglycerols. Eur J Lipid Sci Technol 104:262–270, 2002. About Photochemistry Hippler M. Photochemical kinetics: Reaction orders and analogies with molecular beam scattering and cavity ring-down experiments. J Chem Educ 80:1074–1077, 2003. Jung M.Y., Kim S.K., and Kim S.Y. Riboﬂavin-sensitized photooxidation of ascorbic acid: Kinetics and amino acid effects. Food Chem 53:397–403, 1995. Kim S.K., Jung M.Y., and Kim S.Y. Photodecomposition of aspartame in aqueous solutions. Food Chem 59:273–278, 1997. Logan S.R. Does a photochemical reaction have a reaction order? J Chem Educ 74:1303, 1997. Rae M. and Berberan Santos M.N. A generalized pre-equilibrium approximation in chemical and photophysical kinetics. J Chem Educ 81:436–440, 2004. Toby S. Does a photochemical reaction have a kinetic order? J Chem Educ 82:37–38, 2005. About Kinetics and Mass Transport Cutler A.H., Antal M.J., and Maitland J. Jr. A critical evaluation of the plug ﬂow idealization of tubular ﬂow reactor data. Ind Eng Chem Res 27:691–697, 1988. Haas C.N., Joffe J., Heath M.S., and Jacangelo J. Continuous ﬂow residence time distribution function characterization. J Environ Eng ASCE 123:107–114, 1997. Heppell N.J. Comparison of the residence time distributions of water and milk in an experimental UHT sterilizer. J Food Eng 4:71–84, 1985. Hill C.G. An Introduction to Chemical Engineering Kinetics and Reactor Design. New York: Wiley, 1977. Jung A. and Fryer P.J. Optimising the quality of safe food: Computational modelling of a continuous sterilisation process. Chem Eng Sci 54:717–730, 1999. Levenspiel O. Chemical Reaction Engineering, 2nd ed., New York: Wiley, 1972. Malcata F.X. Modelling of a series of continuously stirred tank reactors for thermal processing of liquid foods. Int J Food Sci Technol 26:535–546, 1991. Ramaya S.V. and Antal M.J. Evaluation of systematic error incurred in the plug ﬂow idealization of tubular ﬂow reactor data. J Energy Fuels 3:105–108, 1989. Sancho M.F. and Rao M.A. Residence time distribution in a holding tube. J Food Eng 15:1–21, 1992. Missen R.W., Mims C.A., and Saville B.A. Introduction to Chemical Reaction Engineering and Kinetics. New York: John Wiley & Sons Inc., 1999. General Textbooks Atkins P.W. Physical Chemistry, 6th edn. Oxford: Oxford University Press, 1999. Cussler E.L. and Moggridge G.D. Chemical Product Design. Cambridge, UK: Cambridge University Press, 2001. Damodaran S., Parkin K.L., and Fennema O.R. Fennema’s Food Chemistry, 4th edn. Food Science and Technology, p. 1144. Taylor & Francis, Boca Raton, FL, 2008. Gardiner W.C. Rates and Mechanisms of Chemical Reactions. New York: WA Benjamin Inc., 1969. Fennema O.R. (Ed.) Food Chemistry, 3rd edn. New York: Marcel Dekker, 1996. Hill C.G. An Introduction to Chemical Engineering Kinetics and Reactor Design. New York: John Wiley & Sons, 1977.

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Laidler K.J. Chemical Kinetics, 3rd edn. New York: Harper & Row, 1987. Marangoni A.G. Enzyme Kinetics. A Modern Approach. Hoboken, NJ: Wiley Interscience, 2003. Maskill H. The Physical Basis of Organic Chemistry. Oxford: Oxford University Press, 1985. Moore W.J. Physical Chemistry. London: Longman, 1972. Olmsted III J. and Williams G.M. Chemistry, the Molecular Science. Dubuque, IA: Wm.C. Brown, 1997. Raff L.M. Principles of Physical Chemistry. Upper Saddle River, NJ: Prentice-Hall Inc., 2001. Tinoco T., Sauer K., and Wang J.C. Physical Chemistry: Principles and Applications in Biological Sciences, 3rd edn. Englewood Cliffs, NJ: Prentice-Hall International, 1995. Sutton R., Rockett B., and Swindells P. Chemistry for the Life Sciences. London: Taylor & Francis, 2000. Voet D. and Voet J.G. Biochemistry, 2nd edn. New York: Wiley & Sons, 1995. Walstra P. Physical Chemistry of Foods. New York: Marcel Dekker, 2003. Westphal G., Buhr H., and Otto H. Reaktionskinetik in Lebensmitteln (Reaction Kinetics in Foods, in German). Berlin: Springer-Verlag, 1996.

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5 Temperature and Pressure Effects

5.1

Introduction

The effect of temperature is one of the most important effects to consider when discussing kinetics in any branch of science and food science is no exception. Many foods are heat treated, for various reasons, and the effect of heat treatment on food quality is a very important issue. Shelf life of food is also strongly affected by temperature. Therefore, knowledge on how the kinetics of reactions is affected by temperature changes is essential. An emerging technological option to preserve foods is by high pressure. Many research papers are currently published on this topic, and it is for this reason that we spend some attention to kinetics of reactions induced by high-pressure treatment. Most food scientists would tend to use Arrhenius’ law and derive an activation energy from it. Arrhenius derived his famous equation empirically, and it was later put in a theoretical perspective, especially for gas reactions, based on the collision theory, which incorporates time via molecular velocities and the number of favorably oriented high-energy collisions. Arrhenius’ equation appears to ﬁt many reactions and is therefore frequently used. Although it may be a perfectly good choice in many cases (but not in all), it seems appropriate to start the discussion at a somewhat more fundamental level by explaining relevant aspects of transition state theory, also referred to as the activated complex theory or absolute rate theory. This theory bridges the gap between thermodynamics and kinetics by postulating an equivalence between energy E and frequency of atomic motions n, making it possible to deduce rate data from energy data (using the Planck expression E ¼ hPn, with hP Planck’s constant). This book is, however, not the appropriate place to discuss the transition state theory in great detail, but the basics will be addressed. The objective of this chapter is to explain the basic principles (mainly transition state theory) behind the effects of temperature and pressure and then to discuss the implications for kinetics as studied in foods.

5.2

van’t Hoff Equation

We begin the discussion with the effect of temperature on the thermodynamic equilibrium constant Keq, deﬁned in Chapter 3. It was mentioned there already several times that temperature is important when considering the thermodynamics of chemical reactions. The effect of temperature on the equilibrium constant Keq is expressed in the so-called van’t Hoff equation. Combining Equation 3.113 with Equation 3.38 results in 5-1

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5-2

ln Keq ¼

DH DS þ RT R

(5:1)

reminding the reader that the position of equilibrium (and hence the value of Keq) depends on both the standard enthalpy and entropy changes of the reaction. The temperature dependence of Keq is found by differentiating Equation 5.1 with respect to temperature d ln Keq DH ¼ dT RT 2

(5:2a)

d ln Keq DH ¼ d(1=T) R

(5:2b)

or

These expressions are known as the van’t Hoff equation and show that the temperature dependence of the equilibrium constant is determined by the difference in enthalpy between the products and reactants in their standard states for 1 mol of reaction. Thus, in contrast to the magnitude of Keq, the change in Keq with temperature is only determined by enthalpy, not by entropy (on the assumption that DH and DS are temperature independent; this assumption is not justiﬁed for hydrophobic reactions where hydration=dehydration is of importance). A most important result is that the equilibrium constant decreases with temperature for an exothermic reaction (negative DH), while it increases for an endothermic reaction (positive DH), a manifestation of Le Chatelier’s principle, discussed in Chapter 3. As an example, Figure 5.1 shows a van’t Hoff plot for the mutarotation of lactose: a-lactose ! b-lactose The enthalpy change calculated from the slope of the plot turns out to be 1250 J=mol. The mutarotation is therefore an exothermic reaction and the equilibrium shifts toward a-lactose when the temperature increases. The enthalpy value derived is in line with the value reported in handbooks.

0.6 y = 150.3x –0.05 0.5

ln Keq

0.4 0.3 0.2 0.1 0 0.0024 0.0026 0.0028 0.0030 0.0032 0.0034 0.0036 1/T (K−1)

FIGURE 5.1 Van’t Hoff plot for the mutarotation of lactose: a-lactose ! b-lactose between 108C and 908C. The equation shown is the regression equation. Dataset in Appendix 5.1, Table A.5.1.

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Temperature and Pressure Effects

5-3

0.45 y = 400.0x–0.9

ln Keq

0.4

0.35

0.3 0.0031

0.00315

0.0032 1/T

0.00325

0.0033

0.00335

(K−1)

FIGURE 5.2 Van’t Hoff plot for the mutarotation of glucose: a-glucose ! b-glucose between 308C and 458C. The equation shown is the regression equation. Dataset in Appendix 5.1, Table A.5.2.

A similar analysis for glucose is shown in Figure 5.2. The enthalpy change calculated from the slope of the plot turns out to be 3322 J=mol. Like with lactose, it is an exothermic reaction and the equilibrium shifts toward a-glucose with increasing temperature. However, the reader is advised that enthalpies derived from van’t Hoff plots can be quite inaccurate, especially when there is only a limited number of data points as in Figure 5.2 and when the values of the equilibrium constants are not much different. Consequently, the value of 3322 J=mol derived for the standard enthalpy from the regression line in Figure 5.2 is substantially different from the value reported in Table 3.12 at 258C, namely 1150 J=mol. This is probably due to the fact that the temperature range used to construct Figure 5.2 is very limited, namely 308C–458C, hence there is very little difference in the equilibrium constants at the various temperatures, and this leads to very inaccurate estimates, as shown. Enthalpy values derived via calorimetry are to be preferred. For a reversible reaction it was shown that the forward and reverse rate equal each other at equilibrium and that Keq ¼ kf=kr, (Equation 4.14). As follows from Equation 5.2, both kf and kr should have an exponential dependence on 1=T, or one of them has and the other is independent of T. We will discuss next the temperature dependence of rate constants as derived via transition state theory and subsequently Arrhenius’ law.

5.3

Transition State Theory

The transition state theory forms a theoretical basis on which more practical equations (such as Arrhenius’ law) can be based. Reaction kinetics in solution has some important aspects that are worth discussing. Reactions in a gas are due to isolated encounters between individual molecules, but this is not possible in solution. The reason is that reactant molecules interact continuously with solvent molecules. If a reaction has taken place, the products will tend to diffuse away, but because of the surrounding solvent molecules, this will take some time, and perhaps in the meantime something can happen to the products or reactants. This is called the ‘‘cage effect.’’ A typical consequence is that encounters last longer in solution (say 1011 s) than in the gas phase (typically about 1013 s).

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5-4

Transition state theory is well suited for reactions in solutions and is not concerned with rates of encounters (like in gas reactions) but considers thermodynamic and statistical mechanics principles to predict how many combinations of reactants will be present in the so-called transition state. This is a type of high-energy state in which molecules can be present in an unstable but activated condition, in which they will undergo some molecular change to yield products. Consider the reactants A, B that are transformed into products P, Q via a transition state ABz, as follows: A þ B Ð ABz ! P þ Q

(5:3)

Figure 5.3A shows schematically how the potential energy of a pair of molecules changes with the reaction coordinate, i.e., the path along the potential energy curve. The reaction coordinate indicates the state of the molecules in the transition from reactants to products. At this point it is perhaps useful to remark that the reaction coordinate does not refer to a state of the macroscopic system, it only refers to the behavior of a pair of molecules. It therefore does not make sense to plot free energy as a function of the reaction coordinate, as is sometimes done (remember that thermodynamic parameters such as

Potential energy

AB ‡: transition state

Reactants A, B

Products P, Q (A)

Reaction coordinate

G

G AB‡

AB‡

∆G ‡

∆G ‡ A+B

A+B

P+Q ∆G > 0

∆G < 0 P+Q

(B)

(C)

FIGURE 5.3 Schematic presentation of the potential energy of reactants A, B, transition state ABz, and products P, Q along the reaction coordinate (A). Standard free energies G for various possible states of reactants, products, and intermediates, where the standard free energy difference DG between reactants and products is either negative (B), or positive (C). The activation standard free energies DGz are always positive. Note that there is no label on the x-axis in the case of the diagrams in B and C.

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free energy refer to huge ensembles of molecules); it should be potential energy. Plotting free energy would violate the property that it cannot show a maximum as the reaction progresses, as shown in Figure 3.16. In other words, the reaction coordinate should not be confused with the extent of reaction j discussed in Chapter 3. An alternative way of expressing energy proﬁles is to plot the standard free energy referring to 1 mol of the states in which molecules or complexes are in, but then without a label on the x-axis so as to avoid the suggestion that free energy is plotted as a function of reaction progress. Figure 5.3B and C is such alternative ways of expressing energy proﬁles. Note that DG can be positive: it implies that the equilibrium constant for that particular change is smaller than unity. It can of course also be negative, implying that the equilibrium constant is higher than unity. In contrast, a reaction can never have a positive DrG, or rather a positive dG=dj; if it does the reaction will run in the reverse way. This is extensively discussed before and the reader is referred to Chapter 3 for more details. The standard free energies for activation DGz are always positive, meaning that the equilibrium constant that describes the position of the equilibrium is always much smaller than unity, expressing that the number of molecules in the activated state is very low compared to the molecules in the nonactivated state. It is, however, possible that some molecules pass over an energy barrier when they have acquired sufﬁcient kinetic energy. This becomes clear, at least qualitatively, from the Maxwell–Boltzmann distribution, showing the distribution of the average kinetic energy of molecules (Figure 5.4). An important point is that the shape of this distribution becomes wider at higher temperature, so that more molecules are able to overcome an energy barrier at higher temperature. Coming back to Figure 5.3A, when molecules start to interact, their potential energy increases, and a maximum is reached in the activated, or transition, state. It decreases again when products are formed. The population of molecules in the transition state is very small as compared to the number of reactant or product molecules. It is assumed that there is some type of equilibrium between reactants and the activated complex (as shown in Equation 5.3), and also between products and activated complex (not shown in Equation 5.3). However, if the activated complex is formed from reactants it is assumed that the activated complex must move on to form products; similarly, the activated complex formed from products must turn into reactants. At complete equilibrium, the forward rate and the reverse rate are equal, as discussed in the previous chapter, and then the concentration of activated complex formed from reactants equals that of the activated complex formed from products. If we next consider the condition that the equilibrium is

Number frequency

T1

T2

Velocity

FIGURE 5.4 Schematic picture of the Maxwell–Boltzmann distribution of the number frequency of molecules having a certain velocity at two temperatures, T1 < T2.

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disturbed, e.g., by removing products, the concentration of activated complex molecules formed from products will be reduced but not the concentration of activated complex formed from reactants. Clearly, this is not an equilibrium as usual, and therefore it is called quasiequilibrium. This is one of the postulates of transition state theory and many modiﬁcations have been proposed. We will not go into these details, but it is remarkable to note that the basic tenets of the original transition state theory are still valid. The rate at which the equilibrium depicted in Equation 5.3 is established is assumed to be fast compared with the rate of conversion of ABz to P, Q, so the position of the equilibrium is not perturbed signiﬁcantly. To be sure: ABz is not an intermediate that can be isolated readily; rather, it represents the conﬁguration of molecules at the moment of collision. The interesting aspect of transition state theory is that it connects kinetics and thermodynamics. On the one hand, the quasiequilibrium between the transition state and the reactant molecules is considered as a thermodynamic equilibrium, so that we can postulate a dimensionless thermodynamic equilibrium constant Kz based upon activities (cf. Equation 3.100): [ABz ] a yz ¼ c Kz ¼ aA aB [A] [B] yA yB c c z

(5:4)

As discussed in Chapter 3, aA ¼ yA[A]=c, with yA the activity coefﬁcient for the molar scale. (A thermodynamic constant is dimensionless, as discussed in Chapter 3, so the concentrations are made dimensionless by dividing by the standard concentration c.) On the other hand, the formation of products out of activated complex is treated using the law of mass action (discussed in Chapter 4) and the rate is considered accordingly to be proportional to the concentration of activated complex r / [ABz ] ¼ kz [ABz ]

(5:5)

Thus, the formation of products is considered to be unimolecular characterized by a rate constant kz. Incidentally, one may wonder whether we should not use the activity of the activated complex rather than the concentration in Equation 5.5, but because of the postulate of the quasiequilibrium between reactants and activated complex it should be concentration indeed. As a reminder, this postulate is that all activated complex molecules, once formed out of reactants, should move to products, and therefore it is the number of molecules that is important, not their activity. Considerations based on statistical mechanics result in kz ¼

kB T hP

(5:6)

so that, by combining all this with Equation 5.4, the rate of product formation r is r¼

d[P] kB T kB T z yA yB [ABz ] ¼ K [A] [B](c )1 z ¼ kz [ABz ] ¼ y dt hP hP

(5:7)

where kB is the Boltzmann’s constant (1.3807 3 1023 J K1), hP is the Planck’s constant (6.626 3 1034 J s), and T as usual the absolute temperature (K). kz has dimension of frequency (s1). An interesting feature is the appearance of the activity coefﬁcients of reactants and that of the activated complex in the rate equation. This is very relevant for reactions in solutions, as we shall see in later chapters, much less so for reactions in the gas phase where the activity coefﬁcients are usually close to 1. In solutions this would be true only for ideal solutions, which are in practice only very diluted solutions Comparing Equation 5.7 with a ‘‘normal’’ rate equation for a bimolecular reaction, such as the one in Equation 4.50, shows that for ideal systems the rate constant kid can be expressed as

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kid ¼

kB T z 1 K (c ) hP

(5:8)

However, for nonideal solutions the relation becomes the observed bimolecular rate constant kobs kobs ¼

kB T z 1 yA yB yA yB K (c ) ¼ kid z z y y hP

(5:9)

As shown in Equation 3.113, the thermodynamic equilibrium constant relates to the standard Gibbs energy change, and consequently the enthalpy and entropy, of activation, as follows: z DGz DS DH z ¼ exp exp K ¼ exp RT R RT z

(5:10)

Combining Equation 5.8 with Equation 5.10 gives then as a general equation kid ¼

z kB T DS DH z exp (c )1Dm exp R RT hP

(5:11)

A similar equation would be found for the nonideal case by combining Equation 5.9 with Equation 5.10. The factor (c)1Dm is necessary to obtain the right units for rate constants. c is the concentration in the standard state, chosen as 1 mol dm3 as discussed in Chapter 3, and Dm is the molecularity (in this case (D)m ¼ 2, see Equation 5.3). The above derivation is based on a bimolecular reaction mechanism. For monomolecular reactions, the same reasoning can be followed, by assuming that the activated complex is formed because of frequent collisions with solvent molecules, related to the above mentioned cage effect. This is called the Lindemann postulate. Equation 5.11 has the correct units for a rate constant of any order because of the factor (c)1Dm, the concentration in the standard state to which the thermodynamic parameters are referred. Coming back again to the discussion on thermodynamics in Chapter 3, DGz should be seen as the standard Gibbs free energy change which would occur if 1 mol of reactants is completely converted into 1 mol of transition state (at the speciﬁed temperature and standard state). In other words, it is the change in the value of free energy between these two states at the extreme ends of a possible process; it does not vary with the progress of the reaction as the free energy of a reaction does (which does decrease!). Equation 5.11 is sometimes referred to as the Eyring equation, after one of the developers of the transition state theory. The importance of this equation is that it relates the effect of temperature on the reaction rate constant to fundamental terms of enthalpy and entropy changes. If, for instance, a high enthalpy of activation exists, this would make the reaction quite slow at moderate temperatures, but this may be compensated by a large activation entropy, whereby the reaction can still proceed at a measurable rate. A striking example of such a phenomenon is the unfolding of proteins, to be discussed in more detail in Chapter 10. This indeed requires a high activation enthalpy because of the high number of bonds being broken simultaneously upon unfolding but, at the same time, the entropy of the unfolded chain increases enormously. In other words, high activation enthalpies and entropies are characteristic for protein unfolding. On the other hand, bimolecular reactions usually have a negative activation entropy (entropy of the two reactants is lost because of bond rearrangements and bond formation). The energy released and needed in breaking and forming bonds results usually in a moderate activation enthalpy. Monomolecular reactions are usually characterized by a moderate activation entropy (either slightly negative or positive, depending on intramolecular changes, the exception being protein unfolding) and an activation enthalpy depending on the type of mechanism.

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5-8 −34 −34.5

ln (khp / kBT )

−35 −35.5 −36 −36.5 −37 −37.5 2.35

2.4

2.45

2.5 103/T

2.55

2.6

2.65

(K–1)

FIGURE 5.5 Eyring plot depicting the temperature dependence of deamidation reactions of caseinate solutions; results are shown for three different initial concentrations. Dataset is given in Appendix 5.1, Table A.5.3.

As it happens, most chemical reactions, though not all, will have a moderate activation entropy so that differences in rate constants between chemical reactions are for the most part determined by differences in activation enthalpy. As an example, reaction rate constants were determined for the heat-induced deamidation in aqueous solutions of caseinate in the range between 1108C and 1458C. The resulting logarithmic plot (according to the natural logarithm of Equation 5.11) is shown in Figure 5.5. Usually, the activation parameters are estimated via linear regression of the logarithmic plot as given in Figure 5.5. For statistical reasons to be discussed in Chapter 7, it is better to use nonlinear regression. An estimate of the activation enthalpy from these data via nonlinear regression is DHz ¼ 92.0 13.6 kJ mol1 and DSz ¼ 69.9 13.8 J mol1 K1 (95% conﬁdence interval). The negative activation entropy is consistent with a bimolecular reaction of hydrolysis of amides. The activation enthalpy and entropy are usually assumed to be independent of temperature, which in general is probably not true, but for the heat treatment of foods the temperature range is mostly not so large on the absolute temperature scale, so the approximation may then hold. A notable exception is, again, protein unfolding in an aqueous environment, because interaction with water comes into play. Upon unfolding, hydrophobic groups are exposed and cause increased structuring of water. There is thus also a contribution of enthalpy and entropy changes of the solvent water that may oppose the positive enthalpy and entropy for protein unfolding. The difference in heat capacity between unfolded and folded (native) proteins is quite large, resulting in temperature dependency of (activation) enthalpy and entropy. In general, when it comes to hydrophobic bonds, their temperature dependence is quite intricate. A very schematic impression is given in Figure 5.6, illustrating that hydrophobic bonds increase with temperature (i.e., free energy becomes more negative), especially between 08C and 608C, but also at higher temperature.

5.4

Arrhenius’ Law

Arrhenius’ law was derived empirically, but it has proven to be very worthwhile in chemical kinetics. Arrhenius’ law states that

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20 15

T∆S

Energy (kJ mol−1)

10 5 ∆Η 0 −5 −10

∆G

−15 −20

0

50 T ⬘(⬚C)

100

FIGURE 5.6 Highly schematic drawing of the change in free energy (DG), the change in enthalpy (DH), and the change in entropy (TDS) with temperature for hydrophobic bond formation.

Ea k ¼ A exp RT

(5:12)

where A is a so-called preexponential factor (sometimes called the frequency factor), and Ea is the activation energy. The dimension of A should be the same as that of the rate constant k; it therefore does have units of frequency only in the case of a ﬁrst-order reaction. It is very instructive to compare Arrhenius’ law, Equation 5.12, with the expression derived from transition state theory Equation 5.11. Obviously, Ea relates to the activation enthalpy DHz and the exact relationship is found as follows. From Equation 5.8 it follows that ln k ¼ ln

kB Kz þ ln T þ ln 1m hP (c )

(5:13)

hence d ln k d ln K z =(c )1m ¼ T þ d(1=T) d(1=T)

(5:14)

and combining the temperature effect on Kz as displayed in Equation 5.2 gives d ln k DH z ¼ T R d(1=T)

(5:15)

From the Arrhenius Equation 5.12 it follows that d ln k Ea ¼ R d(1=T)

(5:16)

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and consequently combining Equation 5.15 with Equation 5.16 results in DH z ¼ Ea RT

(5:17)

The observant reader may ﬁnd this result unexpected because of the deﬁnition of enthalpy given in Equation 3.17 H ¼ E þ PV

(5:18)

PDV ¼ RTDn

(5:19)

From the general gas law it follows that

with Dn the change in number of moles, so that it would follow that DH z ¼ Ea þ DnRT

(5:20)

However, Dn should be seen here as the change in number of moles in going from reactants to the activated state. Hence, for bimolecular reactions Dn ¼ 1, so that Equation 5.17 is indeed seen to be correct. For monomolecular reactions, Dn ¼ 0, so that in that case the activation energy equals the activation enthalpy. As mentioned, the activation energy can be seen as the energy barrier that molecules need to overcome in order to be able to react. As shown in Figure 5.4, the proportion of molecules able to do that increases with temperature, which qualitatively explains the effect of temperature on rates. The Eyring and Arrhenius equation give a quantitative account. The preexponential factor A is seen to be related to z kB T DS exp the activation entropy DSz A ¼ and this comparison makes the preexponential factor R R A much more comprehensible. The physical meaning of A as such seems to be experienced as somewhat vague, which probably accounts for the fact that the factor A very often is not reported as a result in food science literature. It gives, however, as much useful information as Ea does. An interpretation of A is that it represents the rate constant at which all molecules have sufﬁcient energy to react (i.e., Ea ¼ 0). In principle one could even envisage a negative activation energy, namely if molecules attract each other. This could be the case for positively and negatively charged reactants. Another difference between Arrhenius’ and Eyring’s expressions is that the temperature T appears in the preexponential factor in Eyring’s equation (Equation 5.11). This has a consequence in the way results are presented and analyzed. Very often, Arrhenius law is presented as a plot of ln k versus 1=T, which should result in a straight line (if the relationship holds). With Eyring’s relationship, ln(k=T) versus 1=T should be plotted (as is done in Figure 5.5). We would like to remark here that it is not a good idea to derive the activation energy parameters from linear regression of ln k or ln(k=T) versus 1=T because of the weighting of data points through logarithmic transformation; rather, nonlinear regression should be used, as is discussed in Chapter 7 on kinetics and statistics. Another remark in this respect is that the twostep procedure of ﬁrst deriving rate constants and then regressing them versus temperature results usually in very wide conﬁdence intervals if only three to four temperatures have been studied, as is frequently the case. A better approach is to substitute the rate constant in the appropriate rate equations using Equation 5.11 or Equation 5.12 and perform a nonlinear regression. In this way, all data are used to estimate the activation parameters at once and an estimate of these parameters of much higher precision is obtained. This is called global ﬁtting and will also be demonstrated in Chapter 7. It probably remains a good idea to present Arrhenius’ or Eyring’s expression in the form of a plot of ln k or ln(k=T) versus 1=T because any deviation of the data from these expressions becomes immediately apparent. In doing so,

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however, the values of the parameters estimated by nonlinear regression should be used to construct the plot. Discrepancies between experimental data and Arrhenius’ and Eyring’s relationship are indeed possible and it is the responsibility of the researcher to check this. In the case that they are not applicable (for instance, because an undetected change in mechanism occurs at the higher temperatures), the resulting parameter estimates are worthless. So, the ﬁrst step should always be to check the validity of the laws of Arrhenius=Eyring, and only if they appear to be correct the next step would be the estimation of the activation parameters. Obvious as this may seem, this rule is not always obeyed. Temperature dependence of complex reactions. It is essential to realize that the concept of transition state theory or Arrhenius law is strictly speaking only valid for elementary reactions. In the case of a complex reaction in which an observed rate constant is actually composed of several elementary ones the meaning of an activation energy determined from such rate constants becomes a bit blurred. Let us consider, for instance, the reaction k1

A þ B ! C k2

(5:21)

k3

C þ D ! P If this is the correct mechanism, the rate of formation of product P, d[P]=dt is d[P] ¼ k3 [C] [D] dt

(5:22)

However, if we can determine only component P, we are not in the position to estimate rate constant k3 because then we would need to determine compounds C and D as well (if we can do that we can apply multiresponse modeling to be discussed in Chapter 8). Let us suppose for the moment that we can only observe the formation of component P, and we could try to model it with a pseudo-ﬁrst-order reaction, for instance. The observed rate constant will then be a composite of the three elementary rate constants k1, k2, and k3. By postulating that the intermediate compound C is following a steady state after the initial start up of the reaction, we can investigate how the observed rate constant is related to the elementary ones. The steady state implies that d[C]=dt 0 d[C] ¼ k1 [A] [B] k2 [C] k3 [C] [D] ¼ 0 dt

(5:23)

Hence [C] ¼

k1 [A] [B] k2 þ k3 [D]

(5:24)

Substituting Equation 5.24 in Equation 5.22 yields d[P] k1 k3 [A] [B] [D] ¼ dt k2 þ k3 [D]

(5:25)

Now we can investigate some possibilities to see how that works out for the rate equation. First, let us assume that k2 k3[D]. It then follows that d[P] ¼ k1 [A] [B] dt

(5:26)

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We then have a ‘‘normal’’ second-order rate equation, and the rate of formation of P is completely determined by the rate of disappearance of components A and B. The observed rate constant will correspond to the elementary rate constant k1 and the temperature dependence should be Arrheniuslike. Another possibility could be that k2 >> k3[D]. It then follows that d[P] k1 k3 [A] [B] [D] ¼ k0 [A] [B] [D] ¼ dt k2 k1 k3 k0 ¼ k2

(5:27)

and we end up with a third-order rate equation where the rate constant k0 is clearly a composite one. The question is now how temperature affects this composite rate constant. If the three elementary rate constants k1, k2, k3 each obey Arrhenius law, we can apply Equation 5.16 to ﬁnd

d ln k0 E0 d ln (k1 k3 =k2 ) d( ln k1 þ ln k3 ln k2 ) Ea1 þ Ea3 Ea2 ¼ a¼ ¼ ¼ d(1=T) R R d(1=T) d(1=T)

(5:28)

In words, the activation energy determined from the observed rate constant is composed of the activation energies for the underlying elementary reactions. This can lead to unexpected results such as that a reaction decreases in rate with increasing temperature. A negative activation energy is not well conceivable for an elementary reaction (unless it concerns two oppositely charged reactants), but for a composite reaction this can happen if Ea2 > Ea1 þ Ea3 according to Equation 5.28. The question is, of course, when it will happen that a situation occurs as in the above example that k2 >> k3 [D]. In order to investigate this, we performed some simulations based on the mechanism depicted in Equation 5.21. The requirement Ea2 > Ea1 þ Ea3 implies that the preexponential factor must be very high to compensate for the higher activation energy (otherwise k2 will be of no signiﬁcance). Table 5.1 shows a possible condition where this would happen. The reaction was simulated via numerical integration of the differential equations representing the rate equations, using the values shown in Table 5.1 to calculate the rate constants via Arrhenius’ equation. The initial concentrations of A and B were arbitrarily set at 100 units and that of component D at 150 units. It was indeed found that the intermediate C showed steady-state behavior and with the numerical values chosen it was also true that k2 >> k3 [D]. It was then assumed that only compound P was experimentally accessible, so that the simulated experiment consisted of concentration–time proﬁles for various temperatures; Figure 5.7 shows some results. These proﬁles were ﬁtted to a ﬁrst-order rate equation; a reasonable ﬁt was obtained (but not perfect because the actual mechanism is more complex than a ﬁrst-order reaction as shown in Equation 5.21). The so-derived rate constants were then plotted according to the Arrhenius equation (see Figure 5.8). It is indeed found that the rate constants decrease with increasing temperature at the higher temperatures. Admittedly, this may be a rather extreme example because of the requirement that k2 >> k3 [D]. The purpose is, however, to show that the activation energy determined from observed reaction rate constants may not reﬂect a single energy barrier. Thus, one should be aware of this when determining activation parameters from observed rate constants that could be composed of several elementary rate constants, certainly in foods where it is difﬁcult to study elementary reactions. TABLE 5.1

Values Used in the Arrhenius Equation for Simulation of the Complex Reaction in Equation 5.21 Reaction Step 1

Reaction Step 2

Reaction Step 3

A

1 3 1010 dm3 mol1 s1

1 3 1026 s1

1 3 1010 dm3 mol1 s1

Ea

80 kJ mol1

150 kJ mol1

60 kJ mol1

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90

T = 353 K

80

T = 413 K

[P] (arbitrary units)

70

T = 333 K

60 50 40

T = 313 K

30 20 T = 293 K

10 0 0

5

10

15

20

25

Time (arbitrary units)

FIGURE 5.7 Simulated concentration proﬁles for compound P in the reaction mechanism displayed in Equation 5.21 using the numerical values shown in Table 5.1.

Reparameterization. It is possible to reparameterize the Arrhenius or Eyring equation; it is actually desired from a statistical point of view as will be discussed in Chapter 7. A very simple reparameterization is to introduce a reference temperature Tref. The basis for this arises from the application of Equation 5.12 at two temperatures T1 and T2, assuming that the preexponential factor and Ea do not depend on temperature

0 −1 −2

ln k

−3 −4 −5 −6 −7 −8 −9 0.002

0.0025

0.003 1/T

0.0035

0.004

(K −1)

FIGURE 5.8 Arrhenius plot of the ﬁrst-order rate constants found from the simulated concentration proﬁles shown in Figure 5.7 using the numerical values displayed in Table 5.1.

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Ea k1 ¼ A exp RT1 Ea k2 ¼ A exp RT2 If one arbitrarily chooses a reference temperature, say T2 ¼ Tref, one can combine these two equations: k Ea 1 1 ¼ exp R T Tref kref

(5:29)

The actual result of this is that the preexponential factor is replaced by a rate constant at some reference temperature. The reference temperature should preferably be chosen in the middle of the studied temperature regime. Magnitudes of rate constants. It is perhaps instructive to return brieﬂy to the actual values that rate constants can take. Table 5.2 shows orders of magnitude for rate constants, depending on conditions. The very large effect of activation energy on the rate of a reaction is apparent. In fact, without activation barriers, reactions would be so fast that foods would spoil immediately, and there would be no such thing as food technology, or life for that matter. It is just another way of saying how important kinetics is for processes that are studied by life sciences, including of course food science. Generally, the temperature effects as laid down in the Arrhenius and Eyring relationships seem to ﬁt quite well to chemical reactions in foods. As mentioned above, it should be realized, however, that observed rate constants are more often than not reﬂecting more than one elementary reaction, and one has to be careful with interpretation. Some typical values for activation enthalpies=energies and entropies for possible reactions in foods are given in Table 5.3. Several physical reactions are less temperature-dependent and often diffusion controlled; diffusioncontrolled reactions are discussed in Chapter 4. Transition state theory does not apply actually to physical reactions (such as coalescence, aggregation) because there are no molecular rearrangements. However, physical phenomena do usually have an energy barrier (due to, for instance, electrostatic repulsion), which provide stability to colloidal systems. Hence the concept of a kind of activation energy does apply but not with a temperature dependence as occurs for chemical reactions. The effect of temperature will be for the larger part on the rate of encounters. Sometimes, activation energies are reported for physical phenomena such as the temperature dependence of diffusion or viscosity. This seems to be impossible,

TABLE 5.2 Orders of Magnitude for Rate Constants of Bimolecular Reactions in Aqueous Solutions at 258C Order of Magnitude of Rate Constant (dm3 mol1 s1)

Conditions No diffusion limit and no barriera Diffusion limit, no activation energy

1014 1010

No diffusion limit Activation energy 25 kJ mol1 1

Activation energy 50 kJ mol

1

Activation energy 100 kJ mol a

1010 105 104

This is in fact the value of the preexponential factor in the Arrhenius equation, corresponding to the hypothetical situation that T ! 1.

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TABLE 5.3 Order of Magnitudes for Some Typical Food Reactions for Activation Enthalpy (DHz), Activation Energy (Ea), Activation Entropy (DSz), and the Preexponential Factor (A)

Type of Reaction

DHz=Ea (kJ mol1)

DSz (J mol1 K1)

A s1 (First Order) dm3 mol1 s1 (Second Order)

First-order chemical reaction

50–150

0

1013 to 1014

Second-order chemical reaction

50–150

50

1011 to 1012

200–500

500

1030 to 1040

Protein denaturation Microbial inactivation (Enzyme) catalyzed reactions

a

200–500 10–50

a

500 100 to þ50

1030 to 1040 1011 to 1012

Photochemical reactions

0–10?

?

?

Radical reactions

0–10?

?

?

Note: ? indicates not really known. a Not really an activation energy=enthalpy=entropy but a temperature coefﬁcient (see text).

since there is nothing to activate and there is no reaction. As discussed earlier, the point is that the temperature dependence of, for instance, diffusion apparently obeys Arrhenius’ law in several systems, but the parameter that comes out of it does not have the physical meaning of an activation energy.

5.5

Empirical Relations to Describe Temperature Dependence

As mentioned in the previous section, the relations derived from transition state theory, and to some extent the Arrhenius equation as well, are only valid for elementary reactions. In cases where temperature dependences need to be described for nonelementary reactions, one could use empirical relationships. There are also other Arrhenius-like equations proposed in the literature that could be used just as well, but which are not commonly used. The following equation would do equally well as the Arrhenius equation: B k ¼ A exp T

(5:30)

with A and B as ﬁt parameters without a physical meaning. Although this seems undesirable, one has to realize that sometimes parameters do not really have a physical meaning. For instance, if one determines an activation energy for microbial inactivation, what does it mean if an activation energy of, say 300 kJ mol1, has been derived? A mole of bacteria is somewhat hard to envisage. It would actually be better to use Equation 5.30 for phenomena that do show Arrhenius-like behavior but do not really reﬂect a deﬁned chemical reaction. For instance, as mentioned in the previous section, the effect of temperature on diffusivity can often be described using the Arrhenius equation, but there is no activation energy for molecular mobility (though there may be barriers), and therefore it does not make much sense to report an activation energy for diffusion; temperature coefﬁcients A and B like in Equation 5.30 seem more appropriate. Another example of an alternative for the Arrhenius equation is a log–logistic relationship k ¼ m0 ln (1 þ exp [c(T Tc )])

(5:31)

c (8C1), m0 (-), and Tc (8C1) are empirical ﬁt constants, and in many cases it can be assumed that m0 ¼ 1. Such an equation accounts for phenomena with a high rate at high temperature and a slow one at low temperature. The Arrhenius equation, of course, does the same, but an equation such as Equation 5.31 does not need the concept of activation energy. Figure 5.9 shows an example of the ﬁt of the log– logistic equation (Equation 5.31) for degradation of riboﬂavin in heated spinach.

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0.00008 0.00007 0.00006

k (s−1)

0.00005

0.00004 0.00003 0.00002 0.00001 0

0

20

40

60

80

100

120

140

T ⬘(⬚C)

FIGURE 5.9 Fit of the log–logistic equation (Equation 5.31) to rate constants describing the degradation of riboﬂavin in heated spinach. The ﬁt parameters are m0 ¼ 1, c ¼ 0.020, Tc ¼ 6038C. Dataset in Appendix 5.1, Table A.5.4.

There are also empirical models that describe temperature dependence of viscosity in foods undergoing a glass transition. The Williams–Landel–Ferry (WLF) model is often used to model such dependence. It is reported that Arrhenius=Eyring models do not function well for such cases. The WLF model is developed for polymers but nowadays also applied to foods that undergo glass transitions. The WLF model is actually not comparable to the Eyring equation, because it does not consider transition states; it is of empirical nature and attempts to model viscosity as a function of temperature. The WLF model is hv ln hv,g

! ¼

C1 (T 0 Tg0 ) C2 þ (T 0 Tg0 )

(5:32)

C1 (dimensionless) and C2 (8C) are parameters of the WLF model, Tg0 is the glass transition temperature (8C), and hv,g is the viscosity at the glass transition temperature. The WLF model will be discussed in more detail in Chapter 14.

5.6

Activation Energy and Catalysis

The important phenomenon of catalysis was discussed in Chapter 4. It was indicated that a catalyzed reaction does not differ in terms of the equilibrium position with respect to the uncatalyzed reaction (hence the change in standard Gibbs energy remains the same), but it increases the rate (which was loosely translated as the activation energy barrier being lowered). Now that we have discussed the concept of activation energy in terms of transition state theory and the Arrhenius equation, it is perhaps

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Standard Gibbs energy

Ea, uncatalyzed

Activation energy

Ea, catalyzed A ∆G

B

FIGURE 5.10 Incorrect representation of the lowering of the activation energy of a catalyzed reaction. A þ C ! B þ C (C represents catalyst).

useful to reconsider the effect of catalysis on activation energy in more detail. A general misconception is that a catalyst just lowers the activation energy barrier of the same elementary step as that of the uncatalyzed reaction as depicted in Figure 5.10. This is deﬁnitely not the case because a catalyzed reaction always consists of more than one step, namely the reaction of the catalyst with the reactants and the release of the catalyst from the activated complex (as a results of which products are formed and the catalyst is regenerated). So the situation becomes more complex, and could be something like the one depicted in Figure 5.11. A catalyzed reaction has thus different reaction proﬁles than its uncatalyzed counterpart. A catalyst not only catalyses the forward reaction but also the reverse reaction. It may be useful to come back brieﬂy to the concept of a rate-limiting step as discussed in Chapter 4. Sometimes, it is

Standard Gibbs energy

Standard Gibbs energy

A‡

AC ‡1

AC ‡2

∆G ‡ ∆G ‡1 A

∆G ‡2

A ∆G

B

Uncatalyzed reaction

B

Catalyzed reaction

FIGURE 5.11 Likely representation of the change in activation energy due to catalysis. The uncatalyzed reaction is A ! B, the catalyzed reaction A þ C ! B þ C (C represents catalyst).

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attempted to identify rate-limiting steps from the steps having the highest activation energy. This does not always lead to correct conclusions. Activation energy relates to a rate constant, not to a rate. It should be remembered that a rate is determined by the product of rate constant and concentration, and not just by the height of an activation energy barrier.

5.7

Parameters Used in Food Science

The parameters that have been discussed so far, orders, rate constants, activation parameters, etc., are actually all that is needed in (chemical) kinetics. However, it has become the habit to use several other kinetic parameters in food science. They originate from days gone by when it was necessary to derive parameters and models to describe (mainly microbial) changes in foods during processing and storage when no use was made of modern reaction kinetics. All these parameters can be related to the more fundamental parameters that we have discussed so far. We give a brief overview of these parameters so that the reader can see how they relate to the fundamental parameters discussed above. The parameter Q10 describes the temperature dependence of a reaction as the factor by which the reaction rate is changed when the temperature is increased by 108C. If we link that to the reaction rate constant, it can be expressed as Q10 ¼

kTþ5 kTþ10 kT5 kT

(5:33)

If the Arrhenius equation holds, it can be shown that 10Ea Q10 ¼ exp RT 2

(5:34)

The parameter is thus seen to be a rather poor measure of the activation energy, and a serious drawback is its quite strong temperature dependence especially at higher Ea (Figure 5.12). Another drawback of the Q10 parameter when linked to the activation energy is that it does not incorporate the preexponential factor or the activation entropy. So, it makes only sense to use the Q10 parameter for reactions that do not differ too much in activation entropy=preexponential factor. 80 70 Ea = 300 kJ mol–1 60

Q10

50 40 30 Ea = 200 kJ mol–1

20 10

Ea = 100 kJ mol–1

0 0

50

100

150

T⬘(⬚C)

FIGURE 5.12

Temperature dependence of the parameter Q10 as a function of the underlying activation energy.

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40 35 Ea = 100 kJ mol–1 30

Z (⬚C)

25 20

Ea = 200 kJ mol–1

15

Ea = 300 kJ mol–1

10 5 0 0

50

100

150

200

T ⬘(⬚C)

FIGURE 5.13 Temperature dependence of the parameter Z as a function of the underlying activation energy.

Another parameter to describe temperature dependence is Z, which expresses the increase in temperature that would produce an increase in rate by a factor of 10. Z is deﬁned as Z¼

2:303RT 2 10 ¼ Ea log Q10

(5:35)

Like the parameter Q10, Z is temperature-dependent which restricts its use (Figure 5.13). Z is frequently used in bacteriology to describe inactivation of microorganisms. Also used is the parameter D, especially in thermobacteriology. It is the decimal reduction value, the time needed to reduce a concentration by a factor of 10. D is nothing else than an inverse rate constant. For a ﬁrst-order reaction: D¼

2:303 k

(5:36)

9 c0 k

(5:37)

and for a second-order reaction: D¼

A plot of log D versus T 0 (in 8C) is usually taken to be a straight line (for a limited temperature range), see Figure 5.14. D relates to the Z value, like k is related to Ea: log D ¼ log Dref

0 T 0 Tref Z

(5:38)

0 (often chosen as 2508F for historical Dref is the reference value of D at the reference temperature Tref reasons, which is equal to 121.18C). Equation 5.38 is referred to as the TDT curve (thermal death time curve) or the Bigelow model.

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2 log D

1 0 −1 −2

Z T ⬘(⬚C)

FIGURE 5.14

Schematic example of a TDT curve and interpretation of the Z-value.

As shown, all these parameters can be linked to the more fundamental kinetic parameters. They still serve a purpose. On the one hand, they are usually estimated in real foods and as such reﬂect a time– temperature dependence characteristic (not pretending it is something like an activation energy) that can be used for engineering purposes; less so, however, for understanding behavior at the molecular level. On the other hand, the parameters are used commonly by regulatory agents in food safety programs in relation to thermal treatments. We come back to them brieﬂy in Chapter 13 when we discuss thermobacteriology. Typical temperature effects for reactions in foods. When the effect of temperature on reactions in foods has been established, preferably in the form of the parameters discussed if it concerns elementary reactions, i.e., activation energy=enthalpy and activation entropy=preexponential factor, the value of the parameters needs some discussion. Occasionally, there seems to be some misunderstanding regarding interpretation of activation parameters. For instance, if a high activation energy is found, the conclusion is sometimes drawn that the reaction will proceed slowly or difﬁcult. This is not necessarily true, because the reaction may proceed quite fast at very high temperature if the value of the preexponential factor A or, equivalently, DSz is high. Furthermore, if a high activation energy goes along with a high preexponential factor, the reaction may still proceed at a noticeable rate. The point is that a high activation energy indicates a strong temperature dependence, that is to say it will run very slowly at low temperature, but relatively fast at high temperature. Relevant for foods is that chemical reactions (e.g., the Maillard reaction) have a ‘‘normal’’ activation energy of about 100 kJ mol1, whereas inactivation of microorganisms can be characterized by a high activation energy, say, 300 kJ mol1 (even though, as already mentioned, it is incorrect to express it in this way; we will come back to this in Chapter 13). Figure 5.15 illustrates this. Such differences in activation energy are exploited in processes such as high-temperature short-time heating (HTST) and ultrahigh-temperature treatment (UHT). These processes are designed by choosing such time–temperature combinations that desired changes are achieved (microbial inactivation) while undesired changes (chemical reactions leading to quality loss) are minimized as much as possible. Another important consequence for foods is that reactions with relatively low activation energy will continue at a measurable rate at low temperatures, for instance during storage, leading to a limited shelf life. Quite different results are obtained with protein denaturation and microbial inactivation. (Microbial inactivation is according to some authors due to enzyme, i.e., protein, denaturation. It is questionable whether this is the sole cause of inactivation. We will come back to this in Chapter 13 on kinetics of microbial inactivation.) Protein denaturation is characterized by a high activation enthalpy=energy and

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ln k

Microbial inactivation

Chemical reaction

UHT region

1/T

FIGURE 5.15 Schematic presentation of the temperature dependence of a chemical reaction and microbial inactivation. The UHT region is characterized by time–temperature combinations that induce enough microbial inactivation and as little as possible chemical reactions.

this is compensated for by a high activation entropy=preexponential factor. As a result, the temperature dependence of such reactions is very high, much higher than that of chemical reactions. A potential pitfall in the study of denaturation of proteins is the way in which protein denaturation is studied, as discussed in Chapter 10. Frequently, for instance, the protein aggregation that is measured results from protein denaturation. The resulting kinetic parameters are then a combination of unfolding and aggregation. It may happen that at the lower temperatures unfolding is the rate-limiting step, while at the higher temperatures aggregation becomes the rate limiting step. As a consequence, a change in temperature dependence will be seen. It may even happen that the rate of aggregation is the rate-limiting step throughout and then the kinetics of aggregation is established rather than kinetics of denaturation. With biochemical reactions, i.e., enzyme-catalyzed reactions, moderate temperature dependence is found, as is to be expected for catalyzed reactions. However, with enzyme-catalyzed reactions, enzymes become inactivated above a certain temperature, and the catalyzed reaction comes effectively to an end. Most enzymes relevant in food tend to become inactivated between 508C and 808C, though some notably heat-resistant enzymes are known. The same goes for microbial growth: ﬁrst there is an increase with temperature but eventually microbes start to die. A highly schematic picture of the effect of temperature on microbial growth and enzyme action alike is in Figure 5.16; it should be noted that the actual response to temperature can be time-dependent. In the case of microorganisms, there is also a minimum temperature below which there is no growth. Photochemical reactions and radical reactions are not or only weakly temperature-dependent because the changes at the molecular level do hardly depend on thermal energy. Both types of reactions are of importance in foods, as discussed to some extent in Chapter 4. Photochemical reactions cause for instance oxidation of vitamins, they may activate certain enzymes, and they may cause ﬂavor defects. Radical reactions are most notable for oxidation reactions (of unsaturated fats, of vitamins).

5.8

Enthalpy=Entropy Compensation

An effect called enthalpy=entropy compensation, or isokinetic effect, is claimed to exist for some reactions in foods. The phenomenon was ﬁrst described in organic chemistry and would arise from a systematic variation in solvent composition, reactant molecule change (as in homologous series), pH, or

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Temperature optimum

Ln (rate)

Activity

Inactivation

1/T

FIGURE 5.16 Schematic presentation of the effect of temperature on growth and activity of enzymes and microorganisms and their inactivation.

dielectric strength. If DHz is plotted versus DSz in such situations, a straight line through the origin is found with a slope that has the dimension of absolute temperature, and this temperature is called the isokinetic temperature. The relationship is called enthalpy=entropy compensation. The mechanistic signiﬁcance is that at the isokinetic temperature all medium effects (such as pH and solvent effects) on the free energy change would vanish. This is so because a straight line through the origin implies that DH z ¼ Tiso DSz

(5:39)

with Tiso the isokinetic temperature and since DGz ¼ DH z TDSz

(5:40)

it follows that DGz ¼ 0 at T ¼ Tiso. Examples in food for which enthalpy=entropy compensation is claimed are protein denaturation, disaccharide hydrolysis at various pH values, ascorbic acid degradation at various water activities, degradation of chlorophylls, and chlorophyllides and carotenoids stability with change in solvent. There is, however, considerable debate in the literature on the question whether a compensation effect actually exists or merely is a statistical artefact. The problem is that when enthalpy and entropy values are measured from real data (i.e., containing experimental error) these parameters are highly correlated in a statistical sense, resulting from the propagation of errors (see also Chapter 7). The compensation effect does thus not necessarily imply a chemical causation. It has been shown that estimates of DGz and DHz do not suffer from this error correlation problem if evaluated at the harmonic mean of the range of experimental temperatures. The harmonic mean Thm is the reciprocal of the mean of the reciprocal temperatures Thm ¼

1 1 T

(5:41)

So, if there still appears to be correlation between DGz and DHz evaluated at Thm, a true chemical effect is found. We will not further go into details here, but the reader should be warned, when encountering reports on enthalpy=entropy compensation effects, that it could be a statistical artefact rather than a real chemical phenomenon, unless the researcher has worked with Thm. The problem of statistical correlation between parameters such as DHz and DSz is further discussed in Chapter 7.

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Variable Temperature Kinetics

In most of the experiments reported in literature, temperature was kept constant, in other words, these are isothermal experiments (in practice, it is difﬁcult to achieve this because heating-up and coolingdown times play a role). The temperature dependence of the rate constant is then estimated by ﬁtting the obtained rate constants at various temperatures using Equation 5.11 or Equation 5.12. However, it is also possible to vary the temperature in a controlled way, and to follow the concentration of the component of interest as a function of the varying temperature, which is in turn a function of time. Such experiments form the basis for variable temperature kinetics, also called nonisothermal kinetics. Let us take the general rate law equation (Equation 4.58) as a starting point

dc ¼ k(t)cn dt

(5:42)

We now have to write k(t) because the rate constant is a function of the varying temperature and hence a function of time. In other words, the Arrhenius equation becomes Ea k(t) ¼ A exp RT(t)

(5:43)

Integration of Equation 5.42, after some rearrangement, now results in ðc

1 dc ¼ A cn

c0

ðt

Ea dt exp RT(t)

(5:44)

0

One has to know the dependence of T on t, T(t), in order to be able to solve this equation. This dependency could be a temperature varying linearly with time T(t) ¼ T0 þ at

(5:45)

where the coefﬁcient a gives the rate of temperature change, or a quadratic change T(t) ¼ T0 þ at þ bT 2

(5:46)

or a sinusoidal change a 2pt 3p sin þ 2 L 2

(5:47)

T(t) ¼ Th þ (T0 Th ) exp (Jt)

(5:48)

T(t) ¼ Tm þ or an increase described by an exponential

In these equations T(t) represents temperature as a function of time, T0 the starting temperature, Tm the average temperature, and Th is the ﬁnal temperature. The parameter J in Equation 5.48 accounts for the heat transfer coefﬁcient and speciﬁc heat and mass of the product ﬂowing through a heat exchanger. To be sure, other equations than the ones given here are possible, as long as they describe the T-t proﬁle correctly. It is important to realize that the change in concentration is quite different for isothermal kinetics and nonisothermal kinetics. For a ﬁrst-order isothermal reaction it looks as depicted in Figure 5.17A, but for a linearly changing temperature the same ﬁrst-order reaction would look like the curve in Figure 5.17B. In the latter case, the reaction is slow at ﬁrst because the temperature is low but as the temperature rises the reaction rate increases, until the end when it decreases because the reactant becomes depleted (Figure 5.17C).

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1

0.8 0.6 0.4 0.2

0

250

C 0.8

200

T 0.6

150

0.4

100

0.2

50 0

0 0

5 10 Time (arbitrary units)

15

⬚C

Concentration (arbitrary units)

Concentration (arbitrary units)

1

(A)

21.10.2008 10:40pm Compositor Name: MSubramanian

0 (B)

5 10 Time (arbitrary units)

15

5

Rate (arbitrary units)

4 3 2 1 0 0 (C)

5 10 Time (arbitrary units)

15

FIGURE 5.17 Simulation of an isothermal ﬁrst-order reaction at 1438C (A), a nonisothermal ﬁrst-order reaction for Ea ¼ 100 kJ mol1, A ¼ 1 3 1010 s1 and a linear temperature increase from T 0 0 ¼ 1008C at a rate of 68C h1 (B), and the reaction rate at the nonisothermal conditions (C).

On the assumption that Ea and A do not depend on temperature and that one knows how temperature varies with time, one can derive the activation parameters directly from the concentration curve obtained from a nonisothermal experiment. An example of such a study is given in Figure 5.18 for sucrose hydrolysis at high temperature and pH 3.8, which is assumed to be a ﬁrst-order reaction. It is thus not really necessary to do isothermal studies in order to obtain activation parameters from the Eyring or Arrhenius equation, even though this is almost always done. It is true though that one needs numerical procedures and nonlinear regression techniques in order to obtain the desired estimates. This will be discussed in Chapter 7. Varying temperature kinetics is signiﬁcant for heat exchange processes such as pasteurization and sterilization. As an example, two heating proﬁles are depicted in Figure 5.19 for a UHT heating process. Incidentally, such a heating proﬁle can be described with Equation 5.48, typical for heating via an isothermal heat source. Usually one only accounts for the holding time, but the heating up and cooling down periods can sometimes be considerable so that reactions take place at a noticeable rate during these periods and should not be neglected. To account for this, it is convenient to introduce a so-called equivalent time teq to take heating-up and cooling down periods into account. This can be done by numerically integrating the right-hand side of

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10

140

9

T⬘

[Sucrose]

7

120

6 ⬚C

Sucrose (g dm−3)

8

5 4 100

3 2 1

80 200

0 0

50

100

150

Time (min)

FIGURE 5.18 Hydrolysis of sucrose (~) at pH 3.8 with varying temperature. The solid line indicates the ﬁt for a ﬁrst-order reaction in which the Arrhenius equation is incorporated; the dashed line indicates the temperature change. The estimates of the parameters via nonlinear regression are c0 ¼ 9.8 g dm3, A ¼ 2.9 3 1010 s1, and Ea ¼ 93.4 kJ mol1. Dataset in Appendix 5.1, Table A.5.5.

Equation 5.44, for instance by the trapezoid rule or Simpson’s rule, or using mathematical software. The numerical outcome is then equated to the same change that would have occurred in an isothermal process at a constant holding temperature Th ðt A 0

Ea Ea exp teq dt ¼ A exp RT(t) RTh

(5:49)

160 140 120

T ⬘ (⬚C)

100 80 Holding time

60 40 20 0 0

10

20

30

40

Time (s)

FIGURE 5.19 Table A.5.6.

Heating proﬁle during UHT heating up to 1208C (~) and 1408C (*). Dataset in Appendix 5.1,

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TABLE 5.4 Equivalent Times (teq), Holding Time (th), and Effective Heating Time (teff ¼ teq þ th) as a Function of Activation Energy Ea for the Heating Proﬁle and Holding Time at 1408C as Shown in Figure 5.19 Ea (kJ mol1)

teq (s)

th (s)

teff (s)

50

10.4

10

20.4

100

7.6

10

17.6

150

6.0

10

16.0

200

5.9

10

15.9

250 300

5.2 3.5

10 10

15.2 13.5

350

3.1

10

13.1

Thus, the equation can be solved for teq. As indicated by Equation 5.49 the equivalent time does neither depend on the preexponential factor nor on the order of the reaction, but it is essential to realize that the outcome does depend on the value of the activation energy. It is thus not allowed to generalize equivalent times for other reactions than the one that is studied. To illustrate this fact, equivalent times were calculated for the heating proﬁle as depicted in Figure 5.19 for 1408C (Table 5.4). As can be seen, the effective heating time is more than doubled for a low activation energy of 50 kJ mol1, reﬂecting the fact that reactions with a low activation energy can progress at a measurable rate at low temperature, i.e., at the heating-up and cooling-down periods. The effect becomes less for a higher activation energy, but cannot be neglected for a heating proﬁle such as the one in Figure 5.19. Table 5.4 demonstrates that chemical reactions, which have an activation energy between 50 and 100 kJ=mol, are more sensitive to nonisothermal treatments than reactions having a higher activation energy (such as protein denaturation). Another possible complication in heat exchangers is residence time distribution. If the ﬂow inside the exchanger is not turbulent, there may be considerable spread in the time during which elements of the heated material are subject to the heat treatment. In the case of turbulent ﬂow however, one can in most cases assume plug ﬂow, i.e., a more or less constant heating time for every part of the heated material. This is the preferred condition for heated foods, otherwise some parts of the heated foods are underheated, while other parts are possibly overheated. A serious limitation of variable temperature kinetics discussed hitherto is that the analysis is only valid for a single reaction. If other reactions start to interfere above a certain temperature, which is quite normal when working with foods, the above given analysis is no longer valid because it assumes only one reaction. A possible solution to this problem is to take these interfering reactions into account, and model them simultaneously. This is the domain of multiresponse modeling, to be discussed in Chapter 8. Still, for real foods this may be too complicated, not so much because of the fact that more reactions take place, but because these reactions may change the conditions and thus the course of the reaction. In other words, the occurrence of the next phase in the reaction may have been inﬂuenced by what has happened before. What is left then for engineering purposes is an empirical approach. Nonisothermal kinetics for empirical models. In the previous chapter, empirical models were introduced as an alternative to mechanistic (or presumed mechanistic) models. It is possible to model variable temperature situations also with these kinds of models without the use of the Arrhenius or Eyring equation, following a series of papers of Dr. Peleg and coworkers. As an example, the Weibull model was introduced as an empirical model in Equation 4.76 ct ¼ exp (bW t aW ) c0

(5:50)

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ct/c0

Slope = dc/dt T1

T2

t∗

t∗

t∗

Time

FIGURE 5.20 Schematic picture showing the relation between the time t*, momentary rate ct=c0, and isothermal rate at temperature T1 and T2.

This equation describes the change in ct=c0 as a function of time at otherwise constant conditions, such as a constant temperature. What we are looking for is an expression that describes the change in ct=c0 when the temperature is not constant but changing. As a ﬁrst step we can ﬁnd an expression for the rate of change of ct=c0 by differentiating Equation 5.50 at constant temperature

dct c0 dt

¼ bW aW t aW 1 exp (bW t aW )

(5:51)

T

By making now the parameters aW and bW temperature dependent we ﬁnd for the change in the ratio ct=c0 as a function of temperature dct ¼ bW (T(t)) aW (T(t)) t aW (T(t))1 exp (bW (T(t))t aW (T(t)) ) c0 dt

(5:52)

In order to be able to solve this differential equation for ct=c0 we need an expression for ct=c0 in the righthand side of Equation 5.52. This can be found as follows. Consider the time t* that corresponds to the momentary ratio ct=c0 which can be found from Equation 5.50 1 ct 1aW B c0 C t* ¼ @ A bW

0

ln

(5:53)

This relation is schematically depicted in Figure 5.20 and, of course, this is valid at any temperature but since bW is temperature dependent, t* will be different at a different temperature for the same ct=c0, or put differently, for the same value of t* ct=c0 will be different at different temperatures. By substituting Equation 5.53 in Equation 5.52, the latter equation becomes an ordinary differential equation in ct=c0 that can be solved, in principle, by numerical integration using appropriate software* * The software used for the calculations shown here was Athena Visual Studio v.11. See www.athenavisual.com. The calculations can also be done in a spreadsheet: http:==people.umass.edu=mgcorrad=RealTimeNutrientDegradation.xls

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700

[Ascorbic acid] (mg/100 g)

650

600

550

500

450

400

0

10

20

30 40 Time (min)

50

60

70

FIGURE 5.21 Degradation of ascorbic acid in the fruit amla, described by the Weibull model with ﬁxed parameter aW ¼ 0.5 at 508C (^), 608C (D), 708C (&), 808C (X), 908C (*), 1008C (&), 1208C (*). Dataset in Appendix 5.1, Table A.5.7.

aW (T(t)) 1 ct 1 aW (T(t)) dct ct c0 C B ¼ bW (T(t)) aW (T(t)) @ exp ln A c0 dt bW (T(t)) c0 0

ln

(5:54)

The dependence of the parameters aW and bW on temperature can be described by any ad hoc empirical model that is able to capture the relation, and the same goes for the dependence of temperature on time. The logistic model introduced in Equation 5.31 (with m0 ¼ 1) appears to perform well to describe the temperature dependence of bW, but to be sure: any other model may be used. Let us see how this approach works out with an example. Equation 5.50 was applied to the heatinduced degradation of ascorbic acid in the tropical fruit amla; the shape factor had the same value of aW ¼ 0.5 at all temperatures studied: see Figure 5.21. Consequently, parameter aW can be assumed temperature independent in this case and was ﬁxed at its value aW ¼ 0.5, leaving the parameter bW to be estimated at each temperature. Its temperature dependence was subsequently modeled according to Equation 5.31 (Figure 5.22). The parameter estimates found for the ﬁt in Figure 5.22 were used to model the fate of ascorbic acid in amla subject to three different nonisothermal proﬁles (representing three different heating methods, namely open pan cooking, pressure cooking, and a fuel-efﬁcient ‘‘ecocooker’’), following Equation 5.54 (Figure 5.23). In order to give a feel for the accuracy of the predictions the prediction bands for the model are also indicated; these prediction bands were estimated from the imprecision in the temperature dependence of parameter bW; since we are using nonlinear models here, these prediction bands are only approximate. Prediction bands are further discussed in Chapter 7. The model predictions shown in Figure 5.23 are certainly not perfect but they show the right trend; only for the open pan cooking results, the predictions are less than actually observed. Incidentally, note that these are true model predictions (as opposed to model ﬁts) because the model is based on independent isothermal measurements while the data points in Figure 5.23 are obtained from different nonisothermal measurements. The irregular shape of the concentration curves is due to the fact that the Weibullian shape factor aW had the value of 0.5 in this particular case. Another example links to Figure 5.9 where the temperature dependence for the ﬁrst-order

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0.06

0.05

bW

0.04

0.03

0.02

0.01

0

40

60

80

100 T ⬘ (⬚C)

120

140

FIGURE 5.22 Fit of the model in Equation 5.31 to the parameter bW for the data displayed in Figure 5.21. Fit parameters are k ¼ 0.017 8C1, Tc ¼ 291.2 8C (m0 ¼ 1, ﬁxed).

rate constant was shown for the heat-induced degradation of riboﬂavin in spinach; since this was a ﬁrstorder reaction, the Weibullian shape factor aW ¼ 1 and the rate constant corresponds to the parameter bW. The results of nonisothermal predictions and experiments (having the same temperature proﬁle as in Figure 5.23) are given in Figure 5.24. As in the previous example, the predictions look reasonable, though not perfect. In this case, there is a mismatch between the prediction for the ecocooker and the experiment. Similar equations can be set up for the other empirical models introduced in Chapter 4. Equation 4.73, for instance, is a hyperbolic model and Equation 4.74 a limited exponential model. Suppose we have the hyperbolic model displayed in Equation 4.73a for a formation reaction: c ¼ c0 þ

k1 t k2 þ t

(5:55)

The isothermal rate of the formation of a compound expressed as concentration c can be found by differentiating the model with respect to t dc k1 k2 ¼ dt T (k2 þ t)2

(5:56)

To model nonisothermal conditions, we can look at the state of c at time t* and determine from that the isothermal rate; t* is calculated from the model equation (Equation 5.55) t* ¼

k2 (T) [c c0 ] k1 (T) [c c0 ]

(5:57)

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Kinetic Modeling of Reactions in Foods

120

740

140

700

100

700

120

660

80

620

60

580

40

500

5

10

15

60 580

500

40 20

0

5

10 Time (min)

(B)

740

120

700

100

660

80

620

60

580

40

540

20

500

15

0 20

0 0

(C)

T ⬘(⬚C)

80 620

540

0 25

20

Time (min)

[Ascorbic acid] (mg/100 g)

0

100

660

T⬘ (⬚C)

20

540

[Ascorbic acid] (mg/100 g)

740

T ⬘(⬚C)

[Ascorbic acid] (mg/100 g)

5-30

(A)

21.10.2008 10:40pm Compositor Name: MSubramanian

5

10

15 20 25 Time (min)

30

35

FIGURE 5.23 Nonisothermal predictions for the degradation of ascorbic acid (.) in amla in an open pan (A), in a pressure cooker (B), and in an ecocooker (C). The solid lines are the model predictions for the concentration proﬁle, the dashed lines the approximate 95% prediction bands. The temperature proﬁle refers to the right y-axis. Dataset in Appendix 5.1, Table A.5.7.

Equation 5.56 can be converted to a rate at nonisothermal condition by substituting the expression for t* dc k1 (T(t)) k2 (T(t)) ¼ dt k2 (T(t)) [c c0 ] 2 k2 (T(t)) þ k1 (T(t)) [c c0 ]

(5:58)

We have now an expression for the rate as a function of changing temperature based on an isothermal model. This equation can be solved numerically for any desired temperature proﬁle to obtain the proﬁle of c if we know the dependence of the parameters k1 and k2 on temperature. Another empirical model is the limited exponential model equation (Equation 4.74) c ¼ c0 þ (c1 c0 ) [1 exp (k1 t)]

(5:59)

c1 is the asymptotic value of c when t ! 1 and k1 the ‘‘rate constant.’’ For this model, the isothermal rate is

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120

0.208 40 0.204

T ⬘(⬚C)

0.212

[Riboflavin] (mg/100 g)

0.216

80

60

(A)

5

10 15 Time (min)

20

0.212

80 60

0.208

40 20 0.200

0 25

0

10 Time (min)

(B)

15

0 20

120

0.22 0.215

[Riboflavin] (mg/100 g)

5

100

0.21 80 0.205 60 0.2

T ⬘(⬚C)

0

100

0.204

20

0.2

140 120

100

0.216 [Riboflavin] (mg/100 g)

0.220

T ⬘(⬚C)

0.22

40 0.195 20

0.19 0.185 0

10

(C)

20 Time (min)

30

0 40

FIGURE 5.24 Nonisothermal predictions for the degradation of riboﬂavin (.) in spinach in an open pan (A), in a pressure cooker (B) and in an ecocooker (C). The solid lines are the model predictions for the concentration proﬁle, the dashed lines the approximate 95% prediction bands. The temperature proﬁle refers to the right y-axis. Dataset in Appendix 5.1, Table A.5.8.

dc ¼ k1 (c1 c0 ) [ exp (k1 t)] dt T

(5:60)

and t* is t* ¼

1 c0 c1 ln c c1 k1

(5:61)

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and the nonisothermal rate equation becomes dc ¼ k1 (T)(c1 c0 ) dt

(5:62)

Again, if we know how temperature T changes with time t, as well as the dependence of k1 on T, the concentration change can be calculated for any temperature proﬁle using numerical integration. So, in conclusion, variable temperature kinetics can be done for every model, provided we know the time–temperature proﬁle and the temperature dependence of the parameters, either via Arrhenius or Eyring relations, or via empirical relations such as Equation 5.31. This is especially very useful for accelerated shelf life testing, and to study the effect of varying temperature in a food chain during distribution and storage. The same approach can be applied to variable temperature kinetics for microbial kinetics; in such cases the Arrhenius=Eyring equation does not make sense because the processes studied in microbial growth and death are not single reaction events like a chemical reaction is. This is further discussed in Chapters 12 and 13.

5.10

Effect of Pressure

High-pressure treatment of foods is an emerging technology. It implies that foods are brought under high pressure (up to 1000 MPa), and as a result bacteria can be killed and some enzymes inactivated. Because this can be done at relatively low temperature, there is little or no heat damage and the preserved foods retain their freshness. This research area is strongly in development, and here we state only some basic principles. From a thermodynamic point of view, the effect of pressure can be dealt with as follows. First, we start with considering a reaction in the gaseous phase, and we state the general reaction as in Equation 3.1 nA A þ nB B ! nP P þ nQ Q The equilibrium constant KP expressed via partial pressures is (cf. Equation 3.102) P n P P n Q P

Q

Po Po KP ¼ nA nB PA PB Po Po

(5:63)

Po is the standard state pressure of 1 bar. As a reminder, based on Equation 3.113 the following relation holds Dr G ¼ RT ln KP

(5:64)

This equation shows that KP refers to standard states at Po ¼ 1 bar, so it is by deﬁnition independent of pressure. This does not mean, however, that pressure has no effect on equilibria. This should become clear from the following reasoning. We can express the equilibrium constant also in terms of mole fractions, KX KX ¼

(XP )nP (XQ )nQ (XA )nA (XB )nB

(5:65)

For a perfect gas the following relation holds between partial pressure Pi and total pressure P: Pi P ¼ Xi o Po P

(5:66)

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Combining Equation 5.65 with Equation 5.66 gives P nP P nQ P

Q

Po Po K X ¼ n A n B PA PB o P Po

nA þnB nP nQ P Po

(5:67)

Combining Equation 5.63 with Equation 5.67 results in (see also Equation 3.104) KX ¼ KP

Dn P Po

(5:68)

Dn ¼ nP þ nQ nA nB

(5:69)

By increasing the pressure, KX must change by a factor (P=Po)Dn in order to keep KP constant. So, the result of this exercise is to show that the mole fractions of the components in the equilibrium mixture do depend on the total pressure P, despite the fact that KP does not. Just another way of expressing this is by differentiating Equation 5.68 with respect to pressure

@ ln KX @P

¼ T

@ ln KP @P

Dn T

@ ln P @P

(5:70) T

Using the perfect gas law PDV ¼ DnRT, and using the fact that KP is independent of pressure so that the ﬁrst term on the right-hand side of Equation 5.67 equals zero, Equation 5.70 can also be written as

@ ln KX @P

¼ T

Dn DV ¼ P RT

(5:71)

This equation is not only valid for equilibria in the gas phase but also for equilibria in solution. DV is the volume change accompanying 1 mol of reaction with all substances in the standard state. This is actually a statement of Le Châtelier’s principle: equilibrium phenomena that are subject to a decrease in volume upon reaction will move from left to right upon an increase in pressure and vice versa. If DV can be considered pressure independent, it can be estimated from the dependence of KX on pressure P according to Equation 5.71, analogous to the van’t Hoff equation for temperature effects (Equation 5.2). High-pressure treatment thus favors reactions that result in a volume decrease. It affects mainly noncovalent bonds, which implies that low-molecular weight compounds are not really sensitive to pressure but high-molecular weight components (biopolymers) are because they are stabilized by noncovalent bonds. Also dissociation of weak acids in water is enhanced by pressure. Consequently, pressurization of acetate, citrate, and phosphate buffers is accompanied by a large negative volume change upon dissociation and this leads to a signiﬁcant acidiﬁcation of such buffer solutions. Furthermore, crystallization phenomena are also affected by pressure. Especially hydrophobic bonds and ionic bonds are very sensitive to pressure, hydrogen bonds less so. This means that the quaternary and tertiary structure of proteins will be disrupted but not so much the secondary structure when subject to high pressure. In other words, proteins (and hence enzymes) will denature, which could also be the basis for microbial inactivation as a result of high-pressure treatment. Also, aggregation and gelation of proteins may occur as a result of pressure-induced denaturation. When applying pressure, its transmission is uniform and virtually instantaneous, independent of vessel size and geometry (the isostatic principle), unless gas is present because then the gas can be compressed. Furthermore, an increase in pressure (at constant T) results in an increase in the degree of ordering of molecules (the microscopic ordering principle).

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For this book, we are mainly interested in describing the effect of high pressure on the resulting kinetics. By analogy with the effect of temperature, the activation volume DV# (the difference in molar volume of the activated complex and the reactant) and the preexponential factor AP are introduced DV # P k ¼ AP exp RT

! (5:72)

Analogous to the Arrhenius equation, a reference rate constant at a reference pressure can be chosen to eliminate the preexponential factor DV # (P Pref ) k ¼ Kref exp R

! (5:73)

The terms DV#, DH#, DS#, DG# are related DV # @T

! ¼ P

@DS# @P

!

@DH # @DV # ¼ DV # T T @T

(5:74)

! P

Currently, most studies are done on inactivation of microorganisms via high pressure. We will come back to this in Chapter 13. Furthermore, the effect of high pressure on proteins and enzymes has been studied extensively. As an example, Figure 5.25 shows pressure-induced inactivation of an enzyme,

1

0.8

[A]/[A]0

0.6

0.4

0.2

0 0

50

100

150

Time (min)

FIGURE 5.25 Inactivation of polygalacturonase in tomato juice at 208C at 400 MPa (^) and 450 MPa (D) according to a ﬁrst-order model (solid lines). Dataset in Appendix 5.1, Table A.5.9.

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Temperature and Pressure Effects

5-35

−5

ln k

−6

−7

−8

−9 300

400

500

600

P (MPa)

FIGURE 5.26 Example of the use of (the logarithm of) Equation 5.73 applied to inactivation of polygalacturonase in tomato juice at 158C and Pref ¼ 525 MPa. The activation volume is estimated at 43.5 cm3 mol1, and kref at 0.002 s1. Dataset in Appendix 5.1, Table A.5.10.

polygalacturonase (PG) in tomato juice. Figure 5.26 shows how Equation 5.73 can be used to estimate the activation volume DV# for the same example. Since both pressure and temperature have an effect on protein conformation, it has also been studied how the two conditions combine. A mathematical model that takes this into account is the following, based on thermodynamic and kinetic considerations: DV0# DS# Da# (P Pref ) þ 0 (T Tref ) (P Pref )(T Tref ) RT RT RT DCp DB# T 2 T ln 1 þ Tref (P Pref ) þ RT 2RT Tref

ln k ¼ ln kref

(5:75)

In this equation Da# is the thermal expansivity factor (cm3 mol1 K1), DB# the compressibility factor (cm6 J1 mol1), DV0# (cm3 mol1), and DS#0 (J mol1 K1) the volume and entropy change between native and denatured states at Tref and Pref, respectively, DCp the heat capacity at constant pressure (J mol1 K1). These parameters need to be estimated from experimental data. A possible problem is the large number of parameters to be estimated and therefore a simpliﬁed version of Equation 5.75 has been proposed (by simply omitting DB# and DCp)

ln k ¼ ln kref

DV0# DS# Da# (P Pref ) þ 0 (T Tref ) (P Pref )(T Tref ) RT RT RT

(5:76)

With this model it can be predicted what the combined effects of temperature and pressure are. Figure 5.27 gives a so-called isorate contour plot, again for the example of inactivation of polygalacturonase. It shows the temperature–pressure combinations that lead to the same extent of inactivation of the enzyme.

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600

500

P (MPa)

400

300

200

100

0 273

293

313 T (K)

333

353

FIGURE 5.27 Isorate contour plot for inactivation of polygalacturonase in tomato juice (characterized by a rate constant k ¼ 0.00144 s1) as a function of temperature and pressure. Dataset in Appendix 5.1, Table A.5.11.

5.11

Concluding Remarks

This chapter has discussed the effects of temperature and pressure on kinetics, as both these conditions are of major importance for food technology. Parameters such as activation energy, enthalpy, and entropy are frequently used. One should, however, realize that these parameters pertain to elementary reactions. The concept of transition state theory and activation energy becomes somewhat blurred when applied to the complex reactions taking place in foods. One should be careful with interpretation of such derived parameters, and it would probably be better to use at least the term apparent activation energy. Empirical models are available to handle situations in which the concept of transition state theory is not applicable. Nonisothermal cases can be handled both with Arrhenius=Eyring models as well as empirical models. In any case, knowledge on temperature and pressure coefﬁcients is extremely important when modeling food quality attributes, and this chapter has described the tools for this.

Appendix 5.1 TABLE A.5.1

Datasets Used for Examples in This Chapter Mutarotation of Lactose (Figure 5.1)

0

T (8C)

Keq

1=T (K1)

ln Keq

10

1.62

0.003534

0.48

15

1.6

0.003472

0.47

20

1.59

0.003413

0.46

25 30

1.58 1.56

0.003356 0.0033

0.46 0.45

35

1.55

0.003247

0.44

40

1.56

0.003195

0.44

50

1.52

0.003096

0.42

60

1.5

0.003003

0.40

75

1.46

0.002874

0.38

90

1.44

0.002755

0.36

Source: From Roetman, K. and Buma T.J. Temperature dependence of the equilibrium b=a ratio of lactose in aqueous solution. Neth Milk Dairy J 28:155–165, 1974.

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TABLE A.5.2

Mutarotation of Glucose (Figure 5.2) 0

Temperature T (8C)

Keq

30

1.503

35

1.467

40

1.447

45

1.409

Source: From Le Barc H.N., Grossel J.M., Looten P., and Mathlouthi M. Kinetic study of the mutarotation of D-glucose in concentrated aqueous solution by gas-liquid chromatography. Food Chem 74:119–124, 2001.

TABLE A.5.3 Reaction Rate Constants for Deamidation of Caseinate during Heating from 1108C to 1458C (Figure 5.5) T 0 (8C)

103=T

145

2.392

35.705

2.392

35.815

ln(khP=kBT)

2.392

35.97

140

2.421 2.421

35.199 35.272

2.421

35.336

130

2.481

35.73

120

110

2.481

35.803

2.481

35.895

2.545

36.444

2.545

36.499

2.545 2.611

36.609 37.011

2.611

37.103

2.611

37.149

Source: From Metwalli A.A.M. and Van Boekel M.A.J.S. On the kinetics of heat-induced deamidation and breakdown of caseinate. Food Chem 61:53–61, 1998.

TABLE A.5.4 Rate Constants for the Degradation of Riboﬂavin in Heated Spinach (Figure 5.9) T (8C)

k 3 105 (s1)

50

1.33E þ 00

60

1.83E þ 00

70 80

2.66E þ 00 3.17E þ 00

90

5.17E þ 00

100

5.33E þ 00

120

6.67E þ 00

Source: From Nisha P., Singhal R.S., and Pandit A.B. A study on the degradation kinetics of riboﬂavin in spinach (Spinacea oleracea L.). J Food Eng 67:407–412, 2005.

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TABLE A.5.5 Results for Nonisothermal Kinetics of Sucrose Hydrolysis, pH 3.8 (Figure 5.18) Time (min)

T (8C)

Sucrose Concentration (g L1)

0

100.0

0

100.0

10.0 9.833

5

100.5

9.772

5

100.5

9.603

20

102.3

9.18

20

102.3

9.011

30

105.568

8.923

30 40

105.568 106.494

8.517 9.004

40

106.494

8.396

55

108.761

8.614

55

108.761

8.275

65

111.869

7.917

65

111.869

7.545

80

115.322

6.715

80 90

115.322 117.754

6.546 6.594

90

117.754

6.019

105

120.186

5.188

105

120.186

5.189

120

123.466

5.121

120

123.466

3.952

130

126.409

5.034

130 140

126.409 128.849

3.83 2.561

140

128.849

2.358

155

131.788

1.73

155

131.788

1.527

165

135.726

2.521

165

135.726

2.318

180

137.166

0.643

180

137.166

0.474

Source: From Silva, C.L.M., Oliveira, F.A.R., Lamb, J., Torres, A.P., and Hendrickx, M., Int. J. Food Sci. Technol., 29, 227, 1995.

TABLE A.5.6 (Figure 5.19) Time (s)

Dataset for Heating Proﬁles in a UHT Heat Exchanger

T (8C) Final temperature 1208C

T (8C) Final temperature 1408C

0

20

20.563

4

77.465

80

5.429

88.169

102.254

6.714

93.803

115

8

101.127

123.099

9.286

105.634

126.479

10.571

109.577

129.859

11.857 13.143

112.394 115.085

133.239 135.366

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TABLE A.5.6 (continued) Exchanger (Figure 5.19) Time (s)

Dataset for Heating Proﬁles in a UHT Heat

T (8C) Final temperature 1208C

T (8C) Final temperature 1408C

15.714

115.211

137.183

15.714

116.901

137.183

17.143

118.028

138

18.286

118.028

138.873

19.571

118.592

139.437

20.714

118.592

139.437

22

118.592

139.437

23.286 25.429

118.592 119.155

139.437 140

25.571

119.155

139.437

27

119.155

140

28.429

119.155

139.437

29.857

119.155

140

33

61.127

61.127

33.286

55.493

55.493

33.571 35.429

49.859 42

49.859 42

36

30

30

37

23.38

22.817

Source: From Pompei C. and Rossi M. Use of a model solution for the evaluation of heat damage in milk treated in an ultra high-temperature heat exchanger. J Agric Food Chem 42:360–365, 1995.

TABLE A.5.7 Isothermal and Nonisothermal Heating Effect on Ascorbic Acid Content of the Fruit Amla (Figures 5.21 and 5.23) Isothermal, mg=100 g: Time (min)

508C

608C

708C

808C

908C

1008C

1208C

0 5

686

686

686

686

686

686

686 629

10

660

654

647

643

610

595

15

589 569

20

647

632

625

616

582

560

30

632

622

613

600

573

546

40

629

616

602

585

562

539

50

623

609

595

578

556

532

60

619

604

592

573

528

504

551

Nonisothermal, mg=100 g standard deviation (n ¼ 3): Time (min)

Open Pan

Pressure Cooking

Ecocooker 686 5

0

686 5

686 5

5

636 3

667 6

10

612 5

20 30

634 3

676 9

580 5

15 590 11

641 5 605 4

Source: From Nisha P., Singhal R.S., and Pandit A.B. A study on degradation kinetics of ascorbic acid in amla (Phyllanthus emblica L.) during cooking. Int J Food Sci Nutr 55:415–422, 2004.

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TABLE A.5.8 Isothermal and Nonisothermal Heating on Riboﬂavin Content of Spinach (Figure 5.24) Isothermal, mg=100 g: Time

508C

608C

708C

808C

908C

1008C

1208C

0

0.215

0.215

0.215

0.215

0.215

0.215

0.215

10

0.213

0.213

0.215

0.214

0.214

0.206

0.202

20

0.212

0.21

0.213

0.213

0.209

0.201

0.195

30

0.210

0.208

0.21

0.209

0.203

0.194

0.187

40

0.209

0.207

0.207

0.204

0.197

0.186

0.178

50

0.207

0.204

0.203

0.201

0.194

0.182

0.172

60

0.205

0.201

0.198

0.196

0.189

0.177

0.166

Nonisothermal, mg=100 g standard deviation (n ¼ 3): Time (min)

Open Pan

Pressure Cooker

Ecocooker

0

0.215 0.002

0.215 0.002

0.215 0.002

5

0.215 0.002

0.215 0.002

10 15

0.211 0.001

0.208 0.003 0.205 0.000

20

0.206 0.004

0.214 0.004 0.205 0.002 0.190 0.005

30

Source: From Nisha P., Singhal R.S., and Pandit A.B. A study on degradation kinetics of riboﬂavin in spinach (Spinacea oleracea L.). J Food Eng 67:407–412, 2005.

TABLE A.5.9 Results for High-Pressure Inactivation of Polygalacturonase in Tomato Juice (Figure 5.25) t (min)

C=CO at 400 MPa

C=CO at 450 MPa

0

1.0

1.0

5 10

0.800737

0.67781 0.367879

15

0.311403

20

0.64118

30

0.136847 0.037712

40

0.573753

60

0.211072

80

0.15988

100 120

0.178669 0.114559

0.033746

Source: From Fachin, D., Van Loey, A.M., Nguyen, B.L., Verlent, I., Indrawatti, and Hendrickx, M.E., Innovative Food Sci. Emerg. Technol., 4, 135, 2003.

TABLE A.5.10 Rate Constant for Inactivation of Polygalacturonase at 158C in Tomato Juice as a Function of Pressure (Figure 5.26) P (MPa)

k (s1)

400

0.00019

450

0.000532

500

0.001502

550

0.002782

Source: From Fachin, D. and Van Loey, A.M., Innovative Food Sci. Emerg. Technol., 4, 135, 2003.

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TABLE A.5.11 Parameter Values for Isorate Contour Plot of Inactivation of Polygalacturonase (Figure 5.27) Parameter DV0# (cm3 mol1 ) DS#0 (J mol1 K1 ) Da# (cm3 mol1 K1 ) 0

kref (s1)

Value 39.55 175.8 0.849 0.000343

Source: From Fachin, D. and Van Loey, A.M., Innovative Food Sci. Emerg. Technol., 4, 135, 2003. Note: Rate constant is kept constant at 0.00144 s1, temperature and pressure varied to achieve this ﬁxed value using Equation 5.63.

Bibliography and Suggested Further Reading General References Atkins P.W. Physical Chemistry, 6th ed. Oxford: Oxford University Press, 1999. Carlos Aledo J. Metabolic pathways: Does the actual Gibbs free-energy change affect the ﬂux rate? Biochem Mol Biol Ed 29:142–143, 2001. Cruickshank F.R., Hyde A.J., and Pugh D. Free energy surfaces and transition state theory. J Chem Ed 54:288–291, 1977. Hammes G.G. Thermodynamics and Kinetics for the Biological Sciences. New York: Wiley Interscience, 2000. Hamori E. Building a foundation for bioenergetics. Biochem Mol Biol Ed 30:296–302, 2002. Harris D.A. A uniﬁed approach to enzyme catalysis. Biochem Ed 14:2–6, 1986. Kost D. and Pross A. The misuse of energy proﬁles. Educ Chem 14:87–88, 1977. Laidler K.J. Chemical Kinetics, 3rd ed., New York: Harper & Row, 1987. Laidler K.J. Just what is a transition state. J Chem Ed 65:540–542, 1988. Logan S.R. The meaning and signiﬁcance of ‘‘the activation energy’’ of a chemical reaction. Educ Chem 23:148–150, 1986. Núnez de Castro I. and Alonso F.J. Energy diagrams for enzyme catalyzed reactions: A confusing point in the textbooks. Biochem Ed 25:87–89, 1997. Missen R.W., Mims ChA., and Saville B.A. Introduction to Chemical Reaction Engineering and Kinetics. New York: John Wiley & Sons, Inc., 1999. Peleg M., Engel R., Gonzalez-Martinez C., and Corradini M.G. Non-Arrhenius and non-WLF kinetics in food systems. J Sci Food Agric 82:1346–1355, 2002. Robinson P.J. Dimensions and standard states in the activated complex theory of reaction rates. J Chem Ed 55:509–510, 1978. Walstra P. Physical Chemistry of Foods. New York: Marcel Dekker, 2003. Winzor D.J. and Jackson C.M. Interpretation of the temperature dependence of equilibrium and rate constants. J Mol Recognit 19:389–407, 2006. Wolfenden R., Snider M., Ridgway C., and Miller B. The temperature dependence of enzyme rate enhancements. J Am Chem Soc 121:7419–7420, 1999. High Pressure Balny C. and Masson P. Effects of high pressure on proteins. Food Rev Int 4:611–628, 1993. Fachin D., Van Loey A.M., Nguyen B.L., Verlent I., Indrawatti O., Hendrickx, M.E. Inactivation kinetics of polygalacturonase in tomato juice. Innovative Food Sci Emerg Technol 4:135–142, 2003.

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Hendrickx M., Ludikhuyze L., Van den Broeck I., and Weemaes C. Effects of high pressure on enzymes related to food quality. Trends Food Sci Technol 9:197–203, 1998. Hinrichs J., Rademacher B., and Kessler H.G. Reaction kinetics of pressure-induced denaturation of whey proteins. Milchwissenschaft 51:504–509, 1996. Indrawati O., Van Loey A.M., Ludikhuyze, L.R., and Hendrickx, M.E. Soybean lipoxygenase inactivation by pressure at subzero and elevated temperatures. J Agric Food Chem 47:2468–2474, 1999. Matser A.M., Krebbers B., Van den Berg P., and Bartels P.V. Advantages of high pressure sterilization on quality of food products. Trends Food Sci Technol 15:79–85, 2004. Isokinetic Effect Banks B.E.C., Damjanovic V., and Vernon C.A. The so-called thermodynamic compensation law and thermal death. Nature (London) 240:147–148, 1972. Connors K.A. Chemical Kinetics. The Study of Reaction Rates in Solution. New York: VCH Publishers, 1990. Krug R.R., Hunter W.G., and Grieger-Block R.A. 1. Some fundamental statistical problems associated with the analysis of van ’t Hoff and Arrhenius data. J Phys Chem 80:2335–2341, 1976. Krug R.R., Hunter W.G., and Grieger-Block R.A. Enthalpy-entropy compensation. 2. Separation of the chemical from the statistical effect. J Phys Chem 80:2341–2351, 1976. Labuza T.P. Enthalpy=entropy compensation in food reactions. Food Techn 34(2):67–77, 1980. Madamba P.S., Driscoll R.H., and Buckle K.A. Enthalpy-entropy compensation models for sorption and browning of garlic. J Food Eng 28:109–119, 1996. McBane G.C. Chemistry from telephone numbers: The false isokinetic relationship. J Chem Ed 75:919– 922, 1998. Ozilgen M., Durukan A., and Ulgen N. Enthalpy-entropy and frequency factor-activation energy compensation relations for microbial death in fruit juices. Lebensmittel Wissenschaft und Technologie 24:378–381, 1991. Rhim J.W., Nunes R.V., Jones V.A., and Swartzel K.R. Appearance of a kinetic compensation effect in the acid-catalyzed hydrolysis of disaccharides. J Food Sci 54:222–223, 1989. Rhim J.W., Jones V.A., and Swartzel K.R. Kinetic compensation effect in the heat denaturation of whey protein. J Food Sci 55:589–592, 1990. Ulgen N. and Ozilgen M. Kinetic compensation relations for ascorbic acid degradation and pectinesterase inactivation during orange juice pasteurisation. J Sci Food Agric 57:93–100, 1991. Nonisothermal Reaction Kinetics Alibrandi G., D’Aliberti S., and Pedicini R. Computer simulation of variable-parameter kinetic experiments. Chem Educator 6:185–191, 2001. Corradini M.G. and Peleg M. A model of non-isothermal degradation of nutrients, pigments and enzymes. J Sci Food Agric 84:217–226, 2004. Corradini M.G. and Peleg M. Prediction of vitamins loss during non-isothermal heat processes and storage with non-linear kinetic models. Trends Food Sci Technol 17:24, 2006. Corradini M.G. and Peleg M. Shelf-life estimation from accelerated storage data. Trends Food Sci Technol 18:37–47, 2007. Deindoerfer F.H. and Humphrey A.E. Analytical method for calculating heat sterilization times. Appl Microbiol 7:256–264, 1959. Deindoerfer F.H. and Humphrey A.E. Principles in the design of continuous sterilizers. Appl Microbiol 7:264–270, 1959. Dolan K.D. Estimation of kinetic parameters for nonisothermal food processes. J Food Sci 68:728–741, 2003. Labuza T.P. A theoretical comparison of losses in foods under ﬂuctuating temperature sequences. J Food Sci 44:1162–1168, 1979.

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Nunes R.V., Rhim J.W., and Swartzel K.R. Kinetic parameter evaluation with linearly increasing temperature proﬁles-integral methods. J Food Sci 56:1433–1437, 1991. Peleg M., Engel R., Gonzalez-Martinez C., and Corradini M.G. Non-Arrhenius and non-WLF kinetics in food systems. J Sci Food Agric 82:1346–1355, 2002. Peleg M., Corradini M.G., and Normand M.D. Kinetic models of complex biochemical reactions and biological processes. Chemie Ingenieur Technik 76:413–423, 2004. Rhim J.W., Nunes R.V., Jones V.A., and Swartzel K.R. Determination of kinetic parameters using linearly increasing temperature. J Food Sci 54:446–450, 1989. Silva C.L.M., Oliveira F.A.R., Lamb J., Torres A.P., and Hendrickx M. Experimental validation of models for predicting optimal surface quality sterilization temperatures. Int J Food Sci Technol 29:227–241, 1995. WLF Models Sopade P.A., Halley P., Bhandari N., D’Arcy B., Doebler C., and Cafﬁn N. Application of the WilliamsLandel-Ferry model to the viscosity-temperature relationship of Australian honeys. J Food Eng 56: 67–75, 2003. Peleg M. On the use of the WLF model in polymers and foods. Crit Rev Food Sci Nutr 32:59–66, 1992. Roos Y.H. Phase Transitions in Foods. New York: Academic Press, 1995.

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6 Charge Effects

6.1

Introduction

The presence of charged particles can have large implications for the systems in which these particles are present, and this is certainly also true for foods. Foods contain dissociated salts, acids, and bases; biopolymers such as proteins and polysaccharides carry charge as well, depending on pH. Reactions between charged particles depend on a particular way on the charge but also on the properties of the medium in which they are present. Many fundamental investigations have been carried out in the ﬁrst half of the twentieth century by famous scientists. It is essential to understand the behavior of ions to understand properties of foods, and this includes kinetics where ions can exert a major effect. The activity concept is indispensable to understand the behavior of ions, so that is what we start with. Thus, the present chapter builds further upon the activity concept introduced in Chapter 3. After discussing the sometimes peculiar aspects of ion activities, we will take a look at the consequences for kinetics between charged particles.

6.2

Models for Ion Activities

The activities of ions have been studied in much detail and remain a topic of interest. It is an important topic for foods too, but it is not a simple one. We need to pay detailed attention to it in order to understand the intricacies involved. To start with, Figure 6.1 shows an example of the experimentally determined molal activity coefﬁcient of three simple salt solutions as function of their molality. Foods, of course, will usually not contain just one salt but a mixture of salts. The complications with mixtures of salts have been dealt with in the literature for the case of milk and, incidentally, also for nonfood systems such as seawater. First, however, we start with considering a solution of just one salt. Two phenomena are apparent from Figure 6.1, namely that salt solutions show strong deviation from ideality even at low concentrations, the more so for higher valency and, second, that the activity coefﬁcient can increase again at high concentration, and can actually become larger than unity at higher molalities (not shown here). An important phenomenon with species like salts is that they dissociate in aqueous solution, be it partly or completely. This means that we should be looking at the activity coefﬁcients of the ionic species rather than the salt as a whole. How can we do that? As a reminder, the chemical potential of a component is unique for a given solution at a given temperature and pressure, but the chemical potential in the standard state depends on the concentration scale adopted, and so does the activity. This is extensively discussed in Chapter 3. If we have electrolytes it would be desirable to refer to the chemical potentials of the separate 6-1

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1 0.9 0.8 Molal activity coefficient, g

NaCl 0.7 0.6 0.5 CaCl2 0.4 0.3 0.2 CuSO4 0.1 0 0

0.2

0.4

0.6

0.8

1

Molality (mol kg−1)

FIGURE 6.1 Experimentally determined molal activity coefﬁcients of aqueous NaCl (^), CaCl2 (&), and CuSO4 (~) solutions as a function of molality at 258C. The lines are just to guide the eye. Dataset in Appendix 6.1, Table A.6.1.

ionic constituents, but this is hypothetical because we cannot add only one type of ion to a solution because of the requirement of electroneutrality. We therefore need to go into somewhat more detail for the case of electrolytes. We start very general. Suppose we have 1 mol of a salt CpAq that can dissociate into p moles of cation C having valency zþ and q moles of anion A having valency z: Cp Aq Ð pCzþ þ qAz

(6:1)

For reasons of electrical neutrality, it is required that p zþ ¼ jq z j

(6:2)

So, for NaCl, p ¼ q ¼ 1, and zþ ¼ 1, z ¼ 1, and for Ca3(PO4)2, p ¼ 3, q ¼ 2, zþ ¼ 2, z ¼ 3. The chemical potential of the salt CpAq can be written as z þ RT ln aCzþ þ q m mCp Aq ¼ p m Az þ RT ln aA Czþ p q z þ qm ¼ pm Az þ RT ln ((aCzþ ) (aA ) ) Czþ

(6:3)

Thus, we can write for the activity of salt CpAq: aCp Aq ¼ (aCzþ )p (aAz )q

(6:4)

Using the molality scale, a ¼ g m, we can transform this equation into aCp Aq ¼ (g Czþ )p (gAz )q (mCzþ )p (mAz )q

(6:5)

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Remembering that upon complete dissociation: mCzþ ¼ pmCp Aq

(6:6)

mAz ¼ qmCp Aq it follows that aCp Aq ¼ (gCzþ )p (gAz )q (pmCp Aq )p (qmCp Aq )q

(6:7)

¼ (gCzþ )p (gAz )q pp qq (mCp Aq )pþq

Since we cannot measure the activity coefﬁcients of separate ions, the so-called mean ion activity coefﬁcient g is introduced: (g )pþq ¼ (gCzþ )p (gAz )q

(6:8)

p 1=(pþq) g ¼ g þ g q

(6:9)

aCp Aq ¼ pp qq (g mCp Aq )pþq

(6:10)

or

Equation 6.7 then changes into

Next to the salt activity we can also deﬁne a mean salt activity: 1

p

q

a ¼ (aCP Aq )pþq ¼ (aCzþ aAz )1=(pþq) ¼ (pp qq )1=(pþq) (g mCp Aq )

(6:11)

and a mean ionic molality: m ¼ (pq qq )1=(pþq) mCp Aq

(6:12)

These equations are needed for further analysis: it relates the (mean) activity of a salt to its concentration. As mentioned, this was a general derivation. To make it somewhat more comprehensible, Table 6.1 shows some examples for various types of salts using Equations 6.10 and 6.11. The experimental activity coefﬁcients (as shown, for example, in Figure 6.1) reﬂect actually the mean ion activity coefﬁcients of the salt; they are calculated from the measured activity of a salt solution and the known stoichiometric concentration. TABLE 6.1 Examples of Activity Coefﬁcients and Activities of Various Electrolytes Assuming Complete Dissociation as Compared to the Neutral Solute Sucrose Type

Example

Nonelectrolyte

Sucrose

1:1, 2:2

NaCl, CaSO4

2:1 1:2 1:3

CaCl2 Na2SO4 Na3PO4

g

Mean Salt Activity

— ðg Czþ g Az Þ1=2 1=3 g zþ g 2 z C2 A 1=3 g zþ g z C3 A 1=4 gCzþ g Az

— gm 4

1=3

gm

4 gm 271=4gm 1=3

Activity a of Solute Dissolved at Molality m gm g2 m2 4g 3 m3 4g 3 m3 27g 4 m4

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0.8 0.7 0.6

g

0.5 0.4 0.3 0.2 0.1 0 250

350

450

550

T (K)

FIGURE 6.2 Effect of temperature on ion activity coefﬁcient for aqueous solutions of NaCl at mole fraction 0.026 (^), of Na2SO4 at mole fraction 0.026 (~), and of MgSO4 at mole fraction 0.011 (&). The lines are just to guide the eye. Dataset in Appendix 6.1, Table A.6.2.

Incidentally, ion activity coefﬁcients depend on temperature. Figure 6.2 shows an example for the ion activity coefﬁcients of NaCl, Na2SO4, and MgSO4. In the temperature range relevant for foods, say 250–400 K, the recorded changes are noticeable but not drastic. Empirical models have been developed to account for the temperature dependence. We will not discuss them any further here. Now that we have deﬁned activity equations for electrolytes we can take a closer look at theories for activities of charged species.

6.2.1

Debye–Hückel Type Models

Obviously, ionic activity coefﬁcients refer to positively or negatively charged ions but these are not experimentally accessible, as mentioned. Theory has been developed to calculate ionic activity coefﬁcients and this has resulted in the famous Debye–Hückel (DH) theory. In this theory, deviation from ideality is accounted for by calculating electrostatic effects with reference to a central ion while treating all other ions as point charges. The resulting expression is a relation between the natural logarithm of the activity coefﬁcient (ln g) and the charges z, a function called the Debye screening parameter k, the distance of closest approach dR, and several fundamental parameters: ln g ¼

jzþ z je20 k 8p«0 «r kB T 1 þ kdR

(6:13)

The distance of closest approach dR is the sum of the effective radii of a pair of interacting ions. In the DH theory the ions are taken to be all of the same size and then dR equals the diameter of the ions. It is usually estimated to be between 0.3 and 0.6 nm. Furthermore, e0 is the elementary charge, «0 is the permittivity of

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vacuum, and «r is the relative permittivity of the medium.* Dielectric properties are seen to have an effect on activity coefﬁcients. For foods, this is relevant if other solvents such as ethanol are present; the relative permittivity of ethanol is 24.3 as opposed to 78.4 for water at room temperature. The parameter k (dimension m1) is deﬁned as k¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ X ri zi2 4pLB

(6:14)

i

where ri is the number density of ion i (number of particles per m3) and the sum is over all ionic species in solution. The relation between ri and concentration ci (in mol dm3) is of course: ri ¼ 103 ci NAV

(6:15)

The parameter LB (which is twice the so-called Bjerrum distance, dimension length) is deﬁned as LB ¼

e20 4p«0 «r kB T

(6:16)

In water at room temperature LB 0.7 nm. Next, the total ionic strength IT is introduced, deﬁned as IT ¼

1X 2 ci z i 2 i

(6:17)

Its dimension is the same here as that of the concentration ci (mol dm3); it can also be expressed in molality (mol kg1 solvent), or in number densities. The summation in this equation is taken over all ions present. It is assumed that salts are completely dissociated into ions, which is why it is called total ionic strength. The reason why we stress this becomes apparent below when we discuss the phenomenon of ion association. In the equation shown, ionic strength has units of molarity. For instance, a solution of 0.01 M CaCl2 has an ionic strength of 1 IT ¼ (0:01 22 þ 2 0:01 12 ) ¼ 0:03 mol dm3 2

(6:18)

It is perhaps instructive to explain very brieﬂy where the concept of ionic strength originates from. The role of ionic strength is that it accounts for the local charge density near an ion in solution; the mean P charge density is, of course, zero for a solution as a whole for reasons of electroneutrality, i.e., i ci zi ¼ 0. The expression displayed in Equation 6.17 derives from an expansion of the Poisson–Boltzmann equation in the D–H theory. Ionic strength does not appear directly in the fundamental thermodynamic equation that describes how free energy varies with, for instance, temperature and pressure. Therefore, it needs to be speciﬁed, for instance when using stoichiometric equilibrium constants. These do depend on ionic strength because activity coefﬁcients depend on ionic strength. Combining all these expressions yields a relation between k (m1) and ionic strength IT (mol dm3): k¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8pLB 1000NAV IT

(6:19)

* According to IUPAC, the relative permittivity of a medium «r is deﬁned as «m=«0 with «m the permittivity of the medium. For water, for instance, «r is 78.4 at 208C. The dielectric constant is the old name for relative permittivity.

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By combining Equation 6.19 with Equation 6.13 we can ﬁnd an expression in which ion activity coefﬁcients are a function of ionic strength, f(I). The natural logarithm in Equation 6.13 is converted to the base-10 logarithm in most textbooks. To avoid confusion we will also do that; the conversion factor, ln(10) ¼ 2.303, is included in the constants to follow. For an anion we ﬁnd 2 f (I) log g ¼ z

(6:20)

2 f (I) log gþ ¼ zþ

(6:21)

For a cation, a similar equation holds

For the mean ionic activity coefﬁcient the equation becomes log g ¼ jzþ z j f (I)

(6:22)

The function f(I) can take on various forms. The simplest one is used in the so-called limiting DH law: pﬃﬃﬃﬃ f (I) ¼ ADH IT

(6:23)

while the function for the extended DH law is: f (I) ¼

pﬃﬃﬃﬃ ADH IT pﬃﬃﬃﬃ 1 þ BDH dR IT

(6:24)

Two new parameters are introduced here. ADH and BDH are temperature-dependent constants. The parameter ADH is given by the following equation (if ionic strength is expressed in mol dm3):

ADH

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2pNAV 1000L3B 1 1 (dm3 :mol )2 ¼ ln 10

(6:25)

The numerical value for ADH at 258C for aqueous systems is 0.51, a value that is frequently used in the DH equation, but it should not be forgotten that the parameter does have units, as indicated. The parameter BDH is k BDH ¼ pﬃﬃﬃﬃ (kg1=2 mol1=2 m1 ) IT

(6:26)

The numerator in Equation 6.24 accounts for the effect of long-range Coulomb forces while the denominator signiﬁes how these are modiﬁed by the short-range interactions between ions. Table 6.2 lists numerical values of ADH and BDH as a function of temperature for aqueous solutions (note that the relative permittivity is also temperature dependent: it decreases with temperature). The value of the parameter dR is uncertain and acts as an adjustable parameter. Scatchard has proposed to take the product BDHdR as 1.5 dm3=2 mol1=2 (hence, taking dR as 0.46 nm) so that: pﬃﬃﬃﬃ ADH IT pﬃﬃﬃﬃ f (I) ¼ 1 þ 1:5 IT

(6:27)

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TABLE 6.2 D–H Constants, Relative Permittivity «r, and Bjerrum Length LB as a Function of Temperature for Aqueous Solutions at a Pressure of 1 bar «r

LB (nm)

ADH (dm3 mol1)1=2

BDH 3 1010 (m1 mol1=2 (dm3)1=2)

0

87.74

0.696

0.491

0.325

5

85.76

0.700

0.494

0.325

10

83.83

0.703

0.498

0.326

15

81.94

0.707

0.502

0.327

20

80.1

0.711

0.506

0.328

25 30

78.3 76.54

0.715 0.719

0.510 0.515

0.329 0.330

35

74.82

0.724

0.520

0.331

40

73.15

0.729

0.525

0.332

50

69.91

0.739

0.536

0.334

75

62.42

0.768

0.568

0.341

T 0 (8C)

Others have proposed unity for BDHdR. This results in the equation of Güntelberg: pﬃﬃﬃﬃ ADH IT pﬃﬃﬃﬃ f (I) ¼ 1 þ IT

(6:28)

The last two equations are a bit awkward in the sense that the numerical values BDHdR ¼ 1.5 and 1, respectively, have a dimension of (dm3)1=2 mol1=2 and this is not clear by looking at the equation. A modiﬁcation of the Güntelberg equation was suggested by Guggenheim: f (I) ¼

pﬃﬃﬃﬃ ADH IT b0 I T pﬃﬃﬃﬃ 1 þ IT zþ z

(6:29)

where b0 is an adjustable empirical parameter. Again, there seems to be a dimensional problem here. The DH constants have units that actually depend on the unit of the ionic strength because the right-hand side terms in Equations 6.20 through 6.22 need be dimensionless. If ionic strength is expressed in molality, the units change to (kg mol1)1=2 for ADH and to (kg1=2 mol1=2 m1) for BDH. If concentration would be expressed in the SI unit mol m3, the parameter ADH changes numerically by a factor of 10001=2 from 0.510 to 0.0161 at 258C. In fact, it would be better to deﬁne ionic strength as a dimensionless parameter: IT ¼

1 X ci 2 z 2 i c i

(6:30)

where ci is molarity (mol dm3) and c the standard state molarity of 1 mol dm3, as a result of which ADH and BDH would have no units. However, this is not done in most cases in the literature and to avoid confusion we will conform to the current practice. However, the reader should be aware of this possible problem of units. For foods, it would make more sense actually to use molality than molarity because it is somewhat hard to imagine what a liter solution means in products like, say, cheese. For real solutions, molarity can be used as well, of course. For example, the ionic strength of milk is about 0.08 M, and in general, the ionic strength for most foods will be between 0.001 and 1 M or m. When using buffers in model systems that attempt to mimic a speciﬁc food, it should be realized that two different buffers having the same

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Kinetic Modeling of Reactions in Foods 0.2

0.5

0

0

−0.2

−0.5 Log g ±

Log g ±

6-8

−0.4

−1

−0.6

−1.5

−0.8

−2

−1 (A)

0

0.5

1

1.5 2 IT (M)

2.5

3

3.5

−2.5

0

(B)

1

2

3

4

5

IT (M)

FIGURE 6.3 Experimental mean ion activity coefﬁcients as a function of total ionic strength IT in mol dm3 for NaCl (A) and CaCl2 (B). Dotted line: limiting DH, Equation 6.21; hyphenated line: Güntelberg, Equation 6.28; solid line: Davies, Equation 6.31. Dataset in Appendix 6.1, Table A.6.3.

concentration and pH could have a markedly different ionic strength. This phenomenon can have a large kinetic effect as is discussed below. It can have also a strong effect on enzyme activity, for instance. Equation 6.23 is the so-called limiting DH theory because it is only valid in the limit of IT ! 0, i.e., for very diluted solutions (say IT < 1–10 mM). The extended DH equation, the Güntelberg and Guggenheim equations are valid for, say, IT < 100 mM. A frequently used equation in the literature is the so-called Davies equation, which is an empirical relation based on the DH theory and is in fact a modiﬁcation of the Guggenheim equation (Equation 6.29): f (I) ¼ ADH

pﬃﬃﬃﬃ IT pﬃﬃﬃﬃ 0:2IT 1 þ IT

(6:31)

The Davies equation seems to give a good ﬁt to experimental values, also at higher ionic strength. Figure 6.3 shows experimental mean activity coefﬁcients for NaCl and CaCl2 and the ﬁt by the limiting DH equation, the extended DH equation, and the Davies equation. Indeed, the Davies equation performs best of these three equations. Table 6.3 summarizes the various DH equations. However, the reader is advised that the equations displayed in Table 6.3 do not reﬂect the actual ion activity coefﬁcients if ion pairs are present. However, before we discuss this, we ﬁrst dive into the phenomenon that is also shown in Figure 6.3, namely that the ion activity coefﬁcient increases again at higher ionic strengths. This can be explained by a more recent theory called the mean spherical approximation.

6.2.2

Mean Spherical Approximation Theory

The MSA approach is better suited to describe nonideality of electrolyte solutions in concentrated systems (i.e., >0.1 M) than the DH equations can; it can, for instance, explain the increase in mean activity coefﬁcient at the higher concentrations. The DH theory neglects the ﬁnite size of ions and this becomes a problem in more concentrated solutions because it leads to an overestimation of the effects of

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TABLE 6.3

Overview of the Various D–H Expressions Based on f (I) ¼

pﬃﬃﬃﬃ ADH IT b0 IT pﬃﬃﬃﬃ 1 þ BDH dR IT zþ z

Equation pﬃﬃﬃﬃ f (I) ¼ ADH IT pﬃﬃﬃﬃ ADH IT pﬃﬃﬃﬃ f (I) ¼ 1 þ BDH dR IT pﬃﬃﬃﬃ ADH IT pﬃﬃﬃﬃ f (I) ¼ 1 þ 1:5 IT pﬃﬃﬃﬃ ADH IT pﬃﬃﬃﬃ f (I) ¼ 1 þ IT pﬃﬃﬃﬃ ADH IT b0 IT pﬃﬃﬃﬃ f (I) ¼ 1 þ IT zþ z pﬃﬃﬃﬃ IT pﬃﬃﬃﬃ 0:2IT f (I) ¼ ADH 1 þ IT

0

dR ¼ 0, b ¼ 0 dR 6¼ 0, b0 ¼ 0 BDHdR ¼ 1.5 kg1=2 mol1=2, b0 ¼ 0 BDHdR ¼ 1 kg1=2 mol1=2, b0 ¼ 0 BDHdR ¼ 1 kg1=2 mol1=2, b0 6¼ 0 BDHdR ¼ 1 kg1=2 mol1=2, b0 ¼ 0.2jzþzj

Name D–H limiting law (DHLL) Extended D–H equation Scatchard equation Güntelberg equation Guggenheim equation Davies equation

ion–ion interactions, especially for polyvalent ions. In the MSA approach, the ﬁnite size of all solute species, charged or uncharged, is taken into account. This accounts for excluded volume effects, usually called the hard sphere contribution, which is basically an entropic effect and accounts for how space can be occupied by spheres. Also, the decrease in dielectric permittivity of the solution with increasing concentration of solute needs to be considered; this latter effect leads to an increase in the strength of ion–ion interactions with increasing volume fraction of solutes. MSA calculations need to be done in the molarity scale, so if comparisons are to be made with experimental ionic activity coefﬁcients, which are usually tabulated in the molality scale, recalculations are necessary. How to do this has been discussed in Chapter 3. The resulting expression for the single ionactivity coefﬁcient is made up of two contributions; for instance, for the molar activity coefﬁcient yi: ln yi ¼ ln yies þ ln yihs

(6:32)

The advantage of the MSA is that it leads to relatively simple, though lengthy, analytical expressions. The electrostatic part due to the ionic atmosphere is indicated by the superscript ‘‘es,’’ the hard sphere part due to the ﬁnite size of solute species is indicated by the superscript ‘‘hs.’’ We will not list the complete equations here; some are given in Appendix E. Simpliﬁed expressions do exist, however. The most basic one for the electrostatic part, for equal ion sizes, reads ln yies ¼ LB

zi2 G 1 þ GdR

(6:33)

with G¼

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 2kdR 1 2dR

(6:34)

The screening parameter G in the MSA thus replaces the screening parameter k in the DH theory. However, (2G)1 is greater than k1 because the ﬁnite diameter accounted for in the MSA results in the

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fact that ions cannot approach each other so closely as point charges can in the DH theory. In the limit of zero ionic strength I, G ! 0.5k. In the framework of the MSA theory, ionic strength is deﬁned as 1X 2 rz 2 i i i

I¼

(6:35)

One equation for the hard sphere contribution for equal-sized spheres is 2

hs ln yihs ¼ ln y ¼

6J 3J 2J þ 2þ 1 J (1 J) (1 J)3

J¼

(6:36)

pdi3 X rj 6 j

(6:37)

The summation in Equation 6.37 is over all solute species, including neutral species, and Equation 6.37 represents in fact the volume fraction of solutes. The single ion-activity coefﬁcients can be expressed, of course, as mean ion activity coefﬁcients using Equation 6.9. The central parameter in the electrostatic part of the MSA model is thus the screening parameter G. The ionic strength parameter I is enclosed indirectly (via k) in this screening parameter. The interested reader is referred to some selected references given at the end of this chapter for more detailed information. The above given MSA equations are the simplest possible. They constitute the so-called restrictive, primitive MSA; restrictive because all ion sizes are assumed equal and primitive because the solvent is considered a dielectric continuum that only manifests itself via its permittivity, it ignores the discrete nature of the solvent molecules (this is the so-called McMillan–Mayer framework, see Appendix E). Figure 6.4 shows a calculation of the electrostatic and hard sphere contribution for a 1:1 electrolyte and 2:2 electrolyte using Equations 6.33 and 6.36, assuming aqueous solutions at 258C, ion sizes of 0.5 nm, and complete dissociation. The ﬁgure shows nicely that in the range 0.1–0.5 M the contribution of the 1.6

1.5 ln(y hs)

1.2

0.5 ln(y hs)

0.8

ln ( y)

ln (y)

−0.5 0.4 ln(y es)+ln(y hs)

0

ln(y es) + ln(y hs)

−2.5

ln(y es)

−0.4

−1.5

ln(y es) −0.8 0 (A)

0.5 1 c (mol dm−3)

1.5

−3.5 (B)

0

0.5 1 c (mol dm−3)

1.5

FIGURE 6.4 Hard sphere (hs) and electrostatic (es) contribution to the mean ionic activity coefﬁcient on the molar scale for a 1:1 electrolyte (A) and a 2:2 electrolyte solution (B), calculated with the restricted MSA theory. Note the difference in the y-axis scale in A and B.

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hard sphere effect starts to become noticeable, and it is for this reason that the MSA theory is superior to the extended DH theory and related equations for ionic strengths above 0.1 M. In the DH model the presence of a salt is characterized only by its charge, whereas in the MSA model a salt is distinguished by its diameter and its charge. However, the results shown in Figure 6.4 are only meant to illustrate trends. Several improvements can and should be made: .

. .

. .

Dependence of the permittivity of the solvent on concentration must be taken into account. The dielectric constant of a solution decreases when the salt concentration increases by reducing the density of solvent dipole moments. Allowance should be made for different sizes of the various cations and anions. Ion sizes depend on salt concentration. It should be realized that the relevant ion diameter in the MSA theory includes the hydration sphere. This hydration sphere decreases with increasing salt concentration. The effect of the solvent should be taken into account, resulting in the nonprimitive MSA. A very important aspect that needs to be considered is the phenomenon of ion association. This should also be done, in principle at least, when using the DH equations but the ion association effect is less pronounced at IT < 0.1 M, so neglecting it in these cases has relatively less impact (though not negligible anymore for high valency ions) than in the regions above IT > 0.1 M where the MSA is very useful. The phenomenon of ion association is discussed in Section 6.3.

For 1:1 electrolytes, the primitive MSA is applicable to an ionic strength of about 0.3 M. The nonprimitive MSA can be valid up to a few molar. The adjustable parameter in MSA is the ionic diameter. A relatively new way to calculate activity coefﬁcients is via Monte Carlo simulations based on statistical thermodynamics. Like the DH model ions are treated as hard, charged spheres in a continuum with a certain permittivity. Unlike the DH theory where the distribution of particles is approximated with a continuous charge distribution, the distribution of the particles is simulated by moving them randomly in a box, and all interionic particle interactions are taken into account. This topic is, however, beyond the scope of this book; a few references are given at the end of this chapter. Table 6.4 summarizes some guidelines on when to use the various models for activity coefﬁcients.

6.2.3

Pitzer Equations

In order to overcome the limitations of the DH equations for higher ionic strengths than 0.1 M, Pitzer proposed to use virial expansions of the DH equations, as was basically also proposed in the Guggenheim equation shown above. The virial coefﬁcients are supposed to account for speciﬁc interactions between ions that occur at the higher ionic strengths (say above 0.1 M). These equations are therefore also referred to as speciﬁc interaction theory (SIT). Because they account for interactions between ions, these equations should be useful for multicomponent electrolyte solutions. A general equation is: pﬃﬃﬃﬃ X X pﬃﬃﬃﬃ 2 IT g pﬃﬃﬃﬃ þ mi mj Bgij þ mi mj mk Cijk ln 1 þ 1:2 IT þ 1 þ 1:2 IT 1:2 ij ijk

ln g i ¼ z2 A

TABLE 6.4 Overview of Models Describing the Relations between Ionic Activity Coefﬁcients and Ionic Strength IT Ionic Strength IT < 0.01 M

Activity Coefﬁcient Estimation via DHLL equation

IT < 0.1 M

Estimation via extended DH, Güntelberg, or Guggenheim equation

0.1 M < IT < 0.3 M IT > 0.3 M

Primitive mean spherical approximation, Davies equation Nonprimitive mean spherical approximation

(6:38)

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At 258C, A ¼ 0.392, and the Bij and Cijk parameters are related to binary ions i and j and to ternary ions i, j, k. respectively. These parameters account for speciﬁc ion interactions, as mentioned, and are based on experimental measurements made at the same ionic strength. There are many variants of Pitzer equations. They all contain a lot of parameters that do not really refer to physical properties. Moreover, the large number of parameters is a drawback from a modeling point of view. Nevertheless, they are used extensively in practice because they offer a solution to the practical problem of how to estimate activity coefﬁcients when theoretical models are not really valid. Some references on the use of Pitzer equations are given at the end of this chapter.

6.3

Ion Pairing Models

Quantitative studies of electrolyte solutions have shown that at very low concentrations nonideality can be accounted for satisfactorily by the mean activity coefﬁcient and the cause of this nonideality is the existence of long-range electrostatic interactions. The DH theory as well as the MSA assumes complete dissociation of electrolytes into solvated ions, and this assumption appears to be valid only at strongly diluted solutions. At higher concentrations much stronger deviations come into play due to short range Coulombic interactions. These interactions can only be explained by assuming ion association, also called ion pairing. In contrast to common belief it is thus not true that salts are completely dissociated into ions. The reason why it is important to consider these aspects is that we are interested in activities of free ionic species, and the mean ionic activity coefﬁcients do not reﬂect these when ion pairs are formed. This is so because activity coefﬁcients are calculated from measured activities on the one hand and from concentrations that are calculated assuming complete dissociation on the other hand. If ion pairs are formed this assumed concentration is no longer correct. Fortunately, we can correct for this if we are able to estimate the extent of ion association. Cations and anions can associate, for instance, Naþ and SO42 can associate into the ion pair NaSO4 and Naþ and Cl into NaCl0. Such behavior is expected especially for polyvalent electrolytes in aqueous solutions (but also for monovalent ions as we shall see) and in nonaqueous solutions of lower dielectric permittivity. Ion pairs are physically associated entities, moving together and having a ﬁnite but noticeable lifetime. An ion pair lifetime is of the order of 2 ns, which is about 20 times larger than the decay of a diffusion-controlled encounter (Chapter 4). Ion pairs can be measured experimentally via advanced spectroscopic techniques and conductivity measurements, for instance. The result is that at any instant a given fraction of ions is paired and although an individual ion pair does not last long, the timeaveraged concentration of ion pairs will remain constant (at constant conditions such as temperature, ionic strength, pressure). Consequently, we can apply concepts from equilibrium thermodynamics to the phenomenon of ion pairs. Suppose we have a solid salt CpAq(s) that we add to water; we also assume that the salt will dissolve completely in the water so that there is no CpAq(s) left after a while. We can then write Cp Aq (s) ! pCzþ þ qAz

(6:39a)

Suppose then that part of the dissolved ions form ion pairs as Czþ þ Az Ð CAzþ þz

(6:39b)

We thus have free ions as well as ion pairs in solution. We assume here only association into doublets and not into triplets or higher associations; so in the case of CaCl2 we assume CaClþ ion pairs but not CaCl20 ion pairs. This is not to say that the latter are not formed, but we omit it here for simplicity. If ion pairs are formed, the association into doublets will occur the most abundantly. The activity of an electrolyte

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forming ion pairs can be expressed in two equivalent ways (expressed here as molarity, but this could be equally well molality): aCp Aq ¼ (yC,T cC,T )p (yA,T cA,T )q ¼ (yC,F cC,F )p (yA,F cA,F )q

(6:40)

The subscripts ‘‘T’’ and ‘‘F’’ indicate the total (stoichiometric) and the free ion concentration, respectively. The validity of Equation 6.40 can be proven by a thermodynamic reasoning, which we do not repeat here. Equation 6.40 allows us to calculate the free ion activity coefﬁcient if we know the total (or mean) activity coefﬁcient and the concentration of ion pairs. Of course, the following mass balance relation holds: cC,T ¼ cC,F þ cC,ip cA,T ¼ cA,F þ cA,ip

(6:41)

The subscript ‘‘ip’’ refers to ion pairs. A simple example may be illustrative. Suppose that we have a solution of CA (p ¼ q ¼ 1, zþ ¼ z ¼ 1) of 1 M, and that the mean ion activity coefﬁcient of this solution 2 is 0.60 (remembering that gþ C g A ¼ g , Equation 6.8). The question is now what the activity coefﬁcient of the free ions is if ion pairs are formed. The activity is then: aCp Aq ¼ (0:60 1:0)2 ¼ 0:36

(6:42)

Suppose we know that the concentration of ion pairs CA0 is 0.1 M, and consequently the concentration of free ions is 0.9 M as follows from Equation 6.41. Using Equation 6.40 we then can calculate that: (yC,F 0:9)(yA,F 0:9) ¼ 0:36 0:36 ¼ 0:44 yC,F yA,F ¼ (y,F )2 ¼ 0:81 y,F ¼ 0:67

(6:43)

The mean free ion activity coefﬁcient g,F is deﬁned, of course, following Equation 6.9. We thus ﬁnd that the activity coefﬁcient of 1 M solution of CA would be 0.60 if no ion pairs are present while that of the free ions is 0.67 if 10% ion pairs would be formed. In other words, the activity coefﬁcient of the free ions is higher than that of the mean activity coefﬁcient of the solution as a whole. Equation 6.40 shows that the total activity coefﬁcient is a function of the free activity coefﬁcient as well as the ratio of free to total ion concentration. The problem is thus that, if ion pairs are formed, the actual mean free ion activity coefﬁcient is not directly comparable to the ones that are estimated from experiments. This is so because experimentally the activity of an electrolyte solution is measured (not the activity coefﬁcient!), and subsequently the mean ion activity coefﬁcient is calculated from this activity and the known stoichiometric concentration. However, if the concentration of free ions is lower because of the presence of ion pairs then this calculation is in error. Put in a different way, when ion pairs are present, calculations based on the stoichiometric electrolyte concentration overestimate the effects of nonideality due to ions. Furthermore, the ionic strength is lower than calculated. Of course, we can correct for this if we are able to estimate the amount of ion pairs present. Amount of ion pairs. Thus, we are left with the question how to calculate the amount of ion pairs formed in electrolyte solutions. Since the ion pairs are in equilibrium with the free ions as depicted in Equation 6.39b, the association of free ions into ion pairs can be described by a thermodynamic equilibrium constant, in this case called an association constant Kipth : Kipth ¼

aCA(zþ þz ) aCzþ aAz

(6:44)

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TABLE 6.5 Some Association Constants for Ion Pairs at 258C. Dataset in Appendix 6.1, Table A.6.4 Salt

Ion Pair

Association Constant Kipth

0

NaCl

NaCl

0.6

NaNO3

NaNO30

0.3

Na-acetate

Na-acetate0

0.5

Na2SO4

NaSO4

5.0

Na2S2O3

NaS2 O3

4.0

KCl

KCl0

0.6

KNO3

KNO30

0.6

K2SO4 CaCl2

AppKSO41 CaClþ

Ca(NO3)2

CaNO3þ

Ca(ðH2 PO4 )2

CaH2 PO4þ

CaHPO4

CaHPO40

Ca3(PO4)2

CaPO4

CaSO4

CaSO40

10 6 2 10 500 106 190

Ca(lactate)2

Ca(lactate)þ

Ca(butyrate)2 Ca-oxalate

Ca(butyrate)þ Ca(oxalate)0

30 4 1000

Kipth is a thermodynamic constant because it is deﬁned in terms of activities. The magnitude of this association constant is to a ﬁrst approximation determined by the valences z and it becomes quite high for valences higher than 1 (in the order of 102 to 103). In other words, ions with valences >1 tend to associate strongly into ion pairs. Table 6.5 shows some examples to give an impression of the order of magnitude of the effect. We can also consider activity coefﬁcients of ion pairs. If they carry no charge, the activity coefﬁcient will be close to unity in dilute solution. However, if they carry a charge, the activity coefﬁcient will be, most likely, different from unity. Values for thermodynamic association constants can be more or less generalized, as shown in Table 6.6. Equation 6.44 can be rewritten in terms of a stoichiometric constant Kipst ; the reason to do this is that we normally work with concentrations (molarity or molality), so Kipst is the parameter that is experimentally accessible. To make this step, we need to work with activity coefﬁcients: Kipth ¼

yip [CA(zþ þz ) ] yCA(zþ þz ) ¼ Kipst 2 ¼ Kipst K y y [Czþ ] [Az ] yCzþ yAz

TABLE 6.6 Orders of Magnitudes for Thermodynamic Association Constants as a Function of the Valencies of the Ions Constituting a Salt at Room Temperature Valencies

Example

Kipth

zþ ¼ 1, z ¼ 1

NaCl, KCl

0.1–1

zþ ¼ 1, z ¼ 2

Na2SO4

1–10

zþ ¼ 2, z ¼ 1

Ca(NO3)2, CaCl2

1–10

zþ ¼ 2, z ¼ 2

CaHPO4

100–500

zþ ¼ 1, z ¼ 3

Na3PO4

500–1000

zþ ¼ 2, z ¼ 3

Ca3(PO4)2

106

(6:45)

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where we have taken molarity as the concentration scale. This equation shows that the extent of ion pair formation in terms of concentration expressed via Kipst depends on the thermodynamic constant Kipth as well as on the activity coefﬁcients. Note that the stoichiometric association constant is not constant, in contrast to its thermodynamic counterpart, because it depends on the activity coefﬁcients that depend in turn on the value of the concentration. So, if activity coefﬁcients change, Kipst has to change as well because Kipth is a true constant. As an example from the literature, stoichiometric ion association constants were derived from dielectric relaxation spectra. The stoichiometric association constant was related to ionic strength via a Guggenheim type function: log Kipst ¼ log Kipth

pﬃﬃﬃﬃ 2ADH jzþ z j IT 3=2 pﬃﬃﬃﬃ þ Bb IT þ Cb IT 1 þ IT

(6:46)

Bb and Cb are empirical ﬁt constants. Figure 6.5 shows the ﬁt of this equation to measured data for the ion pair sodium malonate. Such measurements can be used to estimate Kipth . Figure 6.5 shows very clearly that Kipst is strongly dependent on ionic strength; in other words, Kipst is only a constant at constant ionic strength. In the literature, three ways are offered to calculate the amount of ion pairs formed, via the mass action law (MAL), via a method developed by Pytkowicz and coworkers, and via a special form of the MSA theory, called the binding MSA (BIMSA) or associating MSA (AMSA).

6.3.1

Mass Action Law

The ﬁrst method is as follows. Let us deﬁne a parameter aip such that a fraction (1 aip) of the cations forms ion pairs. Consequently, aip ¼ 1 corresponds to the absence of ion pairs and aip ¼ 0 to complete association into ion pairs. Then, a solution of a salt CpAq at a molar concentration c will result in a concentration of

1.5 1 0.5

log K stip

0 −0.5 −1 −1.5 −2 −2.5 −3

0

0.5

1

1.5

2

2.5

3

IT (mol dm−3)

FIGURE 6.5 The (logarithm of the) stoichiometric association constant Kipst as a function of ionic strength for the ion pair sodium malonate. The line indicates the ﬁt of Equation 6.46 to the data. Dataset in Appendix 6.1, Table A.6.5.

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TABLE 6.7 Relation between the Free Ionic Activity Coefﬁcient y,f and the Stoichiometric Activity Coefﬁcient y

Type of Salt

Concentration of Ion Pairs in a Salt Solution of Stoichiometric Concentration c

Example

(1 aip)c

Formula y ¼ aip

p¼q¼1

NaCl, MgSO4

p ¼ 1, q ¼ 2

CaCl2

(1 aip)c

y,f ¼ y

4 aip (1 þ aip )2

p ¼ 2, q ¼ 1

Na2SO4

2(1 aip)c

y,f ¼ y

!1=3 1 (aip 6¼ 0:5) a2ip (2aip 1)

y,f

!1=3

Cations: paipc Ion pairs: p(1 aip)c Anions: qc p(1 aip)c As a next step, the chemical potential of all the solute species (free ions and ion pairs) is considered, from which eventually an expression for the activity coefﬁcient can be deduced. We give here the result of this exercise without derivation; it relates the mean free ionic activity coefﬁcient y,f to the mean stoichiometric (or total) activity coefﬁcient y as follows: y,f ¼ y

pp qq p (paip ) [q p(1 aip )]q

1=(pþq) (6:47)

Table 6.7 shows examples for some typical salts. This table shows clearly that the free ion activity coefﬁcient is higher than the mean stoichiometric ionic activity coefﬁcient (which is the one that is deduced from experiments) if ion pairs are formed (aip < 1). This important point is frequently neglected, but it can lead to considerable error as we shall see. Recall that the DH and MSA type equations given above assume that complete dissociation takes place, so that these equations do not differentiate between a free and a stoichiometric activity coefﬁcient. Equation 6.47 and Table 6.7 show that if the degree of ion pair association is known, the free ion activity coefﬁcient can be calculated from the experimentally determined y, and, in reverse, if the free ion activity coefﬁcient would be known, the degree of ion association can be determined. The relation between aip and the stoichiometric association constant Kipst for a salt CpAq is Kipst ¼

[CAip ] p(1 aip )c ¼ [Czþ ] [Az ] (paip c)(qc p(1 aip )c)

(6:48)

For symmetrical electrolytes, such as NaCl and MgSO4 (p ¼ q ¼ 1), the relation between the thermodynamic association constant, the parameter aip, the mean ion activity coefﬁcient, and the concentration becomes (cf. Equations 6.45 and 6.48) Kipth ¼

(1 aip )yip (1 aip )yip ¼ 2 a2ip cyþ y a2ip cy

(6:49)

Usually, the activity coefﬁcient for the ion pair yip is taken as unity if the ion pair is uncharged. In more concentrated solutions this approximation may not hold. For unsymmetrical electrolytes, such as CaCl2,

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the activity coefﬁcient of the ion pair, e.g., CaClþ, can be taken, to a ﬁrst approximation, as that of the anion, e.g., Cl. In the case of a 2:1 electrolyte as CaCl2, the relation becomes Kipth ¼

(1 aip ) a2ip cyþ

(6:50a)

In the case of a 1:2 electrolyte such as Na2SO4 with ion pair NaSO4 it becomes Kipth ¼

(1 aip ) a2ip cy

(6:50b)

The interesting phenomenon is that ion association depends on ionic strength because activity coefﬁcients depend on ionic strength, while ionic strength depends in turn on ion association. If ion pairs are formed, ionic strength decreases; to account for that, the effective ionic strength Ie is introduced X 1 X 2 2 zf ,i cf ,i þ zip,j cip,j Ie ¼ 2 i j

! (6:51)

(on the molar scale). The subscript ‘‘f ’’ refers to free ions. This can be rewritten for the salt CpAq forming ion pairs as Ie ¼

1 2 2 2 [qc p(1 aip )ci ] þ zip p(1 aip )ci zþ paip ci þ z 2

(6:52)

Equation 6.52 shows that the effective ionic strength will be less than the total ionic strength if ion association occurs, the more so when the ion pair carries no charge. In order to calculate the effective ionic strength we need to know the concentration of ion pairs and the charge on the ion pairs, and to be able to calculate this we need activity coefﬁcients (via DH-related equations or the MSA approach). This calls for an iterative calculation. A pseudo-algorithm is given in Figure 6.6. A worked example may help to illustrate the implications. Suppose we have a 1:1 univalent salt, say NaCl, at a concentration of 0.1 mol dm3. From Table 6.5 we assume the association constant Kipth to be 0.6. The ionic strength equals the molar concentration in the case of a completely dissociated 1:1 electrolyte, so as a ﬁrst guess we take IT ¼ 0.1 mol dm3. The mean ionic activity coefﬁcient is then calculated using, for instance, the extended DH equation (Equation 6.24, which is valid up to 0.1 M). From these data, the stoichiometric association constant Kipst can be calculated using Equation 6.45. Using Equation 6.47 next, this results in an estimate of the extent of ion pair formation via the parameter aip. Then we can calculate the concentration of free ions and from that the effective ionic strength Ie. This gives the starting point for the next iteration and usually after four to ﬁve iterations the calculation converges to a ﬁnal value. Figure 6.7 shows some calculated results for the extent of ion pair formation where we have used the extended DH equation (actually the Scatchard equation (Equation 6.27), but using the effective ionic strength rather than total ionic strength). This equation should be valid up until an effective ionic strength of 0.1 M, while for higher concentrations the MSA model should be used. It shows that the effect of ion pair formation deﬁnitely cannot be neglected, especially not for polyvalent ions. It also implies that the effective ionic strength Ie can be substantially lower than calculated as if all ions are dissociated, i.e., total ionic strength IT. Figure 6.8 shows what can happen at higher electrolyte concentration; this is calculated via the MSA model because the ionic strength is higher than 0.1 M. It is seen that the extent of ion pairing decreases

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Electrolyte concentration

Calculate total ion concentration

Calculate total ionic strength IT

Calculate y±,f

Kipth

st

Calculate Kip

Calculate degree of ion association aip

Calculate free ion and ion pair concentration

Calculate effective ionic strength Ie Difference previous iteration k3 so that KM ¼

k2 k1

(9:13)

From Scheme 9.2 it follows that k2 [E] [S] ¼ ¼ Ks k1 [ES]

(9:14)

and so it appears that KM equals the dissociation constant of the ES complex Ks if k3 is much smaller than k2. In that case, a high value of KM indicates a weak binding of E to S and a low value of KM the opposite. It is important to realize that, in this case, KM does not give information about the catalysis process because the actual chemical change happens only after the formation of the ES complex. If the ﬁrst step in Scheme 9.2 is assumed to be irreversible (i.e., the Van Slyke equation, k2 ¼ 0), the parameter KM is a ratio of forward rate constants rather than a (pseudo) equilibrium constant. It thus appears that the original Van Slyke and Michaelis–Menten equations are in fact limiting cases of the steady-state assumption introduced by Briggs and Haldane. The maximal rate vmax is related to the turnover number of an enzyme, which is by deﬁnition the number of substrate moles converted into product per unit time per mole of enzyme when the enzyme is fully saturated with enzyme. From Equation 9.9 it follows that the turnover number equals k3, also called kcat. Turnover numbers of most enzymes are usually in between 1 and 104 per second. The rate is then determined by the actual catalyzed chemical reaction; kcat is a constant, while vmax depends on the total enzyme concentration ET. Note, however, that k3, or kcat, the turnover number, is not the only rate constant involved in determining the maximal rate of an enzyme. This becomes apparent when combining Equations 9.1 and 9.5:

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v¼

kcat [E] [S] KM

(9:15)

This shows that the ratio kcat=KM is an apparent second-order rate constant. This quantity is therefore a measure of the catalytic or enzyme efﬁciency. Combining Equation 9.15 with Equation 9.13 shows that: kcat k3 k1 ¼ KM k2 þ k3

(9:16)

This equation reveals that the ratio kcat=KM is limited by the rate constant k1, which cannot be higher than the diffusion-controlled limit. As discussed in Chapter 4, the value of bimolecular diffusion-limited rate constants is in the order of 108 to 1010 dm3 mol1 s1 (because enzymes are large molecules, k1 is usually estimated to be 108 dm3 mol1 s1) and this is then also the upper limit for kcat=KM. Michaelis–Menten kinetics for reversible reactions. So far, we have only considered reactions in one direction, from reactant to product. This implies in thermodynamic terms that the equations given are suitable for reactions which are strongly exergonic, i.e., have a large negative DrG or, equivalently, a large Keq (>>1). However, reactions for which DrG is close to zero or positive may behave differently in the sense that the reverse reaction becomes prominent. When initially only reactant is present, the reaction will take off in the direction of P, but as soon as product is formed this will have an effect on the rate. When the reaction is started in the presence of product, the rate will depend on the initial concentrations of reactants and products on the one hand and the value of DrG on the other hand and the rate may actually be zero. For foods, the situation is quite realistic that reaction products are already present when the enzyme reaction takes off. As a result, the reaction rate may become zero (or even negative from the product formation point of view). An example to illustrate this effect is the isomerization of glucose-6-P into fructose-6-P, catalyzed by the enzyme phosphoglucoisomerase. The DrG for this reaction is þ1840 J mol1. Figure 9.8 shows how dG=dar (see Equation 3.98) is affected by the concentrations of reactants and products. The reader is reminded that reactions will take place only when dG=dar < 0, as discussed in Chapter 3; this requirement is, of course, also true for enzyme reactions. The Michaelis–Menten equation as displayed in Equation 9.10 does not take this into account and would predict a ﬁnite rate for the situation that dG=dar > 0 and this is physically impossible. It is therefore essential to take reversible reactions into account as shown in Scheme 9.3. The Michaelis–Menten equation for the rate of product formation under steady-state conditions as in Scheme 9.3 can be derived as v¼

[ET ](k1 k3 [S] k2 k4 [P]) k2 þ k3 þ k1 [S] þ k4 [P]

(9:17)

The maximum forward (from reactant to product) and maximum reverse rate (from product back to reactant) leads to the following corresponding maximum rates:

k1 E+S

k3 E+P

ES k2

SCHEME 9.3 Reversible enzymatic reaction mechanism.

k4

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10,000

10,000 dG/dar (J mol−1)

15,000

dG/dar (J mol−1)

15,000

5,000

5,000

0

−5,000 −10,000 (A)

0

−5,000

0

0.1

0.2

0.3

0.4

0.5

[Glucose-6-P] (M)

−10,000

0

0.1

(B)

0.2 0.3 [Glucose-6-P] (M)

0.4

0.5

FIGURE 9.8 dG=dar for the transformation of glucose-6-P to fructose-6-P as a function of the substrate concentration glucose-6-P in the presence of reaction product concentration [fructose-6-P] ¼ 0.01 M (A) and [fructose6-P] ¼ 0.1 M (B).

vmax,f ¼ k3 [ET ]

(9:18)

vmax,r ¼ k2 [ET ]

(9:19)

The Michaelis constant KM is as in Equation 9.4, and the one for the reverse reaction is k3 þ k2 k4

(9:20)

(vmax,f =KM )[S] (vmax,r =KP )[P] 1 þ [S]=KM þ [P]=KP

(9:21)

Kp ¼

Combining the previous expressions results in v¼

This expression reduces to the common Michaelis–Menten equation (Equation 9.10) if we set [P] ¼ 0. In fact, Equation 9.21 appears to be the general form of the Michaelis–Menten equation and the commonly used one (Equation 9.10) is a special case in which the reverse reaction is neglected. Nevertheless, Equation 9.10 appears to be applicable to many enzyme systems, which then must imply that vmax,f >> vmax,r and KP >> KM. A more elaborate analysis about reversible reactions is to consider the possibility that was already shown in Scheme 9.1. This mechanism leads to an expression for the rate similar to one in Equation 9.21 but with different ones for the maximum rates and Michaelis constants: vmax,f ¼

k3 k5 [ET ] k3 þ k4 þ k5

(9:22)

vmax,r ¼

k2 k4 [ET ] k2 þ k3 þ k4

(9:23)

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KM ¼

k2 k4 þ k2 k5 þ k3 k5 k1 (k3 þ k4 þ k5 )

(9:24)

KP ¼

k2 k4 þ k2 k5 þ k3 k5 k6 (k2 þ k3 þ k4 )

(9:25)

When equilibrium is reached, the forward and reverse rates are balanced, the net rate becomes zero, and the concentrations do not change anymore (though the actual reaction itself, of course, does not stop). For reaction Scheme 9.3, the equilibrium constant can be expressed as (compare Equations 3.100 and 4.14) Keq ¼

[P]eq k1 k3 ¼ [S]eq k2 k4

(9:26)

For the reaction depicted in Scheme 9.1 the expression becomes Keq ¼

[P]eq k1 k3 k5 ¼ [S]eq k2 k4 k6

(9:27)

Combining the previous two equations with the expression for the rate, we ﬁnd [P] vmax,f vmax,r [S] Keq v¼ vmax,f [P] KM vmax,f þ vmax,r [S] þ Keq

(9:28)

Combining several previous equations leads to the following expression for the equilibrium constant Keq ¼

vmax,f KP vmax,r KM

(9:29)

This equation is called the Haldane relationship. Its importance is that it shows that the kinetic parameters are not independent of each other and that they are linked via the equilibrium constant. Here we can make a connection with the afﬁnity of a reaction as deﬁned in Chapter 3 (Equation 3.127, see also Equation 4.26) [S] vmax,f KP [S] Af ¼ RT ln Keq ¼ RT ln vmax,r KM [P] [P]

(9:30)

This equation can be useful if one wants to make a connection with irreversible thermodynamics, as discussed in Chapter 3. Enzymatic reactions in the framework of irreversible thermodynamics become increasingly important in the ﬁeld of metabolomics and systems biology. Equation 9.29 can be used to ﬁnd an expression for the forward rate in the case of a reversible reaction:

[P] vmax,f [S] K eq v¼ [P] KM 1 þ þ [S] KP

(9:31)

It is instructive to see how the value of the equilibrium constant affects the rate plot: see Figure 9.9 for a simulation. It clearly shows the importance of considering the equilibrium position when Keq is smaller than, say, 10. A completely wrong rate proﬁle would be predicted if reversibility would be neglected in

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1 0.9

Keq = 50

0.8

K eq = 3

v (arbitrary units)

0.7 Keq = 1

0.6

Keq = 0.5

0.5 0.4

Keq = 0.3

0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

[S] (arbitrary units)

FIGURE 9.9 Simulation of the dependence of rate v on substrate concentration [S] using Equation 9.31 with the following parameters: KM ¼ 0.02, KP ¼ 0.01, [P] ¼ 0.1, vmax,f ¼ 1 (arbitrary units). Keq is varied as indicated in the graph.

such cases. This example shows clearly the importance of considering both thermodynamics (via the equilibrium constant) and kinetics (enzyme kinetic parameters) for the resulting rate equations. A real example of a reversible enzyme-catalyzed reaction is the following. The enzyme enolase catalyses the hydration of phosphoenolpyruvate (PEP) to 2-phospho-D-glycerate (2PG). Initial rates for the conversion of PEP were determined in the presence of the product 2PG at various concentrations. Equation 9.21 was ﬁtted to all the data at once; incidentally, this is an example of global ﬁtting as discussed in Chapter 7: the datasets share common parameters. The parameter estimates are shown in Table 9.3 and the ﬁts in Figure 9.10. Note that the 95% conﬁdence intervals are quite narrow. This good precision is achieved due to the global ﬁtting procedure. Also note that the rate for conversion of PEP can become negative in the presence of 2PG; this means that 2PG is converted to PEP, in other words that the reverse reaction rate is positive. Equation 9.29 allows to calculate the equilibrium constant for this reaction, which happens to be 0.21 and is also obtained with good precision (see Table 9.3). Incidentally, a Lineweaver–Burke plot (discussed in the next section) of the data shown in Figure 9.10 would result in curved plots rather than straight lines (except for the experiment in which [2PG] ¼ 0).

TABLE 9.3 Parameter Estimates for the Parameters in Equation 9.21 Fitted to the Data Shown in Figure 9.10 Estimate

95% Conﬁdence Interval

vmax,f (mM min1)

0.036

0.0004

KM (mM) vmax,r (mM min1)

0.128 0.191

0.007 0.072

KP (mM)

0.142

0.051

Keq (via Equation 9.29)

0.21

0.004

Parameter

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0.04

v (µM min−1)

0.03

0.02

0.01

0

0

1

2

3 PEP (mM)

4

5

6

FIGURE 9.10 Effect of the presence of the product 2PG on the conversion of PEP by enolase. Concentrations of 2PG: ^: 0 mM, ~: 0.05 mM, ~: 0.1 mM, þ : 0.2 mM, : 0.4 mM. The lines are the ﬁts of Equation 9.21 with parameter values as shown in Table 9.3. Dataset in Appendix 9.1, Table A.9.2.

It may be useful to recapitulate the meaning of the various kinetic parameters that we encountered so far. Table 9.4 gives an overview. The discussion so far has been limited to one-substrate cases. Equally important are so-called bi–bi reactions in which two substrates are involved. It concerns especially transferases (catalyzing the transfer of a speciﬁc functional group from one substrate to another) and oxidoreductases (catalyzing oxidation– reduction reactions). Such mechanisms are discussed in Section 9.5.

9.2.1

Linearized Plots

The characteristic constants vmax and KM can be derived from experimentally determined initial rates (v0) as a function of substrate concentration [S] via nonlinear regression, as indicated above. Before the advent of computers this was (and unfortunately still is) done by inverting the Michaelis–Menten equation (Equation 9.10): 1 1 KM 1 þ ¼ v vmax vmax [S]

(9:32)

TABLE 9.4 Overview of the Relation between Various Kinetic Parameters Used in Michaelis–Menten Type Kinetics Mechanism

vmax

kcat=KM

KM

Michaelis–Menten (rapid equilibrium assumption)

k3[ET]

k2=k1

k1k3=k2

Briggs–Haldane (steady-state assumption)

k3[ET]

(k2 þ k3)=k1

(k1k3)=(k2 þ k3)

Reversible Michaelis–Menten Scheme 9.3

k3[ET]

(k2 þ k3)=k1

(k1k3)=(k2 þ k3)

Reversible Michaelis–Menten Scheme 9.1

k3k5[ET]=(k3 þ k4 þ k5)

(k2k4 þ k2k5 þ k3k5)= (k1(k3 þ k4 þ k5)

(k1k3k5)=(k2k4 þ k2k5 þ k3k5)

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1/v

1/vmax Slope = KM/vmax

1/[S] −1/KM

FIGURE 9.11

Schematic example of a Lineweaver–Burke plot.

If Michaelis–Menten kinetics is obeyed, a plot of 1=v versus 1=[S] gives a straight line with an intercept on the y-axis 1=vmax, an intercept on the x-axis of –1=KM, and a slope of KM=vmax. Such a plot is called a Lineweaver– Burke plot and the values of KM and vmax are usually estimated by linear regression, see Figure 9.11. The major objection against this procedure is of a statistical nature: by inverting v and [S], the error structure of the data is disturbed, as a result of which the conditions for linear regression are violated (as discussed in Chapter 7). Consequently, the resulting estimates of vmax and KM can be seriously biased (the seriousness depends on the actual data and the experimental design). Interestingly, Lineweaver and Burke were among the ﬁrst to appreciate this problem and advised statistical weighting to correct for the bias; weighting of the velocity data proportional to v4 should remedy the problem, but unfortunately this is very rarely done. Weighting of the substrate data would not be necessary if they are error-free but they never are, of course. Having nonlinear regression procedures available it is not necessary anymore to invert data; nonlinear regression of Equation 9.10 gives directly the estimates of vmax and KM, as shown above in Figure 9.5. An example is given to illustrate the point. Figure 9.12 shows a Lineweaver–Burke plot for the action of the protease papain on casein. The kinetic values derived from this plot via linear regression of the Lineweaver–Burke plot are KM ¼ 57.2 mM and vmax ¼ 13.0 mM min1. However, nonlinear regression on the untransformed data gives KM ¼ 27.0 mM and vmax ¼ 9.7 mM min1. Figure 9.13 shows the results for the untransformed data, and it is clear that a large bias exists in the parameters obtained via the unweighted Lineweaver–Burke plot. Admittedly, this is a rather extreme example. Apart from the distortion of the error structure of the data due to inverting them, the large bias is also due to a wrong experimental design in this case. More measurement points should have been taken at lower substrate concentrations (well below KM ¼ 27 mM). Other results may show less large deviations, but nevertheless parameters obtained via linear regression of transformed data are in principle biased! Other transformations are used as well besides the Lineweaver–Burke plot. We mention them for the sake of completeness. They were developed to facilitate estimation of kinetic parameters in days gone by when computers were not yet found on every corner of the lab bench. Strangely enough they are still used for this purpose even though that is no longer needed. The reason they are still used extensively is probably because they help to visualize the meaning of kinetic parameters. The Hanes–Woolf plot is obtained by plotting [S]=v versus [S]: [S] 1 KM [S] þ ¼ vmax v vmax

(9:33)

The Eadie–Scatchard plot is obtained by plotting v=[S] versus v: v 1 vmax vþ ¼ KM [S] KM

(9:34)

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0.7 0.6

1/v (M−1 min)

0.5 0.4 0.3 0.2 0.1 0 −0.05

0

0.05 1/[S]

0.1

0.15

(µM−1

)

FIGURE 9.12 Lineweaver–Burke plot for action of papain on casein. Dataset in Appendix 9.1, Table A.9.3.

The Eadie–Hofstee plot is v¼

KM þ vmax [S]

(9:35)

Incidentally, Scatchard plots, much used in physical chemistry, are of the same mathematical form as the Eadie–Hofstee plot, and suffer the same disadvantage for parameter estimation via transformed data. 12

10

v (M/min)

8

6

4

2

0 0

50

100

150

200

[S] (µM)

FIGURE 9.13 Michaelis–Menten plot for the action of papain on casein. The solid line reﬂects the Michaelis– Menten equation with parameters obtained via nonlinear regression, the dotted line reﬂects the Michaelis–Menten equation with parameters obtained via linear regression of the Lineweaver–Burke plot shown in Figure 9.12.

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In conclusion, it should be clear that linearized plots cannot be used for parameter estimation. However, they may be useful to present results (after having estimated the parameters from nontransformed data) because they show whether or not deviations from the expected relationship occur: the human eye is well capable to detect this in a glance for straight lines.

9.3

Enzyme Inhibition

One of the typical difﬁculties of enzyme kinetics is that enzymes are easily inhibited in various ways. Inhibitors can act such that inhibition is competitive, uncompetitive or noncompetitive. Competitive inhibition means that an inhibitor I competes with the substrate for the enzyme, so we have to expand in Scheme 9.4 the uninhibited case depicted in Scheme 9.2: The inhibition constant KI is KI ¼

[E] [I] [EI]

(9:36)

Ks is as given in Equation 9.14. The resulting equation in Michaelis–Menten terminology assuming steady-state=rapid equilibrium condition is v¼

v [S] vmax [S] max ¼ [I] aKs þ [S] þ [S] Ks 1 þ KI

(9:37)

with a¼

1þ

[I] KI

(9:38)

The other possibility is that the inhibitor binds to the ES complex (but not at the active site) such that the enzyme does not convert the substrate anymore, and this is called uncompetitive inhibition (Scheme 9.5): Ks is as given in Equation 9.14 and the inhibition constant KI0 is KI0 ¼

k1 E+S + I k5 KI

[ES] [I] [ESI]

Ks ES

(9:39)

k3

E+P

k2

k4 EI

SCHEME 9.4 Competitive enzyme inhibition mechanism. Ks and KI are the equilibrium dissociation constants for the ES and EI complex, respectively.

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9-17 Ks

k1 E+S

k3 E+P

ES k2

+ I k5

K⬘I

k4 ESI

SCHEME 9.5 Uncompetitive enzyme inhibition mechanism. Ks and KI0 are the equilibrium dissociation constants for the ES and ESI complex, respectively.

The resulting Michaelis–Menten type equation is v¼

vmax [S] v [S] ¼ max 0 [I] Ks þ a [S] Ks þ [S] 1 þ Ki [I] 0 a ¼ 1þ 0 KI

(9:40)

(9:41)

The equilibrium arrows in Schemes 9.4 and 9.5 indicate that inhibition is considered reversible in both cases. In the competitive case, inhibition is maximal at low substrate concentration, and the effect of inhibition disappears at high [S] when the number of substrate molecules is much higher than those of the inhibitor molecules, and so the substrate molecules win the competition. In other words, vmax is not affected by competitive inhibition. The opposite is the case for uncompetitive inhibition. At high [S], all enzyme is bound in the ES form and this complex is then maximally inhibited by I; at low [S], enough enzyme is left to form the ES complex (remember that the reactions are reversible). In fact, competitive and uncompetitive inhibition are two extremes where in the one extreme the inhibitor binds only the free enzyme form and in the other only the enzyme–substrate complex. A mix of both effects can occur also, and this is appropriately called mixed inhibition, and less appropriately noncompetitive inhibition. It should not come as a surprise that the resulting Michaelis–Menten equation is v¼

vmax S aKs þ a0 [S]

(9:42)

A special case of inhibition is product inhibition: this can be competitive as well as uncompetitive; see Scheme 9.6, which is an extension of Scheme 9.2. It implies that the product formed associates with the enzyme and thus hinders its own formation. In the above equations one has then to replace [I] by [P]; the difference is of course that product inhibition can only occur when the reaction is well on its way. It should not be noticeable with initial rate studies. Product inhibition can be quite important in foods,

k1 E+S

ES

k3

k2

SCHEME 9.6 Mechanism to describe product inhibition.

k4 E+P

EP k5

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Increasing [I]

1/v

[I] = 0

1/[S]

FIGURE 9.14 Lineweaver–Burke plot typical for competitive enzyme inhibition. Such a plot should only be used as a diagnosis plot of enzyme inhibition.

which is one reason why initial rate studies are not always relevant for more practical situations. Below we will show a few cases of product inhibition. It is still common practice to show enzyme inhibition in the form of linearized plots, as well as to derive the parameter estimates from such plots. In addition to the already mentioned Lineweaver–Burke plot, a so-called Dixon plot is presented in which 1=v is plotted as a function of several concentrations of I, at ﬁxed substrate concentrations. However, as the reader may have noticed, the author is of the opinion that linearized plots should not be used anymore, at least not for parameter estimation. They are still useful as diagnostic plots for a ﬁrst impression of the mechanism, or to obtain starting values for subsequent nonlinear regression. A Lineweaver–Burke plot for competitive inhibition looks as displayed in Figure 9.14, for noncompetitive inhibition as in Figure 9.15 and for uncompetitive inhibition as in Figure 9.16. A Dixon plot for noncompetitive inhibition looks as shown in Figure 9.17A, and for competitive and mixed inhibition as in Figure 9.17B. For uncompetitive inhibition the lines run parallel as in Figure 9.17C. Another phenomenon that can occur is substrate inhibition. This would become noticeable in the hyperbolic Michaelis–Menten plot if the initial rate ﬁrst increases with substrate concentration but then decreases at higher substrate concentrations (if this is observed, it is of course no proof that it is indeed substrate inhibition, there may also be other reasons for such a behavior). The kinetic scheme 00 for substrate inhibition is shown in Scheme 9.7. The KI accounts for the reversible formation of an unproductive enzyme–substrate complex ESS: 00

KI ¼

[E] [S]2 [ESS]

(9:43)

Increasing [I]

1/v

[I] = 0

1/[S]

FIGURE 9.15 Lineweaver–Burke plot typical for noncompetitive enzyme inhibition. Such a plot should only be used as a diagnosis plot of enzyme inhibition.

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Increasing [I] 1/v

[I] = 0

1/[S]

FIGURE 9.16 Lineweaver–Burke plot typical for uncompetitive enzyme inhibition. Such a plot should only be used as a diagnosis plot of enzyme inhibition.

Increasing [S] 1/v

1/v

[I]

[I]

(B)

Increasing [S]

1/v

(A)

Increasing [S]

(C)

[I]

FIGURE 9.17 Dixon plots for noncompetitive inhibition (A), competitive and mixed inhibition (B) and uncompetitive inhibition (C). Such plots should only be used as diagnosis plots of enzyme inhibition.

E+S + 2S

Ks

ES

kcat

E+P+Q

K ⬙I ESS 00

SCHEME 9.7 Kinetic scheme for substrate inhibition. Ks and KI are the equilibrium dissociation constants for the ES and ESS complex, respectively.

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The rate equation for such a situation is v¼

vmax [S] Ks [S]2 Ks þ [S] þ KI

(9:44)

It is quite likely that every type of inhibition does occur in foods with its many components. Finally, we would like to remind the reader that a situation could occur in which the reaction comes effectively to an end because thermodynamic equilibrium is achieved, as indicated above in Figures 9.9 and 9.10. This may be mistaken as inhibition but it is not, of course. The equations discussed above do not take this effect into account because one studies usually situations far from equilibrium. However, should one want to study situations close to equilibrium one should be aware of this thermodynamic limitation on rates. We now show an example of enzyme inhibition. It concerns polyphenoloxidase (PPO). The literature contains a number of examples of inhibition of this enzyme, as it is very important in relation to enzymatic browning in fruits and vegetables. Unfortunately, most of these reports still use linearized plots for parameter estimation. We would recommend to use nonlinear regression to derive the relevant parameters, for instance as in this example. It opens up more possibilities for model checking and may give better parameter estimates. This example is about the inhibition of polyphenoloxidase by oxalic acid; oxalic acid could be used as an inhibitor of enzymatic browning in foods. Initial rates were determined for a range of substrate concentrations as a function of oxalic acid concentration. The next question was to determine the kinetic mechanism by which oxalic acid inhibits PPO. Three models were tested, namely competitive inhibition, uncompetitive inhibition, and mixed inhibition, and parameters were estimated from all data available (global ﬁtting). The results in Figure 9.18 show clearly that it is deﬁnitely not uncompetitive inhibition, but the difference between a mixed inhibition model and the competitive mechanism model is not immediately clear from the graphs. The model discrimination test via the Akaike criterion shown in Table 9.5 gives preference to the mixed inhibition model, despite the fact that it has one parameter more. Unfortunately, there are no replicates available, so a goodness-of-ﬁt test cannot be performed. The precision of the parameter estimates is not very good; this is because we are trying to estimate four parameters from a limited number of data points. The ﬁts are certainly not perfect. The conclusion from this exercise would be that it is most likely that the inhibition mechanism is mixed inhibition, and the next phase of the modeling step should be to direct experimental design in such a way that the parameters can be estimated more precisely.

9.4

Progress Curves

Concentrations are usually easier to measure than rates, which is one reason why progress curves are of interest. The reaction velocity and substrate concentration vary both continuously during a reaction, and therefore a series of progress curves could contain in principle enough information for the determination of vmax and KM, but also to see whether perhaps another mechanism is active. Furthermore, it is then not necessary anymore to determine initial velocities, which is not trivial. Progress curves can be described by solving differential equations that can be written based on a postulated kinetic scheme. These differential equations can be numerically solved and ﬁtted to experimental progress curves to ﬁnd estimates for the relevant parameters. This is actually nothing more than we have advocated before, particularly in Chapters 4 and 8, but now applied to enzyme reactions. Software to do this is widely available, both commercially but also as freeware or shareware.* The familiar enzymatic * A number of calculations described in this chapter were done using the software program DynaFit (www.biokin. com=dynaﬁt). See references at the end of this chapter. This program derives differential equations based on proposed kinetic schemes. In addition, it performs the usual enzyme kinetic analyses, such as initial reaction rate versus substrate concentration and inhibition. Lineweaver and Dixon plots can be asked as diagnosis plots.

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0.8

0.9

0.7

0.8 0.7 0.6

0.5

v (AU min−1)

v (AU min−1)

0.6

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1 0 (A)

0.1

0

2

4 6 [S] (mM)

8

0

10

0

2

4 6 [S] (mM)

(B)

8

10

0.8 0.7

v (AU min−1)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

2

4

6

8

10

[S] (mM)

(C)

FIGURE 9.18 Enzyme inhibition mechanisms for the effect of oxalic acid on polyphenoloxidase. The lines describe competitive inhibition (A), uncompetitive inhibition (B), mixed inhibition (C). ^: 0 mM oxalic acid, D: 10 mM oxalic acid, þ: 30 mM oxalic acid, &: 50 mM oxalic acid. Dataset in Appendix 9.1, Table A.9.4.

parameters can still be derived if so desired. In fact, the Michaelis–Menten equation (Equation 9.10) is a rate equation that can be integrated. Integrated Michaelis–Menten equation. Recalling that the Michaelis–Menten equation in its simplest form is a rate equation describing the disappearance of substrate S and formation of product P as a function of the substrate concentration S: v¼

d[S] d[P] vmax [S] ¼ ¼ dt dt KM þ [S]

This equation can be integrated and this yields

(9:45)

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Kinetic Modeling of Reactions in Foods

9-22 TABLE 9.5

Parameter Estimates for the Inhibition of Polyphenoloxidase by Oxalic Acid

Model Competitive inhibition

Uncompetitive inhibition

Mixed inhibition

Parameter

Estimate 95% CI

vmax

0.75 0.06

Ks

0.39 0.15

KI

3.85 1.55

vmax

0.89 0.19

Ks

1.10 0.63

KI0 vmax Ks

30.9 18.1 0.78 0.06

SSr

AIC

DAIC

0.0389

144.1

3

0.170

108.7

38.4

0.0301

147.1

0

0.45 16

KI

5.48 2.73

KI0

135.9 127.6

Note: SSr, residual sums of squares; AIC, Akaike information criterion.

KM ln

[S0 ] þ [S0 ] [St ] ¼ vmax t [St ]

(9:46)

Unfortunately, this equation does not give an explicit expression for [St] or [Pt] as a function of time, while the experimental progress curve is, of course, a plot of concentration of reactant and=or product versus time. By rearranging Equation 9.46 one could ﬁnd the parameters KM and vmax from a linear plot: 1 [S0 ] [S0 ] [St ] vmax ¼ þ ln KM t [St ] KM t

(9:47)

This indicates that in principle KM and vmax can be estimated from a single progress curve. However, the earlier objections against estimating parameters from transformed data are also valid here, and use of Equation 9.47 is not a recommended method to ﬁnd KM and vmax. However, it could be an idea to use this method to ﬁnd initial estimates for KM and vmax from a single progress curve, to be used in later estimation procedures. There is, however, a possible pitfall in analyzing progress curves in this way: one has to be sure that (1) the enzyme is stable over the whole of the progress curve, (2) the reversible reaction must be negligible, (3) there must not be product inhibition. This is in contrast with the method of initial rates where one can safely assume that enzyme inactivation or product inhibition will not be noticeable at the very beginning of the reaction. If one of the conditions mentioned is not met, one will ﬁnd parameters that are confounded with other effects. It is therefore advisable to ﬁrst check the validity of the assumptions. A possible check is Selwyn’s test. Selwyn’s test. The so-called Selwyn’s test is useful to check whether the observed time course in change in substrate and=or product concentration is entirely due to the catalyzed reaction and nothing else. If the system is in steady state and the enzyme is stable and there is no inhibition, the product kcat[E0]t should be constant. This implies that the time courses for reactions with various enzyme concentrations [E0] should be superimposable when the time axis is multiplied by [E0]. Figure 9.19 gives an example of such an analysis; this clearly shows for this case that there are no confounding effects taking place. Figure 9.20 shows another case where Selwyn’s test shows that the mechanism is not a just a catalyzed reaction from reactant to product; rather, there could be product inhibition, or inactivation of the enzyme in the course of the reaction. What the complication is needs to be further investigated, of course (this is done below for this particular case).

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9-23

250

250

200

200 [Malate] (µM)

[Malate] (µM)

Enzyme Kinetics

150

100

150

100

50

50

0

0 0

5

(A)

10

15 20 t (min)

25

30

0

2

(B)

4 6 8 [E0]∗ t (nM min)

10

12

FIGURE 9.19 Progress curves for the hydration of fumarate to malate by the enzyme fumarase (A) and Selwyn’s test for this reaction (B). : [E0] ¼ 108 pM, &: [E0] ¼ 216 pM, ~: [E0] ¼ 432 pM; [S] ¼ 253 mM. Dataset in Appendix 9.1, Table A.9.5.

9

9

8

8

7

7

6

6

5 4

5 4

3

3

2

2

1

1

0 (A)

[P] (mM)

[P] (mM)

Analysis of progress curves. The analysis of progress curves is best illustrated by some examples. The ﬁrst case concerns the action of b-galactosidase on a synthetic substrate, for which ﬁve progress curves are available. We take Scheme 9.2 as starting point and assume rapid equilibrium, which implies a diffusion-limited rate constant k1 ¼ 6 3 106 mM min1 and we are left with the task to estimate

0

20

40

60 t (min)

80

100

120

0 (B)

0

20

40

60 80 ∗ −1 (ET) t (µg mL min)

100

120

FIGURE 9.20 Progress curves (A) and Selwyn’s plot (B) for the hydrolysis of 4-nitrophenyl phosphate by alkaline phosphatase. ^: 141 mg mL1 enzyme, &: 282 mg mL1 enzyme, D: 564 mg mL1 enzyme, : 846 mg mL1 enzyme, j : 1128 mg mL1 enzyme. Dataset in Appendix 9.1, Table A.9.6. 3

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9-24

k2 and k3. (Estimation of k1 was not well possible with the present dataset.) The resulting differential equations are d[E] ¼ k1 [E] [S] þ k2 [ES] þ k3 [ES] dt d[S] ¼ k1 [E] [S] þ k2 [ES] dt d[ES] ¼ k1 [E] [S] k2 [ES] k3 [ES] dt d[P] ¼ k3 [ES] dt

(9:48)

These equations were solved numerically and ﬁtted to the data. Note that we do not need a steady-state assumption here because we do not need an analytical expression for the rate if we solve the equations numerically. The result of solving the equations in Equation 9.48 and ﬁtting them to the data via nonlinear regression is in Figure 9.21. The results are not very good overall, there is clearly a misﬁt. The mechanism of product inhibition was therefore introduced, Scheme 9.6, yielding the following set of differential equations: d[E] ¼ k1 [E] [S] þ k2 [ES] þ k3 [ES] k4 [E] [P] þ k5 [EP] dt d[S] ¼ k1 [E] [S] þ k2 [ES] dt d[ES] ¼ k1 [E] [S] k2 [ES] k3 [ES] dt d[P] ¼ k3 [ES] k4 [E] [P] þ k5 [EP] dt d[EP] ¼ k4 [E] [P] k5 [EP] dt

(9:49)

4.5 4 3.5

[PNPG] (mM)

3 2.5 2 1.5 1 0.5 0

0

100

200 300 Time (min)

400

500

FIGURE 9.21 Progress curves for the action of galactosidase on PNPG and ﬁt of the Michaelis–Menten mechanism according to Equation 9.48 (solid lines). Dataset in Appendix 9.1, Table A.9.7.

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Enzyme Kinetics

9-25

4.5 4 3.5

[PNPG] (mM)

3 2.5 2 1.5 1 0.5 0

0

100

200 300 Time (min)

400

500

FIGURE 9.22 Progress curves for the action of galactosidase on PNPG and global ﬁt of the Michaelis–Menten mechanism þ product inhibition according to Equation 9.48 (solid lines). Dataset in Appendix 9.1, Table A.9.7.

The same rate constant was assumed for formation of the ES and EP complex, i.e., k1 ¼ k4 in Scheme 9.5, and this rate constant was ﬁxed at its diffusion limit value of 6 3 106 mM1 min1. This left three rate constants to be estimated from this scheme. The results are in Figure 9.22, and they are much better than the ones displayed in Figure 9.21, though certainly not perfect when looking at the residuals. Since replicate data were not available we cannot test the goodness of ﬁt, unfortunately. Table 9.6 shows the parameter estimates and model discrimination test via the Akaike criterion. The model with product inhibition is clearly favored despite the presence of one parameter more. TABLE 9.6

Parameter Estimates for the Action of Galactosidase on the Synthetic Substrate PNPG

Model

Estimate 95% CI

Parameter

Rapid equilibrium

Rapid equilibrium þ product inhibition

k1 (mM1 min1)

6 3 106 (ﬁxed)

k2 (min1)

1.25 3 107 5.5 3 106

k3 (min1)

1.29 3 104 2.84 3 103

Ks ¼ k2=k1 (mM)

2.08 0.9

vmax (k33[E]) (mM min1)

0.013 0.003

k1 (mM1 min1)

6 3 106 (ﬁxed)

k2 (min1) k3 (min1)

1.30 3 107 4.55 3 106 2.42 3 104 5.0 3 103

k4 (mM1 min1)

6 3 106 (ﬁxed)

1

Ks (mM)

2.17 0.77

vmax (k33[E]) (mM min1)

0.024 0.005

KI (mM)

0.59 0.17

KM (mM)

2.6 0.45

vmax (mM min1) 1

Note: k1 and k4 ﬁxed at 6 3 10 mM criterion. 6

1

AICc

DAIC

260

107

0.115

367

0

3.52 3 106 1.04 3 106

k5 (min )

Michaelis–Menten via initial rates

SSr 0.705

0.024 0.002 6

min . [E] ¼ 10

mM. SSr, residual sums of squares; AIC, Akaike information

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9-26

160 140

v (mM min-1) ⫻ 104

120 100 80 60 40 20 0

0

1

2

3

4

5

[S] (mM)

FIGURE 9.23 Michaelis–Menten plot for initial rates of galactosidase acting on PNPG, as shown in Figures 9.21 and 9.22. The line is obtained via nonlinear regression. Dataset in Appendix 9.1, Table A.9.7.

For comparison, we show in Figure 9.23 the Michaelis–Menten plot obtained from initial rates that were available for the same case. There is, of course, no effect of product inhibition because only initial velocities are used. The KM and vmax estimates are also given in Table 9.6. They compare favorably to the estimates found from the global ﬁt to the progress curves, where Ks and vmax can be calculated indirectly from the rate constants, at least if we assume the product inhibition mechanism. For the plain rapid equilibrium scheme the result for vmax does not match very well, which is another indication that the plain mechanism is not the correct one. The next example is a further analysis of the plots shown in Figure 9.19 about the action of a phosphatase. Selwyn’s test already indicated that this was not a straightforward case of one-substrate irreversible catalysis without any disturbance. Here we analyze two possible causes for this behavior, namely ﬁrst-order enzyme inactivation and product inhibition according to Scheme 9.8 (compare Scheme 9.6). The model to describe the experimental results was in terms of a Michaelis–Menten equation derived by applying the steady-state assumption: d[P] k3 [E] [S]=KM ¼ [S] [P] dt 1þ þ KM KP d[E] ¼ dt

E+S

k1 k2

ES

(9:50)

ki [E] [S] [P] þ 1þ KM KP k3

E+P

k4 k5

EP

ki E⬘

SCHEME 9.8 Kinetic scheme describing enzyme inactivation and product inhibition. E0 symbolizes inactivated enzyme.

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Enzyme Kinetics

9-27

9 8 7

[P] (mM)

6 5 4 3 2 1 0 0

20

40

60 Time (min)

80

100

120

FIGURE 9.24 Global ﬁt of Equation 9.50 (solid lines) to progress plots for the hydrolysis of 4-nitrophenyl phosphate by alkaline phosphate. ^: 141 mg mL1 enzyme, &: 282 mg mL1 enzyme, D: 564 mg mL1 enzyme, : 846 mg mL1 enzyme, 3j : 1128 mg mL1 enzyme. Dataset in Appendix 9.1, Table A.9.6.

Introduction of only the step of enzyme inactivation with rate constant ki without product inhibition (by putting [P] ¼ 0 in Equation 9.50) was unsuccessful; product inhibition without enzyme inactivation (ki ¼ 0) was also unsuccessful: it was impossible to ﬁnd a suitable ﬁt of the model to the data. Only when product inhibition and enzyme inactivation were combined a ﬁt was found: see Figure 9.24 and Table 9.7. The ﬁt shown in Figure 9.24 is reasonable but certainly not perfect, especially not for the higher enzyme concentration. There appeared also to be a statistical problem since the conﬁdence interval for KM could not be estimated. Probably, there is not enough information in the data to extract all parameter values. Nevertheless, the analysis indicates that the mechanism in Scheme 9.8 is not unreasonable and a better experimental design for further analysis would probably solve the statistical problem of estimating the precision of KM. The next example shows how inhibition can be estimated from progress curves. It concerns the action of pepsin on a chromogenic substrate in the presence of various amounts of a pepstatin analog inhibitor. TABLE 9.7 Parameter Estimates Obtained by Fitting Equation 9.50 to the Data Displayed in Figure 9.24 Parameter

Estimate

95% Conﬁdence Interval

k3 (mmol min1 mg1)

0.946

0.035

KM (mM)

2.61

Indeterminate

KP (mM)

1.587

0.191

ki (min1)

0.218

0.008

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9-28

Kinetic Modeling of Reactions in Foods

The task is to ﬁnd the kinetic parameters that describe the inhibition. The substrate concentration was not varied, so a plot like in Figure 9.14 through 9.17 could not be made. The mechanisms of competitive and uncompetitive inhibition were investigated, as displayed in Schemes 9.4 and 9.5. For the competitive mechanism the following differential equations can be derived. d[E] ¼ k1 [E] [S] þ k2 [ES] þ k3 [ES] k4 [E] [I] þ k5 [EI] dt d[S] ¼ k1 [E] [S] þ k2 [ES] dt d[ES] ¼ k1 [E] [S] k2 [ES] k3 [ES] dt d[P] ¼ k3 [ES] dt d[I] ¼ k4 [E] [I] þ k5 [EI] dt d[EI] ¼ k4 [E] [I] k5 [EI] dt

(9:51)

It was not possible to estimate all the rate constants from the available data and so the following assumptions were made: .

.

.

The rate constant for the binding of substrate and inhibitor was assumed to be the same (k1 ¼ k4). Rapid equilibrium was assumed so that we can put k1 equal to the diffusion limit, estimated to be 108 M1 s1 for enzymes. The inhibition constant KI in Scheme 9.4 is written in terms of the rate constants k5 and k4: KI ¼ k4=k5. By making these assumptions, these rate equations could be simultaneously solved using numerical integration and ﬁtted to the available datasets and resulted in parameter estimates for the rate constants k2, k3, and k5.

The ﬁts are shown in Figure 9.25 and the parameter estimates in Table 9.8. Similarly, kinetic equations were derived for the uncompetitive inhibition Scheme 9.5 with k1 ¼ k4: d[E] ¼ k1 [E] [S] þ k2 [ES] þ k3 [ES] dt d[S] ¼ k1 [E] [S] þ k2 [ES] dt d[ES] ¼ k1 [E] [S] k2 [ES] k3 [ES] k4 [ES] [I] þ k5 [ESI] dt d[P] ¼ k3 [ES] dt d[I] ¼ k4 [ES] [I] þ k5 [ESI] dt d[ESI] ¼ k4 [ES] [I] k5 [ESI] dt

(9:52)

This model did not ﬁt well (plots are not shown); model comparison and goodness-of-ﬁt analyses are reported in Table 9.9. It is beyond doubt that the uncompetitive mechanism is not valid at all, while the goodness of ﬁt for the competitive mechanism is quite acceptable. This is perhaps not unexpected because the pepstatin analog is supposed to work according to a competitive mechanism, but the model comparison is shown here to make clear how it can be used to discriminate between models based on progress curve analysis.

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Enzyme Kinetics

9-29

100 90 80 70

[S] (µM)

60 50 40 30 20 10 0

0

1

2 3 Time (min)

4

5

FIGURE 9.25 Global ﬁts of the competitive inhibition kinetic scheme to progress curves of pepsin activity in the presence of a pepstatin analog. [E] ¼ 0.04 mM, [S] ¼ 100 mM, pH 4, T’ ¼ 37 8C, ^: [I] ¼ 0 mM, ~: [I] ¼ 0.05 mM, : [I] ¼ 0.1 mM, þ : [I] ¼ 0.2 mM, *: [I] ¼ 0.3 mM, &: [I] ¼ 0.5 mM, ~: [I] ¼ 1.0 mM, &: [I] ¼ 2.0 mM. Dataset in Appendix 9.1, Table A.9.8.

There are many more things to be said about the analysis of progress curves, but we leave it here. It is hoped that the reader has obtained enough inside information to handle the abundant literature on this topic.

9.5

Kinetics of Two-Substrate Reactions

So far, we have only considered enzyme-catalyzed reactions in which one substrate is converted to one product. However, Table 9.1 showed already that there are more possibilities. Next to one-substrate mechanisms, reactions in which two substrates are converted into products are equally important. It concerns the bi–bi and bi–uni reactions in Table 9.1. One makes usually the distinction between so-called ping-pong mechanisms and sequential mechanisms. A ping-pong mechanism means that one or more products need to TABLE 9.8 Parameter Estimates for the Kinetic Competitive Inhibition Scheme Applied to the Inhibition of Pepsin by a Pepstatin Analog (Figure 9.25) Parameter k4 ¼ k4 (mM1 s1) k2 (s1) k3 or kcat (s1) k5 (s1)

Estimate 100 (ﬁxed) 2113 7514 9.2

95% Conﬁdence Interval — 263 24.5 0.6

Derived Parameters Ks ¼ k2=k1 ¼ 21 mM KI ¼ k4=k5 ¼ 0.09 mM vmax ¼ k3 [E] ¼ 300 mM s1

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9-30

TABLE 9.9 Goodness of Fit and Model Discrimination for Competitive and Noncompetitive Inhibition of Pepsin by a Pepstatin Analog F-Value for Lack of Fit

Mechanism

Sampling Probability

AIC

DAIC

Competitive inhibition

1.00

0.491

162

0

Uncompetitive inhibition

9.427

0.000

230

392

be released before all substrates can react, while a sequential mechanism means that ﬁrst all substrates need to combine with the enzyme before products can be formed. A further distinction can be made into ordered and random sequential mechanisms. Ordered sequential mechanisms imply that there is a speciﬁc order in going from substrates to products, while this is not so for random sequential mechanisms. Let us ﬁrst consider the random sequential bi–bi mechanism. Scheme 9.9 shows the model for this situation. The resulting rate equation for this scheme is v [A] [B] ¼ vmax KsA K AB þ K AB [A] þ K BA [B] þ [A] [B]

(9:53)

vmax ¼ kcat [ET ] With two substrates, one can hold one of the reactants constant and vary the other. The response of the system in such a case helps to establish the kinetic parameters. So, if [A] is kept constant, Equation 9.53 becomes v [B] ¼ 0 0 vmax K þ [B] vmax [A] 0 ¼ BA vmax K þ [A] K A K AB þ K AB [A] K 0 ¼ s BA K þ [A]

(9:54)

Similar expressions hold when [B] is kept constant v

[S] þ [A] v [B] max 0 ¼ AB vmax K þ [B] K A K AB þ K BA [B] K 0 ¼ s AB K þ [B]

0 vmax

E+A + B

¼

KA s

K0

(9:55)

EA + B KAB

K Bs

EAB

EB + A K BA

kcat

E+P+Q

SCHEME 9.9 Scheme for the random sequential bi–bi mechanism for the two substrates A and B. The K symbols represents the equilibrium dissociation constants for the enzyme complexes in the reaction indicated.

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Enzyme Kinetics

9-31

KAB

KA s E+A

EA + B

kcat

EAB

E+P+Q

SCHEME 9.10 The ordered sequential bi–bi mechanism for the two substrates A and B. The K symbols represents the equilibrium dissociation constant for the enzyme complexes in the reaction indicated.

The case for an ordered sequential bi–bi mechanism is depicted in Scheme 9.10. Examples of enzymes acting like this are lipoxygenase and polyphenoloxidase. v [A] [B] ¼ vmax KsA K AB þ K AB [A] þ [A] [B]

(9:56)

vmax ¼ kcat [ET ] When [B] is kept constant, Equation 9.56 becomes v [A] ¼ 0 0 vmax K þ [A] 0 vmax [B] 0 vmax ¼ AB K þ [B] KsAB K AB K 0 ¼ AB K þ [B]

(9:57)

For constant [A] it becomes v [B] ¼ 0 0 vmax K þ [B] 0 vmax ¼ vmax

K0 ¼

KsA K AB [A]

(9:58) þ K AB

The kinetic scheme for a ping-pong bi–bi mechanism is shown in Scheme 9.11. Xanthine-oxidase is an enzyme that shows this mechanism. The resulting rate equation is v [A] [B] ¼ vmax (kcat =k2 )KmA [B] þ KmB [A] þ [A] [B](1 þ kcat =k2 )

(9:59)

vmax ¼ kcat [ET ]

A Km

E+A

k1 k−1

B Km

EA

k2

E⬘+ B

k3 k−3

E⬘B

kcat

E+Q

P

SCHEME 9.11 The ping-pong bi–bi mechanism for the two substrates A and B. The K symbols represents the equilibrium dissociation constant for the enzyme complexes in the reaction indicated.

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Random

Random Ping-pong K⬘

K⬘

Ping-pong

Ordered (A)

Ordered

(B)

FIGURE 9.26 Dependence of the apparent Michaelis constant K0 on concentration of (A) when (B) is held constant (A) and on (B) when (A) is held constant (B) for random-sequential, ordered-sequential, and ping-pong mechanisms.

When [B] is kept constant, Equation 9.59 becomes v [A] ¼ 0 0 vmax K þ [A] v0 [B] 0 ¼ Bmax vmax Km þ [B] (kcat =k2 )KmA [B] K0 ¼ KmB þ [B]

(9:60)

When [A] is kept constant, Equation 9.59 becomes v [A] ¼ 0 0 vmax K þ [A] vmax [A] 0 ¼ vmax (kcat k2 )KmA þ [A] KmA [B] K0 ¼ (kcat =k2 )KmA þ [A]

(9:61)

A relevant example of bi–bi mechanisms for foods is, once again, the oxidation of polyphenols by polyphenoloxidase, where oxygen and the polyphenol are the two substrates. It is not an easy task to ﬁnd out which mechanism is operative. One can get an impression of the kind of mechanism by studying the behavior of K0 as a function of the two substrates A and B for a ﬁrst impression as is very schematically depicted in Figure 9.26. Likely models can then be tested via model discrimination as discussed above for one substrate mechanisms.

9.6

Other Types of Enzyme Kinetics

Cooperative and allosteric enzymes. In so-called cooperative enzymes, there are low- and high-afﬁnity substrate binding sites. If the binding of one substrate molecule results in altered substrate binding afﬁnity, this is called cooperativity. The binding afﬁnity can increase or decrease in principle, but usually it is an increase. If enzyme activity is affected by binding to other sites than the active site, we speak of allosteric enzymes. This binding can be substrate binding (homotropic allosteric response) as well as

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Enzyme Kinetics

9-33

other ligands (heterotropic allosteric response). It is not always possible to distinguish between allosteric effects and cooperativity. Cooperative effects result in sigmoidal curves when v is plotted versus [S] rather than hyperbolic Michaelis–Menten type curves. Michaelis–Menten kinetics is therefore not suitable for cooperative and allosteric effects. A general equation that appears to be suitable to describe cooperativity is the so-called Hill equation: v¼

vmax [S]h KHh þ [S]h

(9:62)

The exponent h is called the Hill coefﬁcient, and expresses the degree of cooperativity in the substrate binding process; the higher h, the higher the cooperativity. KHh is called the Hill constant and is a measure for the afﬁnity of the enzyme for its substrate; however, it does not reﬂect the enzyme–substrate dissociation constant. For h ¼ 1, the Hill constant equals the Michaelis constant. Figure 9.27 shows some simulated curves described by the Hill equation. Note that if h ¼ 1, the Hill equation reduces to the Michaelis–Menten equation (Equation 9.10). If one ﬁnds experimentally sigmoid curves, one could derive the usual parameters plus the Hill coefﬁcient via nonlinear regression. Figure 9.28 shows a real life example. The initial rate–concentration plot is deﬁnitely not hyperbolic, but sigmoidal and therefore the Hill equation was ﬁtted to the data, which gives a reasonable ﬁt, also shown in Figure 9.28. Immobilized enzyme kinetics. Immobilized enzymes are increasingly used, also in food technology. Enzymes can then be used much more efﬁciently, and usually their stability is much enhanced. The reaction is carried out in enzyme reactors. The enzyme is bound on a carrier, which is kept in a reactor, the substrate is fed to the reactor and converted by the enzyme, and the product is removed. This can be done in batch reactors, in plug ﬂow reactors and in continuous-stirred reactors. In addition, membrane

1.2

1

h=4 h=3

v (arbitrary units)

0.8

h=2 h=1

0.6

0.4

0.2

0

0

0.5

1 [S] (arbitrary units)

1.5

2

FIGURE 9.27 Simulated curves using the Hill equation to describe cooperativity. vmax ¼ 1 (arbitrary units) and KM ¼ 0.5 (arbitrary units), the Hill coefﬁcient h is varied (Equation 9.62).

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0.25

v (mM min−1)

0.2

0.15

0.1

0.05

0

0

5

10 15 Cholesterol (mM)

20

FIGURE 9.28 Initial rate of cholesterol oxidase as a function of cholesterol concentration. The line is the ﬁt of the Hill equation with parameters vmax ¼ 0.298 mM min1, KH ¼ 11.8 mM, h ¼ 2.6. Dataset in Appendix 9.1, Table A.9.9.

reactors are also possible. The kinetics of immobilized enzymes do not fundamentally change relative to the nonimmobilized ones, but there may be shifts in parameter values when compared to the nonimmobilized enzyme. Interfacial enzyme kinetics. When there are different phases present in a food, an interfacial area will exist. The most noticeable example is, of course, the presence of fat droplets in oil-in-water emulsions or water droplets in water-in-oil emulsions. Enzyme catalysis can take place at such interfaces, such as lipases hydrolyzing triglycerides. That implies that before the actual catalysis takes place enzymes must bind to the interface, and, incidentally, it may then undergo a conformational change due to the adsorption process. This way of substrate binding is of course different from the enzyme-substrate binding discussed hitherto. One way to tackle interfacial kinetics is via the simple Scheme 9.12. The adsorption of the enzyme to the interface can be described by a Langmuir type equation, with the enzyme coverage parameter q and at equilibrium this results in q¼

E* [E] ¼ * Kd* þ [E] Emax

(9:63)

This equation thus describes the surface coverage by interfacial enzyme as a function of the enzyme concentration in solution. Emax * is the saturation surface concentration of interfacial enzyme. The equilibrium constant describing dissociation from the interface Kd* is (compare Scheme 9.12)

ka E kd

E∗ + S

k1 k2

ES

k3

E+P

SCHEME 9.12 Simpliﬁed scheme representing interfacial kinetics. E symbolizes free enzyme (mol dm3), E* the enzyme bound to the interface (mol m2).

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Kd* ¼

kd ka

(9:64)

The change in interfacial enzyme coverage is described as dq * E*)As kd (E*)As ¼ ka [E](Emax dt

(9:65)

As is the surface area per unit volume (m2 dm3). We can now set up the following equations: v ¼ k3 (E*)As

(9:66)

[E](Emax * E*) E*

(9:67)

[ET ] ¼ [E] þ (E*)As

(9:68)

Kd* ¼

Rearranging and normalization by [ET] leads to v¼

vmax aint Kd* þ aint

(9:69)

* )(1 q)As aint ¼ (Emax

(9:70)

vmax ¼ k3 [ET ]

(9:71)

Thus, we end up again with a Michaelis–Menten type equation (Equation 9.69) describing a hyperbola: see Figure 9.29. Figure 9.30 shows some simulations for various values of the parameters involved to see how these parameters inﬂuence the resulting rate plot. King–Altman procedure. The King–Altman procedure is an ingenious graphically based procedure developed to derive manageable rate expressions for complicated enzymatic reaction schemes. The

12 vmax

10

v (M s−1)

8 6 4 2 0

0 K ∗d

50 As (m2 dm−3)

100

FIGURE 9.29 Initial rate v versus interfacial area per unit volume As describing interfacial enzyme kinetics.

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12 10

12

K ∗d (M)

J = 0.1 Ε ∗max = 0.00001

10−6

Ε *max = 0.00001 K ∗d = 0.00001

10

10−5

0.1 0.4 0.6

8 v (M s−1)

v (M s−1)

8 6 10−4

J=

0.8

6 4

4

2

2 −3

10 0 (A)

0

2

4

6

8

10

0

12

As (m2 dm−3)

0

2

4

(B)

12

10

12

Ε ∗max (mol m−2) 10−4

J = 0.1 K ∗d = 0.00001

10

6 8 As (m2 dm−3)

10−5

v (M s−1)

8

6 10−6

4 2 0

0 (C)

2

4

6

8

10

12

As (m2 dm−3)

FIGURE 9.30 Effect of the parameters Kd* (A), q(B), and E*max (C) on initial rate v versus surface area As for the interfacial kinetics model.

procedure is based on graph theory and mainly used in biochemical engineering. The graphs consist of vertices, representing enzyme species, and arcs, representing interconversion between any two enzyme species; pseudo ﬁrst-order reversible reactions are postulated between the various enzyme forms. Weights are given by the appropriate rate constant. If a rate constant ¼ 0, then the arc from one enzyme to another is not given. The results lead to steady-state rate equations. We do not discuss it further; the interested reader can ﬁnd some references at the end of this chapter.

9.7

Temperature Effects

As was discussed in Chapter 5, temperature has a large effect on reaction rates. This is also true for enzyme-catalyzed reactions, but there are differences. Enzymes being proteins, they may denature and

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10

2

9

1.8

8

1.6

7

1.4

6

1.2

5

1

4

0.8

3

0.6

2

0.4

1

0.2

0 (A)

9-37

ln k

v (µM−1 s−1)

Enzyme Kinetics

0

10

20

30 40 T ⬘(⬚C)

50

60

0

70

0

20

(B)

40 T ⬘(⬚C)

60

80

FIGURE 9.31 Effect of temperature on the activity of the enzyme galactosidase (A) and on the rate constant for the action of invertase on sucrose (B). Dataset in Appendix 9.1, Tables A.9.10 and A.9.11.

loose their enzyme activity. Also the activation energy and entropy is different from the uncatalyzed reaction. Figure 9.31 gives two experimental examples. This ﬁgure clearly indicates that, starting from low temperature, increasing temperature increases the reaction rate but only up to a maximum; after a certain temperature is reached, the activity rapidly declines, due to denaturation (unfolding) of the enzyme. The resulting temperature-activity plot is the balance of higher activity at higher temperature and the loss of active enzyme because of denaturation. If we consider the condition that the enzyme is saturated with substrate (i.e., [S] >> KM) the rate is as expressed in Equation 9.9, v ¼ kcat [ET]. The concentration of active enzyme is the total amount minus the inactive, i.e., denatured form. An equilibrium constant can be deﬁned for the ratio between the concentration of native [EN] and denatured enzyme [ED] in the scheme EN Ð ED: Keq ¼

[ED ] [EN ]

(9:72)

(The intricacies of protein unfolding are discussed in the next chapter.) The mass balance for the enzyme is [ET ] ¼ [EN ] þ [ED ]

(9:73)

In order to describe experimental results as shown in Figure 9.31 we can write the rate as a function of enzyme concentration v(T) ¼ kcat (T)[EN ] ¼

kcat (T)[ET ] 1 þ Keq (T)

(9:74)

In this last equation, both Keq and kcat are made temperature dependent. For the dependence of kcat we can use the Arrhenius equation (Equation 5.12) Ea kcat (T) ¼ A exp RT

(9:75)

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0.6

2.5

0.5

2

In k

ln (activity)

0.4 1.5 0.3

1 0.2

0.5

0 0.0031

0.1

0.0032

0.0033

(A)

1/T

0.0034

(K −1)

0.0035

0 0.0033 0.0034 0.0034 0.0035 0.0035 0.0036 0.0036 0.0037

0.0036

(B)

1/T (K −1)

FIGURE 9.32 Arrhenius plot for the activity of the enzyme galactosidase (A) and on the rate constant for the action of invertase on sucrose (B). Dataset in Appendix 9.1, Tables A.9.10 and A.9.11.

Alternatively the Eyring equation (Equation 5.11) could be used. The temperature dependence of Keq is more complicated 0 0 DHd DCP0 DCP Td DHd0 1 DCP0 (1 þ ln Td ) þ ln T þ Keq (T) ¼ exp RTd R R R R T

(9:76)

This equation combines basically the van’t Hoff equation (Equation 5.2) and the relation between heat capacity DCp0 and enthalpy, Equation 3.16. It is assumed in this derivation that DCp0 is temperature independent over the range where unfolding takes place. Td and DHd0 are the temperature and enthalpy change at the denaturation temperature. Equations 9.75 and 9.76 can be used to ﬁt proﬁles such as the ones in Figure 9.31, provided that the parameter values in Equation 9.76 are known. We now focus on the increase in activity well before it reaches the maximum; to be sure, the position of such a maximum is, of course, enzyme speciﬁc. Figure 9.32 shows an Arrhenius plot of the data shown in Figure 9.31. Table 9.10 shows the parameter estimates obtained via nonlinear regression of the Eyring equation (see Chapter 5). The activation enthalpy is rather low compared to the one for a ‘‘normal’’ chemical reactions (usually around 100 kJ mol1), and this is expected because a low activation energy is a typical consequence of catalysis. The activation entropy is negative because the formation of the enzyme–substrate complex leads to less conformational possibilities. In other words, the magnitude of the rate constant is much higher and the temperature dependence of catalyzed reactions, including

TABLE 9.10 Parameter Estimates Resulting from Nonlinear Regression of the Eyring Equation on the Data Shown in Figure 9.32 Parameter

Galactosidase (95% CI)

Activation enthalpy (kJ mol1)

37 2.7

Activation entropy (J mol1 K1)

112 9

Invertase (95% CI) 9.2 1.4 210 5

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kcatalyzed ln k

Rate enhancement

kuncatalyzed 1/T

FIGURE 9.33 reaction.

Schematic drawing of rate enhancement accomplished by enzymes as compared to the uncatalyzed

enzyme reactions, is substantially lower than for an uncatalyzed reaction. The rate enhancement of enzymes is particularly high at lower temperatures. This is schematically shown in Figure 9.33. A word of caution, however, is necessary here. When it comes to interpretation of the estimates as in Table 9.10, it is not straightforward to relate the parameters to the basic rate constants in, for instance, the Michaelis–Menten Scheme 9.2. The resulting activity is a combination of the various reaction rate constants and concentrations, and the temperature dependence found from an Arrhenius or Eyring plot of enzyme activity cannot be attributed to one single reaction rate constant, or a single elementary step. This situation has been discussed basically in Equations 5.21 through 5.28, and the analysis given there is perfectly applicable to the temperature dependence of enzyme kinetic constants.

9.8

pH Effects

Enzymes being proteins are sensitive to pH and it is quite common in literature to report a pH proﬁle, showing the dependence of enzyme activity on pH. Figure 9.34 gives one of the many examples available, in this case liver enzymes acting on metmyoglobin being much more active in the lower pH range, and a pectinesterase, which seems to have its pH optimum between pH 7 and 9. Many models are available to describe the effect of pH. We will consider only a simple case (proposed by Michaelis in 1911) which shows the general line of thinking. The enzyme is considered to be a dibasic acid with two nonidentical acidic groups, and dissociation can take place as depicted in Scheme 9.13. The dissociation constants Ki are called group dissociation constants. The total enzyme concentration is [ET ] ¼ [HEH] þ [ EH] þ [HE ] þ [E ]

(9:77)

The following equations can be derived for the various enzyme species: [HEH] ¼

[ET ] K1 þ K3 K1 K2 þ þ2 1þ [Hþ ] [H ]

[ET ]K1 [Hþ ] [ EH] ¼ K1 þ K3 K1 K2 þ þ2 1þ [Hþ ] [H ]

(9:78)

(9:79)

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3.5

120

3

100

80 % Activity

v ⫻ 108/M−1 s−1

2.5 2 1.5

60

40 1 20

0.5 0

0 5

(A)

5.5

6

6.5

7

5

7.5

pH

6

7 pH

(B)

8

9

FIGURE 9.34 pH–activity plot for liver enzymes on metmyoglobin reduction (A) and papaya pectinesterase on pectin (B). Dataset in Appendix 9.1, Tables A.9.12 and A.9.13.

[ET ]K3 [Hþ ] [HE ] ¼ K1 þ K3 K1 K2 þ þ2 1þ [Hþ ] [H ]

(9:80)

[ET ]K1 K2 [Hþ ]2 [E ] ¼ K1 þ K3 K1 K2 þ þ 2 1þ [Hþ ] [H ]

(9:81)

K1 K 2 ¼ K 3 K 4

(9:82)

Furthermore,

The above relations are shown graphically in Figure 9.35. −

EH

K1

K2

HEH

E− −

K4

K3 HE−

SCHEME 9.13 Dissociation possibilities for a dibasic acid of an enzyme E. The K-values represent dissociation constants.

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1 0.9

E− −

HEH

Relative concentration

0.8

−

EH

0.7 0.6 0.5 0.4 0.3 0.2 HE−

0.1 0

3

4

5

6

7

8

9

10

pH

FIGURE 9.35 Relative concentrations of enzyme forms with a dibasic acid as a function of pH. Numerical values for dissociation constants are K1 ¼ 13106, K2 ¼ 13108, K3 ¼ 13106.8, K4 ¼ 13107.2.

One can imagine certain pH proﬁles assuming that enzyme activity is reﬂected by concentrations of one of the forms depicted in Figure 9.35. The plot shown for HEH is reﬂected in the behavior shown in Figure 9.34A. If EH is the active form, for instance, the pH dependency will be bell shaped as the curve for [EH]; such behavior is reminiscent of what is shown in Figure 9.34B. The initial rate will in such a case vary according to v¼

kcat [ET ] [Hþ ] K1 K2 =(K1 þ K3 ) þ 1þ K1 þ K3 [Hþ ]

(9:83)

The effect of pH on enzyme activity is often plotted as the logarithm of a kinetic constant versus pH; this is known as a Dixon–Webb plot. Let us consider as a relatively simple case a Michaelis–Menten type equation in which the enzyme as well as the enzyme–substrate complex is protonated (Scheme 9.14). Applying the steady-state assumption results in

vmax [S] 1 þ ([Hþ ]=K2 ) v¼ 1 þ ([Hþ ]=K1 ) þ [S] KM 1 þ ([Hþ ]=K2 )

(9:84)

We thus have an apparent vmax and KM as a function of pH 0 vmax ¼

vmax [Hþ ] 1þ K2

(9:85)

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9-42 k1 E+S

H+

K1

K2

HE

SCHEME 9.14

k3

ES

E+P

H+ HES

Protonation of free enzyme and of the enzyme–substrate complex.

1 0 [Hþ ] 1 þ B K1 C 0 C KM ¼ KM B @ [Hþ ]A 1þ K2

(9:86)

Figure 9.36 shows a Dixon–Webb plot for these apparent kinetic parameters. More complicated expressions can be derived, but we leave it here because the general approach is as given here. Some selected references on this topic are given at the end of this chapter.

9.9

Experimental Design for Enzyme Kinetics

2.5

2.5

2

2

1.5

1.5

log (K ⬘M)

log (v ⬘max)

Experimental design was already discussed in Chapter 7 and the general rules given there also apply, of course, to enzyme experiments. As a reminder, proper experimental design is about how to set up experiments such that most information is extracted from the data. There is considerable literature on experimental design speciﬁcally for enzyme experiments; some references are listed below. As discussed in Chapter 7, an important consideration is what the goal of the experiment is ﬁnding out a particular mechanism (model discrimination), ﬁnding the most precise estimates (parameter estimation), or making predictions. Depending on that goal one can choose a particular design. As shown before, it is

1

0.5

0

1 pK1 = 5

0.5

5

(A)

FIGURE 9.36

6

7

8 pH

9

0

10 (B)

pK2 = 7

3

4

5

6

7

8

9

10

pH

Dixon–Webb plot for the apparent vmax (A) and KM (B) according to the mechanism in Scheme 9.14.

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necessary to have replicate results so that experimental error can be estimated and modeled. As a rule of thumb, for simple Michaelis–Menten kinetics, substrate concentrations should be chosen around the KM value, but not necessarily equally spaced around KM. Prior knowledge about the magnitude of KM is needed, obviously, in order to make such designs. One can also approach the design problem as a decision problem: one has to decide about what are the most useful settings to obtain information. In this sense, this becomes a Bayesian inspired design in maximizing a utility function. This is an interesting new approach; some references are given at the end of this chapter.

9.10

Enzyme Kinetics in Foods

The complex matrix of food is quite a challenge when studying enzyme kinetics: initially enzymes may be separated from substrate and inhibitors due to compartmentalization. Upon processing or because of ripening processes, enzymes may encounter substrate but also inhibitors and activators. However, it is difﬁcult to predict how and what will happen. That means that the only way to ﬁnd out is via experimentation. An interesting discussion arises as to whether such studies should be done in ‘‘clean’’ model systems, or in the foods themselves. When done in clean systems, one learns probably more about the enzyme itself, but less about what the effect in the food will be. In terms of food quality prediction it does not seem to make much sense to work with puriﬁed enzymes and substrates, except when the interest is purely in the mechanistic effect. Having said that, the food matrix is probably too complex to study in its whole, so some sort of model system is needed. On the other hand, when studies are carried out in the food, there are so many possible complications that it will be hard to draw conclusions. The author is of the opinion that both types of studies should be done, starting at the mechanistic level, i.e., with puriﬁed systems, but working towards the food situation. One should mimic the food situation as closely as possible eventually. Typical food problems that can ‘‘disturb’’ enzyme kinetics are . . . . .

. .

Compartmentalization of enzyme and substrate Limiting substrates Mass transfer limitations Presence of inhibitors Conditions for Michaelis–Menten kinetics may not apply in foods: no clean initial velocities, no saturating substrate, possibility of product inhibition and enzyme inactivation Enzymes as well as substrates and products are unstable Effects of ionic strength, pH

One particularly important effect in foods is the presence of seemingly inert molecules. Even though they are inert, they can have a large effect on kinetics. This is the topic of molecular crowding. Molecular crowding effects on enzyme activity. The phenomenon of molecular crowding, to be discussed further in Chapter 14 on reactions in the food matrix, is basically a volume exclusion effect. In a crowded solution, less volume is available to molecules than in a dilute solution and as a result molecules behave highly nonideal in the thermodynamic sense. The effect is obviously much larger for macromolecules. The effect of nonideal behavior can be expressed in activity coefﬁcients. Molecular crowding effects are currently studied in detail for biochemical reactions in cells. No doubt, these effects are equally important in foods. To be sure, the consequence of molecular crowding is that activities of reactants (including enzymes) can be strongly inﬂuenced by cosolutes, also if these cosolutes are inert with respect to the reactants themselves. For proteins the main consequences of molecular crowding are that association of macromolecules is favored. When this happens, there is a reduction in volume and this is thermodynamically favorable in a crowded environment. Crowding also enhances the stability of folded proteins with respect to the unfolded state; this is relevant for stability of enzymes: the denaturation temperature will go up, as is discussed in Chapter 10. If an enzymatic reaction is depending on enzymes that are active as

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9-44

K⬘a

Ka E+S

ES

ES∗

K‡ ES‡

kcat

E+P

SCHEME 9.15 Kinetic scheme showing the various intermediates in enzymatic catalysis in terms of transition state theory. K represents the thermodynamic equilibrium constant for the complexes shown.

associated oligomers, crowding will have a direct effect on enzymatic activity because association is promoted. In contrast, enzymes acting as monomers will lose activity if they become associated. If the association of enzyme and substrate leads to a more compact ES complex, molecular crowding will enhance the formation of the ES complex (i.e., lower the Michaelis constant KM). Then there is also the effect of crowding on diffusion, which is relevant for the enzyme–substrate encounter rate: the more crowded, the more difﬁcult diffusion becomes. A semiquantitative explanation for crowding effects is as follows. The analysis is based on transition state theory discussed in Chapter 5; see also Figure 9.1. We consider a kinetic scheme as displayed in Scheme 9.15. Assuming molar concentrations with activity coefﬁcients y (see Chapter 3) we can write for each thermodynamic equilibrium constant yES [ES] [ES] ¼ Ya yE yS [E] [S] [E] [S]

(9:87)

Ka0 ¼

y * [ES* ] [ES* ] ES ¼ Ya0 yES [ES] [ES]

(9:88)

Kz ¼

yESz [ESz ] [ESz ] ¼ Yz y * [ES* ] [ES* ] ES

(9:89)

Ka ¼

We will concentrate on an initial rate expression and after some algebraic manipulation it can be shown that v ¼ kcat [ESz ] ¼

kcat Ka Ka0 K z [E] [S] Ya Ya0 Y z

(9:90)

This can in turn be written as v¼

kcat Ka Ka0 K z [E] [S] h i K0 Ya þ Ka [S] 1 þ Ya0 Ya0 Y z

(9:91)

a

The vmax,id in ideal conditions (all activity coefﬁcients equal unity) is vmax,,id ¼

kcat Ka0 K z [E] 1 þ Ka0

(9:92)

and this leads to the expression v¼

vmax,id Ka0 K z [S](1 þ Ka0 ) h i K0 Ya þ Ka [S] 1 þ Ya0 Ya0 Y z a

(9:93)

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We now have an expression for the initial rate as function of substrate concentration with an explicit dependence on activity coefﬁcients. For the enzyme rate expressions given in previous sections it has been assumed tacitly that activity coefﬁcients are unity, i.e., thermodynamic ideal behavior was assumed. The task now becomes to ﬁnd expressions for the activity coefﬁcients. A general expression for an activity coefﬁcient is 2 yi ¼ exp42Bi,i [i] þ

X

3 Bi,j [j] þ 5

(9:94)

j6¼i

Bi,i and Bi,j are osmotic second virial coefﬁcients describing physical interactions between pairs of molecules i and between pairs of molecules i and j, respectively. For instance, the activity coefﬁcient for free enzyme is * [ES*] þ BE,ESz [ESz ] þ ] yE ¼ exp [2BE,E [E] þ BE,S [S] þ BE,ES [ES] þ BE,ES

(9:95)

In dilute systems the molar concentrations of the enzyme species are very small, which implies that the term in the exponent in Equation 9.95 is virtually zero, so that the activity coefﬁcient for the enzyme yE is unity. However, when high concentrations of an inert cosolute C are present, the term BE,C[C] cannot be neglected anymore and as a result yE will be different from unity. Similarly, the interaction between the cosolute C and other species cannot be neglected anymore and these terms become dominant to the extent that the following approximation is valid: Ya expb(BES,C BE,C BS,C )[C]c

(9:96)

* BES,C )[C]c Ya0 expb(BES,C

(9:97)

Y z exp [(BESz ,C B*

(9:98)

ES ,C

)[C]]

In the case of impermeable spheres, the second virial coefﬁcient describing physical interaction between components i and j having radii ri and rj, respectively, is Bi,j ¼

4pNAV (ri þ rj )3 zi zj (1 þ kri þ krj ) þ 3 2I(1 þ kri )(1 þ krj )

(9:99)

The distance of closest approach is, obviously, ri þ rj, and the ﬁrst term in the right hand side of this equation describes the volume from which the two molecules exclude each other mutually, the so-called covolume. The second term in the right hand side of the equation describes electrostatic interaction in Debye–Hückel terms (discussed in Chapter 6): z is the charge, k the Debye screening length and I the ionic strength. For uncharged species, the last term disappears, of course. For species that cannot be described as impenetrable spheres, such as random coil polymers, the following expression approximates the second virial coefﬁcient: Bi,C

" 1=2 # 2 2 4 2 2 ¼ Nav r i lC þ p r i lC þ 4 p p ri3 3 3 3

(9:100)

Overlooking all this, several phenomena related to molecular crowding can be explained, at least qualitatively, but perhaps even quantitatively to the extent that the given expressions are reasonable approximations. If we ﬁrst consider the case that the cosolute affects only the interaction between enzyme and substrate (Ya 6¼ 1, Ya0 ¼ 1, Y z ¼ 1), Equation 9.93 results in

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9-46

K0 Vmax Ka (1 þ a [S] Ya v¼ Ka0 [S] 1 þ Ka (1 þ Ya

(9:101)

For KM in the presence of cosolute C can be written KM,C ¼ KM expb(BES,C BE,C BS,C )[C]c

(9:102)

The effect of crowding in this case of thermodynamic nonideality between enzyme and substrate is that there is tighter binding of substrate because (BE,C þ BS,C) > BES,C. vmax is not effected by the cosolute in such a case. If, however, the cosolute has an effect on subsequent steps in the catalysis process, there is an effect on vmax. The effect of cosolutes on maximal velocities is vmax ¼

vmax ,id (1 þ Ya0 ) (Ya0 þ Ka0 )Y z

(9:103)

Hence, vmax will be enhanced in crowding conditions if the formation of ESz goes along with a volume decrease because BES*,C > BESz,C . On the other hand, if the activated ESz complex has the larger volume such that BES*,C < BESz,C , vmax will decrease. With the aid of this semiquantitative analysis of molecular crowding effects can be predicted. Indeed, the predicted effects have also been observed. An example is shown in Figure 9.37 where the activity of the enzyme glyceraldehyde-3-phosphate dehydrogenase is shown as a function of the enzyme concentration. The activity of this enzyme decreases with its concentration. This is explained by the fact that this enzyme is subject to self-association into tetramers and the monomer has a much larger speciﬁc activity than the tetramer; with increasing concentration more monomer associates and therefore the activity 3.5 3 2.5

log v

2 1.5 1 0.5 0 −9

−8

−7 log (cGAPD)

−6

−5

FIGURE 9.37 Activity of the enzyme glyceraldehyde-3-phosphate dehydrogenase (GAPD) as a function of enzyme concentration (in M) in the absence (.) and in the presence of 18 g dm3 of the macromolecular crowder ribonuclease (). Dataset in Appendix 9.1, Table A.9.14.

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2.5

v (mM min−1)

2

1.5

1

0.5

0

0

0.2

0.4 0.6 Sucrose (M)

0.8

1

FIGURE 9.38 Effect of molecular crowding: inﬂuence of sucrose on enzymatic activity of invertase. The solid line indicates the Michaelis–Menten equation with parameters vmax ¼ 2.3 mM min1 and KM ¼ 0.025 M. Dataset in Appendix 9.1, Table A.9.15.

decreases. Addition of macromolecules that do not take part in the enzyme reaction decreases enzyme activity nevertheless as shown in Figure 9.37. It was not only observed for the crowder shown here (ribonuclease) but also with b-lactoglobulin, bovine serum albumin, polyethyleneglycol. This is a typical effect due to macromolecular crowding: addition of inert macromolecules decreases the available volume and induces therefore the self association of the enzyme. Figure 9.38 shows just another simple example where the activity of invertase is inﬂuence by the presence of the substrate sucrose. Of course, the results displayed in Figure 9.38 could also be due to substrate inhibition (since sucrose is the substrate for invertase) as well as of an effect of viscosity increase affecting the encounter rate of enzyme and substrate, but it was ascertained in this case that these factors are not the cause for the effect that enzyme activity markedly decreased at high solute concentration, and the sole explanation remaining is the effect of molecular crowding by sucrose. The main message here is that the food scientist working with enzymes should be prepared that when he=she moves from test tube experiments to real foods that effects of molecular crowding can make orders of magnitudes difference on kinetics for reasons indicated.

9.11

Concluding Remarks

It is clear that the topic of enzymatic reactions in foods is an important but complicated one and the kinetics can be quite intricate. Rate equations for complete reversible reaction mechanisms can be very lengthy and tedious to handle. It is essential to reduce such equations to more simple ones, be it that these simpler equations should still give an adequate prediction at the conditions that the reaction is studied. The most relevant conditions for enzymes are the presence of activators and inhibitors, product and substrate inhibitions, temperature, pH, ionic strength, and the degree of molecular crowding. This chapter has given some tools to handle these situations. However, the reader is advised that the topic is vast and much more can be said about enzyme kinetics, and indeed many textbooks and reviews are available. Some of them are listed in the ‘‘Bibliography and suggested further reading’’ section.

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9-48

Appendix 9.1

Datasets Used for Examples in This Chapter

TABLE A.9.1 Initial Rates of Sucrose Hydrolysis by Invertase as a Function of Substrate Concentration (Figures 9.5 and 9.6) Initial Rate (min1)

Sucrose (M) 0.0292

0.182

0.058

0.258

0.0584

0.265

0.0876

0.311

0.117 0.117

0.33 0.342

0.146

0.349

0.175

0.372

0.205

0.347

0.234

0.371

Source: From Chase A.M., Meier H.Cv., and Menna V.J. The non-competitive inhibition and irreversible inactivation of yeast invertase by urea. J Cell Comp Physiol 59:1–13, 1962.

TABLE A.9.2 [E] ¼ 14.7 nM

Product Inhibition by 2-Phospho-D-Glycerate (2PG) on Enolase Action on PEP (Figure 9.10) PG ¼ 0

PG ¼ 50 mM

PG ¼ 200 mM

PG ¼ 400 mM

1

Rate (mM min )

PEP (mM) 0

0

0.05 0.06

0.0084 0.0125

0.09

0.0154

0.15

0.019

0.25

0.0235

0.35 0.5

PG ¼ 100 mM

0

0.0077 0.0284

0.8

0.0143

0.0012

0.0208

0.012

1 1.2

0.0323

0.0252

0.0182

0.0018 0.0052

2

0.0333

0.0292

0.0246

2.5

0.0338

0.0298

0.0268

0.0195

0.0067

3

0.0341

0.0316

0.0282

0.022

0.0108

3.5

0.0344

0.0319

0.0293

0.0238

0.0141

4

0.0346

0.0298

0.0258

2.2

4.23 4.53

0.0035

0.0261 0.0264

0.0178

5

0.02

5.5

0.0211

Source: From Duggleby R.G. Product inhibition of reversible enzyme-catalysed reactions. Biochim Biophys Acta Protein Struct Mol Enzymol 1209:238–240, 1994.

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Enzyme Kinetics

9-49 TABLE A.9.3 Action of Papain on Casein (Figures 9.12 and 9.13) v (mM min1)

S (mM) 9.5

1.72

19

3.91

28.5

5.2

38

5.28

57

7.75

85.5

7.61

95

7.5

114.5 133

8.03 7.93

171.5

7.7

190

8.465

Source: Data taken from Ritchie, R.J. and Prvan, T. Biochem. Educ., 24, 196, 1996; Tayyab, S. and Quamar, S., Biochem. Educ., 20, 116, 1992.

TABLE A.9.4

Effect of Oxalic Acid on Polyphenoloxidase (Figure 9.18) 0 mM Oxalic Acid

10 mM Oxalic Acid

30 mM Oxalic Acid

50 mM Oxalic Acid

v (AU min1)

v (AU min1)

v (AU min1)

v (AU min1)

Substrate Catechol (mM)

0

0

0

0

0

0.3

0.255

0.1125

0.08125

0.0275

0.5

0.41

0.19

0.125

0.04625

1

0.6125

0.325

0.2375

0.11375

4

0.7375

0.555

0.48125

0.27875

10

0.73125

0.57375

0.5325

0.42875

Source: From Son S.M., Moon K.D., and Lee C.Y. Kinetic study of oxalic acid inhibition on enzymatic browning. J Agric Food Chem 48:2071–2074, 2000.

TABLE A.9.5 Hydration of Fumarate to Malate by Fumarase (Figure 9.19). t (min)

E (pM)

P (mM)

0.5

1.08E-04

6.32

4

1.08E-04

30.04

7

1.08E-04

49.8

10

1.08E-04

67.19

13

1.08E-04

83

16

1.08E-04

98.02

19 22

1.08E-04 1.08E-04

110.67 122.53 133.6

25

1.08E-04

3

2.16E-04

43.48

6

2.16E-04

76.68 (continued )

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9-50

TABLE A.9.5 (continued) Hydration of Fumarate to Malate by Fumarase (Figure 9.19). t (min)

E (pM)

P (mM)

9

2.16E-04

105.14

12

2.16E-04

128.06

15

2.16E-04

147.04

18

2.16E-04

163.64

21

2.16E-04

174.7

24

2.16E-04

184.98

2

4.32E-04

53.75

5 8

4.32E-04 4.32E-04

111.46 150.99

11

4.32E-04

175.49

14

4.32E-04

191.3

17

4.32E-04

200.79

20

4.32E-04

204.74

23

4.32E-04

207.11

Source: Duggleby, R.G., Methods Enzymol., 249, 61, 1995.

TABLE A.9.6 Hydrolysis of 4-Nitrophenyl Phosphate by Alkaline Phosphatase (Figure 9.20 and 9.24) Time (min)

[E] ¼ 141 mg mL1

[E] ¼ 282 mg mL1

[E] ¼ 564 mg mL1

[E] ¼ 846 mg mL1

[E] ¼ 1128 mg mL1

[P] in mM 0 2.0

1.639

2.3 2.35

0.521 0.894

2.36 2.64

1.341 0.297

4.24

2.533

6.11 6.35

3.389 0.594

7.0

1.19

7.6 8.6

2.308 3.463

9.2 10.4

4.357 0.854

10.8

3.909

11.0

1.599

11.4

4.953

12.0

3.089

13.5 14.1

3.312

14.7 15.3 15.4

4.505 1.076 1.97 5.473

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Enzyme Kinetics

9-51

TABLE A.9.6 (continued) Hydrolysis of 4-Nitrophenyl Phosphate by Alkaline Phosphatase (Figure 9.20 and 9.24) Time (min)

[E] ¼ 141 mg mL1

[E] ¼ 282 mg mL1

[E] ¼ 564 mg mL1

16.3 3.721

17.5

2.193 1.223

18.8

6.031

19.1 19.7

5.323 3.981

20.6 21.5

2.341 1.297

22.2

5.694

22.5

4.241

22.55

6.439

23.0 24.0 26.2

2.489 1.371 4.538

26.23

5.954

26.5 26.74

2.674 1.444

28.4 29.8

6.959 1.48

29.8

2.822

29.9 32.1 33.2

4.76 6.399 1.591

33.3

2.933

35.8 36.6

5.056 1.664

38.6

7.403

39.1 42.3 42.5

3.117 6.88 1.737

45.7

5.388

49.6 53.3

3.374 1.845

56.5

7.918

60.2

7.321

63.9

5.68

67.5 70.1 70.9

3.554 8.1 1.913

74.1

7.428

78.1

5.899

81.4 85.1 85.2

[E] ¼ 1128 mg mL1

4.914

16.9 18.7

[E] ¼ 846 mg mL1

3.661 1.983 8.132

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Kinetic Modeling of Reactions in Foods

9-52

TABLE A.9.6 (continued) Hydrolysis of 4-Nitrophenyl Phosphate by Alkaline Phosphatase (Figure 9.20 and 9.24) Time (min)

[E] ¼ 141 mg mL1

[E] ¼ 282 mg mL1

[E] ¼ 564 mg mL1

[E] ¼ 846 mg mL1

89.2

[E] ¼ 1128 mg mL1

7.535

92.9

5.968

96.0

8.203

96.2 99.6

3.694 1.978

99.7 100.7

7.568 8.164

102.8

6.002

104.7

7.566

106.4

3.727

108.0 109.8

6.037 2.012

111.4 115.0

3.726 2.01

Source: Duggleby, R.G., Biochim. Biophys. Acta Protein Struct. Mol. Enzymol., 1205, 268, 1993.

TABLE A.9.7 Galactosidase Action on the Synthetic Substrate PNPG (p-Nitrophenyl a-D-Galactopyranoside) (Figures 9.21 and 9.22) Time (min)

PNPG(mM)

0

0.24

0.48

1

2

4

5

0.231

0.462

0.97

1.951

3.936

10

0.221

0.445

0.94

1.895

3.863

20

0.203

0.413

0.885

1.803

3.741

40

0.168

0.355

0.786

1.616

3.506

60

0.144

0.313

0.7

1.473

3.3

120 180

0.09 0.056

0.226 0.158

0.537 0.405

1.115 0.82

2.782 2.355

240

0.036

0.115

0.312

0.631

2.037

305

0.026

0.08

0.249

0.534

1.87

360

0.012

0.066

0.189

0.44

1.703

420

0.012

0.061

0.183

0.422

1.616

Initial velocities (Figure 9.23) S (mM) 0.24 0.48

v0 mM min1 20.1 37.7

1 2

65 105.6

4

144.6

Source: From Durance T.D., Makhijani S.B.J., and Nakai S. Determination of reaction order by linearization and its application to estimation of Michaelis-Menten parameters. J Food Biochem 10:107–115, 1986.

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Enzyme Kinetics TABLE A.9.8 Time (min)

9-53

Inhibition of Pepsin by a Pepstatin Analog (Figure 9.25)

I¼0 mM

I ¼ 0.05 mM

I ¼ 0.1 mM

I ¼ 0.2 mM

I ¼ 0.3 mM

I ¼ 0.5 mM

I ¼ 1 mM

I ¼ 2 mM

[S] in mM 0

100

100 99.26

100 98.6

100 99.33333

100

100

0.05

98.66

98.66

99.8

0.05

98.13

99.06

98.33

0.05

98

98.6

99.3

98.73333

99.2

99.8

98.66667

98.86

0.1

97.33

97.06

98

98.4

97.8

0.1 0.1

97.46 96.73

97.26 97.93

97.33 98.06

98.53333 98.06667

98.06 98.73

0.5

87.8

89.26

91.4

91.93333

0.5

87.66

88.86

89.13

92.33333

0.5

87.53

88.86

91.13

91.06667

93.06

1

77.13

78.46

81.93

84.6

85.93

1

76.06

78.73

80.26

84.53333

85.66

1

75.66

79.73

82.13

83.86667

87.06

1.5 1.5

66.2 65.66

69.33 69.06

73.06 71.5

77.73333 76.66667

80.2 79.33

1.5

65.46

69.8

73.66

76.6

80.66

2

55.8

60.73

64.66

70.8

74.33

79.2

2

55.46

60.06

63.26

71

73.2

79.2

2

54.93

61.33

65.26

69.73333

74.46

2.5

46.53

52.46

57.06

64.8

68.2

2.5

45.93

51

55.93

64.06667

67.33

2.5 3

45.26 37.93

52.06 44.66

58.13 50.13

63 58.4

3

37.33

43.93

48.06

3

36.67

44.53

51.66

3.5

29.87

37.26

3.5

29.067

36.33

3.5

29.33

4 4 4

100 99.6 99.6

100 100.13 99.33

99

100.13

98.33

99

99.4

98.73

98.33 98.8

99 99.33

92.33

94.2

95.93

98.13

93.06

93.73

95.93

96.86

95.73

97.33

89.06

92.8

96.6

89.46

92.6

94.4

92.13

95.33

89.93 89.8

94.8 92.73

89.46

94.06

86.4

91.86

86.06

91.06

85.66

92

75.06

83

90

75

83.46

88.33

68.66 63.06

70.8

82.86 80.06

89.8 88.13

58.73333

62.06

70.53

57.26667

62.73

43.2

51.8

57.8

41.13

52.33333

56.2

37.73

44.46

50.86667

58.4

23.26

30.73

37.06

46.66667

53.06

22.33333 21.73

30.2 31.33

35.2 39

47.2 45.73333

51.73 52.86

4.5

16.06

24.8

30.93

41.33333

48.53

58.26

71.73

83.13

4.5

16.06

23.8

29.66

41.46667

47

57.86

71.33

80.93

4.5

15.66

25.33

32.66

40.33333

48

72.06

81.6

5

10.4

19.4

26.2

36.86667

43.8

54.26

80.93

5

11.06

18.26

24.06

37.2

42.26

54.06

79.46

5

10.33

19.53

27.26

36

44.4

Source: From examples given in www.biokin.com/dynaﬁt.

84 84.33

79.73

86.86

79.66

87.46

66.06

77.2

86.4

66.2

77

85.06

77.06

85.46

62.06

74.26

85.13

62.06

73.46 73.6

82.53 83.73

80.26

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Kinetic Modeling of Reactions in Foods

9-54

TABLE A.9.9 Cholesterol Oxidase Action on Cholesterol (Figure 9.28) [Cholesterol] (mM)

Initial Rate (mM min1)

0

0

1.43

0.005

1.43

0.01

2.72

0.02

2.72

0.015

3.26

0.02

3.26

0.02

3.95 3.95

0.02 0.03

4.35

0.03

4.35

0.03

4.89

0.03

4.84

0.04

5.28

0.03

5.19

0.04

6.62 6.57

0.04 0.05

7.85

0.06

7.8

0.07

8.25

0.07

8.25

0.08

9.68

0.1

9.68

0.11

10.32 10.32

0.12 0.13

12.94

0.17

12.94

0.18

14.42

0.2

14.42

0.2

15.41

0.2

15.41

0.21

16 16

0.2 0.21

17.88

0.22

17.83

0.22

19.11

0.22

19.11

0.22

19.11

0.23

Source: From Vasudevan P.T. and Zhou T. Kinetics of cholesterol oxidation by cholesterol oxidase. Appl Biochem Biotechnol 60:63–72, 1996.

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Enzyme Kinetics

9-55 TABLE A.9.10 Effect of Temperature on Activity of Galactosidase (Figures 9.31A and 9.32A) Temperature (8C)

Activity (mM s1)

11.1

1.202

16.1

1.549

21.1

2.042

25.5

2.951

30.9

3.548

35.6

4.571

40.5

5.623

45.5 47.5

7.413 6.761

50.6

8.511

52.7

9.120

55.9

8.511

57.0

7.413

59.2

4.786

60.3

2.951

Source: From Hei D.J. and Clark D.S. Estimation of melting curves from enzymatic activity temperature proﬁles. Biotechnol Bioeng 42:1245–1251, 1993.

TABLE A.9.11 Effect of Temperature on Invertase Activity (Figures 9.31B and 9.32B) T=K

k (min1)

276

1.05

277

1.06

284

1.09

290

1.14

293

1.16

298

1.22

304

1.31

310 315

1.45 1.51

320

1.84

325

1.93

330

1.86

333

1.68

337

1.52

338

1.28

339 339

1.22 1.12

339

1.08

341

1.05

342

1.04

342

1.02

Source: From Westphal G., Vogel J., and Pusch D. Prozessberechnung der enzymatischen Saccharosehydrolyse in Abhangigkeit von der Temperatur. Acta Biotechnol 8:357–365, 1988.

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Kinetic Modeling of Reactions in Foods

9-56

TABLE A.9.12 Effect of pH on Enzymatic Reduction of Metmyoglobin (Figure 9.34A) pH

vmax (3108 M s1)

5.06

3.28

5.26

3.01

5.36

2.99

5.39

2.52

5.6

2.1

5.74

2.03

5.8

2.68

5.84 5.9

2.43 2.27

5.91

1.64

5.95

2.08

6.03

1.71

6.08

1.87

6.1

1.76

6.2

1.44

6.5 6.54

0.85 0.85

6.7

0.83

6.75

0.7

6.8

0.67

6.94

0.55

7.19

0.47

7.31

0.53

7.31

0.41

Source: From Mikkelsen A. and Skibsted L.H. Kinetics of enzymatic reduction of metmyoglobin in relation to oxygen activation in meat products. Z Lebensm Unters Forsch 194:9–16, 1992.

TABLE A.9.13 Effect of pH on Papaya Pectinesterase (Figure 9.34B) pH

% Activity

5.0

6.2

6.0 7.0

54.3 95.4

7.5 8.0

99.0 100

8.5

91.8

9.0

88.3

Source: From Fayyaz A., Asbi B.A., Ghazali H.M., Che Man Y.B., and Jinap S. Kinetics of papaya pectinesterase. Food Chem 53:129–135, 1995.

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Enzyme Kinetics

9-57 TABLE A.9.14 Effect of Macromolecular Crowding due to Ribonuclease on Activity of Glyceraldehyde-3Phosphate Dehydrogenase (Figure 9.37) log v (AU min1)

log v with 18 g dm3 Ribonuclease

8.96

2.91

2.86

8.8

2.91

8.64

2.91

log c (M)

8.44

2.87

8.44

2.8

8.26 8.06

2.77 2.69

7.95

2.9

2.54

7.77

2.39

7.64

2.81

7.5

2.77

2.15

7.28

2.83

2.03

6.97

2.71

6.81 6.56

2.36

6.27

2.16

1.81

6.12 6.01

1.57 1.99

5.51

1.47

Source: Minton, A.P. and Wilf, J., Biochemistry, 20, 4821, 1981.

TABLE A.9.15 Effect of Molecular Crowding by Sucrose on Invertase Activity (Figure 9.38) Sucrose [S] (M)

v (mM min1)

0

0

0.01

0.34

0.02 0.04

0.73 1.35

0.06

1.64

0.08

1.73

0.1

1.8

0.13

1.73

0.16

1.67

0.21

1.6

0.26 0.32

1.53 1.44

0.41

1.33

0.49

1.26

0.61

1.24

0.75

1.06

0.9

0.99

Source: Shearwin, K.E. and Winzor, D.J., Arch. Biochem. Biophys., 260, 532, 1988.

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Kinetic Modeling of Reactions in Foods

Bibliography and Suggested Further Reading General Textbooks Alberty R.A. Thermodynamics of Biochemical Reactions. Hoboken, NJ: Wiley Interscience, 2003. Cornish-Bowden A. Fundamental enzyme kinetics. 3rd ed., London: Portland Press, 2004. Leskovac V. Comprehensive Enzyme Kinetics. New York: Kluwer Academic=Plenum Publishers, 2003. Marangoni A.G. Enzyme Kinetics. A Modern Approach. Hoboken, NY: Wiley Interscience, 2003. Segel L.A. Biological Kinetics. Cambridge Studies in Mathematical Biology, Cambridge University Press, 1991. Whitaker J.R. Principles of Enzymology for the Food Sciences. New York: Marcel Dekker, 1994. Whitaker J.R., Voragen A.G.J., and Wong D.W.S. Handbook of Food Enzymology. New York: Marcel Dekker, 2003. About Michaelis–Menten Kinetics Heinrich R., Melendez-Hevia E., and Cabezas H. Optimization of kinetic parameters of enzymes. Biochem Mol Biol Educ 30:184–188, 2002. Hoh C.Y. and Cord-Ruwisch R. A practical kinetic model that considers endproduct inhibition in anaerobic digestion processes by including the equilibrium constant. Biotechnol Bioeng 51:597– 604, 1996. Keleti T. Two rules of enzyme kinetics for reversible Michaelis-Menten mechanisms. FEBS 208:109–112, 1986. Michaelis L. and Menten M.L. Die Kinetik der Invertinwirkung. Biochem Z 49:334–369, 1913. Northrop D.B. On the meaning of KM and V=KM in enzyme kinetics. J Chem Educ 75:1153–1157, 1998. Núnez de Castro I. and Alonso F.J. Energy diagrams for enzyme catalysed reactions: A confusing point in the textbooks. Biochem Educ 25:87–89, 1997. Ratkowsky D.A. A suitable parameterization of the Michaelis-Menten enzyme reaction. Biochem J 240:357–360, 1986. Ritchie R.J. and Prvan T. Current statistical methods for estimating the Km and Vmax of MichaelisMenten kinetics. Biochem Educ 24:196–206, 1996. Schnell S. and Hanson S.M. A test for measuring the effects of enzyme inactivation. Biophys Chem 125:269–274, 2007. Selwyn M.J. Michaelis-Menten data: misleading textbook examples. Biochem Educ 23:138–141, 1995. Tenekedjiev K. and Kolev K. Introduction to interpretation of stochastic parameters. Biochem Mol Biol Educ 30:414–418, 2002. Wachsstock D.H. and Pollard T.D. Transient state kinetics tutorial using the kinetics simulation program KINSIM. Biophys J 67:1260–1273, 1994. About Lineweaver–Burke Plots Chong D.P. On the use of least squares to ﬁt data in linear form. J Chem Educ 71:489–490, 1994. Tayyab S. and Quamar S. A look into enzyme kinetics: Some introductory experiments. Biochem Educ 20:116–118, 1992. Ritchie R.J. and Prvan T. Current statistical methods for estimating the Km and Vmax of MichaelisMenten kinetics. Biochem Educ 24:196–206, 1996. About Progress Curves Duggleby R.G. Analysis of progress curves for enzyme-catalyzed reactions—application to unstable enzymes, coupled reactions and transient-state kinetics. Biochim Biophys Acta Protein Struct Mol Enzymol 1205:268–274, 1993. Duggleby R.G. Analysis of enzyme progress curves by nonlinear regression. Meth Enzymol 249:61–90, 1995.

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Duggleby R.G. Quantitative analysis of the time courses of enzyme-catalyzed reactions. Methods 24:168– 174, 2001. Straathof A.J.J. Development of a computer program for analysis of enzyme kinetics by progress curve ﬁtting. J Mol Catal B: Enzym 11:991–998, 2001. Willeman W.F., Hanefeld U., Straathof A.J.J., and Heijnen J.J. Estimation of kinetic parameters by progress curve analysis for the synthesis of (R)-mandelonitrile by Prunus amygdalus hydroxynitrile lyase. Enzyme Microbial Technol 27:423–433, 2000. About the King–Altman Method Chou K.C. Graphic rules in steady and non-steady state enzyme kinetics. J Biol Chem 264:12074–12079, 1989. King E.L. and Altman C. A systematic method of deriving rate laws for enzyme-catalysed reactions. J Phys Chem 60:1375–1378, 1956. Cornish-Bowden A. Fundamental Enzyme Kinetics. 3rd ed., Chapter 4. London: Portland Press, 2004. Leskovac V. Comprehensive Enzyme Kinetics, Chapter 4. New York: Kluwer Academic=Plenum Publishers, 2003. Software Programs for Enzyme Kinetics Analysis Kuzmic P. Program DYNAFIT for the analysis of enzyme kinetic data: application to HIV proteinase. Anal Biochem 237:260–273, 1996. Web Site: www.biokin.com=dynaﬁt Straathof A.J.J. Development of a computer program for analysis of enzyme kinetics by progress curve ﬁtting. Journal of Molecular Catalysis B: Enzymatic 11:991–998, 2001. Web Site for the computer program Encora: http:==www.tnw.tudelft.nl=live=pagina.jsp?id ¼ ca553f24–64c7–4d5ea344-ce30830894ab&lang ¼ en About Experimental Design for Enzyme Kinetics Dette H. and Biedermann S. Robust and efﬁcient designs for the Michaelis-Menten model. J Am Stat Assoc 98:679–686, 2003. Duggleby R.G. Experimental designs for the distribution-free analysis of enzyme kinetic data. In: Kinetic Data Analysis, Endrenyi L. (Ed.), pp. 169–179. New York: Plenum Press 1981. Lindner P.F.O. and Hitzmann B. Experimental design for optimal parameter estimation of an enzyme kinetic process based on the analysis of the Fisher information matrix. J Theor Biol 238:111–123, 2006. López-Fidalgo J. and Wong W.K. Design issues for the Michaelis-Menten model. J Theor Biol 215:1–11, 2002. Murphy E.F., Gilmour S.G., and Crabbe M.J.C. Effective experimental design: Enzyme kinetics in the bioinformatics era. Drug Discov Today 7:S187-S191, 2002. Murphy E.F., Gilmour S.G., and Crabbe M.J.C. Efﬁcient and accurate experimental design for enzyme kinetics: Bayesian studies reveal a systematic approach. J Biochem Biophys Methods 55:155–178, 2003. Murphy E.F., Gilmour S.G., and Crabbe M.J.C. Efﬁcient and cost-effective experimental determination of kinetic constants and data: The success of a Bayesian systematic approach to drug transport, receptor binding, continuous culture and cell transport kinetics. FEBS Lett 556:193– 198, 2004. About Enzyme Inhibition Hoh C.Y. and Cord-Ruwisch R. A practical kinetic model that considers endproduct inhibition in anaerobic digestion processes by including the equilibrium constant. Biotechonol Bioeng 51:597– 604, 1996. Ochs R.S. Understanding enzyme inhibition. J Chem Educ 77:1453–1455, 2000. Whiteley C.G. Enzyme kinetics: Partial and complete non-competitive inhibition. Biochem Educ 27:15– 18, 1999.

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About Thermodynamics and Enzyme Reactions Alberty R.A. Calculating apparent equilibrium constants of enzyme-catalyzed reactions at pH 7. Biochem Educ 28:12–17, 2000. Alberty R.A. Thermodynamics of systems of biochemical reactions. J Theor Biol 215:491–501, 2002. Alberty R.A. A short history of the thermodynamics of enzyme-catalyzed reactions. J Biol Chem 279:27831–27836, 2004. Alberty R.A. Relations between biochemical thermodynamics and biochemical kinetics. Biophys Chem 124:11–17, 2006. Aledo J.C. Metabolic pathways: Does the actual Gibbs free-energy change affect the ﬂux rate? Biochem Mol Biol Educ 29:142–143, 2001. Aledo J.C. Coupled reactions versus connected reactions. Biochem Mol Biol Educ 35:85–88, 2007. Aledo J.C. and Esteban del Valle A. Glycolysis in wonderland: The importance of energy dissipation in metabolic pathways. J Chem Educ 79:1336–1339, 2002. Bozlee B.J. Reformulation of the Michaelis-Menten equation: How enzyme-catalyzed reactions depend on Gibbs energy. J Chem Educ 84:106–107, 2007. Cannon W.R., Singleton S.F., and Benkovic S.J. A perspective on biological catalysis. Nat Struct Biol 3:821–833, 1996. Cornish-Bowden A. and Hofmeyer J.H.S. Enzymes in context. Kinetic characterization of enzymes for systems biology. The Biochemist April: 11–14, 2005. Edwards R.A. The free energies of metabolic reactions (DG) are not positive. Biochem Mol Biol Educ 29:101–103, 2001. Harris D.A. A uniﬁed approach to enzyme catalysis. Biochem Educ 14:2–6, 1986. Heinrich R., Melendez-Hevia E., and Cabezas H. Optimization of kinetic parameters of enzymes. Biochem Mol Biol Educ 30:184–188, 2002. Hoh C.Y. and Cord-Ruwisch R. A practical kinetic model that considers endproduct inhibition in anaerobic digestion processes by including the equilibrium constant. Biotechonol Bioeng 51:597– 604, 1996. Kjelstrup S., Rubi J.M., and Bedeaux D. Active transport: A kinetic description based on thermodynamic grounds. J Theor Biol 234:7–12, 2005. Rottenberg H. The thermodynamic description of enzyme-catalysed reactions. Biophys J 13:503–511, 1973. Weiss H.M. The roles of acids and bases in enzyme catalysis. J Chem Educ 84:440–442, 2007. Winzor D.J. and Jackson C.M. Interpretation of the temperature dependence of equilibrium and rate constants. J Mol Recognit 19:389–407, 2006. Wolfenden R. Thermodynamic and extrathermodynamic requirements of enzyme catalysis. Biophys Chem 105:559–572, 2003. About Macromolecular Crowding Asaad N. and Engberts J.B.F.N. Cytosol-mimetic chemistry: Kinetics of the trypsin-catalyzed hydrolysis of p-nitrophenyl acetate upon addition of polyethylene glycol and N-ter-butyl acetoacetamide. J Am Chem Soc 125:6874–6875, 2003. Asaad N., Den Otter M.J., and Engberts J.B.F.N. Aqueous solutions that model the cytosol: Studies on polarity, chemical reactivity and enzyme kinetics. Org Biomol Chem 2:1404–1412, 2004. Chebotareva N.A., Kurganov B.I., and Livanova N.B. Biochemical effects of molecular crowding. Biochemistry (Moscow) 69:1522–1536, 2004. Ellis R. and Minton A.P. Join the crowd. Nature 425:27–28, 2003. Hu Z., Jiang J., and Rajagopalan R. Effects of macromolecular crowding on biochemical reaction equilibria: A molecular thermodynamic perspective. Biophys J 93:1464–1473, 2007.

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Lonhienne T.G.A., Jackson C.M., and Winzor D.J. Thermodynamic non-ideality as an alternative source of the effect of sucrose on the thrombin-catalyzed hydrolysis of peptide p-nitroanilide substrates. Biophys Chem 103:259–269, 2003. Lonhienne T.G.A. and Winzor D.J. A potential role for isothermal calorimetry in studies of the effects of thermodynamic non-ideality in enzyme-catalyzed reactions. J Mol Recognit 17:351–361, 2004. Minton A.P. Molecular crowding: Analysis of effects of high concentrations of inert cosolutes on biochemical equilibria and rates in terms of volume exclusion. Methods Enzym 295:127–149, 1998. Minton A.P. The inﬂuence of macromolecular crowding and macromolecular conﬁnement on biochemical reactions in physiological media. J Biol Chem 276:10577–10580, 2001. Minton A.P. How can biochemical reactions within cells differ from those in test tubes? J Cell Sci 119:2863–2869, 2006. Minton A.P. and Wilf J. Effect of macromolecular crowding upon the structure and function of an enzyme: Glyceraldehyde-3-phosphate dehydrogenase. Biochemistry 20:4821–4826, 1981. Patel C.N., Noble S.M., Weatherly G., Tripathy A., Winzor D.J., and Pielak G.J. Effects of molecular crowding by saccharides on a-chymotrypsin dimerization. Protein Sci 11:997–1003, 2002. Ralston G.B. Effects of ‘‘crowding’’ in protein solutions. J Chem Educ 67:857–860, 1990. Shearwin K.E. and Winzor D.J. Substrate as a source of thermodynamic nonideality in enzyme kinetic studies: Invertase-catalyzed hydrolysis of sucrose. Arch Biochem Biophys 260:532–539, 1988.

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10 Kinetics of Protein and Enzyme Denaturation

10.1

Introduction

It goes without saying that proteins are very important in foods. Apart from their nutritional value they play an important role in texture of foods, they can act as emulsiﬁers, foaming agents, and gelling agents. Proteins are characterized by their primary (basically the amino acid sequence), secondary (a-helix and b-sheet), tertiary (3D structure), and quaternary structures (association of subunits). Processing can alter the secondary, primary, and quaternary structure, and this can have large consequences. Enzymes are, of course, also proteins. As shown in the previous chapter, enzymes can have a large effect on food quality for the very reason that they act as catalysts. In the cases that such reactions are undesirable it is, obviously, important to prevent their action. Fortunately, enzymes can be inactivated relatively easily. Traditionally this is done by a heat treatment, but there are also other ways, such as high pressure, change in chemical environment (pH, solvent quality, ionic strength) change. In any case, knowledge of the kinetics of enzyme inactivation is essential, in order to be able to optimize the treatment. Since enzymes are proteins, the mechanism of inactivation is basically protein denaturation, and therefore it makes sense to discuss protein and enzyme denaturation in one chapter.

10.2

Protein Stability

The stability of proteins is so important in foods that a separate section on this topic is warranted. With stability is meant the resistance against denaturation, or unfolding. Denaturation can result in a change of functional properties of proteins, in a loss of enzymatic activity, it may play a part in inactivation of microorganisms (Chapter 13), in an increase in digestibility, in inactivation of antinutritional factors, to name some examples. There is some ambiguity about the question what denaturation is. Denaturation can be described as a change in the native conformation of a (globular) protein due to a change in conditions. Causes for denaturation are heat, a change in pH, solvent quality, adsorption to surfaces, high pressure, to name the most important ones for foods. The most relevant cause for foods is, of course, heat. Denaturation does not result in a change in primary structure, but it may result in a change in secondary, tertiary, and quaternary structure. Denaturation does not necessarily result in complete destruction of all native structures, but it will result in an (partial) unfolding of the protein. Such unfolding is in principle reversible, i.e., upon removal of the cause of the unfolding (in the case of heat 10-1

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by cooling) refolding into the original native structure is possible but also into another conformation. This is the very cause of ambiguity: the term denaturation is also frequently used to indicate irreversible changes in proteins (such as aggregation), but strictly speaking Intermediates this is not correct. What happens after unfolding is that amino acid residues (buried in the interior of the protein hitherto) become exposed and consequently Random coil Folded conformation their reactivity increases and this may result in irreversible changes. Let us take a closer look at unfolding Folding and further reactions. Unfolding As indicated above, denaturation is in principle a reversible process, and equilibrium is postulated FIGURE 10.1 Highly schematic drawing meant to between the native and denatured state (assuming illustrate the principle of cooperativity in protein this is a one step reaction). This implies that both conformation. native and unfolded molecules are present but no unstable intermediate structures. This is plausible because of presumed cooperative effects in bonds that are involved in stabilizing a certain protein conformation. An individual noncovalent bond involved in protein conformation is weak but together they may be stronger than the sum of the individual interactions. A highly schematic picture of this is shown in Figure 10.1. The explanation for such a cooperative effect is that intramolecular internal bonds can be formed much easier than when they would have to be formed intermolecularly between individual molecules. In other words, the thermodynamic activity of intramolecular groups is effectively much higher than their concentration would indicate. Conversely, this means that if such bonds are disrupted in the process of denaturation then the loss of a few bonds will result very easily in the disruption of many bonds. So, we assume equilibrium between native and denatured molecules: k1

N ÐD k2

(10:1)

N symbolizes native protein molecules and D denatured protein molecules (these species can also be partly denatured in the sense that certain protein domains may be unfolded and others not). This equilibrium is considered to be fast such that both states are populated with amounts that allow a thermodynamic treatment. Equilibrium can thus be characterized by a proper thermodynamic equilibrium constant Keq ¼ k1=k2 (compare Equation 4.14). Focusing on temperature effects, the denaturation temperature is by deﬁnition the temperature at which Keq ¼

[D] k1 ¼ ¼1 [N] k2

(10:2)

It is thus the temperature at which the number of unfolded molecules equals that of folded molecules. At temperatures below the denaturation temperature the equilibrium lies toward the left in Equation 10.1, i.e., mainly native molecules, and above the denaturation temperature it is the opposite, i.e., mainly unfolded molecules. How much of both species are present at a certain temperature is governed by the standard Gibbs energy at that particular temperature. As a reminder, the equilibrium constant is related to the standard Gibbs energy, as displayed in Equation 3.113, DrG ¼ RT ln Keq. It is interesting to note that the conformational stability of most proteins (in terms of DG N!D ) is not very high, say in the order of 10–50 kJ mol1, far smaller than for one covalent bond. Figure 10.2 gives a hypothetical but realistic example of DG N!D as function of temperature. If DGN!D < 0, unfolded molecules predominate. The effect of a changing temperature is thus to change the position of the equilibrium.

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∆GN→D (kJ mol−1)

30 20

10

0 Tden

−10

Tden

−20

0

20

40

60

80

T (C)

FIGURE 10.2 Conformational stability expressed as the change in Gibbs energy for the reaction folded ! unfolded of a hypothetical protein as a function of temperature.

An interesting feature following from Figure 10.2 is that at low temperatures also so-called cold denaturation can occur; the reason for this is the disappearance of hydrophobic bonding at low temperatures. We will not discuss this phenomenon further. Equation 10.1 indicates thus a two-state population with unfolded and folded molecules. The unfolding process is believed to be a cooperative transition: the breaking of one or more bonds destabilizes others. Because it concerns many bonds, this requires a large energy input per mole protein DH. If this were the only factor, unfolding would not happen because of this required high energy input. However, unfolding causes at the same time a very large entropy increase. It should be remarked here that this pertains to the entropy of the whole system. Thus, for a protein in an aqueous environment it concerns the entropy of the protein itself (very much increased upon unfolding) and the entropy of the water molecules surrounding the protein (decreased because of the exposure of hydrophobic amino acid residues). On the whole, the entropy gain is quite high for unfolding proteins in water. This then compensates for the high positive enthalpy, so that the overall DG remains moderate, remembering that DG ¼ DH TDS. Figure 10.3 shows a schematic illustration of both conformational and activation standard enthalpy and entropy changes. N‡ N‡, D

J mol−1

D

∆H

+

0

N

‡

∆H T∆S ≈ T∆S ‡ N

N‡ ∆G

‡

∆G N→D

N D

_

FIGURE 10.3 Schematic drawing of standard enthalpy and entropy changes upon protein unfolding from native molecules N to denatured molecules D via the activated complex. The x-axis is not labeled (Chapter 5).

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Keq = [D]/[N] = k1/k2

ln k or ln K

k2 (refolding)

k1 (unfolding)

1/T

FIGURE 10.4 Schematic drawing of the temperature dependence of the reaction rate constant describing unfolding (k1) and refolding (k2) and the equilibrium constant for unfolding (Keq).

With regard to unfolding, it is generally accepted that there is a single rate-limiting step and a single transition state that is considered to be a distorted high energy form of the native state. The activation enthalpy of unfolding is considered to be independent of temperature. The heat capacities of the folded and transition state are assumed to be nearly the same. The temperature dependence of the reaction rate constant k1 describing unfolding follows usually the Arrhenius=Eyring equation quite well, i.e., a plot of ln k1 versus 1=T is linear. This is, however, not so for the reaction rate constant k2 describing refolding; it may decrease with temperature. The reason for this is twofold. First, the heat capacity of the unfolded molecule is much higher than that of the folded molecule due to hydrophobic amino acids that are exposed upon unfolding. Second, the metastable intermediates in the prefolded state are destabilized easier with increasing temperature. Consequently, the temperature dependence of the equilibrium constant is not according to the van’t Hoff equation (Equation 5.2). Figure 10.4 shows a schematic picture. Prior to refolding there will be some transient prefolded state (not a random coil); sometimes these intermediate states are described as molten globules or compact intermediates, but there is no consensus about the actual existence of such species. A rate-limiting step for refolding may be the occurrence in unfolded molecules of trans–cis isomerization of peptide bonds involving proline residues. Peptide bonds in native proteins are in the trans-conformation and isomerization does not easily occur except for bonds involving proline. If trans–cis isomerization occurs, refolding in the native state is obviously hindered. Concerning thermodynamics of unfolding, the following analysis may be helpful. Protein unfolding goes along with a change in heat capacity DCP: the heat capacity of the unfolded protein is higher than that of the folded protein, which is, as mentioned, due to exposure of hydrophobic amino acids residues. The thermodynamic parameters DCP, DG, DH are temperature dependent, so we should write DG(T) ¼ DH(T) TDS(T)

(10:3)

The temperature dependence of the enthalpy change can be written as ðT DCP dT DH(T) ¼ DH(Tden ) þ

(10:4)

Tden

Tden indicates the denaturation temperature. Similarly, for the entropy change we can write: ðT DCP dT DS(T) ¼ DS(Tden ) þ Tden T

(10:5)

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As a ﬁrst approximation we can assume DCP to be temperature independent, so that integration of the previous equations yields DH(T) ¼ DH(Tden ) þ DCP (T Tden ) DS(T) ¼ DS(Tden ) DCP ln

Tden T

(10:6) (10:7)

By deﬁnition, DG(Tden) ¼ 0 and consequently DS(Tden) ¼ DH(Tden)=T, so that we can write Equation 10.3 as TDH(Tden ) Tden DG(T) ¼ DH(Tden ) þ DCP (T Tden ) þ TDCP ln T Tden T Tden ¼ DH(Tden ) 1 þ DCP (T Tden ) þ TDCP ln T Tden

(10:8)

This equation was already used in Chapter 9 (Equation 9.76) while discussing the temperature dependence of enzyme activity. This equation can be rewritten in terms of the equilibrium constant via the relation DG ¼ RT ln Keq and describes therefore the temperature dependence of the equilibrium constant of unfolding. It is an approximate relation because we have assumed that DCP is temperature independent, which is not really true, but it is usually not too bad an approximation if the temperature range is not too large. For a food technologist it is usually more interesting what happens after unfolding because that may have a large impact on food quality. Nevertheless, it is essential to understand unfolding processes in order to understand what happens next. Expanding Equation 10.1 we can envisage a few scenarios. The ﬁrst scenario is to assume equilibrium between N and D and to introduce a further reaction step describing the formation of an irreversibly unfolded molecule I out of the unfolded molecule D, ﬁrst proposed by Lumry and Eyring in the 1950s: k1 N

k3

D

I

k2

SCHEME 10.1

Model 1 (Lumry and Eyring) for protein=enzyme denaturation.

Alternatively, the second reaction may be a bimolecular one: two unfolded molecules react together to form an aggregate: k1 N

k2

D+D

SCHEME 10.2

D

k3

I

Model 2 for protein=enzyme denaturation.

Some proteins are notably heat stable. There may be two reasons for that. The ﬁrst is that the protein has a high conformational stability, in other words that it starts to unfold only at high temperature. The second possibility is that unfolding does occur at relatively low temperature but that the subsequent reaction occurs only very slowly (i.e., very low k3), so that upon cooling the molecule refolds into the native state and it appears as if nothing has happened. Thermal instability of proteins on the other hand implies a relatively fast reaction (i.e., very high k3) leading to an irreversibly unfolded molecule, which may have a large impact on food quality. Table 10.1 gives an overview of possible reactions leading to irreversible changes.

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TABLE 10.1 Possible Reactions Occurring after Unfolding Causing Irreversible Changes in Unfolded Protein Molecules Reaction

Amino Acid Residue Involved

SH=SS exchange

Cysteine, cystine

Deamidation

Asparagine, glutamine

Oxidation

Tryptophan, methionine

Hydrolysis of peptide bonds

All

Trans–cis isomerization

Proline

Aggregation=coagulation

Hydrophobic residues

Dissociation of subunits

Cyst(e)ine, hydrophobic residues

Chemical cross-links (lysinoalanine, lanthionine, histidinoalanine, Maillard-type cross-links)

Phosphoserine, cysteine, lysine, histidine

When it comes to kinetics of denaturation, there are many conﬂicting results in literature. Even for the same protein reported orders may vary from 1 to 2. These conﬂicting results are most likely due to different conditions under which denaturation is studied; denaturation is not a property of the protein alone but also of the environment it is in. However, conﬂicting results may remain even if the conditions are the same because of a nonoptimal experimental design. This leads to estimates of the reaction order with very large conﬁdence intervals so that almost any order would give a reasonable ﬁt. An example of this was shown in Chapter 7 on the denaturation of a-lactalbumin. It is the author’s experience that for many results published in literature about protein (and enzyme denaturation as well) the imprecision in parameter estimates is disappointingly large. Also, all too often, authors force a ﬁrst-order plot through their data without exploring the possibility that another order may be more appropriate. A striking example is given in Figure 10.5 for denaturation of apo-lactoferrin. Clearly, there is a misﬁt for nt ¼ 1. Some authors have reported the order with respect to concentration, nc, rather than the one with respect to time, nt, in relation to protein denaturation; an example was given in Figure 4.16 for the denaturation of b-lactoglobulin. It appears that then more consistent results are obtained because this 200

c (mg dm−3)

160

120

80

40

0 0

1000

2000 Time (s)

3000

4000

FIGURE 10.5 Denaturation of apo-lactoferrin in phosphate buffer at 728C. Solid line is the optimal ﬁt obtained for an order nt ¼ 2.2, the broken line is for a forced ﬁrst-order plot (nt ¼ 1). Dataset in Appendix 10.1, Table A.10.1.

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method uses only initial rates and possible interfering reactions in the course of the reaction are avoided. The difference between the two orders is discussed in Chapter 4. Just another reason for the discrepancies reported in literature is due to the method of analysis used. If one analyzes the remaining native molecules after a heat treatment (by whatever method), one not only determines the remaining native molecules that were not unfolded but also the ones that were actually unfolded but were able to refold in the native state after cooling. The following analysis may illustrate this. Taking Scheme 10.1 as a starting point, the rate of irreversible change is r¼

d[I] ¼ k3 [D] dt

(10:9)

The problem in using this equation is that usually neither [D] nor [I] is measured, but the remaining [N] after cooling; this will include then refolded molecules if [D] refolds back easily upon cooling (large k2 upon cooling). So, the change detected is actually [N þ D], but after cooling we have no way of distinguishing between [N] and [D]. However, if we assume rapid equilibrium between [N] and [D], we can use the equilibrium constant to state something about [D]: Keq ¼

k1 [D] 1 x ¼ ¼ k2 [N] x

(10:10)

where x represents the fraction of [N]=([N] þ [D]). So we can write [D] ¼

Keq ([N] þ [D]) 1 þ Keq

(10:11)

This equation signiﬁes that at temperatures below the denaturation temperature Tden the fraction of [D] of the total protein concentration is low; for instance, the fraction of [D] is 0.09 for Keq ¼ 0.1. At the denaturation temperature Tden, Keq ¼ 1, and the fraction of [D] is half of the total protein concentration, while at temperatures well above Tden, the fraction of [D] is virtually unity. Equation 10.9 can now be written as r¼

Keq d[I] ([N] þ [D]) ¼ kobs ([N] þ [D]) ¼ k3 1 þ Keq dt

(10:12)

To recapitulate, if T >> Tden then Keq >> 1, hence kobs k3, in other words, the observed rate is completely determined by the reaction leading to irreversible changes. Figure 10.6 gives a simulated example of such a situation. If T ¼ Tden, Keq ¼ 1, hence kobs ¼ 0.5 k3. Such behavior is simulated in Figure 10.7. At temperatures well below the denaturation temperature, Keq 1, hence kobs ¼ k3Keq 1, in other words nothing happens because almost all molecules are in the native state and the reaction leading to irreversible changes is very slow anyway at the lower temperature. The same trend would be observed for the reaction depicted in Scheme 10.2, albeit with slightly different kinetics due to the bimolecular reaction. Another scenario is to assume that refolding does not occur (k2 0). In fact, this is the situation discussed in Chapter 4 for a consecutive reaction, Equation 4.48: [N] ¼ [N]0 exp (k1 t) k1 [N]0 [D] ¼ (exp (k1 t) exp ( k3 t)) k3 k1 k1 exp (k3 t) k3 exp (k1 t) [I] ¼ [N]0 1 þ k3 k1

(10:13)

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Kinetic Modeling of Reactions in Foods

10-8

1

c (arbitrary units)

0.8

[I]

0.6

[N+D]

0.4

[D] 0.2 [N]

0 0

20

40 60 80 t (arbitrary units)

100

120

FIGURE 10.6 Simulation of protein denaturation and subsequent reaction leading to irreversible unfolding. Referring to Scheme 10.1, k1 ¼ 1, k2 ¼ 0.01 (a situation similar to T >>Tden) and k3 ¼ 0.01, [N]0 ¼ 1 (arbitrary units). It is assumed that all remaining D refolds back into N after cooling.

Figure 10.8 gives a simulated example of such a situation; if one assumes that renaturation is not possible, also not after cooling, and one can only measure N, no information on reaction rate constants other than k1 can be obtained from such experiments. Only if one would be able to measure D and I, this would give information on k1 and k3. Sometimes, one is able to measure more than one response, for instance by measuring peak areas in gel permeation chromatography. One example of this is shown in Figure 10.9, a study on denaturation and

1

c (arbitrary units)

0.8

[N]

0.6

[N+D]

0.4

[I] [D]

0.2

0 0

20

40 60 80 t (arbitrary units)

100

120

FIGURE 10.7 Simulation of protein denaturation and subsequent reaction leading to irreversible unfolding. Referring to Scheme 10.1, k1 ¼ 1, k2 ¼ 1, and k3 ¼ 0.01 (a situation similar to T ¼ Tden), [N]0 ¼ 1 (arbitrary units). It is assumed that all remaining D refolds back into N after cooling.

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Kinetics of Protein and Enzyme Denaturation

10-9

1 [N] 0.8 c (arbitrary units)

[D] [I]

0.6

0.4

0.2

0

0

20

40 60 80 t (arbitrary units)

100

120

FIGURE 10.8 Simulation of protein denaturation and subsequent reactions leading to irreversible unfolding. Referring to Scheme 10.1, k1 ¼ 0.1, k2 ¼ 0, and k3 ¼ 0.01, [N]0 ¼ 1 (arbitrary units). It is assumed that all remaining D does not refold back into N after cooling.

subsequent aggregation of the potato protein patatin. Another example is given in Figure 10.10 for the denaturation and aggregation of the enzyme b-galactosidase, where it was even possible to measure an intermediate; incidentally, the decrease in native protein coincided with the decrease in enzymatic activity. The simultaneous analysis of residual protein as well as of aggregated protein makes it possible to analyze the rate constants for disappearance of native protein as well as the formation of aggregated

1

0.8

c/c0

0.6

0.4

0.2

0 0

0.2

0.4 0.6 Time (h)

0.8

1

FIGURE 10.9 Denaturation=aggregation of patatin as a function of time after heating at its denaturation temperature 558C. Native protein (*), aggregated protein (*), measured as fraction of total peak area from gel permeation chromatography. Dataset in Appendix 10.1, Table A.10.2.

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Kinetic Modeling of Reactions in Foods

10-10

100 90 80

Peak area (%)

70 60 50 40 30 20 10 0

0

20

40 Time (min)

60

80

FIGURE 10.10 Denaturation=aggregation of b-galactosidase. Native protein (*), denatured protein (&), aggregated protein (~) as percentage of peak areas from gel permeation chromatography. Dataset in Appendix 10.1, Table A.10.3.

protein, in principle at least. In fact, such analyses can be done via multiresponse modeling as discussed in Chapter 8. However, there is an additional difﬁculty when using peak areas as measure for concentration. As a ﬁrst check, of course, the total peak area of all fractions measured should remain constant; if it is not, some component is apparently missing. In the two examples displayed in Figures 10.9 and 10.10 the peak area remains approximately constant. However, if it is assumed that two molecules merge into one aggregated molecule as in a bimolecular reaction, one has to account for this in the kinetic analysis: one aggregated molecule gives the same response in peak area as two nonaggregated molecules. This means that the interpretation of the measured response depends on the assumed mechanism, and that is a bit tricky. For the data in Figure 10.9 several models were tested and the one that came closest to the data was one in which denatured protein D had to be transformed to a reactive form D*: k1 N

k2 k3

D

D∗ + D∗

SCHEME 10.3

D

D∗

k4 I

Kinetic model for denaturation and aggregation of patatin.

The experimental results were obtained at the denaturation temperature of patatin (558C) and rapid equilibrium was assumed between N and D, and consequently high values for k1 and k2 were given, and ﬁxed (i.e., not estimated). The modeling result is shown in Figure 10.11, using the multiresponse

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Kinetics of Protein and Enzyme Denaturation

10-11

1 0.9 0.8 0.7

c/c0

0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4 0.6 Time (h)

0.8

1

FIGURE 10.11 Fit of the model in Scheme 10.3 to the denaturation and aggregation data for patatin at 558C. The same data as displayed in Figure 10.9.

modeling technique (Chapter 8). Although the trend in the data is followed by the model, the ﬁt is certainly not perfect. Another model was tried with the b-galactosidase data. First, the most simple model N ! D ! I was used (Equation 10.13), resulting in the ﬁt shown in Figure 10.12A. The model is overestimating the ﬁt for the aggregated protein. Therefore, it was tested whether a second-order aggregation model gave a better ﬁt using the model shown in Scheme 10.2; the ﬁt is shown in Figure 10.12B. There is not too much difference with Figure 10.12A. Therefore, Scheme 10.3 used for patatin was also applied to the b-galactosidase data (Figure 10.12C). Though there are slight differences, it is remarkable that more or less the same ﬁts are obtained with different models. Most likely, the models are still too simple to catch all the intricacies of protein denaturation. A kinetic analysis of protein denaturation is thus seen to be quite complicated, and the possible kinetic analysis depends strongly on what one can measure experimentally. As a result of all this, departures from simple kinetics readily occur with proteins. This can also become apparent from temperature studies. Figure 10.13 shows the temperature dependence of the observed rate constant kobs for denaturation of a milk protein. Such behavior clearly reﬂects the shift of rate controlling steps if temperature changes. Other reasons for departure from simple kinetics are heterogeneity of proteins, multiple rate-limiting steps in unfolding and subsequent reactions, the reaction leading to irreversibility is neither ﬁrst nor second order (e.g., in the case of aggregation). Since proteins are macromolecules they are sensitive to volume exclusion, or in other words, macromolecular crowding. This phenomenon was already touched upon in Chapter 9 on enzyme kinetics. Qualitatively, it implies that if the surroundings of the protein are crowded, that the denaturation temperature will increase, because it becomes energetically unfavorable to unfold in a crowded environment due to volume exclusion effects. An example is given in Figure 10.14 where the denaturation temperature of the enzyme a-chymotrypsin is given as a function of sucrose concentration. Sucrose is inert with respect to unfolding of the protein molecule and acts purely as a ‘‘crowder,’’ that is to say the observed effect on denaturation temperature is due to volume exclusion effects. The effect is seen to be considerable. It is to be expected that such effects play a role in foods. In other words, if the denaturation temperature of an isolated and puriﬁed protein is studied, it may well be that this does not correspond to

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Kinetic Modeling of Reactions in Foods

10-12

90

90

80

80

70

70

60 50 40

60 50 40

30

30

20

20

10

10

0 (A)

Peak area (%)

100

Peak area (%)

100

0

20

40 Time (min)

60

0

80

0

20

(B)

40 Time (min)

60

80

100 90 80

Peak area (%)

70 60 50 40 30 20 10 0 (C)

0

20

40 Time (min)

60

80

FIGURE 10.12 Kinetic models applied to the b-galactosidase data. First-order consecutive reaction (A), ﬁrst-order denaturation followed by a second-order aggregation (B), and ﬁt of the model shown in Scheme 10.3 (C). Same data as in Figure 10.10.

the actual denaturation temperature of the same enzyme in the food matrix where potentially many crowders may be present. As discussed in Chapter 3, nonideal behavior can be captured conveniently in activity coefﬁcients. This can also be done for proteins in crowded conditions. Crowding effects will manifest themselves in activity coefﬁcients. To give just an example of this, Figure 10.15 shows the activity coefﬁcient of a solution of hemoglobin as a function of its concentration. The effect is seen to be enormous: the activity coefﬁcient at c ¼ 100 g dm3 is about twice its value than that at 1 g dm3 while it is up to about 100 times higher at c ¼ 300 g dm3. Enzymes are, of course, also proteins and therefore the above analysis is also valid for enzymes. Nevertheless, enzymes show additional features that make it worth to discuss these in a little bit more detail, and this is the subject of Section 10.3.

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Kinetics of Protein and Enzyme Denaturation

−1

10-13

Moderate T-dependence (irreversible inactivation)

ln k

−3

Strong T-dependence (unfolding)

−5

−7

−9

−11 0.0022 0.0023 0.0024 0.0025 0.0026 0.0027 0.0028 0.0029 0.003 1/T (K −1)

FIGURE 10.13 Arrhenius plot for denaturation of b-lactoglobulin A (~) and a-lactalbumin (.) in heated milk. Dataset in Appendix 10.1, Table A.10.4.

10.3

General Kinetic Schemes Describing Enzyme Inactivation

Being able to inactivate enzymes is an important goal in food technology because the presence of enzymes can have a large effect on food quality. Traditionally, inactivation is achieved via a heat treatment, but newer methods such as high-pressure treatment and pulsed electric ﬁelds are increasingly 100

65

55

y

T den

60

10

50

45

0

0.2

0.4 0.6 0.8 Sucrose (M)

1

1.2

FIGURE 10.14 Effect of sucrose on the denatur0 ation temperature Tden of a-chymotrypsin in an aqueous solution. Dataset in Appendix 10.1, Table A.10.5.

1

0

50

100

150 200 c (g dm−3)

250

300

350

FIGURE 10.15 Molar activity coefﬁcient y of hemoglobin as a function of its concentration. Dataset in Appendix 10.1, Table A.10.6.

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Kinetic Modeling of Reactions in Foods

10-14

log (activity)

studied. Whatever the technology applied, Activation/inactivation knowledge of inactivation kinetics is indispensable because that opens the possibilities Firs t-or to optimize. When it comes to the mechander ism of enzyme inactivation, it is basically Grace period due to protein denaturation (unfolding) followed by some inactivation reaction that makes refolding impossible, as discussed in Section 10.2. However, refolding may occur when the cause for unfolding is removed, that is to say if the inactivation reaction did Biphasic inactivation not go to completion during the treatment. Several types of inactivation curves are possible. Figure 10.16 shows some schematic Time examples. In the following we discuss some of the mechanisms underlying such FIGURE 10.16 Some types of possible enzyme inactivation behavior. curves. The simplest scheme for protein=enzyme denaturation was already shown in Scheme 10.1 and it is equally applicable to enzyme denaturation. In the case of an enzyme, it is usually assumed that the denatured form D is not active, but that it can refold back to the native, active form N after the treatment. While D can in principle refold back to an active form N, it can also transform into an inactive form I, and it is assumed that this is an irreversible process, so once the I form is reached activity is lost. Mostly, it is assumed that inactivation of enzymes can be described as a ﬁrst-order reaction. An apparent ﬁrst-order reaction, if Scheme 10.1 is applicable, implies that the step with rate constant k3 is rate determining. In many cases, if not all, remaining enzyme activity will be measured rather than inactivated enzyme and this makes sense because that is what counts. Usually, a relative activity is reported by dividing the measured activity a by the initial activity before the treatment a0: arel ¼

a a0

(10:14)

An example of apparent ﬁrst-order behavior is given in Figure 10.17 for heat inactivation and another one in Figure 10.18 for pressure-induced inactivation. Many inactivation curves reported in literature conform more or less to the ones depicted in Figures 10.17 or 10.18. However, the mechanism may be more complicated than in Schemes 10.1 or 10.2 but this may go unnoticed in graphs such as the ones in Figures 10.17 and 10.18. This is due to the fact that enzyme activity is measured, and there are three important aspects to consider. 1. First of all, the activity is measured at an assay temperature different (usually lower) from the treatment temperature. This means that we have to extrapolate from the measuring temperature to the treatment temperature if we want to infer what happens at the treatment temperature. Refolding can take place during cooling to the assay temperature, and thus the situation will be different when the activity is measured. The problem then arises as to how to couple that information to the mechanism: what happens to the enzyme at the molecular level at the treatment temperature. 2. A second possible problem is that the enzyme activity measured (a) is actually the sum of the activities of all active enzyme forms (ai), while these forms may behave differently at different conditions.

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10-15

2

1

1.8

0.9

1.6

0.8

1.4

0.7

1.2

0.6

1

0.5

arel

log (%)

Kinetics of Protein and Enzyme Denaturation

0.8

0.4

0.6

0.3

0.4

0.2

0.2

0.1

0

0 0

200

400 600 Time (s)

(A)

800

1000

0

200

400 600 Time (s)

(B)

800

1000

FIGURE 10.17 An example of apparent ﬁrst-order heat-induced inactivation kinetics of pectin methylesterase from tomato at 69.88C, presented as a logarithmic plot (A) and as relative activity plot (B). The lines represent a ﬁrst-order model. Dataset in Appendix 10.1, Table A.10.7.

a¼

X

ai

(10:15)

i

3. Third, it is quite conceivable that an enzyme is partly denatured while retaining activity, complete or partial. The fact that (partial) denaturation has occurred will then go unnoticed, unless another measurement technique is employed to study the reaction, such as circular dichroism (CD) or differential scanning calorimetry (DSC). This is advisable whenever possible when one wants to state something about an underlying mechanism.

0

1

−0.2

0.9

−0.4

0.8

−0.6

0.7

−0.8

0.6

−1

arel

ln(A/A0)

A major problem in interpretation is that several possible mechanisms all result in apparent ﬁrst-order behavior if one is only able to measure enzyme activity, and it will not be possible to distinguish between

−1.2

0.4

−1.4

0.3

−1.6

0.2

−1.8

0.1

−2

0 0

(A)

0.5

20

40 Time (min)

60

0

80 (B)

20

40 Time (min)

60

80

FIGURE 10.18 An example of apparent ﬁrst-order pressure-induced inactivation kinetics of an a-amylase from Bacillus subtilis at a pressure of 700 MPa, presented as a logarithmic plot (A) and as relative activity plot (B). The lines represent a ﬁrst-order model. Dataset in Appendix 10.1, Table A.10.8.

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Kinetic Modeling of Reactions in Foods

10-16 TABLE 10.2

Overview of Some Possible Kinetic Schemes for Enzyme Inactivation

Kinetic Scheme k1

EÐ D

Remarks

Activity at T Tden

Activity at T > Tden

Only denaturation

Partly, E present in noticeable amounts

No, only D present

Yes, if refolded correctly

Denaturation followed by inactivation

Partly, E present in noticeable amounts

No, only D and I present. Kinetic scheme at T:

Depends on k3, or k03 at T

k2

k1

k3

E Ð D ! I k2

or

k03

D þ D ! I k1

k3

D ! I, or k03

k3

E1 Ð D1 ! I1

2D ! I

k2

k4

k6

E2 Ð D2 ! I2 k5

Activity at Tassay

Isoenzymes E1, E2, biphasic inactivation

Partly, E1 and E2 present in noticeable amounts

No, only D1 and D2, I1 and I2 present Kinetic scheme at T:

Depends on k3 and k6 at T, a ¼ E1 þ E2

k3

D1 ! I1 k6

k1

EÐ E

0

k2

k3

D Ð k 4

D! I k5

Partly denatured E0 shows enzymatic activity: E0 ¼ aE grace period if a < 1, activation period if a > 1.

0

Partly, E and E present in noticeable amounts

D2 ! I2 No, only D and I present Kinetic scheme at T:

Depends on k5 at T, a ¼ E þ E0 ¼ (1 þ a) E

k5

D ! I

Note: Tassay is the assay temperature at which enzyme activity is measured, T is the treatment temperature, Tden is the denaturation temperature. Usually, Tassay T. E, active enzyme; E0 , partly denatured but still active enzyme; D, denatured and inactive enzyme; I, inactive enzyme. The assumption is that refolding has occurred rapidly at the assay temperature (any D present will turn into E). a, measured enzyme activity; a, relative activity (0, can be larger than 1).

these mechanisms, unless other measurements are applied that give additional information. Let us consider some possibilities. Table 10.2 gives an overview, which is by no means exhaustive. An important point to notice from Table 10.2 is that parameters describing unfolding cannot be derived from heating experiments above the denaturation temperature. This can be done only at temperatures around the denaturation temperature, and then the kinetic equations become quite cumbersome. We focus here on some possible inactivation models. In many cases, as mentioned, it is found that inactivation can be modeled as a ﬁrst-order equation. This is actually quite remarkable considering the complex three-dimensional structure of proteins; one would expect perhaps also stable intermediates that are still partially enzymatically active. In many cases, the actual mechanism is probably more complex but conditions lead to apparent ﬁrst-order behavior. Nonetheless, also non-ﬁrst-order inactivation can be observed, perhaps more often than a researcher is willing to see. It is impossible to consider all possible inactivation mechanisms, because the possibilities are endless. As remarked, the simplest mechanism conceivable is irreversible ﬁrst-order inactivation where the rate determining step is rate constant k3 in Scheme 10.1, and neglecting the previous steps: k1

E ! I The equation describing such inactivation is just a ‘‘normal’’ ﬁrst-order equation: arel ¼ exp (k1 t)

(10:16)

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Kinetics of Protein and Enzyme Denaturation

10-17

The next possibility is that there are only two forms, active enzyme and inactivated enzyme, but they can be transformed into each other by a reversible ﬁrst-order reaction: k1

E ÐI k2

This leads eventually to equilibrium with an equilibrium activity [E]1 and the equation describing such a situation is [E] [E]1 ¼ exp ((k1 þ k2 )t) [E]0 [E]1

(10:17)

Then we come to the situation already depicted in Scheme 10.1. The equation describing this mechanism as an overall ﬁrst-order inactivation is arel ¼ exp (k0 t) k1 k3 k0 ¼ k1 þ k2

(10:18)

where the apparent ﬁrst-order rate constant is a composite one. The next possibility is that an enzyme is transformed to another conformation that still has enzymatic activity: b

k1 E

E1

SCHEME 10.4 Model 3 for change in enzyme activity leading to an enzyme form E1 that has a different activity than the native enzyme E. b is a measure for the ratio of speciﬁc activities of E and E1.

Obviously, it is not possible to distinguish the enzyme activity of E and E1 and therefore the measured activity will be a weighted average of the active species: arel ¼

E þ bE1 E0

(10:19)

with b¼

d1 d0

(10:20)

d0 and d1 are the speciﬁc activity of species E and E1, respectively. The next logical extension of this mechanism is k1 E

b E1

k2 I

SCHEME 10.5 Model 4 for enzyme inactivation via an intermediate enzyme form E1 that has a different activity than the native enzyme E.

The equation describing this mechanism is arel ¼

bk1 k1 b 1þ exp (k2 t) exp (k1 t) k2 k1 k2 k1

(10:21)

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Kinetic Modeling of Reactions in Foods

10-18

This is thus a biexponential equation describing biphasic inactivation. A similar biexponential is found for a series–parallel mechanism: k1 E

b E1

k2 I

k3 E

SCHEME 10.6

I

Model 5 for enzyme inactivation via a parallel route.

arel

¼ 1þ

bk1 k1 b exp (k2 t) exp ((k1 þ k3 )t) k2 k1 k3 k2 k1 k3

(10:22)

These are just a few examples of inactivation mechanisms. Several of them lead to apparent ﬁrst-order behavior, sometimes indicated as disguised ﬁrst-order inactivation. Of course, there is also the possibility for second-order inactivation, for instance in the following mechanism where the enzyme dissociates into subunits:

E

k1 k2

E1 2E1

SCHEME 10.7

k3

2E1 I1 I2

Model 6 for enzyme inactivation via an intermediate following a consecutive route.

It is not so easy to ﬁnd analytical solutions for these kinds of mechanisms and it is usually simpler to resort to numerical integration of ODEs. For the mechanism depicted in Scheme 10.7 this would lead to d[E] ¼ k1 [E] dt d[E1 ] ¼ 2k1 [E] k2 [E1 ] 2k3 [E1 ]2 dt

(10:23)

Figure 10.19 shows an example of biphasic inactivation. Several mechanisms are conceivable for biphasic behavior. The simplest one is to postulate the presence of two isoenzymes showing different heat stability. Both isoenzymes are inactivated in a parallel fashion but at different rates. A model for this is arel ¼ a exp (k1 t) þ (1 a) exp (k2 t)

(10:24)

where a represents the fraction of the less resistant enzyme k1 is the rate constant for inactivation of the less resistant enzyme k2 is the rate constant for inactivation of the more resistant enzyme Another model that describes biphasic inactivation is the fractional conversion model, discussed in Chapter 4, which assumes that a constant and stable enzyme activity a1 remains (at least over the period investigated):

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Kinetics of Protein and Enzyme Denaturation

10-19

1

2

0.8 0.6 arel

log arel

1.5

1

0.4 0.5

0

0.2 0 0

2

(A)

4

6

8

10

0

2

4

6

8

10

Time (min)

(B)

Time (min)

FIGURE 10.19 Biphasic inactivation of the enzyme cathepsin D in milk at 62.68C, logarithmic plot (A) and relative activity plot (B) with a ﬁt of Equation 10.24 (solid line) and Equation 10.25 (—-); parameters of model Equation 10.24: k1 ¼ 1.63 min1, k2 ¼ 0.11 min1, a ¼ 0.79, parameters of model Equation 10.25: k ¼ 1.30 min1, a1 ¼ 0.11. Dataset in Appendix 10.1, Table A.10.9.

a a1 ¼ exp (kt) a0 a1

(10:25)

Both models were ﬁtted to the data displayed in Figure 10.19B. Table 10.3 shows the results of model discrimination as discussed in Chapter 7, and the models appear to perform equally well. There are more mechanisms conceivable for biphasic inactivation. The mechanisms in Scheme 10.8 cannot be distinguished from the ones above in a ﬁrst-order plot if one only measures E (the remaining active enzyme) or I (the inactivated enzyme): (a)

E

E1

E2

I

(b)

E

E1

E2

I

(c)

E

E1

I

(d)

E

E1

E2

I

SCHEME 10.8 Various kinetic models that lead to biphasic behavior. E1 and E2 represent partly denatured but still active enzyme forms.

In these equations, E1 and E2 represent partly active enzymes that have undergone some structural changes (i.e., denaturation). TABLE 10.3 Results of a Model Discrimination Study on Inactivation of Cathepsin D in Milk at 62.68C (Figure 10.19B) SSr

log (Posterior Probability)

AICc

DAICc

Model 1 (Equation 10.24)

0.0106

4.485

71.94

0.4

Model 2 (Equation 10.25)

0.014

4.308

72.36

0

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Kinetic Modeling of Reactions in Foods

10-20

Sometimes it may seem that enzyme activity reaches a plateau value, suggesting that an equilibrium state is attained (as in Figure 10.19B where a1 0.1). However, it may also be that the experiment was stopped too early and that further inactivation does occur if one would extend the analysis period. Another possibility is the so-called grace-period inactivation in which the enzyme appears to be stable for a period of time, followed by an inactivation period. This could occur, for instance, with the mechanism in Scheme 10.9 if the reverse reaction in the ﬁrst step cannot be neglected. E

SCHEME 10.9

E1

I

E2

Kinetic model leading to grace-period behavior.

The unfortunate message from all this is that it will not be easy to differentiate between all the possible mechanisms. As remarked before, it may help if additional structural measurements can be made. In fact, this could turn it into a multiresponse model, which allows a much better model discrimination, as we have discussed in Chapter 8. Global ﬁtting. In the quest for suitable models, an interesting approach is available in global ﬁtting of models to experiments. This can be done easily when measurements are done at various temperatures. It may be assumed that the temperature effect on enzyme inactivation can be described by Arrhenius’ or Eyring’s law. A consistent model in which the Arrhenius relation is incorporated should then be applicable at every temperature. The following three examples may demonstrate the usefulness of this approach. Figure 10.20 shows the inactivation of a peroxidase according to a ﬁrst-order reaction at various temperatures. There is a slight difference between the individual and global ﬁts: the individual ﬁts look a little bit better, but overall the differences are not large. Analyzing the Arrhenius parameters via the two-step procedure and via the global ﬁt gave the same estimates: k00 ¼ 0:155 0:011 min1 and Ea ¼ 368 28 kJ mol1. In other words, the ﬁrst-order model seems applicable at every temperature studied.

1

0.8

0.8

0.6

0.6

a/a0

a/a0

1

0.4

0.4

0.2

0.2

0

0 0

(A)

2

4

6 Time (min)

8

10

0

12 (B)

2

4

6

8

10

12

Time (min)

FIGURE 10.20 First-order inactivation model for broccoli peroxidase at 658C (^), 67.58C (~), 708C (*), and 728C (&) via individual ﬁts (A) and via global ﬁtting applying the reparameterized Arrhenius equation (B). Dataset in Appendix 10.1, Table A.10.10.

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Kinetics of Protein and Enzyme Denaturation

10-21

1

0.8

0.8

0.6

0.6 arel

arel

1

0.4

0.4

0.2

0.2

0

0 0

10

20

(A)

30 40 Time (min)

50

60

0

10

20

(B)

30 40 Time (min)

50

60

FIGURE 10.21 Heat inactivation of a yeast invertase at various temperatures modeled via (A) a ﬁrst-order model (solid lines) and (B) the ﬁt of Equation 10.24 representing biphasic inactivation (solid lines). The ﬁts were derived for each temperature individually. (^) 408C, (~) 508C, (*) 558C, (&) 608C. Dataset in Appendix 10.1, Table A.10.11.

The second example of global ﬁtting is given in Figure 10.21 showing the heat inactivation of a yeast invertase. The inactivation data could not be ﬁtted satisfactorily by a ﬁrst-order model (Figure 10.21A). There is a hint of biphasic inactivation, so the model displayed in Equation 10.24 was applied for each temperature, as shown in Figure 10.21B. Judged by the ﬁts obtained the biphasic inactivation model seems to perform well. However, if we apply the global modeling approach by forcing the model in Equation 10.24 to all data at once, as advocated in Chapter 7, we obtained the ﬁt as shown in Figure 10.22 using the model in Equation 10.24.

1

0.8

arel

0.6

0.4

0.2

0

0

10

20

30 Time (min)

40

50

60

FIGURE 10.22 Heat inactivation of a yeast at various temperatures. The solid lines are the result of the global ﬁt of Equation 10.24 to all data at once. (^) 408C, (~) 508C, (*) 558C, (&) 608C. Same data as in Figure 10.21.

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10-22

TABLE 10.4 Parameter Estimates from the Iso-Enzyme Inactivation Model (Equation 10.24) for the Data Displayed in Figure 10.21B k1 95% CI

k2 95% CI

408C

0.052 0.045

0

a 95% CI 0.09 0.03

508C

0.06 0.01

0

0.211 0.019

558C

0.126 0.080

0.005 0.004

0.322 0.13

608C

0.345 0.126

0.076 0.045

0.76 0.21

Clearly, the ﬁt resulting from the global approach is not a good one, and in fact we have obtained evidence that the isoenzyme inactivation model is not the correct one, even though it performs well at each temperature separately. The reason for this apparent discrepancy becomes apparent if we take a look at the parameter estimates displayed in Table 10.4, derived from the individual ﬁts at each temperature separately. These results show ﬁrst of all that the parameters are fairly uncertain, and in some cases even inestimable. Furthermore, it appears that the parameter a is temperature dependent, something one would not expect if the isoenzyme mechanism is the correct one. In the global-ﬁt approach only one overall value for a can be found. The fact that a is not constant has its bearings on the values for the other parameters. Just another way to detect this is to make an Arrhenius plot of the rate constant k1 (Figure 10.23) (this could not be done for k2 since only two values were available). It appears from Figure 10.23 that parameter k1 does not obey Arrhenius’ law (of course, this is actually doing the same as performing a global ﬁt). So, in conclusion, this shows that the isoenzyme inactivation mechanism is not the correct one for this particular case, and one has to search for a better model. In doing so, it would be worthwhile to spend some effort in making a better experimental design to come to better model discrimination and parameter estimation. A third example of complex inactivation kinetics is the following. It concerns a proteinase from Pseudomonas ﬂuorescens, a psychrotrophic bacterium that can be found in milk. It produces rather heat stable proteinases (and lipases for that matter) that can cause problems in UHT milk in the sense that some enzymatic activity remains after UHT treatment, leading to proteolysis resulting into protein destabilization and bitter taste. The simplest model to test is the one in Equation 10.16. Figure 10.24 shows ﬁrst-order plots at three temperatures obtained via global ﬁtting using the Eyring equation. 0 −0.5

ln k1

−1 −1.5 −2 −2.5 −3 −3.5 0.00295 0.003 0.00305 0.0031 0.00315 0.0032 0.00325 1/T (K −1)

FIGURE 10.23 Arrhenius plot for the parameter k1 displayed in Table 10.4.

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Kinetics of Protein and Enzyme Denaturation

Log (residual activity)

0

−0.2

−0.4

−0.6

−0.8 0

50 Time (min)

100

FIGURE 10.24 Global ﬁrst-order plots for the inactivation of a puriﬁed proteinase from P. ﬂuorescens isolated from milk 908C (&), 1008C (^), and 1108C (D). Dataset in Appendix 10.1, Table A.10.12.

10-23

It is clear that the ﬁts are not satisfactory, there is a hint of grace-period behavior and accordingly more complicated models were tested. Two of them are shown in Scheme 10.10 and the resulting plots in Figure 10.25. In both models, the two rate constants k1 and k2 were not estimated because it was assumed that at the temperatures studied there would not be any native E present, since it was known that the denaturation temperature of this particular enzyme was near 558C. Parameter estimates are given in Table 10.5 along with the Akaike criterion, and it appears that the model in Scheme 10.10b performs much better than that in Scheme 10.10a. This is not only indicated by the Akaike criterion, but also by the large conﬁdence intervals for the parameters model in Scheme 10.10a. Although the model in Scheme 10.10b performed best, this does not mean that it is the ultimate model. It is, however, not conﬂicting with the data and it could be used as a starting point for further studies, if so desired.

Regeneration. It is possible that under certain conditions an enzyme regains its activity, partly or completely, after cooling. This must then be due to refolding taking place. Alternatively, it may be that refolding does occur rapidly but in an incorrect way. The incorrectly folded enzyme may then slowly return into a correctly folded way and thereby regain activity. Regeneration has been reported especially for the enzyme peroxidase in several cases and in several foods. Figure 10.26 shows an example for peroxidase in broccoli. It can be seen that there is considerable recovery of enzyme activity after a relatively short time, indicating that the enzyme is quite well capable of refolding back to an active form. The author is not aware of systematic studies on the kinetics of refolding, although this should be of importance to get an idea of how fast refolding can take place, and also the temperature dependence should be interesting, if one is to avoid regeneration of enzymatic activity. Autodigestion of proteases. Some bacterial proteases excreted by psychrotrophic microorganisms are extremely heat stable and their presence in UHT sterilized milk products may cause enzymatic deterioration. However, even though these enzymes are heat stable, inactivation is sometimes observed at

(a)

k3

k1 k2

(b)

k2

I

k4

k1 E

k5 D2

D1

E

k3 k4

k6 D2

D1

I

k5

E1 b

SCHEME 10.10

Two kinetic models to describe inactivation of a proteinase from P. ﬂuorescens.

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0

0

−0.2

−0.2

Log (residual activity)

Log (residual activity)

10-24

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8 0

50 Time (min)

(A)

100

0

50 Time (min)

(B)

100

FIGURE 10.25 Inactivation plots according to Scheme 10.10a (A) and Scheme 10.10b (B) of a puriﬁed proteinase from P. ﬂuorescens isolated from milk at 908C (&), 1008C (^), and 1108C (D). Same dataset as in Figure 10.24. TABLE 10.5 10.10a and b

Parameter Estimates and Akaike Criterion for the Models Shown in Scheme Eyring Parameter Estimates for Rate Constant 95% Conﬁdence Intervals k3: DHz ¼ 267.9 829.2 kJ mol1 DSz ¼ 436.8 511.2 J mol1 K1

Model in Scheme 10.10a

DAICc 12

k4: DHz ¼ 55.4 873.2 kJ mol1 DSz ¼ 161.8 1425.3 J mol1 K1 k5: DHz ¼ 62.5 724.1 kJ mol1 DSz ¼ 140.8 1951.1 J mol1 K1 k3: DHz ¼ 55.8 17.4 kJ mol1

Model in Scheme 10.10b

0

DSz ¼ 157.9 48.1 J mol1 K1 k6: DHz ¼ 221.2 44.4 kJ mol1 DSz ¼ 298.1 129.0 J mol1 K1 b: 0.61 0.17

relatively low temperatures in the range of 508C–608C (sometimes referred to as ‘‘low temperature inactivation,’’ a slightly confusing term because 508C–608C is not really low). This process is properly called autodigestion. Figure 10.27 gives an impression. The kinetic scheme describing autodigestion is

SCHEME 10.11

Step 1:

E

Step 2:

E+D

D E−D

→ E+I

Kinetic scheme describing intermolecular autodigestion of proteases.

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10-25

1

0.8

a/a0

0.6

0.4

0.2

0

0

1

2

3

4

5

Time (min)

FIGURE 10.26 Regeneration of the enzyme peroxidase from broccoli. Enzyme activity measured directly after heating at 758C (*) and enzyme activity after heating at 758C when keeping the enzyme 30 min at 208C before measuring (*). Dataset in Appendix 10.1, Table A.10.13.

The reaction in step 1 in Scheme 10.11 is the reversible unfolding process near the denaturation temperature, leading to the presence of native and denatured enzyme. The native enzymes that are still present can then hydrolyze their unfolded counterparts leading to irreversible loss of active enzyme (and concomitant formation of breakdown products I that are no longer enzymatically active). The results in Figure 10.27 suggest that the denaturation temperature should then be near 558C, the temperature at which both native and enzyme form are present in equal amounts; this was conﬁrmed by DSC. At temperatures below the denaturation temperature (say T 0 < 458C), most of the enzyme molecules will not be unfolded and therefore cannot be attacked by other enzyme molecules. Above the denaturation

1

Residual activity

0.8 0.6 0.4 0.2 0 40

60

80

100

120

140

T (C)

FIGURE 10.27 Example of the phenomenon of autodigestion for a puriﬁed protease from P. ﬂuorescens 228F in aqueous solution: Residual protease activity at 378C after holding the enzyme solution for 20 min at the indicated temperature. Dataset in Appendix 10.1, Table A.10.14.

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10-26

temperature (say T 0 > 658C), all enzymes are unfolded, and therefore there is no enzymatic activity and no hydrolysis takes place and upon cooling they refold into active enzymes. With increasing temperature (say at T 0 > 808C), the unfolded enzyme becomes subject to irreversible reactions so that refolding upon cooling becomes increasingly difﬁcult and this is actually the ‘‘normal’’ thermal inactivation.

10.4

Food Matrix Effects

It can make quite a difference whether enzyme inactivation is studied on isolated enzymes or in foods. Obviously, interpretation is much easier with isolated enzymes, but if the inactivation kinetics is much different from the behavior in a real food it becomes very dangerous, if not impossible to predict something for enzyme behavior in the food. In this section, we highlight just two examples to show the intricacies involved. The ﬁrst example shows the effect of water content on enzyme inactivation; in the second example, we compare the inactivation behavior of the milk protease plasmin in puriﬁed form and in its native environment milk. Effect of water content. Heat stability of enzymes is known to be strongly inﬂuenced by the water content of the system in which the enzyme is present. Usually, the stability increases strongly with decreasing water content. Figure 10.28 shows an example for the enzyme lipoxygenase. The explanation for such behavior is not straightforward but it is probably linked to the phenomenon of volume exclusion. With less water available as solvent, it becomes less favorable for proteins to unfold, and therefore heat stability increases in such situations (i.e., k decreases). Another effect may have to do with the interaction of water with the protein itself resulting in enthalpic and entropic effects; the 0.003

0.025

72°C

76°C

0.0025

0.02 k (s−1)

k (s−1)

0.002 0.0015

0.015 0.01

0.001 0.005

0.0005 0 (A)

0

0.2 0.4 0.6 0.8 Water content (g g–1)

0

1 (B)

0

0.2 0.4 0.6 0.8 Water content (g g–1)

1

0.12 80°C

0.1

k (s−1)

0.08 0.06 0.04 0.02 0 (C)

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Water content (g g–1)

FIGURE 10.28 Heat stability of lipoxygenase in a glucose calcium-alginate gel, expressed as the ﬁrst-order rate constant for inactivation as a function of water content at 728C (A), 768C (B), and 808C (C). Dataset in Appendix 10.1.

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6

0

5

−2

4

−4

3

−6

2

−8

1

−10

0 (A)

10-27

ln k

log D

Kinetics of Protein and Enzyme Denaturation

40

60

80

100 T (C)

120

140

160

−12 0.0024 0.0025 0.0026 0.0027 0.0028 0.0029 0.0030 0.0031 (B)

1/T (K −1)

FIGURE 10.29 Heat inactivation of plasmin in milk, expressed as the logarithm of D-values as a function of temperature (A) and as Arrhenius plot (B). Dataset in Appendix 10.1, Table A.10.16.

consequences of these effects will be difﬁcult to predict. Why there is a minimum observed in the plots shown in Figure 10.28 is unknown. In any case, the implication of this phenomenon is that in dried foods, the heat stability of enzymes may be much higher than in an aqueous environment, and this could have a large effect on the resulting quality. Behavior of plasmin. Plasmin is a milk-indigenous enzyme and is known to be very heat stable in milk. Its temperature dependence as found for inactivation in heated milk does deﬁnitely not conform to expected behavior. Figure 10.29 gives an example; it can be seen that above 1008C the inactivation rate constant (expressed as D-value in a TDT plot, or as rate constant k in an Arrhenius plot) hardly decreases, which is very atypical for protein=enzyme denaturation. The interesting fact is that plasmin, when isolated and puriﬁed, is not heat stable at all (Figure 10.30). The fraction of native enzyme as a function of temperature was estimated by subtraction of the extrapolated activity (Figure 10.30A) from the measured activity. The enzyme is seen to loose its activity at a temperature near 558C; to be sure that unfolding indeed occurs at that temperature, unfolding was conﬁrmed by CD studies. However, the enzyme was not irreversibly inactivated because enzyme activity was regained after holding it for 10 min at 658C, or 13 h at 548C. Inactivation started to become noticeable at higher temperatures between 758C and 908C. Another remarkable effect is that the inactivation of the enzyme is strongly enhanced in the presence of sulfhydryl (SH) compounds, such as cysteine and b-lactoglobulin; Figure 10.31 gives an impression. This is relevant because SH groups are present in milk. However, as shown in Figure 10.29, in milk the enzyme is much more stable and is deﬁnitely not inactivated between 758C and 908C. Therefore, there must be protecting agents present in milk, though there is also b-lactoglobulin present in milk, which should enhance inactivation when its sulfhydryl groups are exposed upon heating. As it happens, the major milk protein casein appears to protect plasmin from thermal inactivation, by a mechanism that is not yet understood; it is likely that plasmin associates with casein molecules. A qualitative model that attempts to explain the inactivation behavior of plasmin in milk is given in Scheme 10.12.

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10-28

1

0

0.9

−1

0.8 Fraction native enzyme

−0.5

ln (activity)

−1.5 −2 −2.5 −3 −3.5

(A)

0.6 0.5 0.4 0.3 0.2

−4 −4.5 2.9

0.7

0.1

3

3.1

3.2

3.3

3.4

0 30

3.5

1000/T (K −1)

40

50

60

70

80

T (C)

(B)

FIGURE 10.30 Activity of puriﬁed plasmin in aqueous solution as function of temperature (A) and calculated fraction of native enzyme as function of temperature (B). Dataset in Appendix 10.1, Table A.10.17. 100

Activity (%)

80

60

40

20

0

0

5

10 15 Time (min)

20

25

FIGURE 10.31 Heat inactivation of plasmin in the absence (*) and presence (*) of cystein (2 mmol dm3) at 708C. Dataset in Appendix 10.1, Table A.10.18. Reaction step 1:

PL

PL∗

Reaction step 2:

PL∗

iPL

Reaction step 3:

β-LG

β-LG∗

Reaction step 4:

PL ∗ + β-LG∗

iPL

Reaction step 5:

β-LG ∗ +

β-LG ∗-cas

Reaction step 6:

PL ∗ + cas

cas

PL ∗-cas

SCHEME 10.12 Qualitative scheme showing the various reactions of plasmin in milk upon thermal treatment. PL ¼ plasmin, PL* ¼ denatured plasmin, iPL ¼ inactivated plasmin, b-LG ¼ b-lactoglobulin, b-LG* ¼ denatured b-lactoglobulin with exposed SH groups, cas ¼ casein.

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10-29

This scheme shows the complexity of inactivation of plasmin in milk: Plasmin is subject to unfolding (step 1), the unfolded form can be inactivated directly (step 2) and, furthermore, it can react in its unfolded state with SH compounds (step 4), which in heated milk are delivered by denatured b-lactoglobulin (step 3); as it happens, b-lactoglobulin also reacts with casein (step 5) and is therefore in competition with casein and, ﬁnally, unfolded plasmin is protected by its association with casein (step 6). This example is meant to illustrate the phenomenon that inactivation behavior of enzymes studied after enzyme puriﬁcation can be completely different from their behavior in the food matrix. Thus, it is dangerous to extrapolate from test-tube experiments to real foods. As shown for plasmin, the enzyme is not very stable in aqueous solution, but it is very stable in its ‘‘natural’’ environment milk.

10.5

Concluding Remarks

This chapter has shown that the kinetic analysis of protein inactivation is not straightforward, especially when only the native or only the denatured form can be measured. When both forms (and possible intermediates) can be measured, the testing of kinetic models becomes easier, though still not easy. For enzymes, analysis is even more complicated due to such phenomena as disguised kinetics, autodigestion, and interaction with components in the food matrix. Notwithstanding these difﬁculties, it remains of utmost importance to be able to describe inactivation by kinetic models. In many cases, one has to be content with apparent kinetics. If one wants to make statements about molecular events, more elaborate techniques such as DSC and CD are needed, next to a kinetic analysis.

Appendix 10.1 Datasets Used for Examples in This Chapter

TABLE A.10.1 Denaturation of Apo-Lactoferrin (Figure 10.5) Time (s)

c (mg dm3)

0

181.9

320

114.8

650

93.3

900

85.1

1200

85.1

1500

66.0

1800 2100

57.5 50.1

2400

50.1

2700

38.0

3000

41.7

3400

41.7

3600

37.1

4000

26.9

Source: From Sanchez L., Peiro J.M., Castillo H., Perez M.D., Ena J.M., and Calvo M. Kinetic parameters for denaturation of bovine milk lactoferrin. J Food Sci 57:873–879, 1992.

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10-30 TABLE A.10.2 Time (h)

Denaturation of Patatin (Figure 10.9)

Fraction Native Protein

Fraction Aggregated Protein

0

1

0

0.03

0.8

0.01

0.07

0.72

0.02

0.1

0.7

0.04

0.13

0.64

0.08

0.17 0.5

0.58 0.36

0.13 0.34

1

0.32

0.52

Source: From Pots A.M., Gruppen H., De Jongh H.J.J., Van Boekel M.A.J.S., Walstra P., and Voragen A.G.J. Kinetic modeling of the thermal aggregation of patatin. J Agric Food Chem 47:4593–4599, 1999.

TABLE A.10.3 Time (min)

Denaturation of b-Galactosidase (Figure 10.10)

Fraction Native

Fraction Intermediate

Fraction Final Aggregate

0

91.5

7.2

1.2

5

64.6

7.8

24.8

10

47.7

6.6

42.3

15

37.3

5.3

54.0

20

29.5

4.4

63.0

30

16.9

2.8

74.0

45

9.1

2.2

82.8

60 75

4.4 1.6

1.6 1.6

89.0 92.2

Source: From Yoshioka S., Aso Y., Izutsu K., and Kojima S. Is stability prediction possible for protein drugs? Denaturation kinetics of beta-galactosidase in solution. Pharm Res 11:1721–1725, 1994.

TABLE A.10.4 Denaturation Rate Constants k of b-Lactoglobulin A and a-Lactalbumin as a Function of Temperature (Figure 10.13) T 0 (8C)

k for b-LG-A

k for a-LA

70 75

0.00015 0.00047

0.00012 0.00055

80

0.00274

0.00156

85

0.00811

0.00181

90

0.02184

0.00288

95

0.03599

0.00341

100

0.05253

0.00566

110

0.07468

0.00971

120 130

0.13365 0.17788

0.01574 0.02597

140

0.26009

0.04108

150

0.37953

0.0704

Source: From Dannenberg, F. and Kessler, H.G., Milchwissenschaft, 43, 3, 1988.

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10-31

TABLE A.10.5 Example of Molecular Crowding on Denaturation: Effect of Sucrose on the Denaturation 0 of a-Chymotrypsin (Figure 10.14) Temperature Tden 0 Tden (8C)

[Sucrose] (M) 0

49.8

0.1

51.0

0.5 0.7

53.9 55.7

0.9

58.1

1

58.1

Source: From Lee J.C., and Timasheff S.N. The stabilization of proteins by sucrose. J Biol Chem 256:7193–7201, 1981.

TABLE A.10.6 Molar Activity Coefﬁcient of Hemoglobin as a Function of Its Concentration (Figure 10.15) c (g dm3)

Molar Activity Coefﬁcient y

1

1

25

1.2

50

1.5

75

2

100 150

2.5 4.7

200

10

250

25

300

90

Source: From Minton, Biochem., 55, 119, 1983.

A.P.,

Mol.

Cell.

TABLE A.10.7 Inactivation of Tomato Pectin Methylesterase (Figure 10.17) Time (s)

Log (%)

0

2

64.171

1.9

183.104

1.6

305.405

1.5

424.337

1.2

546.619

1.0

662.318 784.658

0.9 0.7

903.648

0.5

Source: From Anthon G.E., Sekine Y., Watanabe N., and Barrett D.M. Thermal inactivation of pectin methylesterase, polygalacturonase, and peroxidase in tomato juice. J Agric Food Chem 50:6153–6159, 2002.

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10-32

TABLE A.10.8 Inactivation of a-Amylase from Bacillus subtilis at High Pressure (700 MPa) (Figure 10.18) Time (min)

ln(a=a0)

0

0

10

0.257

20

0.453

30

0.765

40

1.078

50

1.34

60

1.547

70

1.748

Source: From Geeraerd A.H., Herremans C.H., Ludikhuyze L.R., Hendrickx M.E., and Van Impe J.F. Modelling the kinetics of isobaric–isothermal inactivatoin of Bacillus subtilis a-amylase with artiﬁcal neural networks. J Food Eng 36:263–279, 1998.

TABLE A.10.9 Biphasic Inactivation of the Enzyme Cathepsin D in Milk at 62.68C (Figure 10.19) Time (min)

log(% activity)

0

2

0.15

1.88

0.5 1

1.76 1.67

2

1.24

2

1.19

3

1.06

3

0.99

6

0.88

6

0.83

9 9

0.63 0.56

Source: From Hayes M.G., Hurley M.J., Larsen L.B., Heegarrd C.W., Magboul A.A.A., Oliveira J.C., McSweeney P.L.H., and Kelly A.L. Thermal inactivation kinetics of cathepsin D. J Dairy Res 68:267–276, 2001.

TABLE A.10.10 Time (min) 0

Inactivation of Broccoli Peroxidase (Figure 10.20)

a=a0 at 658C 1

a=a0 at 67.58C 1

0.5

1

0.89

0.73

0.75

0.44

0.61

0.25

0.887

3

1 0.62 0.35

4

0.17 0.08

5 6

a=a0 at 728C

0.76

1 2

a=a0 at 708C

0.75 (continued )

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Kinetics of Protein and Enzyme Denaturation TABLE A.10.10 (continued) (Figure 10.20) Time (min)

a=a0 at 658C

10-33

Inactivation of Broccoli Peroxidase a=a0 at 67.58C

7

a=a0 at 708C

a=a0 at 728C

0.54

8

0.70

9

0.45

10

0.66

11

0.38

Source: From Thongsook, T. and Barrett, D.M., J. Agric. Food Chem., 53, 3215, 2005.

TABLE A.10.11 Time (min)

Inactivation of a Yeast Invertase (Figure 10.21)

a=a0 at 408C

a=a0 at 508C

a=a0 at 558C

a=a0 at 608C

0

1

1

1

1

6

0.977

0.946

0.793

0.245

10

0.954

0.901

0.753

0.138

20

0.954

0.847

0.633

0.048

30

0.918

0.824

0.587

0.026

60

0.913

0.796

0.5

0.01

Source: From Vrabel, P., Polakovic, M., Stefuca, V., and Bales, V., Enzyme Microb. Technol., 348, 1997.

TABLE A.10.12 Time (min)

Inactivation of a Puriﬁed Proteinase from P. ﬂuorescens (Figure 10.24)

a=a0 at 908C

Time (min)

a=a0 at 1008C

Time (min)

a=a0 at 1108C

10

0.886

5

0.839

3

0.882

20

0.781

5

0.833

6

0.662

20

0.784

5

0.913

6

0.725

30 30

0.735 0.698

5 10

0.873 0.708

9 9

0.668 0.614

40

0.617

10

0.699

10

0.543

40

0.654

10

0.799

10

0.543

60

0.423

10

0.858

10

0.522

60

0.435

15

0.611

10

0.642

60

0.499

15

0.593

10

0.573

70

0.33

15

0.631

12

0.457

70 80

0.404 0.309

15 20

0.623 0.505

15 15

0.353 0.438

80

0.386

20

0.514

18

0.353

80

0.336

20

0.513

18

0.359

80

0.295

20

0.517

20

0.238

80

0.312

25

0.407

20

0.221

100

0.217

25

0.455

20

0.224 (continued )

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10-34 TABLE A.10.12 (continued) (Figure 10.24) Time (min)

Inactivation of a Puriﬁed Proteinase from P. ﬂuorescens

a=a0 at 908C

Time (min)

a=a0 at 1008C

Time (min)

a=a0 at 1108C

100

0.168

30

0.377

20

0.241

100

0.231

30

0.337

21

0.252

100

0.217

35

0.301

21

0.199

35

0.345

24

0.189

40

0.222

40

0.252

Source: From Schokker, E.P. and Van Boekel, M.A.J.S., J. Agric. Food Chem., 45, 4740, 1997.

TABLE A.10.13 Time (min)

Regeneration of Peroxidase (Figure 10.26)

% Activity, Directly Measured

% Activity, Measured after 30 min at 208C

0.2

87.9

98.0

0.5

81.8

92.9

1

63.6

83.8

2

40.4

77.8

3

30.3

69.7

4

18.2

61.6

Source: From Thongsook, T. and Barrett, D.M., J. Agric. Food Chem., 53, 3215, 2005.

TABLE A.10.14 Autodigestion of a Puriﬁed Protease from P. ﬂuorescens (Figure 10.27) T (8C)

a=a0

40.5

1

45.6

0.98

48.4 49.3

0.83 0.65

51.6

0.13

53

0.075

53.5

0.15

56.9

0.35

57.9

0.42

59.8

0.54

62.1 65.9

0.7 0.89

70.3

0.98

80.6

0.91

90.2

0.67

100

0.155

111

0.17

120

0.05

130

0

Source: From Schokker E.P. and Van Boekel M.A.J.S. Mechanism and kinetics of inactivation at 40–708C of the extracellular proteinase from Pseudomonas ﬂuorescens 22F. J Dairy Res 65:261–272, 1998.

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Kinetics of Protein and Enzyme Denaturation TABLE A.10.15 (Figure 10.28)

10-35

Heat Stability of a Lipoxygenase

Water Content (g g1)

k at 728C

k at 768C

k at 808C

0.167

0.000864

0.00244

0.00619

0.231

0.000316

0.00155

0.00465

0.375

0.000121

0.000625

0.00617

0.5

0.000288

0.6

0.000721

0.00539

0.03

0.75

0.00271

0.0195

0.108

0.0125

Source: From Liou J.K. An approximate method for nonlinear diffusion applied to enzyme inactivation during drying. PhD thesis, Wageningen University, the Netherlands, 1982.

TABLE A.10.16 (Figure 10.29) T0 (8C)

Heat Inactivation of Plasmin D-Value (s)

60

59,520

70

11,700

75 80

660 187

85

126

90

71

100

33

110

23

120

18

140

13

Source: From Saint Denis T., Humbert G., and Gaillard J.L. Heat inactivation of native plasmin. Plasminogen and plasminogen activators in bovine milk: A revisited study. Lait 81:715–729, 2001.

TABLE A.10.17 Plasmin Activity as a Function of Temperature (Figure 10.30) 1000=T

ln(Activity)

2.96

3.9

3 3.05

2.53 1.56

3.09

1.26

3.15

1.18

3.2

1.35

3.25

1.54

3.31

1.84

3.36

2.07

3.42 3.47

2.44 2.84

Source: From Metwalli, A.A.M., de Jongh, H.H.J., and Van Boekel, M.A.J.S., Int. Dairy J., 8, 47, 1998.

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TABLE A.10.18 Effect of Cystein on Heat Inactivation of Plasmin (Figure 10.31) t (min) 0 0.3

% Activity (no cystein) 100

% Activity (2mM cystein) 100 72.14

0.4

68.42

0.5

45.2

0.7

28.17

1

22.6

5

0.31

10

75.23

15 20

68.73 59.75

25

53.25

Source: From Metwalli, A.A.M., de Jongh, H.H.J., and Van Boekel, M.A.J.S., Int. Dairy J., 8, 47, 1998.

Bibliography and Suggested Further Reading About Protein Stability and Protein Denaturation Busti P., Gatti C.A., and Delorenzi N.J. Thermal unfolding of bovine beta-lactoglobulin studied by UV spectroscopy and ﬂuorescence quenching. Food Res Int 38:543–550, 2005. Chaplin L.C. and Lyster R.L.J. Irreversible heat denaturation of bovine a-lactalbumin. J Dairy Res 53:249–258, 1986. Chen X.D., Chen Z.D., Nguang S.K., and Anema S.G. Exploring the reaction kinetics of whey protein denaturation=aggregation by assuming the denaturation step is reversible. Biochem Eng J 2:63–69, 1998. Creighton T.E. Protein folding. Biochem J 270:1–16, 1990. Creighton T.E. Proteins: Structures and Molecular Properties, 2nd ed. New York: Freeman, 1993. Croguennec T., O’Kennedy B.T., and Mehra R. Heat-induced denaturation=aggregation of betalactoglobulin A and B: Kinetics of the ﬁrst intermediates formed. Int Dairy J 14:399–409, 2004. Dannenberg F. and Kessler H.G. Application of reaction kinetics to the denaturation of whey proteins in heated milk. Milchwissenschaft 43:3–7, 1988. Le Bon C., Nicolai T., and Durand D. Kinetics of aggregation and gelation of globular proteins after heatinduced denaturation. Macromolecules 32:6120–6127, 1999. Liang K.K., Hayashi M., Shiu Y.J., Shao J., Yan Y.J., and Lin S.H. Thermodynamics and kinetics of protein folding: A mean ﬁeld theory. Phys Chem Chem Phys 5:5300–5308, 2003. Lumry R. and Eyring H. Conformation changes of proteins. J Phys Chem 58:110–120, 1954. Lyster R.L.J. The denaturation of a-lactalbumin and B-lactoglobulin in heated milk. J Dairy Res 37:233–244, 1970. Privalov P.L. Stability of proteins. Small globular proteins. Adv Protein Chem 33:167–241, 1979. Relkin P. Thermal unfolding of b-lactoglobulin, a-lactalbumin, and bovine serum albumin. A thermodynamic approach. Crit Rev Food Sci Nutr 36:565–601, 1996. Roefs S.P.F.M. and de Kruif C.G. A model for the denaturation and aggregation of B-lactoglobulin. Euro J Biochem 226:883–889, 1994. Seckler R. and Jaenicke R. Protein folding and protein refolding. FASEB J 6:2545–2552, 1992. Tanford C. Protein denaturation. Adv Protein Chem 23:121–282, 1968.

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About Enzyme Inactivation Ahern T.J. and Klibanov A.M. The mechanism of irreversible enzyme inactivation at 1008C. Science 228:1280–1284, 1985. Aymard C. and Belarbi A. Kinetics of thermal deactivation of enzymes; a simple three parameters phenomenological model can describe the decay of enzyme activity, irrespectively of the mechanism. Enzyme Microb Technol 27:612–618, 2000. Daniel R.M. The upper limits of enzyme thermal stability. Enz Microb Technol 19:74–79, 1996. Hei D.J. and Clark D.S. Estimation of melting curves from enzymatic activity temperature proﬁles. Biotechnol Bioeng 42:1245–1251, 1993. Hayes M.C., Hurley M.J., Larsen L.B., Heegaard C.W., Magboul A.A.A., Oliveira J.C., McSweeney P.L. H., and Kelly A.L. Thermal inactivation kinetics of bovine cathepsin D. J Dairy Res 68:267– 276, 2001. Lencki R.W., Arul J., and Neufeld R.J. Effect of subunit dissociation, denaturation, aggregation, coagulation, and decomposition on enzyme inactivation kinetics. 1. First-order behavior. Biotechnol Bioeng 40:1421–1426, 1992a. Lencki R.W., Arul J., and Neufeld R.J. Effect of subunit dissociation, denaturation, aggregation, coagulation, and decomposition on enzyme inactivation kinetics. 2. Biphasic and grace period behavior. Biotechnol Bioeng 40:1427–1434, 1992b. Makki F. and Durance T.D. Thermal inactivation of lysozyme as inﬂuenced by pH, sucrose and sodium chloride and inactivation and preservative effect in beer. Food Res Int 29:635–645, 1996. Metwalli A.A.M., de Jongh H.H.J., and Van Boekel M.A.J.S. Heat inactivation of bovine plasmin. Int Dairy J 8:47–56, 1998. Owusu R.K., Makhzoum A., and Knapp J. The thermodynamic stability of lipases and proteases from psychrotrophic bacteria. Food Chem 39:187–195, 1991. Owusu R.K., Makhzoum A., and Knapp J.S. Heat inactivation of lipase from psychrotrophic Pseudomonas ﬂuorescens P38-activation parameters and enzyme stability at low or ultra-high temperatures. Food Chem 44:261–268, 1992. Owusu R.K. and Berthalon N. A test for the 2-stage thermoinactivation model for Chymotrypsin. Food Chem 48:231–235, 1993. Owusu R.K. and Berthalon N. Determination of enzyme global thermostability from equilibrium and kinetic analysis of heat inactivation. Food Chem 51:15–20, 1994. Polakovic M. and Vrabel P. Analysis of the mechanism and kinetics of thermal inactivation of enzymes: Critical assessment of isothermal inactivation experiments. Process Biochem 31:787– 800, 1996. Rollema H.S. and Poll J.K. The alkaline milk proteinase system: Kinetics and mechanism of heat inactivation. Milchwissenschaft 41:41, 1986. Sadana A. Biocatalysis. Fundamentals of Enzyme Deactivation Kinetics. Englewood Cliffs, NJ: PrenticeHall, 1991. Schokker E.P. and Van Boekel M.A.J.S. Kinetic modeling of enzyme inactivation: Kinetics of heat inactivation at 908C–1108C of extracellular proteinase from Pseudomonas ﬂuorescens 22F. J Agric Food Chem 45:4740–4747, 1997. Schokker E.P. and Van Boekel M.A.J.S. Kinetics of thermal inactivation of the extracellular proteinase from Pseudomonas ﬂuorescens 22F: Inﬂuence of pH, calcium, and protein. J Agric Food Chem 47:1681–1686, 1999. Vrabel P., Polakovic M., Stefuca V., and Bales V. Analysis of mechanism and kinetics of thermal inactivation of enzymes: Evaluation of multitemperature data applied to inactivation of yeast invertase. Enzyme Microb Technol 20:348–354, 1997. Zale S.E. and Klibanov A.M. On the role of reversible denaturation (unfolding) in the irreversible thermal inactivation of enzymes. Biotechnol Bioeng 25:2221–2230, 1983.

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Kinetic Modeling of Reactions in Foods

Enzyme Regeneration Adams J.B. The inactivation and regeneration of peroxidase in relation to the high-temperature-short time processing of vegetables. J Food Technol 13:281–297, 1978. Adams J.B. Review: Enzyme inactivation during heat processing of food-stuffs. Int J Food Sci Technol 26:1–20, 1991. Thongsook T. and Barrett D.M. Heat inactivation and reactivation of broccoli peroxidase. J Agric Food Chem 53:3215–3222, 2005. About the Effect of Molecular Crowding on Protein and Enzyme Stability Agena A.M., Bogle I.D.L., and Pessoas F.L.P. An activity coefﬁcient model for proteins. Biotechnol Bioeng 55:65–71, 1997. Cheung M.S., Klimov D., and Thirumalai D. Molecular crowding enhances native state stability and refolding rates of globular proteins. Proc Natl Acad Sci USA 102:4753–4758, 2005. Davis-Searles P.R., Saunders A.J., Erie D.A., Winzor D.J., and Pielak G.J. Interpreting the effects of small uncharged solutes on protein-folding equilibria. Annu Rev Biophys Biomol Struct 30:271– 306, 2001. Despa F., Orgill D.P., and Lee R.C. Molecular crowding effects on protein stability. Ann NY Acad Sci 1066:54–66, 2005. Eggers D.K. and Valentine J.S. Molecular conﬁnement inﬂuences protein structure and enhances thermal protein stability. Protein Sci 10:250–261, 2006. Ellis R. Macromolecular crowding: Obvious but underappreciated. Trends Biochem Sci 26:597–604, 2001. Ellis R. and Minton A.P. Join the crowd. Nature 425:27–28, 2003. Hall D. and Minton A.P. Macromolecular crowding: Qualitative and semiquantitative successes, quantitative challenges. Biochim Biophys Acta 1649:127–139, 2003. Hu Z., Jiang J., and Rajagopalan R. Effects of macromolecular crowding on biochemical reaction equilibria: A molecular thermodynamic perspective. Biophys J 93:1464–1473, 2007. Minton A.P. Excluded volume as a determinant of macromolecular structure and reactivity. Biopolymers 20:2093–2120, 1981. Minton A.P. The effect of volume occupancy upon the thermodynamic activity of proteins: Some biochemical consequences. Mol Cell Biochem 55:119–140, 1983. Minton A.P. Molecular crowding: Analysis of effects of high concentrations of inert cosolutes on biochemical equilibria and rates in terms of volume exclusion. Methods Enzymol 295:127–149, 1998. Minton A.P. Effect of a concentrated ‘‘inert’’ macromolecular cosolute on the stability of a globular protein with respect to denaturation by heat and by chaotropes: A statistical-thermodynamic model. Biophys J 78:101–109, 2000. Minton A.P. The inﬂuence of macromolecular crowding and macromolecular conﬁnement on biochemical reactions in physiological media. J Biol Chem 276:10577–10580, 2001. Minton A.P. Inﬂuence of macromolecular crowding upon the stability and state of association of proteins: Predictions and observations. J Pharm Sci 94:1668–1675, 2005. Minton A.P. How can biochemical reactions within cells differ from those in test tubes? J Cell Sci 119:2863–2869, 2006. Patel C.N., Noble S.M., Weatherly G., Tripathy A., Winzor D.J., and Pielak G.J. Effects of molecular crowding by saccharides on a-chymotrypsin dimerization. Protein Sci 11:997–1003, 2002. Ralston G.B. Effects of ‘‘crowding’’ in protein solutions. J Chem Educ 67:857–860, 1990. Schnell S. and Turner T.E. Reaction kinetics in intracellular environments with macromolecular crowding: Simulations and rate laws. Prog Biophys Mol Biol 85:235–260, 2004. Shimizu S. and Smith D.J. Preferential hydration and the exclusion of cosolvents from protein surfaces. J Chem Phys 121:1148–1154, 2004.

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Winzor D.J. and Wills P.R. Effects of thermodynamic nonideality on protein interactions. Equivalence of interpretations based on excluded volume and preferential solvation. Biophys Chem 25:243–251, 1986. Winzor D.J. and Wills P.R. Molecular crowding effects of linear polymers in protein solutions. Biophys Chem 119:186–195, 2006. Zimmerman S.B. and Minton A.P. Macromolecular crowding: Biochemical, biophysical and physiological consequences. Annu Rev Biophys Biomol Struct 22:27–65, 1993.

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11 Kinetics of Physical Changes

11.1

Introduction

Physical changes in foods are very important in relation to quality. Many foods are dispersions: they contain (colloidal) particles. Examples are fat globules in an aqueous phase (as in milk), water globules in a fat phase (as in margarine), foam bubbles in an aqueous phase (as in beer foam), fat crystals in an oil phase (as in butter and margarine), protein particles in an aqueous phase (as in milk), starch granules (as in bread), etc. Quality changes arise from changes in such dispersions. Most colloidal systems are inherently unstable in a thermodynamic sense, in other words the systems are not in equilibrium; they may, however, be kinetically stable. For instance, emulsions will tend to separate into an oil and water phase, which is the result of coalescence. Aggregation may occur, of emulsion droplets, of protein particles, of crystals, etc. Sometimes these processes are desirable, for instance, coalescence in the case of churning of cream, coagulation of casein micelles in cheese curd formation, formation of sugar crystals in sugar manufacture, etc. Mostly, however, such changes will result in quality loss. Examples are aggregation and sedimentation of fruit particles in a fruit juice, serum separation in tomato ketchup, creaming of fat globules in milk, sugar crystallization in candy bars, and many more examples could be given. In this chapter we will discuss how some of these changes can be described kinetically. The size scale of particles present in foods may span up to six orders of magnitude; the structural elements range from low molecular weight molecules to high molecular weight compounds to colloids to microorganisms, i.e., from nanometers to some hundred micrometers. In general, the larger the particles, the longer the time scales involved at which something happens (such as a molecular reaction or aggregation of colloidal particles). We need to remark that kinetics of physical changes is not an easy topic to discuss. Most of the available models are developed for dilute, ‘‘clean’’ systems, and foods are all but ‘‘clean’’ and dilute, as remarked before. One has to be careful therefore in applying models to describe physical changes in foods. Much active research is going on in this ﬁeld and it is beyond the scope of this book to discuss all these developments; the interested reader will ﬁnd some references at the end of this chapter. We will limit ourselves therefore to some phenomena that are important for food quality such as diffusion, aggregation and coalescence. Nevertheless, it is stressed that the length of this chapter is inversely proportional to its importance for food quality.

11-1

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11-2

11.2

Kinetics of Diffusion

If there are concentration differences within a system, these differences will tend to disappear over time, and this is an entropy-driven process. This process is commonly known as diffusion, caused by the random translational and rotational motion of molecules and particles. In relation to quality of foods, the migration of water due to diffusion is particularly important. We are interested in how fast these changes occur. Two theories will be discussed here, one the conventional Fickian diffusion, and the other the Maxwell–Stefan (MS) approach which has become more popular the last decades, for reasons that will be discussed. We will start with Fick’s laws because those are probably the familiar ones, and we then move on to show the more general MS approach.

11.2.1

Fick’s Laws

The rate of migration of matter is measured by its ﬂux J (concentration of a species times its velocity, mol m2 s1). This ﬂux is found to be proportional to the concentration gradient (of a compound, or of the number of particles per unit volume): J/

dc dx

(11:1)

If we introduce a proportionality constant in Equation 11.1 called the diffusion coefﬁcient Df we ﬁnd J ¼ Df

dc dx

(11:2)

Equation 11.2 is the ﬁrst diffusion law of Fick. In fact, we introduced this equation already in Equation 3.152 in relation to irreversible thermodynamics. Einstein derived that the root-mean-square distance in a given direction for translational motion is pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃ hx2 i ¼ 2Df t

(11:3)

For spherical particles (Equation 4.159), the Stokes–Einstein equation is Df ¼

kB T 6phv Rp

(11:4)

This equation shows that small molecules and particles move faster than bigger ones. Typical values for the diffusion coefﬁcient Df in water at room temperature are 1.7 3 109 m2 s1 for water molecules, 4.7 3 1010 m2 s1 for sucrose molecules in water, 6 3 1011 m2 s1 for a globular protein as serum albumin, and 4 3 1013 m2 s1 for an emulsion droplet of size 1 mm. The diffusion coefﬁcient depends on size and shape of the molecule=particle, on its interaction with the solvent, and the viscosity of the solvent. Our interest is in describing a situation where the concentration is not homogeneous and we would like to describe the rate of change in such a situation. Consider a situation of a thin slab such as depicted in Figure 11.1. The concentration at x is c at time t and there is an inﬂux J1 at the left-hand side. The amount of molecules M (in moles) that enter the slab with area Aa in an interval dt is M ¼ J1 Aa dt

(11:5)

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Kinetics of Physical Changes

11-3 Area Aa

J1

J2

dx

FIGURE 11.1 Schematic picture to show the ﬂow in and out of a volume element with length dx.

As a result of this inﬂux, there is a concentration change inside the slab: @c J1 Aa dt J1 ¼ ¼ @t Aa dx dt dx

(11:6)

There is of course also an outﬂow J2 at the right-hand side of the slab which is similarly @c J2 Aa dt J2 ¼ ¼ dx @t Aa dx dt

(11:7)

The ﬂux J1 entering the slab at x minus the ﬂux J2 leaving the slab at x þ dx equals accumulation over dx. It then follows that the concentration change is @c J1 J2 ¼ dx @t

(11:8)

Applying Fick’s ﬁrst law gives @c ¼ @t

Df

@c @c þ Df @x x @x xþdx dx

(11:9)

and this can be written as 2 @c @ c ¼ Df @t x @x2 t

(11:10)

This is Fick’s second law, giving the change in concentration c with time at any x as a function of the concentration gradient. For diffusion in all three directions it can be derived that: 2 @c @ c @2c @2c þ þ ¼ Df @t @x2 @y2 @z2

(11:11)

Fick’s second law is a second-order differential equation with respect to space and a ﬁrst-order one with respect to time (see also Appendix A). The solution depends on the boundary conditions, two for space and one for time; those boundary conditions must be speciﬁed. Many books are written with particular

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11-4

solutions of Fick’s equations. With modern software, numerical solutions are another option to solve them relatively easy. We will not discuss this in detail but refer to some references given at the end of this chapter. Two solutions are worth considering though, because they are instructive. One is the calculation of mass transfer via diffusion over a ﬂat boundary with area Aa, where the concentration of a component is maintained at a certain level. The amount transported per unit time is then: rﬃﬃﬃﬃﬃﬃﬃ Df t (11:12) M ¼ 2Aa c p This shows the typical square root relation with time for diffusion, already shown in Equation 11.3. Another equation that is useful is the concentration as a function of the distance x from a ﬂat inﬁnite surface: 3 2 ðy 2 2 c(x) ¼ c(1 erf y) ¼ c41 pﬃﬃﬃﬃ exp (z )dz 5 p (11:13) 0 x y ¼ pﬃﬃﬃﬃﬃ 2 Dt In this equation, erf symbolizes the error function, and z is an integration variable. This equation is, for instance, useful to predict the migration of molecules from a packaging material into a food. It can be derived from Equation 11.13 that x0:5 Df t0:5

(11:14)

x0.5 is the distance over which the concentration is halved, and t0.5 is the time at which that happens. Given the fact that for most solutes in water Df 1010 m2 s1, the latter equation gives an order of magnitude estimation of diffusion effects. A complication in foods is that the diffusion coefﬁcient depends quite strongly on the food matrix. In the presence of macromolecules and networks, diffusion is strongly hindered and the diffusion coefﬁcient can be orders of magnitude lower than the one found in water. This will be further discussed in Chapter 14; see, however, also the MS approach discussed in Section 11.2.2. Fick’s laws are the starting point for all kinds of rate equations, such as moisture loss or uptake, drying kinetics, sorption kinetics, rate of migration of residues from packaging materials into foods, etc. Such situations are thoroughly discussed in books dedicated to food engineering, so we refer the interested reader to these; some references are given at the end of this chapter. Diffusion through a barrier. It can be of great importance for food quality that diffusion of molecules can be inﬂuenced, for instance by packaging or by applying a coating. It concerns diffusion of water, ﬂavor molecules, colored components, oxygen, etc. This situation can be modeled in a simpliﬁed way as follows. Imagine for instance two aqueous layers (phase I) separated by a lipid layer (phase II) as shown in Figure 11.2, and we would like to model how fast a component migrates through the lipid layer into the other compartment. It is assumed that the concentration in phase a is the same everywhere. It is known that the solubility of the component differs in phases a and b, and that it dissolves better in phase a. This can be quantiﬁed in the partition coefﬁcient (to be discussed in more detail later on in this chapter, Equation 11.82): Pa=b ¼

Ca Cb

(11:15)

From Fick’s ﬁrst law it can be derived that the mass ﬂux J through an area Aa at steady state is J¼

Df Aa Pa=b Dc x2 x1

(11:16)

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Kinetics of Physical Changes

11-5

Phase β

Concentration, c

Phase α

Phase α

[c]

[c] [c]

x1

x2

Distance, x

FIGURE 11.2 Simpliﬁed representation of the diffusion of a component from the left compartment (phase a) through a layer (phase b) to the right compartment.

This equation shows how the ﬂux can be inﬂuenced. One possibility is to adjust the thickness of layer x1 x2, another to search for a material in which the effective diffusion coefﬁcient is different in phase b, and yet another to search for materials for which the partition coefﬁcient differs. Equation 11.17 also shows that the diffusion through a very thin layer is hardly hindered, unless the values of Df and Pa=b are such that diffusion is hindered strongly. This is of importance for diffusion through layers and ﬁlms that are very thin: diffusion is then hardly retarded. The ﬁlm model. The ﬁlm model is used extensively in chemical engineering, and also in food engineering. It tackles ﬂow near interfaces. Concentration gradients in the bulk are assumed to be absent because of convective currents and eddies. Near phase interfaces, eddies die out and a small ﬁlm is supposed to exist in which the only transport mechanism is diffusion (Figure 11.3). The thickness dz of such a ﬁlm is roughly 104 m in gases, 105 m in liquids, 104 to 107 m in membranes, and even in solid particles a thin ﬁlm is assumed to exist with a thickness of roughly 0.1 3 diameter of the solid particle. Film ∆z Phase a

No concentration gradients due to convection

Phase boundary

Phase b

Concentration gradient due to diffusion

FIGURE 11.3 Schematic illustration of the ﬁlm model (not to scale).

No concentration gradients due to convection

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Kinetic Modeling of Reactions in Foods

11-6

Mass transfer via packaging materials. Food quality is very much inﬂuenced by how the food is packaged it regulates the interaction with the environment and it is all about mass transfer. Packaging may inﬂuence the mass transfer of water vapor, oxygen, volatile aroma components, and other molecules via . .

.

Permeation: transfer through the packaging material Sorption: penetration and dispersal of molecules coming from the food into the packaging material (a process sometimes referred to as ‘‘scalping’’) Migration: transfer of molecules from the packaging material into the food

Furthermore, packaging can protect against light (basically against photooxidation). If there is a headspace, the relations become even more intricate because it adds a vapor interface on the food side. As for the environment, the factors of importance with respect to food quality are the oxygen content, the light intensity and wavelength, the relative humidity, and the temperature. Assuming that the packaging material does not allow permeation of microorganisms or dirt materials, its permeability to oxygen, light, and moisture is important. A special case is modiﬁed atmosphere and controlled atmosphere packaging, in which the food is contained in a special gas composition to control metabolic changes within the food and to control growth of microorganisms. Composition varies then usually with respect to oxygen and carbon dioxide content, and the permeability with respect to CO2 and nitrogen can then also be important. If changes in the food are inﬂuenced by transfer of light, oxygen and moisture through the package, kinetics could be controlled by mass transfer effects (diffusion-limited reactions, as discussed in Chapter 4). It should be noted that this not only concerns transfer from the environment to the food, but also from the food to the environment. For instance, transport of moisture from bread to the environment may lead to a quality loss because of drying out (physical quality loss). The permeation of gases (oxygen, moisture) will be governed by diffusion and can thus be described by Fick’s ﬁrst law of diffusion. For unsteady state diffusion (concentration is a function of time), the second law of Fick is applicable. In this way, the rate of change in concentration is related to the spatial variation of the concentration at that point. The following example may illustrate. Suppose that we are interested in how fast the migration of a component from the packaging material into the food occurs, a situation that is usually not desired. The concentration change of such a component (in one dimension) can be described by the following equations: @cP @ 2 cP ¼ Df ,P 2 @t @x

@cF Aa @cP ¼ Df ,P @t VF @x x¼lP

(11:17) (11:18)

cP is the concentration of the component in the package having thickness lP cF the concentration in the food having a volume VF Aa is the contact area between the packaging material and the food The boundary conditions at t ¼ 0 are that the initial concentration of the component in the packaging material cP0 has a ﬁnite value while the concentration at time zero in the food is zero, i.e., cF0 ¼ 0; at x ¼ 0, @[email protected] ¼ 0 and at x ¼ lP, cP ¼ PP=F 3 cF with PP=F the partitioning coefﬁcient of the component between the packaging material and the food. The concentration in the food cF is calculated from this last relation. Figure 11.4 shows how the concentration in the packaging material changes as a function of x at various times, while Figure 11.5 shows how the concentration in the food increases with time; for convenience concentration and space were made dimensionless.* * The calculations were done with the software program AthenaVisualStudio v.11, a program capable of handling partial and ordinary differential equations via numerical integration. Web site: www.athenavisual.com

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Kinetics of Physical Changes

11-7 t =0

1.0

t = 10

t = 25

cP

0.6 Food

Package

0.8

0.4 t = 50 0.2

0.0 0.0

0 1 x

t = 75 0.2

0.4

0.6

0.8

1.0

x

FIGURE 11.4 Dimensionless concentration change cP of a component in a packaging material as a function of dimensionless distance x and time t (h). Parameter values are lP ¼ 104 m, PP=F ¼ 1.4, VF ¼ 104 m3, Aa ¼ 0.02 m2, cP0 ¼ 2.3 g dm3, Df,P ¼ 3 3 1014 m2 s1.

With such simulations one could explore how migration can be inﬂuenced by changing parameters as the diffusion constant and the partition coefﬁcient; these parameters can be manipulated by changing the properties of the packaging material. Packaging materials that are permeable to gases are usually polymeric ﬁlms. Diffusion of gases through such materials starts with a dissolution process of the gas molecules in the surface of the ﬁlm, followed by diffusion through the polymer and ﬁnally desorption of the molecule at the surface. In this respect, one uses frequently the permeability coefﬁcient Pc, which is the product of the diffusion coefﬁcient Df and the solubility coefﬁcient S and is a measure of the steady-state transfer rate of a permeant: Pc ¼ Df S

(11:19)

0.020

cF

0.015

0.010

0.005

0.000 0

20

40

60

80

Time

FIGURE 11.5 Dimensionless change in concentration cF of a component in a food migrating from a packaging material as a function of time (h). Same parameter values as in Figure 11.4.

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11-8

The solubility coefﬁcient S refers to the amount of the permeant molecules dissolved in the material at equilibrium conditions.

11.2.2

Maxwell–Stefan Approach

The Maxwell–Stefan (MS) equation is a more general alternative for the Fick equations; in fact, the Fick equation is a limiting case of the MS equation. MS equation is especially useful for multicomponent mass transfer and should be used in such cases because the Fick equation is not well suited for multicomponent diffusion. Here we consider only one-dimensional diffusion; the reader is referred to literature for the case of three dimensions. Why is the MS equation more general than the Fick equation? Fick’s law explains the ﬂux of component i with respect to the mixture; a species should move down to its concentration gradient (Equation 11.2). However, Fick’s law is not able to explain several common phenomena, for instance that in some situations components move against their concentration gradient, or the diffusion of ions, while this can be explained logically by the MS approach. The basic idea of the MS approach is that mass transfer is driven by the potential gradient of a species and limited by friction with the surroundings (which can include a solid matrix); the balance of these two determines the speed with which a species will move. The MS equation allows also other driving forces than just concentration gradients, such as forces due to pressure, centrifugation, and electrical forces. The approach is to take into account the friction between each pair of components (including any solid matrix), and to consider all relevant driving forces as well as to account for viscous ﬂow in heterogeneous media. The basis for this approach is that motion in a mixture is governed by forces working on individual species. In this respect there are driving forces coming from a potential gradient and friction forces arising due to velocity differences. As discussed in Chapter 3, the concept of forces and displacement is also used in thermodynamics and there is actually a link with irreversible thermodynamics. In fact, Fick’s law was already mentioned there as an example in Equation 3.152 and also the MS equation is ﬁrmly rooted in irreversible thermodynamics. In terms of irreversible thermodynamics, the rate of entropy production is a sum of the products of two quantities: the force acting on component i and the relative velocity of the movement of i with respect to the mixture. This is the dissipation due to diffusion. At equilibrium the driving forces have disappeared (are dissipated), and in the case of diffusion this implies that there are no net movements anymore. There are internal and external forces. Internal forces cause motion inside a mixture, external forces cause motion of the mixture as a whole (e.g., gravitational force). In general, the driving force Fi is the negative gradient of the total potential ci: Fi ¼

dci dz

(11:20)

Examples are gravitational, electrical, pressure, and chemical driving forces. The driving force for mass transfer due to molecular diffusion is the negative gradient of the chemical potential of component i over distance z: F¼

dmi d ln ai RT dai ¼ RT ¼ dz dz ai dz

(11:21)

which results effectively in motion of component i relative to the mixture in which it is in. Then there is friction if one component i moves with respect to another component j; for component i the friction is proportional to the concentration of component j and the velocity difference n1n2: zi,j Xj (ni nj )

(11:22)

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Kinetics of Physical Changes

11-9

zi,j is the friction coefﬁcient between components i and j (in N s mol1 m1), Xj is the mole fraction of component j. Friction coefﬁcients in liquids are of the order of 1012 to 1013 N mol1 (ms1)1; these large values arise because of the huge numbers of molecules involved. On the other hand, driving forces are also large as shown in the following example. It concerns the transfer of CO2 in a carbonated liquid such as a soda drink or beer. Use is made of the ﬁlm model introduced above, so we consider mass transfer in a ﬁlm between the bulk liquid and the bubble of thickness dz ¼ 105 m, and let us assume that the mole fraction of CO2 in the drink is 0.003, and that in the bubble 0.001. Assuming ideal behavior, the driving force for mass transfer is according to Equation 11.20 Fi ¼ 2.4 3 108 N mol1 CO2. The concept of driving force and friction force is the basis of the MS equation (after the nineteenth century scientists Maxwell and Stefan) as it expresses the balance between a driving force and the friction due to motion; for a component i surrounded by other components j it reads X zi,j Xj ni nj Fi ¼ (11:23) j6¼i

In words, the MS equation reads that the driving force on a component i in a mixture equals the sum of the friction forces between i and the other components j, while the friction exerted by j on i is proportional to the fraction of j in the mixture and the difference in velocity ni and nj between i and j, respectively. In other words, mass transfer is driven by the potential gradient of a species and limited by friction with the surroundings (which can include a solid matrix); the balance of the two determines the speed with which all this occurs. The velocity referred to here is the diffusive velocity, not the thermal velocity (which is much larger and random). The velocities in the MS equation are differences, they only say something about relative motion inside the mixture, not about the mixture as a whole. To obtain absolute velocities of components, other relations are needed, such as mass balances, stagnant matrices, reaction stoichiometry, equimolar exchange, and the like. Instead of friction coefﬁcients, one commonly uses MS diffusivities indicated by the symbol D to differentiate it from the Fickian diffusion coefﬁcient Df. In the MS equation, the diffusion coefﬁcient, as well as the friction coefﬁcient, are mutual coefﬁcients. In literature one often ﬁnds so-called selfdiffusivities: these are diffusivities of a species with the same physical properties as the solvent. The relation of the MS diffusion coefﬁcient with the friction coefﬁcient is RT Di,j ¼ § 1,2

(11:24)

For liquids, the order of magnitude for D is 109 m2 s1, for gases 105 m2 s1. Sometimes, mass transfer coefﬁcients are used rather than the diffusion coefﬁcient. The relation is D1,2 km 1,2 ¼ (in m s1 ) dz

(11:25)

The order of magnitude for kmi,j in gases is 101 to 102, and in liquids 104 to 105; in packaging materials kmi,j is several orders of magnitude lower, as it should be if the packaging material is to protect the food from, for instance, gas and vapor entering the food from the atmosphere. It is sometimes easier to work with ﬂuxes rather than with velocities, as we saw already for Fickian diffusion. The ﬂux Ji is the product of the velocity of a component and its concentration: Ji ¼ ni ci ¼ ni cXi (in mol s1 m2 )

(11:26)

The MS equation in ﬂux form is Fi cXi ¼

X i

z1,2 Xj Ji Xi Jj

(11:27)

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11-10

The MS diffusion coefﬁcient Df and the Fickian diffusion coefﬁcient D are in general not equal, but they are, obviously, related. An important difference between the MS and Fick equation is the effect of nonideality (in the thermodynamic sense). Deviations from ideality end up in the Fickian diffusion constant Df, whereas in the MS approach they are part of the driving force, since the driving force is based upon the chemical potential: Fi ¼ RT

d ln ai d ln fi Xi ¼ RT dz dz

(11:28)

Comparing this equation with Equation 11.2 results in Df 1,2 d ln f1 ¼ 1 þ X1 D dX1 1,2

(11:29)

It shows how much the Fickian diffusion coefﬁcient can deviate from its MS counterpart in terms of the activity coefﬁcient. Figure 11.6 shows an example for an ethanol–water mixture. It is seen that the MS diffusion coefﬁcient is hardly affected by the composition of the mixture, while the Fick diffusion coefﬁcient is strongly depending on it. This is, as mentioned, because the effect of nonideality is enclosed in the Fickian diffusion coefﬁcient. There are several thermodynamic models, expressing the behavior of activity coefﬁcients as a function of composition. These can be linked to Equation 11.28, if so desired, to get a quantitative expression of the difference in the two diffusion coefﬁcients.

2 Ð1,2

Ð1,2, Df1,2/10−9 m2 s−1

1.6

1.2

Df1,2

0.8

0.4

0

0

0.2

0.4

0.6

0.8

1

X1

FIGURE 11.6 MS diffusion coefﬁcient D1,2 (.) and Fick diffusion coefﬁcient Df1,2 (*) as a function of the mole fraction of ethanol X1 in a binary ethanol–water mixture. Dataset in Appendix 11.1, Table A.11.1.

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Kinetics of Physical Changes TABLE 11.1

11-11

Some Potentials and Driving Forces to be Used in the MS Equation

Type

Potential mi ¼ Constant þ ln (fi Xi ) RT ci Vi P ¼ RT RT

Driving Force d ln fi Xi Fi ¼ RT dz dP Fi ¼ Vi dz

Electrical

ci Ff zi ¼ c RT RT e

Fi ¼ Ff zi

Centrifugal

ci Mi v2 z 2 ¼ RT 2RT

Fi ¼ Mi v2 z

Gravity

ci Mi gz ¼ RT RT

Fi ¼ Mi g

Chemical potential for nonideal mixtures Pressure

dce dz

Note: Ff, Faraday constant (C mol1); g, acceleration due to gravity (m s2); v, angular frequency (s1).

As mentioned, the MS equation allows for other driving forces than chemical potential gradients. Table 11.1 gives an overview of some relevant potentials and driving forces. Such an analysis allows to calculate, for instance, that gravity becomes more important for movement of particles than chemical potential differences if the particle size is larger than about 0.3 mm. Ion mass transfer. Mass transfer of ions can show some peculiar effects. It was already discussed in Chapter 6 that ionic species show strong nonideal behavior and several theories were presented to account for this nonideality. Obviously, the most important difference with mass transfer of neutral solutes is the presence of electrical driving forces. The MS equation in terms of ﬂuxes for such a case is X d ln ai dc RT þ Ff e ¼ Xj Ji Xi Jj cXi RT D dz dz j6¼i i,j

(11:30)

In the case of very dilute solutions, friction between ions and the solvent dominates and if the velocity of the solvent is zero, we obtain the Nernst–Planck equation: Ji ¼ c Di,w

dXi Ff zi Xi dce þ dz RT dz

(11:31)

In considering the ﬁlm model for ion mass transport, it should be realized that such a ﬁlm is much thicker than the electrical double layer (say 104 nm versus a few nanometers), and consequently we can P consider the condition of electroneutrality in the ﬁlm model: i zi Xi ¼ 0 and the no-current condition, implying that nþ ¼ n. When solving the transport equations it appears that there is a small but relevant electrical potential difference across the ﬁlm, this is the so-called diffusion potential: RT Dc ¼ 2Ff

1 k,w

1 kþ,w

n

(11:32)

with Ff the Faraday constant and k,w and kþ,w the mass transfer coefﬁcient. This has the result that anion and cation move with the same velocity, in other words as a single substance: 1 1 1 dX n ¼ 2 kmþ,w km,w X

(11:33)

However, for mixed electrolytes this is no longer true; it may happen that the electrical gradient caused by an electrolyte forces another ion to move against its concentration gradient.

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11-12

7

6

Ð/10−10 m2 s−1

5

4

3

2

1

0 0

50

100

150

KCl (mol m−3)

FIGURE 11.7 MS diffusion constant D of lysozyme as a function of KCl concentration. Dataset in Appendix 11.1, Table A.11.2.

When studying diffusion of proteins, electrical effects are important because proteins are polyelectrolytes. Figure 11.7 shows the diffusion coefﬁcient of the protein (enzyme) lysozyme when it has chloride ions as counterions at 0 M KCl. When KCl is added it is seen that the diffusion coefﬁcient decreases. The explanation is that in the absence of KCl the protein and its counterions must move together in order to maintain electroneutrality and the resulting diffusivity is the average of the protein and the chloride ions. When KCl is added, the role of the protein is taken over by the more mobile potassium ions and the protein need not move anymore along with the chloride ions. Figure 11.8 shows diffusivities of NaCl, in terms of DNaþ,w, DCl,w, and DNa,Cl. It is seen that the mutual diffusivities of the separate ions and water do not depend much on the concentration of NaCl, but the mutual diffusivity of Naþ and Cl do, because there is a lot of friction between the two ions. Another noteworthy fact is that the diffusion coefﬁcients of Hþ and OH are much higher (in the order of 108 m2 s1) than those of other ions (in the order of 109 m2 s1). Also, diffusivities decrease strongly with increasing charge numbers. All in all, diffusion of charged species can be quite peculiar and we cannot do justice to all the intricacies in this book. We just mentioned a few points to draw attention to it. The reader is advised that there is substantial literature on these phenomena and some references are given at the end of this chapter. However, it is reassuring that the phenomena can be tackled by the MS equation, in principle at least. An example about salt diffusion in cheese may be illustrative to conclude this section on diffusion. Gouda type cheese is salted by putting the cheese into a brine solution. As a result, salt diffuses into the cheese while water moves out of the cheese. Attempts have been done to model this via Fickian diffusion, which was to some extent possible, but modeling via the MS approach appeared to be more powerful. For the MS approach to model salt and water diffusion, cheese can be considered as a three component system characterized by three MS diffusion constants:

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Kinetics of Physical Changes

11-13

−8

Ðw,Cl−

−9 log Đi,j (m2 s−1)

Ðw,Na+

−10

−11 −2

Ðw,NaCl

−1.5

−1

−0.5

0

0.5

1

log NaCl (mol dm−3)

FIGURE 11.8 Mutual MS diffusion coefﬁcients of water–Cl, water–Naþ, and water–NaCl in NaCl solutions. Dataset in Appendix 11.1, Table A.11.3.

D1,2: for the components NaCl (species 1)–water (species 2) D1,3: for the components NaCl–cheese matrix (species 3) D2,3: for the components water–cheese matrix The cheese matrix is assumed to consist of protein-fat-amorphous calcium phosphate. For obvious reasons, the diffusional ﬂux of the cheese matrix is assumed to be negligible, which leaves two ﬂux equations for one dimension: "

J1 J2

#

31 2 X2 X3 X1 @ ln f1 þ 6D1,2 D1,3 6 1 þ X1 @X D1,2 7 1 7 6 6 ¼ ct 6 7 6 4 4 X2 X1 X3 5 @ ln f2 þ @X1 D1,2 D1,2 D2,3 2

32

3 dX1 76 76 dz 7 7 7 @ ln f2 54 dX2 5 1 þ X2 @X2 dz @ ln f1 @x2

(11:34)

where ct is the total molar concentration (mol m3) X is the mole fraction J represents the molar diffusional ﬂux of species 1 (NaCl) and 2 (water) with respect to the cheese matrix f is the rational activity coefﬁcient The ﬁrst matrix on the right hand side accounts for the friction between the species and the second matrix for nonideal behavior of the species concerned. We do not discuss here the intricacies involved in estimating activity coefﬁcients and diffusion coefﬁcients in a complex matrix as cheese is, but only show the application of the MS modeling approach in Figure 11.9. The interested reader is referred to the original literature source for details. As shown, the MS approach is well capable to model the diffusion of NaCl and water for cheese immersed in brine, in fact better than was obtained with the Fick diffusion equations.

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Kinetic Modeling of Reactions in Foods 25

45

20

40 Water in cheese (%)

Salt in cheese moisture (%)

11-14

15

10

0

35

30

25

5

(A)

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20

0

0.01 0.02 0.03 0.04 0.05 x (m)

0.06 0.07

0

0.01 0.02 0.03 0.04 0.05 0.06 0.07

(B)

x (m)

FIGURE 11.9 Application of MS diffusion equation (solid line) to predict the salt and water content after 8.1 days due to diffusion of NaCl (A) and water (B) in cheese in contact with brine (20% salt solution); x is the distance from cheese–brine interface. Dataset in Appendix 11.1, Table A.11.4.

11.3

Kinetics of Changes in Dispersity

Many foods are in fact disperse systems. Milk is an oil-in-water emulsion, butter and margarine are an water-in-oil emulsion and also may contain fat crystals, the major milk protein in milk is in the form of aggregates (casein micelles); fruit juices contain particles; ice cream is an intricate dispersion of icecrystals, air bubbles, fat crystals, and casein micelles; bread is a sponge; and the list goes on and on. As mentioned before, such dispersions can be unstable and this will, in that case, lead frequently to quality loss. We will discuss in this section some kinetic equations describing such changes.

11.3.1

Kinetics of Aggregation of Colloids

Aggregation of colloidal particles has a large effect on properties of the system. It can lead to creaming or sedimentation, and to changes in rheological properties. Whether or not particles will aggregate depends on colloidal interaction forces, i.e., electrostatic repulsion and attraction, steric repulsion and van der Waals attraction, depletion interaction, etc. The simplest (but already quite complicated) situation is that of unhindered ﬂocculation of particles that meet each other due to diffusion and immediately stick together. Smoluchowski developed this theory in the early 1900s based on diffusion kinetics. In fact, the outcome was already given in Chapter 4 where it was used to calculate how fast diffusion-limited reactions would be (Equation 4.161). In the case of colloidal particles, this phenomenon is described as fast perikinetic aggregation. The resulting equation for equal-sized particles in a dilute system is

dN 4kB T 2 N ¼ kp N 2 ¼ 4pDf dp N 2 ¼ dt 3hv

(11:35)

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Kinetics of Physical Changes

11-15

where N represents the number of particles t represents time Df is the Fickian diffusion coefﬁcient dp is the particle diameter hv is the viscosity of the suspending medium kp is the second-order perikinetic rate constant Integration of this equation leads to the following simple expression for the number of particles as a function of time: Nt ¼

N0 1 þ kp N0 t

(11:36)

This is, not surprisingly, the same expression as the second-order reaction discussed in Equation 4.53. However, Equation 11.36 expresses a behavior different from a chemical reaction, and it can be used not only to see how the number of primary particles decreases but also the course of dimers, trimers, and so on. A useful transformation in this respect is the introduction of the halving time t0.5: t0:5 ¼

phv dp3 1 1 ¼ ¼ kp N0 4pDf N0 d 8kB Tw0

(11:37)

where w0 is the volume fraction of primary particles, so that Nt 1 ¼ N0 1 þ t t0:5

(11:38)

and Ni (t=t0:5 )i1 ¼ N0 (1 þ t t0:5 )iþ1

(11:39)

This relation is depicted in Figure 11.10. Some interesting features follow from this. Table 11.2 shows some calculations for halving times in water at room temperature, demonstrating the large effect of particle size and volume fraction when encounters are dominated by Brownian motion: smaller particles are much more sensitive to aggregation than larger ones, and obviously, increasing the volume fraction leads to shorter halving times. These numbers are just to show the trends; for real food systems it may be much different, as will be shown below. Besides perikinetic aggregation, there is orthokinetic aggregation. This implies that encounters are brought about by velocity gradients. In the most simple case of simple shear with velocity gradient Gs (s1), the second-order equation becomes

dN 2 3 4 ¼ dp Gs N 2 ¼ w0 Gs N ¼ ko N 2 dt 3 p

(11:40)

where ko represents the orthokinetic second-order rate constant. It is instructive to compare the perikinetic and orthokinetic rate constants:

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Kinetic Modeling of Reactions in Foods

11-16

1 0.9 0.8 0.7 N1

N/N0

0.6

Ni

0.5 0.4 0.3 N2

0.2

N3

0.1

N4

0 0

0.5

1

1.5

2 t/t0.5

2.5

3

3.5

4

FIGURE 11.10 The course of particle aggregation as a result of fast perikinetic aggregation according to Equation 11.39. N1 is the number of primary particles, N2 that of dimers, N3 that of trimers, N4 that of tetramers, Ni that of the total number of particles.

4kB T 3hv 2 3 ko ¼ dp Gs 3 3 ko dp hn Gs ¼ kp 2kB T kp ¼

(11:41)

There is a strong effect of particle size in the case of orthokinetic aggregation. If we have a particle size dp ¼ 3 mm and a very low shear rate, say Gs ¼ 1 s1, it can be calculated that orthokinetic aggregation is already more important than perikinetic aggregation: the ratio ko=kp is then by a factor 3.3 times larger for an aqueous system at 208C. The opposite is true for smaller particles: then perikinetic aggregation is much more important. This means that even the smallest amount of ﬂow can easily lead to aggregation if particle sizes are in the micron range, and this is very relevant in foods. Flow rates of Gs ¼ 0.1 s1 can easily arise, if only because of temperature gradients. As mentioned, the Smoluchowski equation does not take colloidal interaction forces between particles into account, nor hydrodynamic interactions. As it happens, the van der Waals attraction between particles may roughly compensate the hydrodynamic resistance, so that in the absence of repulsive forces,

TABLE 11.2 Halving Times t0.5 (s) as a Function of Volume Fraction w0 and Particle Diameter dP in Water at Room Temperature w0 ¼ 0.001 w0 ¼ 0.01 w0 ¼ 0.1

dp ¼ 0.01 mm

dp ¼ 0.1 mm

dp ¼ 1 mm

dp ¼ 10 mm

1 3 104

1 3 101

1 3 102

105

5

2

1 3 10

104

3

1 3 10

103

1 3 10

6

1 3 10

1 3 10 1 3 10

1 0

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Kinetics of Physical Changes

11-17

the Smoluchoswki equation often holds reasonably well. However, when repulsion exists between particles, this will slow down aggregation. In such a situation, the aggregation rate is conveniently called slow perikinetic and orthokinetic aggregation with rate constants kSp and kSo , respectively. A so-called stability factor W is introduced: ko W kp kSp ¼ W kSo ¼

(11:42)

The question is of course if we can ﬁnd an expression for this stability factor W. This appears not to be easy, though approximate expressions may be derived for simple and well-deﬁned cases. We cannot do justice to it in this book but the following very brief introduction to colloidal stability is appropriate in order to understand some basic concepts of the stability factor W. We do however urge the reader to consult other books more dedicated to this topic, as listed at the end of this chapter. As mentioned, under very strict conditions, W can be related to colloidal interaction as given by the DLVO theory, the ﬁrst colloidal interaction theory developed by Deryagin, Landau, Verweij, and Overbeek in the 1940s, based on electrostatic repulsion and van der Waals attraction. Colloidal particles often carry a charge; in foods this will be mainly due to proteins adsorbed to such particles. Proteins have acid and basic groups that can ionize, depending on pH. The ionized groups are surrounded by ions of opposite charge, the so-called electrical double layer. If two identically charged particles approach each other they will experience repulsion when the electrical double layers start to overlap. The equation for the repulsive interaction energy (in Joules) between two globules with radius Rp as function of the interparticle distance h is VR ¼ 2p«r «0 Rp c20 ln½1 þ exp (kh) 4:5 109 Rp c20 ln½1 þ exp (kh)

(11:43)

where «r is the dielectric constant of the medium «0 is the permittivity of free space c0 is the electric surface potential 1=k is the thickness of the electrical double layer We encountered these parameters already in Chapter 6. The approximation in Equation 11.43 is for conditions in water at room temperature. The thickness 1=k is related to the ionic strength of the solution I, which was already given in Equation 6.19, and can roughly be approximated as 1 0:3 pﬃﬃ k I

(11:44)

with 1=k in nm and I in mol dm3. With respect to attraction between colloidal particles, van der Waals forces come into play. The equation for the attraction energy VA (in Joules) between two spherical particles with radii Rp is VA ¼

AH R p 12h

(11:45)

where AH represents the Hamaker constant, which is material speciﬁc and also depends on the dispersion medium. For instance, for oil emulsion droplets in water AH (4–6) 3 1021 J. According to the DLVO theory, the total interaction energy VT between two particles is the sum of VR and VA. Depending on the magnitude of the Hamaker constant and the surface charge and thickness of electrical double layer,

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300

200 VR

V/kB T

100

0

−100

VT

0

4

8

12

16

20

VA

−200 −300 h (nm)

FIGURE 11.11 Example of a DLVO plot for interaction between two spheres as a function of interparticle distance h with RP ¼ 2 mm, c0 ¼ 16 mV, I ¼ 0.01 M, AH ¼ 1020 J. VT ¼ VA þ VR . The interaction energies VA, VR, VT, are divided by kBT.

particles can attract or repel each other. The distance h over which this occurs is typically of the order of 1–20 nm. Figure 11.11 shows a theoretical example. Though the calculation is exact, the ﬁgure is only meant to indicate some general trends. It shows that particles attract each other at very small distance (the so-called primary minimum where VT < 0), repel each other at short distance (the primary maximum where VT > 0), but may be attracted to each other at slightly larger distance (the secondary minimum where VT < 0). It should be mentioned that the particles cannot easily reach the primary minimum because there may be quite a high energy barrier (in Figure 11.3 about 40 kBT), in other words the particles may be quite stable. The usefulness of the DLVO theory for food colloids is that it can explain trends that are observed experimentally, even though there are many complications in foods. The effect of pH on colloidal interaction is explained by the effect of pH on ionizable groups on the surface, hence inﬂuencing the surface charge. The effect of ionic strength is explained via its effect on the thickness of the electrical double layer; hence adding salt to a colloidal system decreases repulsion and divalent ions more so than monovalent ions. As mentioned, this much-simpliﬁed discussion does no justice to the DLVO theory, but we needed this to be able to explain at least qualitatively effects that are important for kinetics of changes in dispersity. In addition to electrostatic repulsion, there may be so called steric repulsion. This is due to macromolecules adsorbed onto particles; part of the molecules is adsorbed onto the particles, but other parts may stick out in the solution (depending on solvent quality) and when such extending layers interact a repulsive energy is felt by the particles. Again, the theory is much more complicated than this, but it serves to indicate trends. The result is that particles with adsorbed polymers are stabilized in addition to electrostatic repulsion if the polymers are charged. The distance over which steric repulsion is experienced is much larger than electrostatic repulsion. In general, therefore, particles stabilized by polymers are fairly stable, and there is no secondary minimum. Another important type of interaction relevant for foods is depletion interaction. This phenomenon is due to nonadsorbing, dissolved polymers in a system containing colloidal particles (e.g., emulsion droplets). The nonadsorbing polymer must keep a certain distance away from the colloidal particles

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(volume exclusion). Consequently, the polymer concentration is higher in the bulk than in the vicinity of the particles (where a certain volume is depleted of polymer) and this gives rise to a higher osmotic pressure of the bulk solution. If then two colloidal particles come close together (i.e., start to aggregate) part of the depleted layers overlap and the concentration of the polymer in the bulk decreases and so does the osmotic pressure. The system will strive for the lowest possible osmotic pressure, and so aggregation becomes a driving force for this lowering of osmotic pressure. This is conveniently called depletion aggregation. The interaction free energy involved in this phenomenon is roughly proportional to the concentration of the polymer, while it also depends on the interaction between the polymer and the solvent (solvent quality for the polymer). Now, we return to the phenomenon of slow coagulation and the stability factor W. As mentioned, this factor is in principle just a ﬁt factor, it lacks physical meaning in the sense that it cannot be interpreted directly in terms of energy barriers, nor can it be calculated from ﬁrst principles. However, qualitatively it is clear that W must be related somehow to the balance of repulsive and attraction interactions discussed above. Its value can be roughly estimated from experimental observations. Using Equation 11.37, the halving time t0.5 ¼ 1 s for a particle with dp ¼ 1 mm and w ¼ 0.1 (Table 11.2), which means that coagulation is very fast. If we put W ¼ 107 for the same conditions then t0.5 ¼ 4 months. Based on actual observed coagulation times, one can make rough estimates of W. For aggregation of casein micelles during renneting, for instance, the following observation was made. In their native form and in conditions as in fresh milk, casein micelles are stable against ﬂocculation because of steric and electrostatic repulsion. Due to the action of rennet (hydrolysis of k-casein) steric stabilization is lost, and the stability factor reduces considerably such that aggregation approaches fast perikinetic coagulation according to Smoluchowski. Addition of CaCl2 is another way of increasing coagulation; the action of Ca2þ ions decreases the stability factor W because of their effect on ionic strength and the thickness of the electrical double layer. Hence, the effects can at least qualitatively be explained. Based on experimentally determined ﬂocculation times, a rough estimate of the stability factor W is 106 for intact casein micelles, and 103 for renneted micelles. Overall, the above equations are, of course, not directly applicable to foods, where particles do not have equal size, and do interact, while many other kinds of complications arise. It should be realized, therefore, that such equations can only predict trends and, perhaps, an order of magnitude estimation. However, the equations show that slow coagulation has to do with interaction between particles and this is the knowledge that the food technologist could exploit, at least qualitatively.

11.3.2

Kinetics of Creaming or Settling

Creaming and settling can lead to serious quality defects. Examples are the formation of a cream layer on top of milk, or the sedimentation of cocoa particles in chocolate milk, or of fruit particles in a fruit juice. Such changes are usually reversible as they are only a result of the forces of gravity, and they do not affect food safety at all, but they make the food unattractive for the consumer. Therefore, a producer needs to know how he can slow down such processes; in other words, he needs to know the kinetics of such changes. Starting very simple, consider a single globular particle of diameter dp and density rp in a ﬂuid of density rc and viscosity hv. The gravity force Fg (with acceleration due to gravity gg ¼ 9.81 ms2) acting upon such a particle is according to Archimedes: 1 Fg ¼ pdp3 gg (rp rc ) 6

(11:46)

When such a particle moves in a ﬂuid with velocity np, a friction force acts upon the particle: Fs ¼ 3phv dp np

(11:47)

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4.5

0.35

4

0.3

3.5 3

0.25

vS (µm s−1)

vS (µm s−1)

0.4

0.2 0.15

2.5 2 1.5

0.1

1

0.05

0.5

0

0 0

(A)

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0.002

0.004

0.006

0.008

0

0.01

hv (N s m−2)

(B)

2

4

6

8

10

dp (µm)

FIGURE 11.12 Effect of ﬂuid viscosity on the creaming rate of a particle nS with dp ¼ 3 mm and Dr ¼ 60 kgm3 (A), and the effect of particle size on the creaming rate with hv ¼ 1 mN s m2 and Dr ¼ 70 kg m3 (B).

When Fg equals Fs, it follows that nS ¼

gg (rp rc )dp2 18hv

(11:48)

This is the Stokes equation. Depending on the density difference between particle and suspending liquid, the particle will either cream or settle. If we consider a single fat globule in milk with a diameter dp ¼ 3 mm the creaming rate can be calculated as 0.3 mm s1, which corresponds to about 25 mm day1. So, in a few days a fat globule would be able to reach the top of a bottle of milk. Figure 11.12 shows the effects of particle diameter and ﬂuid viscosity graphically. However, the Stokes equation is only valid for homogeneous spheres on which no other forces act than the ones shown, and the liquid should be Newtonian. Usually, these conditions are not fulﬁlled in foods, and consequently the Stokes equation will never predict the exact creaming or sedimentation rate in foods. However, it is very useful in predicting trends. For instance, the effect of particle size is quite strong because of the factor dp2 in the Stokes equation, as shown in Figure 11.12B. This knowledge is used in homogenizing milk: by reducing the particle size by about a factor of 10, creaming in milk is effectively retarded. Furthermore, if one can adjust the density difference, the rate can be inﬂuenced in a desired direction; this will however not be easy in foods. Finally, one can do something about hv and this is one of the reasons to use thickening agents: by increasing the viscosity one can effectively reduce creaming or sedimentation, as indicated in Figure 11.12A. Stokes’ equation also shows how the creaming or sedimentation rate can be enhanced by increasing acceleration in a centrifuge. Some corrections can be applied to the Stokes equation to relax the severe restrictions. One can replace the requirement for a single monodisperse droplet by replacing dp2 by a parameter derived from the particle size distribution (provided it has a reasonably narrow distribution): P ni d5 hd 2 i ¼ Pi i3 i ni di

(11:49)

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0.4

vc (µm/s)

0.3

0.2

0.1

0

0

0.1

0.2

0.3 φ

0.4

0.5

0.6

FIGURE 11.13 Effect of volume fraction w on the creaming velocity nc of a particle with d ¼ 3 mm, hv ¼ 1 mN s m2, and Dr ¼ 70 kg m3 according to Equation 11.51.

Another correction to nS can be applied for systems having moderate volume fractions (up to volume fraction f ¼ 0.05): nc ¼ nS (1 w)n

(11:50)

where the exponent n ranges from 8–9, depending on the polydispersity of the suspension. For systems with higher volume fractions, only empirical relations exist, such as nc ¼

nS (1 w) 5w 1 3 1 þ w exp 3ð1 wÞ

(11:51)

This relation is shown in Figure 11.13 and shows the large impact of volume fraction on the creaming rate. Finally, it should be mentioned that sedimentation or creaming is strongly enhanced by aggregation, which is qualitatively explained by considering that the size of the aggregates will be much larger than the size of the original particles and, in view of the large effect of particle size in Equation 11.48, this enhances the creaming rate enormously.

11.3.3

Kinetics of Coalescence

Coalescence is the phenomenon that oil droplets in an emulsion merge into bigger globules. If this process happens unhindered an emulsion would be destabilized in seconds, i.e., the result would be an oil layer on top of an aqueous phase. This is the thermodynamically stable state of the system. This is clearly unwanted and it is therefore important to know something about the speed of coalescence. Unfortunately, there is not a clear-cut theory yet to describe this. Qualitatively, however, the process is well understood. The ﬁrst thing to happen is that globules must encounter each other (which they do all the time due to Brownian motion) and stay together for a while (which they do not unless there is some

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attraction between them or when they are pushed together in a cream layer or in a sediment). When they stay together for some time, then there is a probability that the thin ﬁlm between the droplets ruptures if the repulsive energy between the droplets can be overcome. There is theory developed for this situation that we will not discuss here. For the kinetics part it sufﬁces to know that there are two stages in coalescence: ﬂocculation followed by actual coalescence. Empirically it is sometimes found that this process can be described roughly by a simple ﬁrst-order model, with N the number of emulsion droplets per unit volume: dN ¼ kN dt

(11:52a)

Nt ¼ N0 exp (kt)

(11:52b)

Or, equivalently

Experimentally, this has to be linked to changes in particle size distributions, which is not a trivial task. Great care should be taken to differentiate between ﬂocculation and coalescence. Flocculated particles can also be seen as coalesced particles by the measuring equipment. So, Equation 11.52 has to be linked to some suitable parameter characterizing particle size or particle number. Figure 11.14 gives an example of applying the logarithmic form of Equation 11.52b on the change in number of emulsion droplets as a function of time. The slope of the line represents the rate constant. It should be realized that this is purely empirical, there is no theory that predicts that it should be ﬁrst order. Figure 11.14 also shows that the relation is certainly not perfect. The stability factor W, if that can be used at all, is estimated to be in the order of magnitude of 106, but it is strongly dependent on the size of the emulsion droplets: smaller droplets are characterized by a larger stability factor.

6

5

ln [N0 /Nt]

4

3

2

1

0 0

100

200 Time (h)

300

400

FIGURE 11.14 Example of a ﬁrst-order kinetic plot for the coalescence of oil-in-water emulsions with butter oil (^) or peanut oil (&). The emulsions were stabilized by 1% whey protein and coalescence was studied at 508C. Dataset in Appendix 11.1, Table A.11.5.

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Qualitatively, the following can be said about the process of coalescence. When two emulsion droplets come close together, they are still separated by a thin ﬁlm of continuous phase. If this ﬁlm is ruptured the two droplets will merge, so processes that determine this ﬁlm rupture will determine coalescence. Basically, this rupture process is a probabilistic event. The probability of rupture will be proportional to the time that the droplets are together and to the surface area of the ﬁlm. So, ﬂattening of the droplets while they are close together will lead to a greater ﬁlm area and hence to a larger probability for coalescence. It is therefore interesting to consider factors that affect this ﬂattening phenomenon. An important characteristic for this appears to be the so-called Weber number We:

We ¼

sext dp2 8gh

(11:53)

The Weber number expresses the ratio between the local stress on a droplet pair and the Laplace pressure of the drop; g represents the interfacial tension. The local stress is the external stress sext multiplied by the stress concentration factor which is in this case dp2=h, h being the smallest distance between the drop surfaces. sext can arise due to colloidal attraction, gravitational or centrifugal stress. If We > 1, a ﬂat ﬁlm is formed, and if We 1 then a ﬁlm is not really formed and coalescence can be ruled out. What factors can now be deduced in relation to coalescence stability? If droplets are smaller, a smaller We is found and the probability for coalescence is less. If the ﬁlm between droplets is thicker, which may happen if there are strong repulsive forces between the droplets, coalescence becomes less likely. Steric repulsion is especially effective in this respect. If the interfacial tension g is higher, We becomes lower, and this lowers the probability for coalescence. The reason for this is that a higher g makes deformation of a ﬁlm more difﬁcult. Incidentally, lowering g is needed to make an emulsion which is one of the functions of a surfactant. As it happens, proteins are very suitable as emulsifying and stabilizing agents. On the one hand, they lower the interfacial tension enough for the emulsion making process, but not too much so that coalescence stability is enhanced. Moreover, proteins provide usually steric as well as electrostatic repulsion (depending on pH). An interesting complication arises when oil partly crystallizes. This phenomenon usually destabilizes an emulsion enormously. The explanation is that fat crystals partly protrude out of the emulsion droplet and rupture the thin ﬁlm between approaching droplets. The two droplets will then only partly coalesce because the crystal structures inside the droplets prohibit complete coalescence. The phenomenon is therefore called partial coalescence. Especially ﬂow has a detrimental effect on emulsion stability of partly crystallized emulsion droplets. This knowledge is exploited in churning: there is an optimal amount of solid fat content to have the highest fat yield. If the solid fat content is too high, there is too little liquid fat for coalescence, if the solid fat is too low there are too few crystals to cause destabilization. In any case, the kinetics of coalescence is strongly inﬂuenced by the presence of fat crystals in the emulsion droplets.

11.3.4

Kinetics of Ostwald Ripening

Ostwald ripening is the phenomenon that larger particles (emulsion droplets, foam bubbles, fat crystals) grow at the expense of smaller ones. It has to do with the curvature of the particles: smaller particles are more curved (have a larger surface area) than large ones. The driving force is the difference in chemical potential of the dispersed material due to the difference in radius. The resulting equation in terms of solubility is (known as the Kelvin equation) S(RP ) 2gVm ¼ exp RP RT S1

(11:54)

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The parameters are Vm the molar volume of the dispersed component S(RP) the solubility of the disperse phase in particle with radius RP S1 the solubility of the dispersed phase in the continuous phase when the interface is planar (RP !1) g the interfacial tension If the solubility of the disperse phase in the continuous phase is high enough to allow mass transfer to take place (via diffusion), the process will become noticeable. The rate of Ostwald ripening is dhRP i3 8Vm S1 gDf ¼ dt 9RT

(11:55)

where < RP > is the mean particle radius. Since the parameters Vm, g, and Df do not vary very much from one system to another, the most important factor is the parameter S1. For oil-in-water emulsions consisting of triglycerides, their solubility in water is negligible, and so Ostwald ripening does not occur readily in O=W emulsions. Water is somewhat better soluble in oil, so that Ostwald ripening does occur in W=O emulsions such as margarine and butter. In aqueous foams, the solubility of gases in water is quite high, so that Ostwald ripening is one of the main causes of foam coarsening and instability (it even has a separate name: disproportionation; the other causes of foam instability are coalescence and drainage). This knowledge can be used for beer foam stability: beer bubbles consisting of nitrogen are more stable than those of carbon dioxide because carbon dioxide is better soluble in water than nitrogen. The rate of Ostwald ripening in foams is described by the de Vries equation: R2p (t) ¼ R20

RTDf S1 g t Pd

(11:56)

This equation describes the fate of a small bubble, initially of radius R0, surrounded by much larger bubbles at a distance d; P is ambient pressure (105 Pa). This equation shows that a bubble will shrink increasingly faster when it becomes smaller. Coming back to the nitrogen and carbon dioxide solubility in beer, it can be calculated from the de Vries equation that for a bubble with radius R0 ¼ 0.1 mm and d ¼ 1 mm, a nitrogen bubble disappears in 3 min and a CO2 bubble in 4 s. Of course, these are only order of magnitude estimations, because, as usual, there are complications. For instance, when the bubble shrinks, the surface load of surfactant becomes higher which may have its bearings on the interfacial tension g and the surface dilational modules, such that the shrinkage rate becomes less. Nevertheless, the example goes to show that Ostwald ripening in foams can occur quite fast.

11.3.5

Kinetics of Gelation of Particles

When colloidal particles aggregate due to Brownian motion (perikinetic aggregation), two scenarios are possible (Figure 11.15).

v1

v2

+

FIGURE 11.15 Two scenarios of aggregation. If the rate of aggregation n1 n2, coalescence takes place according to Smoluchowski (fractal dimension df ¼ 3); if n1 >> n2, fractal aggregation takes place (fractal dimension df < 3) (see text).

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If the two particles immediately merge (coalesce) after aggregation one new particle emerges. If they do not merge so fast, fractal aggregates may develop. Fractal aggregates are not closely packed but rather they have an open, disordered structure. A fractal object like an aggregate has a characteristic fractal dimensionality df that is smaller than the dimensionality of the space in which it exists, which is 3 for a three-dimensional system, 2 for a ﬂat surface. The larger fractal aggregates are, the less dense they are. With a higher df, the structure is more dense. Fractal aggregation has been the topic of much research in the past two decades, and computer simulations have given much insight. We will not discuss this further here, but merely state some results, since our interest is in the kinetic consequences. It can make a big difference in practice whether aggregation follows conventional slow coagulation or according to fractal aggregation, as we shall see. A property of fractal aggregates is that the structure is scale invariant at length scales that are much higher than the individual particle dimensions; this is called a self-similar structure. The relation between the average number of primary particles Np in a fractal aggregate, consisting of primary particles with radius Rp, and the radius of the resulting aggregate Ragg, is Np ¼

Ragg Rp

df (11:57)

This equation deﬁnes the fractal dimension df, which is smaller than 3, except for immediate coalescence. For diffusion-limited aggregation (i.e., fast perikinetic aggregation), df ¼ 1.75–1.80. There is also reactionlimited aggregation, meaning that particles do not immediately stick upon an encounter, but rather rearrange themselves, or diffuse away again. Whether or not this happens is related to the interaction energy between the particles. In such cases, df ¼ 2.05–2.11. The end result of coagulation is either a lowdensity porous sediment that settles (or creams) as a result of gravity, or a particle gel network that ﬁlls all available space. In dilute systems, the onset of coagulation may be due to aggregation of individual particles, but at a later stage, and certainly in more concentrated systems, cluster-cluster aggregation becomes dominant. Particle gels are of importance in foods. Examples are cheese and yoghurt, which are particle gels of casein micelles. If we would have a close-packed, nonfractal, aggregate with radius Ragg, then the number of particles Nmax in such an aggregate is Nmax ¼

Ragg Rp

3 (11:58)

Combining this with Equation 11.57 shows that the average volume fraction of the fractal aggregate wagg is wagg

(Ragg =Rp )df Np ¼ ¼ ¼ Nmax (Ragg =Rp )3

Ragg Rp

df 3 (11:59)

Since df < 3 in the case of fractal aggregation, the consequence is that wagg decreases with increasing Ragg. Gelation will occur when the average volume fraction of the fractal aggregate equals the overall volume fraction of particles, hence when wagg ¼ w, in which case the system is fully packed and the system has gelled. We are interested in the kinetics of aggregation, i.e., the time after which aggregation becomes visible. The aggregation rate (reciprocal time) as depicted in Equations 11.35 and 11.40 appears in that sense to be less important than the aggregation time because at the same initial aggregation rate the aggregation time for one system may be completely different from that of another system, depending on the structures of the aggregates formed. The aggregation rate may increase abruptly after aggregates have

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reached a certain critical size (which may be well below the size of visible aggregates). How to ﬁnd the time to reach such critical aggregation size? Let us consider the relation between the concentration of primary particles and the size of an aggregate: N¼

3 f w Rdf 3 Rd agg 4p 0 p

(11:60)

The change in number of particles with change in aggregate particle size is then: dN 3df df 1 w Rdf 3 Ragg ¼ 4p 0 p dRagg

(11:61)

By combining these equations with Equation 11.35 or Equation 11.37 to ﬁnd a relation between the time for aggregation and the critical size, dt=dRagg may be integrated from the radius of the primary particle to the critical size: Rðcr

tcr ¼ Rp

dt dRagg dRagg

(11:62)

For the case of perikinetic aggregation, all this results in Rðcr

tcr ¼ Rp

phv Df 1 3df df 1 R Ragg dRagg kB T f0 p

(11:63)

When it comes to the effects of aggregation in relation to food quality, the following phenomena are important for foods, namely the formation of visible particles, the formation of a gel, or the occurrence of separation (creaming or settling). Three different critical sizes are worth considering, namely the one for gelation, the one for orthokinetic aggregation, and the one for sedimentation. The smallest value should be used for an estimate of aggregation time. The critical aggregate radius Rcr at which a gel is formed is 1 d 3

Rcr ¼ Rp w0f

(11:64)

This equation shows the large dependence of the critical aggregate size on the volume fraction. It is important to realize that the volume fraction of particles does not change if particles immediately coalesce upon an encounter; also if the particles aggregate into compact aggregates, w does hardly change. On the other hand, w does increase considerably in the case of fractal aggregation. Using the equation for the halving time (Equation 11.37), Equation 11.63 can be written as tcr ¼ t0:5

Rcr Ragg

df

! 1

phv R3p kB T

3 d 3

w0f

df d 3

¼ t0:5 f0f

(11:65)

So, this equation relates the aggregation time to the halving time, which is a rate expression. This equation is very instructive in showing the large difference between tcr and t0.5. If particles would coalesce immediately upon an encounter, df ¼ 3. Particles become visible at a few tenths of a millimeter, say, Rcr 0.2 mm. Suppose we start with particles of Rp ¼ 0.5 mm at 208C, w0 ¼ 0.01, and W ¼ 1. The halving

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time according to Equation 11.37 is then 9.7 s, so that according to Equation 11.65, tcr ¼ 6 3 108 s 20 years. In the case of fractal aggregation, say df ¼ 2, the critical value for Rcr is calculated according to Equation 11.63 and a completely different value of tcr ¼ 9.77 3 104 s 1 day is found. This shows the enormous effect of fractal aggregation. But more importantly, it also shows that prediction of aggregation time from aggregation rates can be wildly misleading because t0.5 is the same in both cases. To be sure, these numbers should be taken as orders of magnitude, not as absolute values. For orthokinetic aggregation, the critical aggregate size can be approximated as Rcr ¼

kB T 4Gs hv

1 3

(11:66)

An approximate estimate for the aggregation time due to orthokinetic aggregation is then

tcr ¼

p 4wGs

4pGs R3p

!3df 3

(11:67)

kB T

In the case of immediate coalescence upon an encounter (df ¼ 3) in orthokinetic coagulation, it can be derived that p 4hGs R3cr ln tcr ¼ kB T 4wGs

(11:68)

These equations tell us that the orthokinetic aggregation time for coalescing particles is independent of the size of the primary particles, and only slightly dependent on the critical size of the aggregate. For fractal aggregates, the aggregation time appears to be independent of Rcr but dependent on the size of the primary particles. Finally, an expression for the size at which sedimentation becomes noticeable is Rc ¼

3kB T 2pr 3df Drgg

1 df þ1

(11:69)

In this last equation, Dr is the density difference between ﬂocs and continuous phase and gg is the acceleration due to gravity. The equations stated above are derived under various assumptions. The most important ones are that it is for dilute systems, monodisperse, spherical particles and unhindered aggregation. To some extent, these assumptions can be relaxed but the reader should be aware of the fact that the equations will not be correct when directly applied to foods. However, it is reassuring that the theory (that we have touched upon only very brieﬂy) is able to predict at least some important phenomena. The following approximate equations have been derived. For perikinetic aggregation an approximate expression is tg

1

phR3p df 33 6df 3df þ w 2df þ 3 df þ 6 kB T 0

(11:70)

and an approximate expression for the gel time due to orthokinetic aggregation is tg

1 pdf w 1 4Gs (3 df ) 0

(11:71)

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TABLE 11.3 Calculated Aggregation Times as a Function of Volume Fraction w0, Fractal Dimensionality df, and Velocity Gradient Gs Gs (s1)

w0

df

tc or tg (s)

Equation Used

0.0001

3

0

6 3 1010

Equation 11.63

0.0001

1.8

0

6 3 104

Equation 11.65

0.0001

3

0.1

1 3 105

Equation 11.67

0.0001

1.8

0.1

3 3 102

Equation 11.67

0.01

1.8

0

6 3 101

Equation 11.65

0.01

1.8

0.1

6 3 101

Equation 11.65

Note: Rp ¼ 20 nm, hv ¼ 1 mPa s. For df ¼ 3, visible aggregation at Ragg > 0.2 mm, for df ¼ 1.8, gelation occurs.

As a summary, Table 11.3 shows the calculated aggregation times for various situations to show the large differences that can arise.

11.3.6

Kinetics of Crystallization

Crystallization of all kinds of materials can happen in foods. Examples are the formation of lactose crystals in ice cream, causing an unpleasant mouthfeel, crystallization and recrystallization of fat crystals in butter or margarine, crystallization of sugars in powders, retrogradation of starch (which is in fact crystallization of amylose), leading to such phenomena as staling of bread. The rate at which these changes occur is obviously of importance. However, the process of crystallization is a complex one. For crystallization from a homogeneous solution, the process can be distinguished generally in nucleation followed by crystal growth. The driving force for nucleation and subsequent crystallization is the difference in chemical potential between nucleus, or crystal, and solution. This can be expressed in supersaturation. Nucleation can be homogeneous and heterogeneous. Homogeneous nucleation occurs in the absence of a surface, while heterogeneous nucleation takes place at foreign surfaces (called catalytic impurities). The homogeneous nucleation rate depends very strongly on temperature, the heterogeneous nucleation rate much less so. If nuclei grow above a certain critical size, crystal growth can start. Crystal growth rate is basically a balance between the adsorption of molecules to a surface and the release of molecules from that surface. There are quite a few factors that inﬂuence crystal growth. First of all, the amount of supersaturation, and therefore the temperature, the shape of the crystals, the presence or absence of similar molecules that may adsorb and thus limit further adsorption, the heat of fusion that causes the temperature to rise locally. Crystal growth can be diffusion limited, or reaction limited. A general model that encompasses all the stages does not exist, though there are abundant (and competing) theoretical models available describing both the kinetics of nucleation and crystal growth. We will not discuss these here because of the many complexities involved. There is however a model that seems to be of general use, called the Avrami model, that describes crystallization for nucleation and growth if a homogeneous solution is rapidly cooled to a constant temperature. It is a semiempirical model, the equation of which is w(t) ¼ 1 expðkt nA Þ

(11:72)

w is the volume fraction of crystals in the solution. Depending on the values of the two parameters, the rate constant k and the Avrami exponent nA (usually between 0.5 and 4), the equation shows an induction period (representing the nucleation phase), a rapid increase (crystal growth) and a slowing down of the rate (when supersaturation becomes less at the end of the process). Figure 11.16 shows an example for fat crystallization.

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40 35

Solid fat (%)

30 25 20 15 10 5 0

0

50

100

Time (min )

FIGURE 11.16 Kinetics of crystallization of hardened palm oil at 408C (data points) as described by the Avrami equation (solid line). Parameter k ¼ 2.64 3 105, nA ¼ 2.67. Dataset in Appendix 11.1, Table A.11.6.

The Avrami equation has also been applied to the kinetics of retrogradation of starch. Once again, it should be realized that the Avrami equation is an empirical equation without a physical meaning.

11.4

Kinetics of Texture Changes

Many literature reports can be found describing the kinetics of texture change, for instance, in vegetables. Undoubtedly, texture changes are important for food quality. Sometimes, they are desired, as in cooking potatoes, sometimes undesirable, as in overripened fruits. Frequently, kinetics of texture changes is modeled as ﬁrst-order kinetics. First of all, it should be clear that texture changes in fruits and vegetables are the result of many molecular changes in cell wall components, such as pectin molecules. However, texture is measured by some rheological technique, not as molecular change. Consequently, it makes no sense to describe the dependence of apparent rate constants on temperature via the Arrhenius equation and to report the results as activation energy in kJ mol1. Nonetheless, one can ﬁnd many results, reporting for instance activation energy for texture degradation of cooked potatoes. It would be interesting to see a mole of potatoes! Frequently, kinetics of texture changes seems to follow a biexponential decay (Figure 11.17). It has been proposed in literature to model physical changes via a fractional conversion ﬁrst-order model rather than a normal ﬁrst-order equation. Fractional conversion would be suitable to correlate the extent of a chemical reaction with the measurement of a physical property. The fractional conversion fc is deﬁned as the ratio between what has reacted and what can potentially react. Mathematically this is translated into fc ¼ c0 is the concentration at time zero c1 is the concentration at time inﬁnity

c0 ct c0 c1

(11:73)

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140

Compression force (N)

120 100 80 60 40 20 0 0

20

40

60 t (min)

80

100

120

FIGURE 11.17 Texture (measured as compression force in N) of potato cubes as a function of time after cooking at 808C. Dataset in Appendix 11.1, Table A.11.7.

It follows that 1 fc ¼ 1

c0 ct ct c1 ¼ c0 c1 c0 c1

(11:74)

And so for a ﬁrst-order reaction this translates into ct c1 ¼ exp (kt) c0 c1

(11:75)

If c1 ! 0 we arrive again at the familiar ﬁrst-order equation. When we are not measuring concentrations but a physical property, we have F0 as the physical property at t ¼ 0, F1 the physical property at t ¼ 1, Ft the physical property at time t F0 Ft F0 F1 Ft F 1 1 fc ¼ F0 F1

(11:76)

Ft ¼ F1 þ (F0 F1 ) exp (kt)

(11:77)

fc ¼

The ﬁrst-order model is then

If this relation holds, texture change would be characterized by a single rate constant, but of course, one has to estimate the physical property at F1 at t ¼ 1, so it is a model with three parameters if F0 is estimated as well. It is not clear beforehand whether F1 is temperature dependent or not. It should be

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100 90 80

Failure stress (MPa)

70 60 50 40 T = 90C 30 T = 103C 20 10

T = 120C

0 0

2

4

6 t (min)

8

10

12

FIGURE 11.18 Kinetics of texture changes of potatoes as a function of cooking temperature (908C, 1038C, and 1208C) and time. The lines are the fractional conversion model (Equation 11.77) or the Weibull model (Equation 4.76); the ﬁts of the lines are indistinguishable. The texture parameter measured was failure stress. Dataset in Appendix 11.1, Table A.11.8.

realized that such a ﬁrst-order model is just an empirical model without a physical background, and other empirical models such as the Weibull model (Equation 4.76) may be equally applicable. Both the Weibull model (Equation 4.76) and Equation 11.77 were tested on some data describing texture change of potatoes as a function of cooking (Figure 11.18). Both models gave an equally good ﬁt and cannot be distinguished visually. Kinetics of water uptake. Water uptake by foods is important for rehydration of foods but it can also lead to quality loss if products take up water. This can be a problem, for instance, with cereals immersed in a liquid like milk: the cereals may become soggy in a short while, which is a texture problem. Or, a product can lose its crispiness by water uptake during storage. It can also be a problem when several components are mixed that have a different water activity, such as cereals mixed with raisins. Such water uptake can in principle be modeled via mass transfer models discussed earlier in this chapter, but it also possible by empirical models, of which one is the Weibull model again: aW t Mt M0 ¼ (M1 M0 ) 1 exp bW

(11:78)

Another one is the so-called Peleg model for water sorption: Mt M0 ¼ (M1 M0 )

t kþt

(11:79)

An example of both models is shown for the kinetics of uptake of water by cereals by whole milk in Figure 11.19.

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300

Mt −M0 (g H2O/100 g initial weight)

250

200

150

100

50

0

0

2000

4000 t (s)

6000

8000

FIGURE 11.19 Kinetics of water uptake by cereals immersed in whole milk at 558C modeled by the Weibull model (—, Equation 11.78) and the Peleg model (- - -, Equation 11.79). Dataset in Appendix 11.1, Table A.11.9.

Both models are seen to describe the experimental values well. Once again, these are empirical models that are nevertheless very useful, and when product composition is varied a possible relation can be found between product composition and parameters of the model. Incidentally, the data shown in Figure 11.19 are seen to be heteroscedastic (variance increases with water uptake) and this might call for a transformation to induce homoscedasticity, as discussed in Chapter 7, if one wants to ﬁnd the most probable estimates of the parameters. Here, we do not perform this exercise because the principal purpose is to show the various models.

11.5

Partitioning Phenomena

As remarked before, many foods are heterogeneous and may contain more than one phase. The possibilities are combinations of the following: . . . .

Aqueous phase Lipid phase Headspace (usually air) Solid phase

Components may divide themselves over the various phases. Although not a phase in the strict sense, micelles formed from surfactants have, to some extent, the same properties as phases, and micelles, vesicles, and microemulsions could be called a pseudophase. The fact that components may partition over various phases has an impact on kinetics because of the following reasons: . .

Concentrations of reactants and products will be affected If there is partitioning of carboxylic acids, pH changes may occur

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If there is partitioning of volatile compounds, this will effect their release from the food (upon storage but also upon cooking and eating) and will affect ﬂavor perception If there is partitioning of antioxidants, this will have an effect on oxidation kinetics

All in all, the phenomenon of partitioning in foods on kinetics deserves attention. Partitioning is, of course, a physical phenomenon, which is why it is discussed in this chapter.

11.5.1

Partition Coefﬁcients

To describe partitioning quantitatively, a partition coefﬁcient is introduced, which is linked to activities of the solute. If a solute is free to move between phases, then the activity in each phase will be the same when equilibrium is attained: fia Xia ¼ fib Xib

(11:80)

Here, Xi is the mole fraction of solute i, fi the dimensionless (rational) activity coefﬁcient, and a and b refer to two different phases (e.g., oil and water, or air and water). The thermodynamic partitioning coefﬁcient is deﬁned as x Pa=b ¼

Xia Xib

¼

fib fia

(11:81)

This equation shows that, if uneven distribution of a component over two phases occurs, this is in fact due to different activity coefﬁcients in these two phases. In actual practice, one uses the conventional partitioning coefﬁcient in terms of concentrations: Pa=b ¼

cai cbi

¼

yib yia

(11:82)

(As remarked in Chapter 3, the numerical values of activity coefﬁcients depend on the way they are deﬁned, so the value of yi may be different in Equation 11.82 than fi in Equation 11.81.) All solute–solute and solute–solvent interactions are accounted for in the activity coefﬁcients.

11.5.2

Partitioning of Volatiles

The release of components from the food matrix is important for several reasons, a very important one being the release of volatiles that act as ﬂavor compounds.* This can happen during processing and storage, and, most importantly, during eating (mastication). Many research papers deal with this topic, and models have been presented that consider mass transfer as a function of temperature and mechanical treatment, and obviously, partitioning has an important part to play. In fact the partitioning coefﬁcient is the thermodynamic measure that indicates the maximum possible extent of partitioning, and it also accounts for the inﬂuence of chemical composition of the medium. Often, the partitioning behavior of volatile compounds is characterized by the so-called log P value; this is in fact the logarithm of the n-octanol=water partitioning coefﬁcient. These log P values are not directly transferable to foods because n-octanol has different characteristics than the common fat phase in food, but the values may be converted to those for oils. log P values are frequently reported in literature * The term ﬂavor release is often used. Strictly speaking, it is better to use the term volatile release because a volatile causes only a ﬂavor when it reacts with a receptor in the nose or mouth and is perceived as a ﬂavor in the brain.

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TABLE 11.4 log P Values of Some Selected Compounds at 258C Compound

log P

Diacetyl

1.3

Acetaldehyde

0.2

Ethyl acetate

0.7

Hexanal

1.8

2-Octanone

2.4

Ethyl hexanoate

2.8

Linalool

3.5

to indicate the hydrophobicity of the compound. A positive value indicates a hydrophobic compound (the higher, the more hydrophobic) and a negative one a hydrophilic compound. Incidentally, log P values can be estimated also from the molecular structure of a molecule using appropriate software. Table 11.4 shows some log P values. If a volatile compound partitions between a liquid phase l and a vapor phase v, the partition coefﬁcient can be deﬁned as the ratio between the concentration in the vapor phase cv (g dm3) and that in the liquid phase cl (g dm3): according to Equation 11.82 Pv=l ¼

cv : cl

(11:83)

The reader is reminded that in Chapter 3 Henry’s law was introduced in Equation 3.55, describing the relation between the partial pressure Pi of a component in the vapor phase above a solution containing that compound in mole fraction Xi: Pi ¼ kH,i Xi

(11:84)

where kH,i is Henry’s constant. In relation to partitioning of volatile compounds, one uses in the literature frequently the ‘‘activity coefﬁcient at inﬁnite dilution,’’ indicated by the symbol g1 i . It should be realized that in that case one does not use Henry’s law as a reference but rather Raoult’s law. This may be a bit confusing because it was explained in Chapter 3 that the activity coefﬁcients of solutes at inﬁnite dilution approach unity by taking Henry’s law as reference. However, one is free to choose reference states, so it is equally valid to use Raoult’s law, and one uses in fact the symmetrical convention here, introduced in Chapter 3. To remind the reader of the concept, Figure 11.20 shows graphically how the activity coefﬁcient is deﬁned in the symmetrical convention (i.e., by using Raoult’s law as a reference). However, we prefer to use the symbol f to indicate activity coefﬁcients at the mole fraction concentration scale; the symbol g was chosen for the molality scale. We shall thus refer to the activity coefﬁcient at inﬁnite dilution as fi1 for Xi ! 0 as indicated in Figure 11.20. The relation between fi1 and Henry’s constant is fi1 ¼

kH,i Pi

(11:85)

Pi is the saturated vapor pressure of component i. fi1 is in fact a measure for the afﬁnity of component i for the solution in which it is dissolved. The higher this value, the less afﬁnity it has for the solution. In general, the activity coefﬁcient as a function of its mole fraction will be as indicated in Figure 11.21, if Raoult’s law is taken as reference, which implies that fi ! 1 when Xi ! 1.

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Henry ’s law P

Pi

ai =

P∗i

fi =

ai = fiXi

Pi

P iX ∗

Region of infinite dilution (Xi 0)

: Pi

=

i

aw

’s l

lt aou

Pi

P oi = P∗i

Pi

R

Pi

Xi

0

1

FIGURE 11.20 Illustration of Raoult’s and Henry’s law and the activity coefﬁcient fi according to the symmetrical convention; for Xi ! 0, fi ! fi1 .

f 1∞

f

f1 f 2∞ f2

1

X1

0

1

FIGURE 11.21 General shape of the relation between the activity coefﬁcients f and the mole fraction of compound 1 in a binary liquid mixture, showing non-ideal behaviour fi1 is found at inﬁnite dilution, i.e., when Xi 0.

Table 11.5 shows values of this activity coefﬁcient for some typical compounds. As expected, the more hydrophobic a component is, the higher its activity coefﬁcient extrapolated from inﬁnite dilution, and the higher its volatility. In fact, if the magnitude of the activity coefﬁcient at inﬁnite dilution is much larger than 1000, the following approximate relation holds: Xi

1 fi1

(11:86)

Obviously, a link can be made between partition coefﬁcients and Henry’s law because they both refer to the same phenomenon. It is assumed that the vapor behaves as an ideal gas so that the general gas law is valid and it is assumed that at inﬁnite dilution the amount of moles of the volatile can be neglected. Then, for a volatile component 1 dissolved in a solvent 2, it can be derived that: Pv=l ¼

kH,i PT =RT P1 n2 =Vl

(11:87a)

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TABLE 11.5 Some Typical Values for fi1 , the Activity Coefﬁcient Extrapolated from Inﬁnite Dilution in Aqueous Solutions, at 258C fi1

Compound Diacetyl

1.1

Acetaldehyde

4.2

Ethanol Butanol

4.8 67

Ethyl acetate

86

Benzaldehyde

559

Hexanal

1310

Linalool

14000

Note: Compiled from various sources.

PT is the total pressure, P1 the partial pressure of component 1, and n2 the amount of moles of component 2 (the solvent), Vl is the volume of the liquid phase. By combining this with Equation 11.85 an alternative equation is Pv=l ¼

fi1 Pi PT =RT P1 n2 =Vl

(11:87b)

For an aqueous solution and a vapor phase of 1 L and at 208C, this results in Pv=l ¼

kH,i 1:38 108 [Pa]

(11:88)

For a partition coefﬁcient between vapor and liquid oil, taking a molecular weight of 730 Da and a density of 910 kg m3 for oil, the relation would be Pv=l ¼

kH,i 3:1 106 [Pa]

(11:89)

The temperature dependence of partitioning of volatiles can be derived from the Clausius–Clapeyron equation, which describes actually the temperature dependence of the vapor pressure for a pure compound: Pio

¼

o Pref

DHvap 1 1 exp R T Tref

(11:90)

It seems reasonable to assume that the same relation holds for the partial pressure but with the enthalpy of hydration replacing the enthalpy of vaporization:

DHhydr 1 1 Pi ¼ Pi,ref exp R T Tref

(11:91)

The enthalpy of hydration represents the amount of energy released when 1 mol of volatile dissolves in an inﬁnite amount of water. Since the partition coefﬁcient is related to the partial pressure of component i, it may be assumed that the temperature dependence of the partition coefﬁcient is similar:

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0.016 0.014

Pair/aq, Pair/oil

0.012 0.010 0.008 0.006 0.004 0.002 0 0

20

40

60

T (C)

FIGURE 11.22 Effect of temperature on partition coefﬁcients. Methyl acetate in water (&), ethylacetate in water (*), and isopentylacetate in coffee oil (~). Dataset in Appendix 11.1, Table A.11.10.

DHhydr 1 1 Pv=l ¼ Pv=l,ref exp R T Tref

(11:92)

With increasing temperatures, partition coefﬁcients increase as well because of the increase in saturation vapor pressures. Figure 11.22 shows some examples. Figure 11.23 shows the partitioning of methanethiol between the headspace and an aqueous phase, and the headspace and a fat phase at equilibrium; methanethiol is a volatile related to cheese ﬂavor. The linear relation found is in line with Henry’s law. The activity coefﬁcient fi1 may depend also on the presence of other solutes. Figure 11.24 gives some examples for the effect of sucrose. The effects are quite large: notice the logarithmic scale. The increase in fi1 indicates less favorable interaction of the volatile with the aqueous phase. It is perhaps instructive to show the effect of the presence of a fat phase in a food on the ﬂavor composition of the headspace, because the ﬂavor proﬁle of low fat foods may be quite different from their high-fat equivalents. If we denote the total amount of a volatile present in a closed system as NT, the mass balance is NT ¼ cf Vf þ caq Vaq þ cv Vv

(11:93)

where cf, caq, and cv stand for the concentration of the volatile in the fat, aqueous and vapor phase, respectively Vf, Vaq, and Vv for the volume of the fat, aqueous, and vapor phase Using the appropriate partition coefﬁcients, we can transform Equation 11.93 as follows: NT ¼

Pf =aq Vaq Vf cv þ þ cv Vv Pv=aq Pv=aq

(11:94)

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12

Headspace (nmol dm−3)

10

8

6

4

2

0 0

5

10 15 Solution (n mol dm–3)

20

FIGURE 11.23 Partitioning of methanethiol between headspace and a solution consisting of 0.1 M sodium acetate þ37.5 g L1 NaCl, pH 5.2 (^) and between headspace and liquid milk fat (&). Dataset in Appendix 11.1, Table A.11.11.

So, if we know the volumes of the three phases and the partition coefﬁcients, we can calculate the distribution of volatiles over the various phases. Table 11.6 shows a simple calculation for several volatiles, for an imaginary full-fat cheese and a low-fat cheese, just to show the differences of the presence 6

5

log f ∞i

4

3

2

1

0 0

0.2 0.4 0.6 Mass fraction sucrose (kg kg–1)

0.8

FIGURE 11.24 Dependence of log fi1 for ethylacetate (^), butylacetate (&), and n-hexylacetate (*) on sucrose concentration in aqueous sucrose solutions. Dataset in Appendix 11.1, Table A.11.12.

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TABLE 11.6 Concentration of Volatiles in the Vapor Phase (cv) for a Full-Fat Cheese (Vf ¼ 3 mL, Vaq ¼ 4 mL) and a Low-Fat Cheese (Vf ¼ 1.5 mL, Vaq ¼ 5.5 mL) Volatile

Pf=aq

Pv=aq

Pv=f

cv in Full-Fat Cheese (nmol mL1)

cv in Low-Fat Cheese (nmol mL1)

H2S

3.5

1.31

0.41

32.2

35.5

Methanethiol

6.13

0.14

0.025

5.6

8.1

14.25

0.17

0.014

3.4

5.6

0.0006

0.0014

0.11

0.10

Acetaldehyde

0.6

0.47

2-heptanone

0.03

0.06

Dimethylsulﬁde Diacetyl

0.43

Note: Vv is taken as 20 mL and the total amount of the volatile is taken as 1000 nmol.

of a fat phase. In reality, things are more complex because there may be no equilibrium, and there may be adsorption of volatiles onto proteins, for instance. Nevertheless, Table 11.7 shows the large impact that the amount of fat can have on the partitioning of volatiles, a factor that needs to be taken into account in the design of foods. The equations given so far about partitioning are shown as being time independent, and they have been discussed to show the importance of the partitioning coefﬁcient. In the following, we will also consider the effect of time, and hence kinetics of partitioning. Release kinetics. As argued above, thermodynamic parameters give an idea about the extent of partitioning. Mass transfer on the other hand determines the rate at which partitioning occurs. Hence, both thermodynamic as well as kinetic factors determine the ultimate release or partitioning rate. Several models have been developed that describe mass transfer. We will only discuss a few of them here. References for more models are given at the end of this chapter. For situations in which there is no ﬂow of the product or the headspace, static diffusion models are used, implying only molecular diffusion. Static diffusion, i.e., without any ﬂow or movement of the material and the vapor around it, is not very realistic in practical terms, but it gives insight in the effect of key parameters. An equation describing the total amount of release of a component from a liquid l to vapor v, based on Fick’s diffusion laws, reads as follows: Mm (t) ¼

2Fc0l

rﬃﬃﬃﬃﬃ Df pt

(11:95)

Mm represents the total amount of volatiles that is released. The factor F in this equation is qﬃﬃﬃﬃﬃﬃ F¼

Pv=l 1 þ Pv=l

Df ,v Df ,l

qﬃﬃﬃﬃﬃﬃ

(11:96)

Df ,v Df ,l

TABLE 11.7 Numerical Values for Parameters Used in Equation 11.118 for the Fit of the Model to the Experimental Data in Figure 11.30 Dimethylpyrazine

Diacetyl

Acetaldehyde

Pv=l

Parameter

5.7 3 105

3.9 3 104

2.7 3 102

0.025

km

0.0191

2.7 3 103

8.98 3 104

9.97 3 105

c0

l

Dimethylsulﬁde

0.00837

0.000961

0.0011

0.000846

wv

1.17 3 106

1.17 3 106

1.17 3 106

1.17 3 106

Aa

0.001

0.001

0.001

0.001

Vv

50 3 106

50 3 106

50 3 106

50 3 106

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Relative release rate 105 (m s−1/2)

3.5 3 2.5 2 1.5 1 0.5 0

−5

−4

−3

Pv/l

−2

−1

0

FIGURE 11.25 Relative release rate (Equation 11.97) as a function of partition coefﬁcient values Pv=l. Df,l ¼ 109 m2 s1, and Dv ¼ 2 3 105 m2 s1.

This factor represents the driving force for the diffusion process. The partition coefﬁcient Pv=p appears to be the most important factor, determining differences in release rates for volatiles. Figure 11.25 gives an impression by plotting the relative release rate nr per unit area: nr ¼

Mm (t) pﬃﬃﬃﬃﬃ pt 2c0l

(11:97)

Figure 11.25 shows that the value of the partition coefﬁcient has a large effect on the release rates. Not unexpectedly, the higher the partition coefﬁcient, the higher the release rate. The differences in diffusion coefﬁcients are not so large for the various volatiles, so that parameter does not cause large differences between volatiles. Once again, the reader is advised that this situation pertains to static diffusion, which may not be very realistic for foods, but the analysis gives insight in the factors that play a role. Convective mass transfer (or eddy diffusion) models are developed for cases where a liquid phase in the product and the air phase are moving, one way or another. The transport from one phase to another is considered to take place via an interfacial layer. In general, the driving force for transport of volatiles over an interface is determined by concentration gradients and mass transfer coefﬁcients in the various phases. This transport can be described in differential equations. For solid foods that are in contact with saliva, a model is developed based on stagnant-ﬁlm theory, in which it is assumed that in the interfacial layer only molecular diffusion occurs (Figure 11.26). The equation describing the total amount Mm of volatile transported over the interface Aa reads dMm Df ,saliva ¼ Aa (t) cint saliva csaliva dt Ls

(11:98)

The differential equation describing the concentration of a volatile in the saliva (csaliva) as a function of time becomes km Aa (t) dcsaliva Df ,saliva ¼ Aa (t) cint (Psaliva=food cfood csaliva ) saliva csaliva ¼ dt Ls Vsaliva Vsaliva

(11:99)

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Concentration cfood

Solid food

Saliva

int csaliva

csaliva Ls x Stagnant interfacial layer

FIGURE 11.26 Schematic picture showing the release of a volatile from a solid food to saliva via diffusion. cfood, cint saliva , and csaliva are the concentrations of a volatile in the food, the interfacial layer, and saliva, respectively. Ls is the thickness of the stagnant interfacial layer.

where Df is the diffusion coefﬁcient of the volatile in the liquid phase Ls is the thickness of the stagnant layer Aa is the surface area of the interface Psaliva=food is the partition coefﬁcient between saliva and food Because Ls cannot be measured Df,saliva and Ls are replaced by a mass transfer coefﬁcient km. An important, saliva complication that will happen during eating is that the surface area Aa is not constant but a function of time, and some function has to be assumed to account for this. In the case of a volatile released from a liquid food to the vapor phase via convection, models can be based upon the so-called penetration theory, in which it is assumed that there is mass transport in the interfacial layer via eddy diffusion. This is supposed to happen via a volume element of the bulk phase that enters (penetrates) the interfacial layer and stays there for a short time te, during which exchange between gas and liquid phase takes place. During this period equilibrium is assumed in the interfacial layers through diffusion, so that the volatile compound can reach the gas layer before the volume element moves back to the bulk liquid. Figure 11.27 gives a schematic representation. The differential equation describing mass transfer is rﬃﬃﬃﬃﬃﬃﬃ dMm Df ,l int cv Aa cl cl ¼ km Aa cl ¼2 dt pte Pv=l

(11:100)

The parameter te cannot be measured independently, and therefore the mass transfer coefﬁcient km is introduced again: km ¼ 2

Df pte

12 (11:101)

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Concentration cl

c int l Liquid food

Headspace

c int v cv

Penetration of a bulk volume element

x Interfacial layers

FIGURE 11.27 Schematic picture showing the release of a volatile from a liquid food to a headspace via penetration int of a bulk volume element in the interfacial layer. cl, cint l , cv , and cv are the concentrations of a volatile in the liquid food, the interfacial layer at the liquid side, the interfacial layer at the vapor side, and in the vapor phase, respectively.

The solution in terms of the change in the concentration in the headspace is

! Pv=l c0,l Vl km Aa 1 exp 1 þ t cv (t) ¼ Pv=l Vv Pv=l Vv Vl þ1 Vl

(11:102)

The assumption made in this derivation is that the vapor phase is thoroughly mixed so that cint v cv. Two extreme situations can be derived from this equation. The ﬁrst is the initial release rate: cv (t ! 0) ¼

c0,l Aa km t Vl

(11:103)

The second is the situation at equilibrium: cv (t ! 1) ¼

Pv=l cg,0 t Pv=l Vv þ1 Vl

(11:104)

This last equation is in fact describing partitioning at equilibrium. Figure 11.28 shows two examples of release kinetics where this model is applied. It has also been shown that release kinetics is slowed down when the viscosity is increased by low molecular compounds but hardly so when the viscosity is decreased by high molecular weight compounds, such as pectin (provided that pectin does not bind the volatiles). It is interesting to note that the initial release is mainly determined by the mass transfer coefﬁcient, while the ﬁnal headspace concentration is determined by the partition coefﬁcient, which explains the initial differences observed in Figure 11.28 for linalool and ethylhexanoate. Effect of volatile adsorption. The situation becomes even more complex when substances are present that have an effect on the concentration of the volatile in the aqueous solution. This can happen if, for instance,

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100

Percentage of equilibrium concentration

90 80 70 60 50 40 30 20 10 0 0

2000

4000 6000 Time (s)

8000

FIGURE 11.28 Release kinetics of linalool (*) and ethylhexanoate (~) from an aqueous solution containing 0.1% NaCl. The concentrations are expressed as % of the ﬁnal equilibrium concentrations. The lines are calculated using Equation 11.102 with parameters Vv ¼ 111.63 cm3, Vl ¼ 24 cm3, Aa ¼ 17.9 cm2, Pv=l ¼ 0.0094 for linalool, and Pv=l ¼ 0.001 for ethylhexanoate, km ¼ 0.0004 cm s1 for linalool, and km ¼ 0.000092 cm s1 for ethylhexanoate. Dataset in Appendix 11.1, Table A.11.13.

volatile molecules bind with proteins or polysaccharides; we consider this binding a physical process, not a chemical one because in the latter case the volatile would not be released anymore. Another possibility is the presence of lipid phases, notably emulsions. Most volatiles are lipophilic and will tend to dissolve easily in lipid phases. As a result of all this, the partitioning of volatiles will be disturbed as compared to an aqueous phase without lipids or biopolymers. A general discussion about such phenomena is as follows. If we consider the concentration of a volatile in a product, the following mass balance is valid: ctotal ¼ cb þ cu

(11:105)

where ctotal is the total concentration of the volatile in the product cb is the concentration of the binder–volatile complex BI–F cu is the concentration of the volatile F that is unbound Suppose that the concentration of the binding substance BI is cbi, then we have the following situation at equilibrium: k1

BI þ F ! BI---F k

(11:106)

2

The equilibrium constant is then K¼

cu cbi cb

(11:107)

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Assuming that only a small fraction of the binder is complexed with the volatile, the equilibrium concentration of the binder can be put equal to its initial concentration. Rearranging and combining Equations 11.105 and 11.107 results in cu ¼

ctotal 1 þ Kcb

(11:108)

So, this equation shows clearly that the concentration of the unbound volatile in solution is reduced by a factor 1=(1 þ Kcb) and this must then also be true for the partition coefﬁcient. The above discussion is valid for the equilibrium situation in the aqueous phase. However, in the case of the presence of a headspace, the equilibrium will be disturbed and we have to consider the kinetics of these changes. The following equation applies to that situation: dcu ¼ k1 cb k2 cu cbi dt

(11:109)

The resulting model allows for binding of volatile components to food constituents such as proteins and polysaccharides: Kb is the binding constant, and cb the concentration of the ligand. The factor hD is the gas–liquid mass transfer coefﬁcient. te is the mean contact time of an element of solution with the gas phase; it is taken proportional to stirring rate. # Aa km 1 Vv 1 i 1 exp þ t cv (t) ¼ h V Vv Pv=l Vl 1 þ Kb cb 1 þ KPbv=lcb þ Vgl "

cv,0

(11:110)

Volatile release from emulsions. The equilibrium headspace above an oil-in-water emulsion (subscripts oil, aq and em, respectively) can be calculated, as was already shown above in Equation 11.94: cem ¼ (1 w)caq þ wcoil

(11:111)

cv ¼ Pv=em cem b(1 w) þ wPoil=aq c

(11:112)

The partition coefﬁcient for the gas-emulsion equilibrium is Pv,em ¼

Pv=aq 1 þ (Poil=aq 1)w

(11:113)

Equation 11.112 can be written as Pv=em c0,em

cv (t) ¼ P

v=em Vv

Vl

þ1

1 exp 1 þ

! Vem k m Aa t Pv=em Vv Vem

(11:114)

Dynamic headspace dilution. As remarked before, equilibrium situations are not easily encountered with foods. It is therefore of interest to consider situations where equilibrium is disturbed. One such situation is when the headspace above a food is disturbed by an air ﬂow, for instance when a package is opened. For the situation as displayed in Figure 11.29, the following relation can be derived. It concerns the blowing of a gas above a solution as a result of which volatiles are removed from the solution. For

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φv

11-45

civ

civ Vg

Interface gas

Area Aa

Interface liquid

cil

Vl

cil (A)

(B)

Concentration

FIGURE 11.29 Schematic representation of a cell in which a gas ﬂow blows with ﬂow wv over a solution containing a volatile in concentration cil in the bulk (A). The volume of the gas compartment is Vv, that of the solution Vl. The concentration of the volatile in the gas phase is civ. The concentration proﬁle at one moment in time is depicted (B). The concentration changes in the interface with area Aa and it is assumed that there is no concentration gradient in the bulk.

simplicity, the conditions are chosen in such a way that the concentrations of the volatile in the bulk gas phase and liquid phase can be considered constant, while there is a concentration gradient in the interface of liquid and water. A mass balance can be set up between the release of the volatile from the liquid phase and removal by the gas ﬂow: dcvi ¼ Ji Aa fv cvi dt Ji ¼ km Pi,v=l cli cvi

Vv

(11:115) (11:116)

Ji is the ﬂux. The mass transfer coefﬁcient is a function of the transfer from the bulk liquid to the interface and from the interface to the bulk gas, and since this is a process in series the following relation holds (Equation 4.86): Pi,v=l 1 1 ¼ þ km km,v km,l

(11:117)

It is assumed that there is equilibrium at the interface, and the following differential equations can be derived: dcvi km Aa km Aa þ fv v l ¼ Pv:l ci ci dt Vv Vv

(11:118)

dcli km Aa ¼ Pv=l cvi dt Vl

(11:119)

At the start of the experiment (t ¼ 0) it follows that: Pi,v=l cli ¼ cvi

(11:120)

If the time of experiment is not too large and Vl not too small, it can be assumed that c1i is constant, and a g change in ci can be predicted. Figure 11.30 shows an example of the match between experiment and

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1

0.8

c/c0

0.6

0.4

0.2

0

0

100

200 Time (s)

300

400

FIGURE 11.30 Headspace concentrations of various volatiles upon dilution of the headspace by air ﬂow. Lines are the ﬁt by the model displayed in Equation 11.118 solved numerically with the parameters shown in Table 11.7. Dimethylpyrazine (D), diacetyl (*), acetaldehyde (.), dimethylsulﬁde (&) in aqueous solutions. Dataset in Appendix 11.1, Table A.11.14.

model as displayed in Equation 11.118, which was solved numerically. The data used for the calculation are shown in Table 11.7. Overall, the model ﬁts reasonably well. To conclude this section on release kinetics of volatiles, we have discussed several models that seem to be able to ﬁt experimental data for various conditions. However, the ultimate challenge is to be able to construct models that are also capable to describe release of volatiles in the mouth where things become much more complicated. Chewing deforms the food, which has an effect on the mass transfer kinetics, there is dilution with saliva, and there is air ﬂow by inhalation and exhalation. This research area is developing quickly and models have been presented also for such cases. We will not discuss them here any further, but some references are given at the end of this chapter.

11.5.3

Partitioning of Weak Acids

Partitioning of weak acids (such as lactic acid, acetic acid, benzoic acid, sorbic acid) in fat-containing foods over aqueous and oil phases can have large effects on their potential as preservative. We can envisage the following situation for the oil and aqueous phase: þ HAoil ! HAaq ! H þA

(11:121)

A partition coefﬁcient Poil=aq can be derived for the distribution of the undissociated form over the oil and the aqueous phase: Poil=aq ¼

[HA]oil [HA]aq

(11:122)

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If we know the volume of oil (Voil) and that of the aqueous phase (Vaq), we can quantify the amount of oil as volume fraction f: f¼

Voil Voil þ Vaq

(11:123)

The mass balance for the acid reads (the subscript ‘‘T’’ indicates the total amount of acid): [HA]T Vaq ¼ [HA]oil Voil þ [HA]aq Vaq þ [A ]Vaq

(11:124)

Combining this with Equation 11.123 gives [A ] ¼ [HA]T

w [HA]oil [HA]aq 1w

(11:125)

After some algebraic rearrangement the result is [HA]aq ¼ [HA]T

1 þ Poil=aq

1 w þ 10pHpKa 1w

(11:126)

This equation is in fact a modiﬁcation of the Henderson–Hasselbalch equation, and shows the fraction of undissociated form present in the aqueous phase as a function of pH and volume fraction of oil, given the pKa and partition coefﬁcient for the acid involved. Figure 11.31 shows an example of the effect for benzoic acid, and it is obvious that a large fraction of undissociated benzoic acid is partitioning into the oil phase. This can have large consequences if benzoic acid is used as a preservative in low pH foods where it will only be effective in the aqueous phase in its undissociated (protonated) form.

1 f = 0.0

[HA]aq/[HA]T

0.8

0.6 f = 0.2 0.4 f = 0.4

0.2

f = 0.6 0 2

4

6 pH

FIGURE 11.31 Effect of volume fraction of oil (f) and pH on the fraction of undissociated benzoic acid in the aqueous phase. Calculated according to Equation 11.126 using pKa ¼ 4.19, Poil=aq ¼ 5.73.

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A solute–solute interaction that is relevant for foods in this respect is the dimerization of carboxylic acids in nonaqueous phases, which is due to hydrogen interactions. Suppose we have an aqueous phase (subscript aq) and a lipid phase (subscript oil) and a carboxylic acid HA, we can envisage the following equilibria: Partitioning: (HA)aq Ð (HA)oil

(11:127)

Dimerization: 2(HA)oil Ð (HA)2,oil

(11:128)

Hence, Poil=aq ¼

[HA]oil [HA]aq

(11:129)

A practical equilibrium constant can be given for the dimerization: Kc ¼

[(AH)2 ]oil [AH]aq

(11:130)

This leads to the following expression: [HA]oil þ 2[AH2 ]oil 2 ¼ Poil=aq þ 2Poil=aq Kc [HA]aq [HA]aq

(11:131)

This equation shows that both partitioning as well as dimerization have an effect on the ﬁnal distribution of components over the various phases. Figure 11.32 gives an example concerning the distribution of benzoic acid and shows how Equation 11.131 is obeyed experimentally.

8 7 6

coil/caq

5 4 3 2 1 0

0

2

4

caq

6

8

10

FIGURE 11.32 Distribution of benzoic acid over water and sunﬂower oil at 258C. caq is the concentration in the aqueous phase, coil is the concentration in the oil phase. Dataset in Appendix 14.1, Table A.11.15.

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In relation to kinetics, these partitioning phenomena are important because they affect the concentrations of possible reactants and products.

11.6

Concluding Remarks

To conclude this chapter on physical changes, it can be stated that it is difﬁcult to predict these in a quantitative way. This is partly due to the fact that the available theory is developed for very ideal systems, and partly to the fact that foods are so incredibly complex with respect to physical structure. Nevertheless, one can ﬁnd many studies in which kinetics of physical changes are described. A case in point is the kinetics of texture changes. Texture is an important quality aspect, related to changes in cell wall components, changes that are brought about by enzymatic activity, but also due to moisture changes, breakdown of starch and protein during the ongoing metabolism after harvest. It becomes an arduous task to derive kinetics in which all such changes are taken into account. So, the approach taken is of an empirical nature: one uses a rheological technique to describe changes, and ﬁts a kinetic equation to the data. While this is deﬁnitely useful for description of phenomena observed, one should be careful with interpretation of such results in molecular terms.

Appendix 11.1 Datasets Used for Examples in This Chapter TABLE A.11.1 MS Diffusion Coefﬁcient and Fick Diffusion Coefﬁcient for Ethanol–Water Mixtures (Figure 11.6) MS D (m2 s1)

Fick D (m2 s1)

0

1.71

1.72

0.03

1.68

1.5

0.11 0.15

1.52 1.39

1.01 0.79

0.21

1.45

0.69

0.26

1.51

0.63

0.31

1.55

0.6

0.41

1.59

0.64

0.5

1.65

0.74

0.6

1.64

0.84

0.68 0.79

1.64 1.69

1.02 1.25

0.88

1.7

1.44

0.96

1.65

1.55

1

1.65

1.65

Mole Fraction Ethanol

Source: From Wesselingh, J.A. and Krishna R., Mass Transfer in Multicomponent Mixtures, Delft University Press, Delft, the Netherlands, 2000.

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TABLE A.11.2 MS Diffusion Coefﬁcient of Lysozyme as a Function of KCl (Figure 11.7) KCl (mol m3)

2 1 D (m s )

0

5.5

1

4.14

2.5

2.89

5

2.18

50

1.39

150

1.33

Source: From Wesselingh, J.A. and Krishna R., Mass Transfer in Multicomponent Mixtures, Delft University Press, Delft, the Netherlands, 2000.

TABLE A.11.3 Mutual MS Diffusion Coefﬁcients in NaCl Solutions (Figure 11.8) log c (M)

log Dwater–Cl

log Dwater–Naþ

8.67

8.88

2

log D,

water–NaCl

1.01

8.68

8.88

10.59

0.67 0.47

8.67 8.67

8.88 8.89

10.35 10.22

0.12

8.69

8.9

9.99

0

8.69

8.93

9.89

0.21

8.71

8.94

9.77

0.33

8.75

8.98

9.68

0.44

8.78

9.03

9.64

0.52

8.82

9.09

9.59

Source: From Wesselingh, J.A. and Krishna R., Mass Transfer in Multicomponent Mixtures, Delft University Press, Delft, the Netherlands, 2000.

TABLE A.11.4 Diffusion of NaCl and Water for Cheese Immersed in a Brine (Figure 11.9) Salt data: x (m)

% Salt in Cheese Moisture

0.001 0.0025

7.76 8.12

0.0035

9.66

0.0043

10.56

0.0053

11.82

0.0069

12.64

0.0084

13.99

0.0096

14.62

0.0104 0.0118

14.98 15.7

0.0127

15.16

0.0137

16.25

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TABLE A.11.4 (continued) Diffusion of NaCl and Water for Cheese Immersed in a Brine (Figure 11.9) Salt data: x (m)

% Salt in Cheese Moisture

0.0147

16.88

0.0159

17.24

0.0169

17.24

0.0182

18.05

0.0196

18.86

0.0204 0.0224

18.59 18.32

0.0224

19.04

0.0231

19.68

0.0243

19.49

0.0249

20.31

0.0259

19.86

0.0278

20.4

0.0286 0.03

20.58 21.3

0.0312

21.48

0.0325

21.84

0.0335

21.3

0.0345

21.84

0.0363

22.29

0.0371

22.47

0.038 0.0388

22.38 22.11

0.0396

22.56

0.0412

22.56

0.0433

22.83

0.0453

22.65

0.0473

22.92

0.049

22.83

0.0518 0.0537

22.92 23.29

0.0559

22.83

0.0573

22.83

0.059

22.74

0.061

22.74

0.0625

22.83

0.0649

22.83

Water data: x (m)

% Water

0.0008

40.91

0.0037

38.58 (continued )

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TABLE A.11.4 (continued) Diffusion of NaCl and Water for Cheese Immersed in a Brine (Figure 11.9) Water data: x (m)

% Water

0.0051

37.77

0.0076

36.31

0.0098

34.85

0.0135

33.03

0.0159

32.23

0.0182 0.0224

30.69 29.67

0.0247

28.72

0.0273

28.07

0.0335

26.75

0.0369

26.17

0.0418

25.73

0.0496

25.15

0.0541 0.0643

25.07 25.15

Source: From Payne, M.R. and Morison, K.R., Int. Dairy J., 9, 887, 1999.

TABLE A.11.5 First-Order Kinetic Plot for Coalescence of Emulsions (Figure 11.14) ln N0=Nt Time (h)

Butter Oil

Peanut Oil

0

0

0.87

0.12

0

10

0.28

0.22

23 31

0.23 0.3

0.15 0.31

47

0.41

0.56

55

1.31

1.13

80

1.8

0.79

90

2

0.66

112

2.43

0.88

124

2.89

1.26

148 175

2.64 3.12

1.52 1.58

200

3.05

1.44

220

2.46

250

2.43

275

2.91

300

3.05

Source: From Das K.P. and Kinsella J.E. Droplet size and coalescence stability of whey protein stabilized milk fat peanut oil emulsions. J Food Sci 58:439–444, 1993.

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TABLE A.11.6 Crystallization Kinetics of Hardened Palm Oil at 408C (Figure 11.16) Time (min)

% Solid Fat

0

0

2.5

0

6.6

0.3

9.5

0.7

11.9

1.1

15.2

1.4

19.0

2.5

21.0 24.3

3.2 4.7

27.6

5.5

30.1

7.3

35.9

10.7

40.4

13.6

45.0

17.3

50.4

20.7

55.7 59.9

24.0 27.0

64.8

29.6

71.0

31.4

74.7

32.5

80.5

33.2

85.0

34.3

90.4

34.3

94.9

34.6

Source: From Singh, A.P., Bertolli C., Rousset P.R., and Marangoni A.G. Matching Avrami indices Achieves similar hardnesses in palm oil-based fats. J Agric Food Chem 52:1551–1557, 2004.

TABLE A.11.7 Texture of Potato Cubes as a Function of Time after Cooking at 808C (Figure 11.17) Time (min)

Compression Force (N)

0

118.59

3

100.68

5 6

89.38 79.95

10

60.16

15

52.58

20

37.48

30

33.64

40

21.32

50

18.42

60 75

21.16 10.7

90

12.47

105

12.35

Source: From Stoneham, T.R., Lund, D.B., and Tong, C.H., J. Food Sci., 65, 968, 2000.

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TABLE A.11.8 Kinetics of Texture Changes of Potatoes as a Function of Cooking Temperature and Time (Figure 11.18) Time (min)

Failure Stress (N)

908C 0

100

2 3

61.3 47.39

5

37.42

6

29.12

7

22.47

10

17.75

12

13.37

1038C 0

100

0.8

61.35

1.6

36.57

2

28.63

2.5

22.34

3.0 3.5

17.37 13.05

4

10.06

4.5

7.4

5

6.06

6.0

4.7

7

4.01

9

3.58

11 12

2.52 2.49

1208C 0

100

0.2

61.38

0.5 0.6

36.61 29.02

0.7

22.41

0.9

16.8

1.0

13.49

1.2

9.52

1.5

6.21

1.9

4.54

2.3 2.7

3.53 2.53

3.5

2.5

4.9

2.44

6.8

2.03

Source: From Rahardjo, B. and Sastry, S.K., Trans. I. Chem E., 71, 235, 1993.

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TABLE A.11.9 Kinetics of Water Uptake by Cereals Immersed in Whole Milk at 558C (Figure 11.19) Time (s)

Mt M0 g water=100 g Initial Weight

290 290

78.1 86.8

520

102.4

520

112.8

1050

133.8

1050

144.2

1580

149.5

1580

161.6

1550 2080

172 161.7

2080

173.9

2590

163.6

2590

184.4

2590

193

3140

167.2

3120

181

3120 3650

191.4 181.1

3650

191.5

4200

169.1

4180

198.6

4180

207.3

4760

191.8

4730

202.2

4730 5260

209.1 200.6

5260

209.2

5260

226.6

5790

188.5

5790

200.7

6320

207.7

6270

218.1

6270 6850

245.9 178.4

6850

195.7

6820

209.6

7380

216.6

7350

225.3

Source: From De Fatima Machado, M., Oliveira, F.A.R., and Cunha, L.M., Int. J. Food Sci. Technol., 34, 47, 1999.

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TABLE A.11.10 Effect of Temperature on Partition Coefﬁcients (Figure 11.22) Methyl Acetate in Water

Ethyl Acetate in Water

Isopentyl Acetate in Coffee Oil

25

0.00527

0.00694

0.000198

30

0.00672

0.0092

35

0.00872

T0 (8C)

0.00027 0.000359

40

0.0152

50

0.000809

Source: From Kieckbusch, T.G. and King, C.J., J. Agric. Food Chem., 27, 504, 1979.

TABLE A.11.11 c in Solution (nmol dm3)

Partitioning of Methanethiol (Figure 11.23) c in Headspace above Aqueous Solution (nmol dm3)

0

0

0.12

0.3

c in Headspace above Liquid Milk Fat (nmol dm3)

0.36

0.21

0.72 0.96

0.76 1.06

1.08

0.3

1.92

0.58

2.16

3.61

2.22

0.85

2.46

0.64

3.66

0.85

4.49 4.85

6.27

1.91

7.01

10.06

7.61

1.76

7.79

10.12

8.15

2.76

16.9

4.03

Source: From Van Boekel M.A.J.S. and Lindsay R.C. Partitioning of cheese volatiles over vapour, fat and aqueous phases. Neth Milk Dairy J 46:197–208, 1992.

TABLE A.11.12 Dependence of Activity Coefﬁcients of Volatile Components on Sucrose Concentration (Figure 11.24) Sucrose (kg kg1) 0 0.2

log fi1 Ethyl Acetate

log fi1 Butylacetate

log fi1 Hexylacetate

1.82 1.93

3.13 3.24

4.39 4.54

0.4

2.1

3.43

4.82

0.6

2.3

3.71

5.3

0.7

2.49

Source: From Bruin S. and Luyben K.C.A.M. Drying of food materials: A review of recent developments. In: Mujumdar A.S. (ed.) Advances in Drying, Vol. 1, pp. 155–215. Hemisphere Publishing Corporation, Washington, 1980.

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TABLE A.11.13 Release Kinetics of Linalool and Ethylhexanoate (Figure 11.28)

Time (s)

Ethylhexanoate % of Equilibrium Concentration

Linalool % of Equilibrium Concentration

0

0

12

0

23.68

13

6.96

25 69

32.87

80

49.03

11.42

110

17.83

180

27.3

215

72.15

260

32.6

300

74.38

400 410

82.19

42.91

1800

80

1850

85.57

4500

95.68

7250

99.94

7500

100

9000

100

Source: From Juteau A., Doublier J.L., and Guichard E. Flavor release kinetics from i-carrageenan matrices: A kinetic approach. J Agric Food Chem 52:1621–1629, 2004.

TABLE A.11.14 Headspace Concentrations of Volatiles upon Dilution of the Headspace (Figure 11.30) Time (s) 0 3.6

Dimethylpyrazine c=c0 1 0.99

5.2

Diacetyl c=c0 1

Acetaldehyde c=c0

Dimethylsulﬁde c=c0

1

1 0.96

0.88

0.84

0.84

0.74

0.95

6.8

0.97

11.7

0.96

13.4

0.84 0.79

16.5 19.6

0.95

23.1 26

0.93

0.75

31

0.8

32.7

0.73

35.9

0.93

45

0.93

0.69 0.72

0.64

49

0.48

52

0.61

58

0.59 (continued )

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TABLE A.11.14 (continued) Headspace Concentrations of Volatiles upon Dilution of the Headspace (Figure 11.30) Time (s)

Dimethylpyrazine c=c0

Diacetyl c=c0

Acetaldehyde c=c0

Dimethylsulﬁde c=c0

60

0.41

63

0.71

66

0.56

68

0.34

73

0.93

80 85

0.28 0.7

93

0.24 0.21 0.5

106

0.94

112

0.71

118

0.13

126

0.46

146.5

0.69

156 176

0.1 0.43

0.93

209

0.69

0.1

0.69

0.09

235 265

0.93

0.67

0.38

0.09

0.38

0.1

300

0.68

0.38

0.09

350

0.67

0.38

0.09

0.67

0.37

0.09

400

0.94

Source: From Marin M., Baek I., and Taylor A.J. Volatile release from aqueous solutions under dynamic headspace dilution conditions. J Agric Food Chem 47:4750–4755, 1999.

TABLE A.11.15 Partitioning of Benzoic Acid over Aqueous and Oil Phase (Figure 11.32) caq (mmol dm3)

coil=caq

1.73

4.76

1.78

4.64

3.23 3.33

5.22 5.12

4.53

5.67

4.55

5.54

5.71

6.03

5.91

5.82

6.63

6.58

6.75

6.46

7.64 7.64

6.99 6.86

9.38

7.51

9.46

7.4

Source: From Wedzicha B.L., Zeb A., and Ahmed S. Reactivity of food preservatives in dispersed systems. In: Dickinson E. (ed.) Food Polymers, Gels and Colloids, London, pp. 180–193. Royal Society of Chemistry, 1991.

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Bibliography and Suggested Further Reading General References about Physical Changes Aguilera J.M. and Lilford P. Food materials science. Principles and practice. In: Food Engineering Series, Barbosa-Cánovas G.V. (Ed.). New York: Springer, 2008. Atkins P.W. Physical Chemistry, 6th ed. Oxford: Oxford University Press, 1999. Dickinson E. An Introduction to Food Colloids. Oxford: Oxford University Press, 1992. McClements D.J. (Ed.) Understanding and Controlling the Microstructure of Complex Foods. Cambridge: Woodhead, 2007. Walstra P. Physical Chemistry of Foods. New York: Marcel Dekker Inc., 2003. Walstra, P. and Van Vliet T. Dispersed systems: Basic considerations. In: Fennema’s Food Chemistry, 4th ed. Chapter 13, Damodaran, S., Parkin, K., and Fennema, O.R. (Eds.), Boca Raton, FL: CRC Press, 2008. General Food Engineering References Bird R.B., Stewart W.E., and Lightfoot E.N. Transport Phenomena, 2nd ed. New York: John Wiley, 2002. Fryer P.J., Pyle D.L., and Rielly C.D. Chemical Engineering for the Food Industry. London: Blackie Academic Professional, 1997. Heldman D.R. and Lund D.B. Handbook of Food Engineering. New York: Marcel Dekker, 1992. Singh R.P. and Heldman D.R. Introduction to Food Engineering. 3rd ed. London: Academic Press, 2001. About Fickian Diffusion Crank J. The Mathematics of Diffusion. 2nd ed. Oxford: Clarendon Press, 1990. About Maxwell–Stefan Diffusion Payne M.R. and Morison K.R. A multi-component approach to salt and water diffusion in cheese. Int Dairy J 9:887–894, 1999. Taylor R. and Krishna R. Multicomponent Mass Transfer. New York: Wiley, 1993. Wesselingh J.A. and Krishna R. Mass Transfer in Multicomponent Mixtures. Delft, the Netherlands: Delft University Press, 2000. About Flocculation Ives K.J. The Scientiﬁc basis of ﬂocculation. Alphen aan de Rijn, the Netherlands: Sijthoff and Noordhoff, 1978. Overbeek J.T.G. Kinetics of ﬂocculation. In: Colloid Science, Vol. I. Irreversible Systems, Kruyt H.R. (Ed.). Amsterdam, the Netherlands: Elsevier, 1952. Rajagopalan R. Stability of colloidal dispersions: A thermodynamic approach. Water Sci Technol 27:117–129, 1993. About Fractal Aggregation Bremer L.G.B. Fractal aggregation in relation to formation and properties of particle gels. PhD thesis, Wageningen University, the Netherlands, 1992. Bremer L.G.B., Walstra P., and Van Vliet T. Estimations of the aggregation times of various colloidal systems. Colloids Surf A 99:121–127, 1995. Dickinson E. On gelation kinetics in a system of particles with both weak and strong interactions. J Chem Soc Faraday Trans 93:111–114, 1997. Meakin P. Fractal aggregates. Adv Colloid Interface Sci 28:249–331, 1988. Weitz D.A., Lin M.Y., and Lindsay H.M. Universality laws in coagulation. Chem Intell Lab Systems 10:133–140, 1991.

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Kinetic Modeling of Reactions in Foods

About Emulsions McClements D.J. Food Emulsions: Principles, Practices and Techniques. Boca Raton, FL: CRC Press, Taylor & Francis, 2005. Walstra P. Emulsion stability. In Encyclopedia of Emulsion Technology, Vol. 4, Chapter 1, Becher P. (Ed.) New York: Marcel Dekker Inc., 1996. Walstra P. Emulsions. In: Fundamentals of Interface and Colloid Science, Vol. V. Soft Colloids, Chapter 8, Lyklema J. (Ed.). London: Academic Press, 2005. About Packaging Han J.H. Innovations in Food Packaging, Food Science and Technology, International Series, p. 496. Amsterdam: Elsevier, 2005. About Release of Volatiles De Roos K.B. and Graf E. Nonequilibrium partition model for predicting ﬂavor retention microwave and convection heated foods. J Agric Food Chem 2204–2211, 1995. Gosh S. and Coupland J.N. Inﬂuence of food microstructure on ﬂavor interactions. In: Understanding and Controlling the Microstructure of Complex Foods, McClements D.J. (Ed.). Cambridge: Woodhead, 2007. Harrison M. and Hills B.P. A mathematical model to describe ﬂavour release from gelatine gels. Int J Food Sci Technol 167–176, 1996. Harrison M., Hills B.P., Bakker J., and Clothier T. Mathematical models of ﬂavor release from liquid emulsions. J Food Sci 62:653–664, 1997. Harrison M. and Hills B.P. Mathematical model of ﬂavour release from liquids containing aroma-binding macromolecules. J Agric Food Chem 45:1883–1890, 1997. Hills B.P. and Harrison M. Two-ﬁlm theory of ﬂavour release from solids. Int J Food Sci Technol 30:425–436, 1995. Klooster J.R., Druaux C., and Vreeker R. Air-liquid partition coefﬁcients of aroma volatiles in frozen sugar solutions. J Agric Food Chem 53:4503–4509, 2005. Overbosch P., Afterof W.G.M., and Haring P.G.M. Flavor release in the mouth. Food Rev Int 7:137–184, 1991. Rabe S., Krings U., and Berger R.G. In vitro study of the inﬂuence of physiological parameters on dynamic in-mouth ﬂavour release from liquids. Chem Senses 29:153–162, 2004. Seuvre A.M., Philippe E., Rochard S., and Voilley A. Kinetic study of the release of aroma compounds in different model food systems. Food Res Intern 40:480–492, 2007. Van Ruth S.M. and Roozen J.P. Delivery of ﬂavours from food matrices. In: Food Flavour Technology, Chapter 6, Taylor, A.J. (Ed.). Shefﬁeld, England: Shefﬁeld Academic Press, 2002. About Kinetics of Water Sorption and Moisture Migration De Fatima Machado M., Oliveira F.A.R., and Cunha L.M. Effect of milk fat and total solids concentration on the kinetics of moisture uptake by ready-to-eat breakfast cereal. Int J Food Sci Technol 34:47–57, 1999. De Fatima Machado M., Oliveira F.A.R., Gekas V., and Singh P. Kinetics of moisture uptake and soluble-solids loss by puffed breakfast cereals immersed in water. Int J Food Sci Technol 33:225– 237, 1998. Fu Y.C., Tomg C.H., and Lund D.B. Moisture migration in solid food matrices. J Food Sci 68:2497–2503, 2003. Nisha P., Singhal R.S., and Pandit A.B. Kinetic modelling of texture development in potato cubes (Solanum tuberosum L.), green gram whole (Vigna radiate L.) and red gram splits (Cajanus cajan L.). J Food Eng 76:524–530, 2006.

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Rahardjo B. and Sastry S.K. Kinetics of softening of potato tissue during thermal treatment. Trans I ChemE 71:235, 1993. Ramesh M.N., Sathyanarayana K., and Girish A.B. Kinetic modeling of vegetable cooking at 1008C. J Texture Studies 257–264, 1996. Risbo J. The dynamics of moisture migration in packaged multi-component food systems. I. shelf life predictions for a cereal-raisin system. J Food Eng 58:239–246, 2003. Risbo J. The dynamics of moisture migration in packaged multi-component food systems II: Analytical solutions and comparison to experimental moisture transfer rate results. J Food Eng 58:247–252, 2003. Rizvi A.F. and Tong C.h. Fractional conversion for determining texture degradation kinetics of vegetables. J Food Sci 62:1–7, 1997. Stoneham T.R., Lund D.B., and Tong C.H. The use of fractional conversion technique to investigate the effects of testing parameters on texture degradation kinetics. J Food Sci 65:968–973, 2000. Verlinden B.E., Debarsy T., Debaerdemaeker J., and Deltour R. Modelling the mechanical and histological properties of carrot tissue during cooking in relation to texture and cell wall changes. J Texture Studies 15–28, 1996. About Activities Carelli A.A., Crapiste G.H., and Lozano J.E. Activity coefﬁcients of aroma compounds in model solutions simulating apple juice. J Agric Food Chem 39:1636–1640, 1991. Fanelli A. Explaining activity coefﬁcients and standard states in the undergraduate physical chemistry course. J Chem Educ 63:112–114, 1986. Kieckbusch T.G. and King C.J. Partition coefﬁcients for acetates in food systems. J Agric Food Chem 27:504–507, 1979. Letcher T.M. and Battino R. An introduction to the understanding of solubility. J Chem Educ 78:103– 111, 2001. About Partitioning of Weak Acids Wilson D.G., Wilson D.R., and Waspe C.R. Weak acids: Dissociation in complex buffering systems and partitioning into oils. J Sci Food Agric 80:471–476, 2000.

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12 Kinetics of Microbial Growth

12.1

Introduction

The microbial quality of food is of utmost importance. It is in fact the ﬁrst and foremost task of a food technologist to produce safe food. The risk of contamination and growth of pathogenic microorganisms is much larger than that of toxic or mutagenic substances. Foods are very vulnerable to microbes because the required nutrients for microbial growth are usually present in sufﬁcient amounts. In addition, foods may contain spoilage-inducing microorganisms. Thus, it is very important to be able to calculate how fast microorganisms will grow in a food. Having such knowledge is the basis for prediction of shelf life in relation to microbial activity, calculation of risks, and optimization of fermentation processes. Microbial growth is typically a kinetic problem, so it is appropriate to discuss some principles in this book. However, at the same time microbial kinetics is different from chemical kinetics discussed hitherto. In previous chapters, kinetics was related mostly to molecular events but with microbial kinetics it is about changes in numbers of cells, and, though superﬁcially the equations may look the same, the underlying principle is different: it is about population dynamics. Growth, resulting in the increase of number of cells, is the result of uncountable molecular events, and requires a different treatment. The kinetics of microbial cell growth depends on the extracellular chemical environment (pH, ionic strength, water activity, presence of nutrients, temperature) and on the intracellular environment (mainly resulting in numerous enzymatic reactions). The resulting growth is the interplay between extracellular and intracellular conditions. Predictive microbiology, as this ﬁeld of research is called, has been booming the past two decades. However, the reader is advised that this ﬁeld is rapidly expanding and too large to do justice to in this chapter. Next to some books that review the state of the art, food microbiological journals contain many articles on this topic, so the interested reader is referred to the literature if he=she is interested in more details; some selected references are given at the end of this chapter. There are quite a few growth models, some of which include the (presumed) effect of pH, water activity and temperature, mainly via empirically derived polynomials because mechanistic insight is not (yet) available. The topic is under development and rapidly evolves. At this stage it is useful to distinguish between kinetic models and stochastic models. Kinetic models are suitable for describing changes in numbers of cell, in other words, the extent of growth, whereas stochastic models are apt to describe probabilities of growth or no growth, or even death. Stochastic models are useful for quantitative microbial risk assessment (QMRA), a topic that we will not address in this book. We limit ourselves here to some typical kinetic growth models. We only give a short basic introduction in Section 12.2; as 12-1

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mentioned above, there is so much development in this area currently that it makes no sense to present the state of the art in much detail. One distinguishes between primary, secondary, and tertiary growth models in predictive microbiology. Primary models describe the growth or inactivation of microorganisms (as changes in numbers of cells), secondary models describe how kinetic parameters from the primary models depend on environmental conditions (such as pH, water activity, etc.), and tertiary models are models that attempt to describe all responses of microorganisms to their environment in experts systems or decision support systems.

12.2

Primary Growth Models

Growth models are developed with the aim to have a tool available that allows prediction of the number of microorganisms as a function of conditions as prevalent in food (notably water activity, aw, temperature, pH, and other effects, such as compounds that stimulate or inhibit growth). Incidentally, it is questionable whether aw is directly related to microbial growth; it may well be that aw is related to another mechanism, but it seems to act as a good indicator, and it is therefore modeled frequently as an independent factor. Predictive microbiology is complicated both by variability and uncertainty: we cannot yet apply kinetics based on mechanistic insight, due to lack of knowledge concerning microbial physiology in relation to the food matrix, and adaptive responses of the microbes to changing conditions. One can express microbial growth as the change in population size N (the absolute number of microorganisms per gram or per milliliter) or as the logarithm of the relative population size, ln(N=N0) or log(N=N0). The growth rate is by deﬁnition dN(t)=dt, i.e., the absolute increase in cell concentration per unit time. The speciﬁc growth rate, i.e., the increase in cell concentration per unit time per cell is: dN(t) d ln N(t) m(t) ¼ dt ¼ N(t) dt

(12:1)

Usually, a microbial growth curve is of a sigmoidal nature (Figure 12.1) and a model should then of course be cast in such a form. The ﬁrst phase, in which no growth occurs, is called the lag phase, Exponential phase

Lag phase

Stationary phase

ln N/N0

As = ln N/Nmax

m max

l

Inflection point

Time

FIGURE 12.1 Schematic depiction of a microbial growth curve. l is the lag time, mmax the maximum speciﬁc growth rate (the tangent in the inﬂection point), As the asymptotic value.

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characterized by a lag time l, in which microorganisms have to adapt to the medium in which they are present. Some damaged cells will die, some will recover, and some will start to grow. This results in a more or less stable (or sometimes slightly declining) number of microorganisms. The lag phase is difﬁcult to model, it depends strongly on the prior history of the cells. After this adaption phase, the exponential phase develops characterized by the maximum speciﬁc growth rate mmax ¼ m(t), the tangent in the inﬂection point of a ln N versus t plot, while the lag time l is deﬁned as the x-axis intercept of this tangent, see Figure 12.1. This is followed by a stationary phase (due to the exhaustion of nutrients, or the formation of toxic products such as alcohol or lactic acid) in which an asymptotic value As (expressed as logarithm of number of microorganisms) is reached. Eventually the number of microorganisms may decline in the so-called dying off phase, but this is not considered here. When the asymptotic phase is reached, foods are so much spoiled that they will not be consumed anymore. We will now consider some equations that are used to describe microbial growth curves. In fact, all these models can be related to one generic growth model, and can be considered as special cases of the generic growth model. We will not consider these relations here.

12.2.1

Differential Equations

Just as with chemical rate equations, ordinary differential equations can be set up to describe microbial growth, but as remarked in the introduction to this chapter it is about population growth, not about chemical reactions. The logistic function A logistic function is able to describe the S-shaped curve shown in Figure 12.1. An application of the logistic function is the so-called Verhulst equation (described by Verhulst in the nineteenth century) to model population growth. It is based on the two assumptions that the growth rate dN(t)=dt is proportional to (1) the momentary existing population size N(t), and to (2) the remaining resources available to the existing population. Expressed mathematically this results in dN(t) N(t) ¼ k N(t) 1 dt Nmax

(12:2)

Nmax represents the number of cells that the system can support (carrying capacity) that is to say that the population starts at N0 and asymptotically reaches Nmax. The proportionality constant k represents the growth rate constant (dimension time1). Equation 12.2 is basically a rate equation. When N(t) Nmax, the ﬁrst term in the right-hand side of Equation 12.2 dominates (i.e., growth is not limited by depletion of resources), and when N(t) ! Nmax the second term dominates (i.e., growth becomes limited by depletion of resources). Equation 12.2 can be integrated: N(t) ¼

Nmax N0 exp (kt) Nmax þ N0 ( exp (kt 1))

(12:3)

Equation 12.3 describes the characteristic shape shown in Figure 12.1 but note that the number of cells is obtained as a function of time (Figure 12.2A). However, this shape disappears when the logarithm of N is taken (Figure 12.2B).* In microbiology, it is common to work with logarithmically transformed numbers because the numbers of cells span several orders of magnitude; moreover, taking logarithms appears to stabilize the variance because of the heteroscedasticity of bacterial cell counts (Chapter 7). It is probably best to

* Some of the graphs in this chapter were produced using the freeware available on the Web as Mathematica ﬁles or as Excel ﬁles: http:==www-unix.oit.umass.edu=aew2000=

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900,000

14

800,000

13

700,000

12 11 10

500,000

ln N (t)

N (t )

600,000

400,000 300,000

8 7

200,000

6

100,000

5 4

0 0 (A)

9

10

20 30 Time (h)

40

50

0

(B)

10

20

30

40

50

Time (h)

FIGURE 12.2 Simulated growth curve using the integrated form of the Verhulst Equation 12.3 when N(t) is plotted (A) or when ln N=N0 is plotted (B). N0 ¼ 1000, k ¼ 0.5 h1, Nmax ¼ 106.

consider microbial growth data as log-normally distributed. Taking logarithms, however, has an impact on the equations. We ﬁnd for the logistic equation the following transformation: dY(t) Y(t) ¼ k Y(t) 1 dt YAs N(t) Y(t) ¼ ln N0 Nmax YAs ¼ ln N0

(12:4)

In Equation 12.4 the natural logarithm base e (ln) is used, but this can be equally well the logarithm base 10 (log). The reader is advised that although there is no principal distinction between ln and log, there is a numerical difference: ln N ¼ ln10 3 log N 2.303 3 log N. Sometimes growth is expressed as ln N and sometimes as log N and the experimental data should of course be expressed in the same way as in the model; as obvious as this may seem, mistakes are made because of this. It is important to realize that integration of Equation 12.4 leads to much longer apparent ‘‘lag times’’ than integration of Equation 12.2: see Figure 12.3, which should be compared to Figure 12.2B. As an extension of the logistic Verhulst model one can also postulate that the momentary growth rate is not proportional to N(t) and [1 N(t)=Nmax] but to N(t)a and [1 N(t)=Nmax]b: dN(t) N(t) b ¼ k N(t)a 1 dt Nmax

(12:5)

and the same can be done for the logarithmic version: dY(t) Y(t) b ¼ k Y(t)a 1 dt YAs

(12:6)

This adds more ﬂexibility to the models. If a < 1, this implies that the microorganism does not use its full capacity to grow exponentially, while a > 1 implies that it exceeds this capacity. If b > 1, this means that

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8 7

Y(t)=ln(N(t)/N0)

6 5 4 3 2 1 0

0

10

20 30 Time (h)

40

50

FIGURE 12.3 Simulated growth curve using the integrated form of Equation 12.4. N0 ¼ 1000, k ¼ 0.5 h1, YAs ¼ 7 (corresponding to Nmax ¼ 106 ).

the microorganism is more sensitive to resource depletion than for the case b ¼ 1 while the situation that b < 1 implies that it is less sensitive to this. There is, however, not an analytical solution to these last two equations, they can only be solved via numerical integration. Freeware is offered however on the web to do this, see footnote on p. 12-3. The Baranyi–Roberts model. The Baranyi–Roberts model is actually based on the logistic model discussed above. However, as shown in Figure 12.2B the logistic model is not able to ﬁt situations in which a long lag time is prominent. Therefore, a factor is introduced in the Baranyi–Roberts model to cope with longer lag times and in doing so, an attempt was made to base it on semimechanistic principles. It is semimechanistic because it uses an adjustment function that accounts for the physiological state of the cells. It is reasoned that the speciﬁc growth rate m depends on the amount of an intracellular compound q(t) . The change in this described by a function q(t). The adjustment function takes on the form q(t) þ 1 function over time is described by: dq ¼ mmax q(t) dt

(12:7)

When the function q(t) increases, implying that the amount of the critical substance per cell increases so that it becomes actually no longer rate limiting, the adjustment function thus approaches 1. In this way the transition from the lag phase to the exponential phase is characterized, as in the exponential phase the cells reach their maximal growth rate. For the transition from the exponential phase to the stationary phase the following differential equation is introduced: dN q(t) N m ¼ mmax N 1 dt 1 þ q(t) Nmax

(12:8)

The exponent m characterizes the curvature before the stationary phase is reached. For m ¼ 1 Equation 12.8 becomes similar to the logistic Equation 12.2. Note that Equation 12.8 is not completely similar to the modiﬁed logistic equation in Equation 12.5. The two differential equations (Equations 12.7 and 12.8)

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lnN

mmax = dlnN/dt

h0

l

Inflection point

t

FIGURE 12.4 Schematic picture of a growth curve and the meaning of the parameters mmax, l, and h0 in the Baranyi–Roberts model.

describe the dynamic growth of cells. Note that Equation 12.8 does not use the logarithmic ratio but the number of cells directly. A differential equation valid for the logarithm of cell concentration is dY(t) 1 ¼ m (t)ð1 exp (Y(t) YAs )Þ dt 1 þ exp (Q(t)) max Q(t) ¼ ln q(t)

(12:9)

Y and YAs are deﬁned in Equation 12.4. A transformation of q(t0) for conditions at a constant environment for the cells is 1 ¼ mmax l h0 ¼ ln 1 þ q(t0 )

(12:10)

Thus, a link is proposed between the lag time and the maximum speciﬁc growth rate (in a constant environment): the lag time l is inversely proportional to mmax with h0 the proportionality constant. Figure 12.4 illustrates the meaning of the parameters. Incidentally, the Baranyi–Roberts model as given here is a simpler version; the original one is more complicated, as it contains six parameters.

12.2.2

Algebraic Equations

Algebraic models represent static models, i.e., models that are only a function of time, not of other variables. The integrated Verhulst Equation 12.3 is an example of this. Monod model. One of the oldest (primary) growth model is the Monod model: mX ¼ where m is the speciﬁc growth rate X is the concentration of biomass

mm S X Ks þ S

(12:11)

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S is the concentration of substrate Ks is a saturation constant mm is the maximum speciﬁc growth rate The Monod equation resembles the Michaelis–Menten equation discussed in Chapter 9. The Monod equation is mainly used in fermentation kinetics. It is less to suited for food microbiology because the transition from the exponential to the stationary phase as described by Monod’s model is not so relevant for microbial growth in foods: at such microbial levels the food will be spoiled. Lag exponential. The lag exponential model is a very simple model in which it is posed that at t < l no growth takes place, while at t > l growth takes place exponentially: ln N ¼ ln N0 at t < l ln N ¼ ln N0 þ m (t l)

(12:12)

No account is given of Nmax, and so it is only applicable for growth that does not slow down. The lag exponential model can be used to estimate the order of magnitude of growth, but obviously it has its limitations. Shifted logistic function. A modiﬁed version of the logistic equation is the shifted logistic function: Y(t) ¼ YAs

1 1 1 þ expðk(tc t)Þ 1 þ expðktc Þ

(12:13)

Here, the parameters have no direct physical meaning; it is a purely empirical model. Nevertheless, the parameters can be interpreted loosely as follows. YAs corresponds more or less to the asymptotic value As (i.e., ln Nmax=N0), k measures the steepness of the growth around the inﬂection point and is therefore related to the speciﬁc growth rate, and tc locates the inﬂection point. Modiﬁed logistic equation. In the literature, more frequent use is made of the modiﬁed logistic equation, in which parameters are reparameterized in the microbial interpretable parameters As, mmax, and l: ln

N ¼ N0

As 4mmax 1 þ exp (l t) þ 2 As

(12:14)

Figure 12.5 shows the ﬁt of the shifted logistic Equation 12.13 and the reparameterized modiﬁed logistic Equation 12.14 to some experimental data. Parameter estimates are given in Table 12.1. Both models seem to give a reasonable ﬁt, but based on the Akaike criterion (introduced in Chapter 7) the shifted logistic equation performs signiﬁcantly better, at least with this dataset. Incidentally, one could also estimate the initial number N0 if so desired. One then has to ﬁt Equation 12.14 to data in the form of ln N instead of ln N=N0 and estimate ln N0 as an extra parameter. Modiﬁed Gompertz model. Another frequently used model is the Gompertz equation (developed by Gompertz in 1825 to predict human mortality as a function of age): y ¼ a exp [exp (b ct)]

(12:15)

with a, b, c parameters of the model. The modiﬁed Gompertz equation has been reparameterized into the relevant microbiological parameters: ln

N m e ¼ As exp exp max (l t) þ 1 N0 As

(12:16)

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12-8

11

11

9

9

7

7

In N/N0

13

In N/N0

13

5

5

3

3

1

1

−1

0

50

100

(A)

150 200 250 Time (h)

300

350 400

−1

0

(B)

50

100

150 200 250 Time (h)

300

350

400

FIGURE 12.5 Fit of the shifted logistic Equation 12.13 (A) and the reparameterized modiﬁed logistic Equation 12.14 (B) to growth of L. monocytogenes at 58C. Dataset in Appendix 12.1, Table A.12.1.

with the following reparameterizations for the parameters a, b, c of the original Gompertz equation: a ¼ As m e b ¼ max l þ 1 As mmax e c¼ As

(12:17)

in which As is the natural logarithm of the asymptotic value of the relative population size, As ¼ ln (Nmax=N0), e ¼ exp(1) ¼ 2.718 Also, it is worth noting that the dependent variable y has been substituted by the (natural) logarithm of the relative population, as was done with the logistic equation. This three-parameter-model has proven to be adequate in many tested cases for isothermal growth. Figure 12.6 gives an example. The original logistic and Gompertz model can be considered as mechanistic models, but using the logarithm of number of cells, rather than the number of cells directly, turns the modiﬁed logistic and Gompertz model into empirical models. When used with logarithmic ratios the term ‘‘modiﬁed’’ signiﬁes this fact. The modiﬁed Gompertz model is asymmetric about the point of inﬂection which occurs at 1=e of the distance between the lower and upper asymptote, in contrast to the modiﬁed logistic equation which is symmetric about the inﬂection point, which occurs at 1=2 of the distance between the lower and upper asymptote. The rate obtained from the modiﬁed Gompertz equation is always the maximum speciﬁc growth rate mmax and occurs at an arbitrary inﬂection point; consequently, this may lead to an

TABLE 12.1 Parameter Estimates 95% Conﬁdence Intervals and Akaike Criterion (AIC) of the Shifted Logistic Equation and the Modiﬁed Logistic Equation Applied to the Growth of L. monocytogenes at 58C Model Shifted logistic equation Modiﬁed logistic equation

AIC 50.5

0

38.0

12.5

YAs ¼ 11.9 0.6

tc ¼ 164.6 5.9 h

k ¼ 0.019 0.003 h

As ¼ 11.2 0.4

l ¼ 75.6 10.7 h

mmax ¼ 0.060 0.006 h1

Note: Same data as in Figure 12.5.

DAIC

1

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14 12

ln(N/N0)

10 8 6 4 2 0

0

10

20 30 Time (h)

40

50

FIGURE 12.6 Example of the ﬁt of the modiﬁed Gompertz equation to growth of Salmonellae in a laboratory medium. Fit parameters: As ¼ 13.14, l ¼ 4.37 h, mmax ¼ 0.70 h1. Dataset is given in Appendix 12.1, Table A.12.2.

underestimation of the growth rate. Another disadvantage of the modiﬁed Gompertz equation is that at t ¼ 0, Y ¼ ln N=N0 6¼ 0; only for t ! 1 Y ! 0. Integrated Baranyi model. Integration of the differential equations in the Baranyi model given above at constant conditions of T, pH, aw, etc., leads to 0 ln

1

B N exp (mmax AB (t)Þ 1C C ¼ mmax AB (t) lnB A @1 þ Nmax N0 exp ln N0

(12:18)

in which AB (t) ¼ t þ

1 mmax

lnðexp (mmax t) þ exp (h0 exp (mmax t h0 )Þ

(12:19)

with h0 given in Equation 12.10. Equation 12.18 thus describes isothermal growth at constant conditions, like the earlier modiﬁed logistic and modiﬁed Gompertz equations, with parameters mmax, Nmax, N0, and h0 that can be estimated from growth curves. Figure 12.7 gives an example of the ﬁt to experimental data. The above discussed models were tested on a dataset in a model discrimination study. The dataset concerned the growth of Clostridium perfringens in ground beef. Table 12.2 shows the parameter estimates, Table 12.3 the results of the model discrimination study, and Figure 12.8 the resulting ﬁts. As explained in Chapter 7, the goodness of ﬁt (which can only be done if replicates are available) indicates the sampling probability of obtaining a greater ratio than the one obtained if the null hypothesis is true that the model performs well. If this probability is high, say higher than 0.01, there is no statistical reason to reject the model because of lack of ﬁt. The model with the lowest Akaike criterion and the highest posterior probability performs the best in the statistical sense. Though slight differences can be seen in the various ﬁts in Figure 12.8, all models seem to describe the data points quite well. This is conﬁrmed by the goodness-of-ﬁt result in Table 12.3; there is no serious lack of ﬁt with any of the models: all sampling probabilities are substantially higher than 0.01. Nevertheless, it is also clear that the goodness of ﬁt differs between the models and this goes along with the two

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14 12

lnN/N0

10 8 6 4 2 0

0

5

10 15 Time (h)

20

25

FIGURE 12.7 Fit of the Baranyi model (—) to growth of C. perfringens at 308C in boneless ham with ﬁt parameters mmax ¼ 1.15 h1, As ¼ 12.93, h0 ¼ 6.19. Dataset in Appendix 12.1, Table A.12.3.

TABLE 12.2 Parameter Estimates 95% Conﬁdence Intervals for the Models Tested on the Growth of C. perfringens in Ground Beef at 308C Modiﬁed Gompertz Model

Shifted Logistic Model

mmax ¼ 2.0 0.3 h1

k ¼ 0.5 0.1 h1

As ¼ 15.3 0.6 l ¼ 3.8 0.7 h

Modiﬁed Logistic Model

Baranyi–Roberts Model

mmax ¼ 2.0 0.4 h1

mmax ¼ 1.6 0.3 h1

As ¼ 15.0 0.6

As ¼ 14.8 0.6

l ¼ 4.1 0.8 h

h0 ¼ 5.3 2.1

YAs ¼ 15.3 0.6 tc ¼ 7.7 0.4 h

l ¼ 3.2 0.8 ha a

Calculated from h0 ¼ mmax 3 l.

TABLE 12.3

Results of Model Discrimination Test (See Table 12.2 and Figure 12.8) Modiﬁed Gompertz Model

Shifted Logistic Model

Modiﬁed Logistic Model

Baranyi–Roberts Model

AIC

3.01

2.89

4.3

7.1

DAIC

0

5.9

7.31

10.11

log (PB) Goodness of ﬁt

12.6 0.526

13.3 0.241

13.5 0.193

13.8 0.121

Note: AIC, Akaike criterion; log(PB), logarithm of the posterior probability. Goodness of ﬁt: Sampling probability of greater ratio. Results are presented from best to worst performing model.

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18 16 14

ln N/ N0

12 10 8 6 4 2 0 −2

0

10

20 Time (h)

30

40

FIGURE 12.8 Fit of the shifted-logistic (), modiﬁed logistic (-.-.-), modiﬁed Gompertz (. . . . ), and Baranyi– Roberts model (- - - -) to a dataset describing growth of C. perfringens in beef at 308C. Dataset in Appendix 12.1, Table A.12.4.

model discrimination tests applied, the Akaike criterion and the posterior probability criterion, both discussed in Chapter 7. The differences are not negligible and preference is given to the modiﬁed Gompertz model, while the Baranyi–Roberts model seems to perform to a lesser extent, at least from a statistical point of view. Of course, these results cannot be generalized to a general statement about the performances of these models. The results are only valid for this particular dataset. Many more datasets need to be investigated to come to a general conclusion. This exercise is only done here to show the possibilities of model discrimination and the application of techniques discussed earlier in the book to microbial growth models. As shown in Chapter 7, it is quite instructive to study the statistical characteristics of parameter estimates via bootstrapping and Monte Carlo simulations. This is certainly true for parameter estimates for microbial growth models as these are based on variable and possibly heteroscedastic data. Since the methods are exactly the same as given in Chapter 7 we will not repeat that exercise here.

12.3

Secondary Models

It is of course desirable to be able to predict what will happen if, for instance, temperature is changed. Figure 12.9 shows an example of the effect of temperature on the growth curves of C. perfringens on cooked boneless ham. It can be seen that the upper asymptote is hardly affected by temperature, but both the maximum growth rate and the lag time are. By doing measurements at various temperatures, as shown in Figure 12.9, one can get an impression of the temperature dependence of microbial growth. However, it is of course much easier to cast this dependence in a mathematical equation, just as the temperature dependence of a chemical reaction rate constant can be cast in the form of the Arrhenius equation or the Eyring equation, discussed in Chapter 5. However, as mentioned before, the temperature dependence of microbial growth cannot be described by the Arrhenius or Eyring equation because microbial growth is not a simple elementary reaction. If an Arrhenius plot of microbial growth is made anyway, it is frequently nonlinear. Therefore, empirical models are used.

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16 14 12

ln N/N0

10 8 6 4 2 0 −2

0

5

10 15 Time (h)

20

25

FIGURE 12.9 Growth curves of C. perfringens in cooked boneless ham at 308C (&), 358C (.), and 458C (~). The ﬁts shown are the ﬁts by the Baranyi–Roberts model. Dataset in Appendix 12.1, Table A.12.5.

These secondary models, as they are called, describe the dependence of m, l, and As on environmental conditions (temperature, pH, etc.). For the effect of temperature on mmax, for instance, an empirical square root relationship is proposed (also called the Ratkowsky square root model): mmax ¼ [b1 (T Tmin )]2

(12:20)

Tmin is the minimum temperature at which growth is observed, and b1 is a ﬁt parameter (dimension 8C time1). Equation 12.20 is based on the observation that the square root of the speciﬁc growth rate varies linearly with temperature at lower temperatures. The equation only uses temperature differences so it does not matter whether the units are in K or in 8C. Figure 12.10 shows an application of this model to data obtained for growth of C. perfringens in ham and Table 12.4 shows the parameter estimates. The ﬁt as shown in Figure 12.10 and the goodness-of-ﬁt measure are excellent as well as the precision of the parameter estimates. It should be remembered though that these parameters can only be used for the range on which they are based. Especially extrapolation to higher temperatures would be very dangerous as the speciﬁc growth rate will drastically decrease at some temperature. At higher temperatures, i.e., around and above the optimum and maximum temperature, the expanded Ratkowsky model is more appropriate: mmax ¼ fb2 ðT Tmin Þð1 exp½c2 (T Tmax )Þg2

(12:21)

with Tmin, Tmax, the minimum and maximum temperature for growth (in K or 8C), respectively, and b2 and c2 ﬁt parameters. A modiﬁcation of the expanded square root model, now known as the Zwietering equation, was proposed to prevent positive values of mmax at T > Tmax: mmax ¼ [b3 (T Tmin )]2 [1 expðc3 (T Tmax )Þ]

(12:22)

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5 4.5 4

m max /h−1

3.5 3 2.5 2 1.5 1 0.5 0 20

25

30

35

40

45

50

T (C)

FIGURE 12.10 Fit of the Ratkowsky model (Equation 12.20) to a dataset describing the maximum speciﬁc growth rate of C. perfringens as a function of temperature in ham. Dataset in Appendix 12.1, Table A.12.6.

It should be noted that these equations are only valid inside the interval TminTmax, in other words, only positive growth rates can be used in the estimation procedure. Figure 12.11 gives an example of the ﬁt of the Zwietering Equation 12.22 and the Ratkowsky model to the maximum speciﬁc growth rate of Listeria monocytogenes in milk, while Table 12.5 shows the parameter estimates. The ﬁt for the Ratkowsky equation is not all that well as can be seen in Figure 12.11, and it also follows from the goodness-of-ﬁt measure. The reason is obvious: the growth rate starts to decline at the higher temperatures, which cannot be taken into account in the Ratkowsky equation. The ﬁt for the Zwietering model seems to be OK, and this is substantiated by the goodness-of-ﬁt measure. There is a problem, however, with the estimate of parameter c3 as its conﬁdence interval is larger than the estimate itself, and the conﬁdence interval of Tmax is also quite high. It is quite clear that there is not enough information in the data to estimate Tmax properly, or put differently: the experimental design was not tailored to the use of this model to estimate Tmax. Just another example is given in Figure 12.12 and Table 12.6. In this case, there is no lack of ﬁt, as in the previous example with the Zwietering model, but the parameter estimates are much more precise, showing the beneﬁt of a well-designed experiment, if the purpose was to estimate the growth rate also above the optimum temperature. For the asymptotic value As the following equation could be used: As ¼ b4 [1 expðc4 (T Tmax )Þ]

(12:23)

TABLE 12.4 Parameter Estimates 95% Conﬁdence Intervals and Goodness of Fit of the Ratkowsky Equation 12.20 to the Data Shown in Figure 12.10

Estimate

b1 (8C h1)

Tmin (8C)

Goodness of Fit

0.020 0.007

10.6 2.1

0.991

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0.4 0.35 0.3

m max/h−1

0.25 0.2 0.15 0.1 0.05 0

0

5

10

15

20 25 T (C)

30

35

40

FIGURE 12.11 Fit of the Ratkowsky model (- - -, Equation 12.20) and the Zwietering model (—, Equation 12.22) to experimental data obtained for the maximum speciﬁc growth rate of L. monocytogenes in milk as a function of temperature. Dataset in Appendix 12.1, Table A.12.7. TABLE 12.5 Parameter Estimates 95% Conﬁdence Intervals and Goodness of Fit of the Ratkowsky Equation 12.20 and the Zwietering Equation 12.22 to the Data Shown in Figure 12.11 b1=b3 (8C h1) b1 ¼ 0.060 0.004 b3 ¼ 0.020 0.007

c3 (8C1)

Tmin (8C)

—

4.9 3.2

0.1 0.2

04

Tmax (K)

Goodness of Fit

—

0.037

41 14

0.57

1.4 1.2

m max/h−1

1 0.8 0.6 0.4 0.2 0

0

10

20

30

40

50

T (C)

FIGURE 12.12 Speciﬁc growth rate mmax as a function of temperature for growth of Lactobacillus plantarum in a MRS medium modeled according to Equation 12.22. Dataset in Appendix 12.1, Table A.12.8.

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TABLE 12.6 Parameter Estimates 95% Conﬁdence Intervals and Goodness of Fit of the Zwietering Model (Equation 12.22) Applied to the Speciﬁc Growth Rate Parameter mmax as Function of Temperature for Growth of Lactobacillus plantarum in a MRS Medium

Estimate

b3

c3

Tmin (8C)

Tmax (8C)

Goodness of Fit

0.041 0.007

0.16 0.06

43

44 0.3

0.352

However, usually the parameter As is not strongly temperature dependent (as can be seen in Figure 12.9 for instance) and may be considered approximately constant in many cases. The lag time is temperature dependent and several expressions are derived as secondary models. One example is: l ¼ exp

p1 0 T q1

(12:24)

which is a hyperbola model, with T 0 in 8C. Another one is the extended hyperbola model: l¼

p2 T 0 q2

m (12:25)

As an example of a secondary model for the lag time, Equations 12.24 and 12.25 were ﬁtted to experimental data describing lag time as a function of temperature for the growth of Salmonella typhimurium on chicken meat. The results are in Figure 12.13 and Table 12.7.

50 45 40 35

l (h)

30 25 20 15 10 5 0

0

10

20

30

40

50

T (C)

FIGURE 12.13 Fit of a secondary model for the lag time (Equation 12.24) as a function of temperature for the growth of S. typhimurium on chicken meat. Dataset in Appendix 12.1, Table A.12.9.

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TABLE 12.7 Parameter Estimates 95% Conﬁdence Intervals and Goodness of Fit for a Secondary Model for the Lag Time (Equation 12.24)

Estimate

p1 (8C)

q1 (8C)

Goodness of Fit

28.9 2.1

0.43 0.56

0.336

The goodness of ﬁt seems to be in order, and the imprecision for parameter estimate p1 is acceptable, but the one for q1 is much too high. It can be seen from the ﬁt in Figure 12.13 that the residuals are not well behaved at T 0 > 208C. A further statistical analysis showed that the data were not normally distributed: see Figure 12.14 for a normal probability plot (Chapter 7). A log transformation removed this problem but the ﬁt of the model actually got worse (not shown), and the goodness-of-ﬁt parameter was well below 0.0001. As a next attempt the extended hyperbola model in Equation 12.25 was modeled. Again, a log transformation was needed to stabilize the variance, and the resulting ﬁt and probability plot is shown in Figures 12.15 and 12.16, respectively, while the parameter estimates are shown in Table 12.8. The normal probability plot of the transformed data looks much better though there is still a hint of nonnormality (right skewed, see Figure 7.16). The ﬁt and the residuals are well behaved and the parameter estimates shown in Table 12.8 are also satisfactory, even though the extended hyperbola lag time model has one parameter more than the hyperbola equation (as a reminder: more parameters give a better ﬁt but less precise parameter estimates). This chapter shows that it is worthwhile to use residuals and normal probability plots as diagnostics, and to take action if the diagnosis is that something is not right. Schoolﬁeld model. An attempt has been made to model the temperature dependence of mmax based on mechanistic reasoning, the Schoolﬁeld model. The reasoning is that only one enzyme reaction is rate

3

2

Residuals

1

0

−1

−2

−3 −3

−2

−1

0 z-score

1

2

3

FIGURE 12.14 Normal probability plot for the data and the model shown in Figure 12.13.

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4.5 4 3.5 3

ln l

2.5 2 1.5 1 0.5 0 −0.5

0

10

20

T (C)

30

40

50

FIGURE 12.15 Fit of the extended hyperbola model (Equation 12.25) to the logarithm of lag time for growth of Salmonella on chicken. Same data as in Figure 12.13 but logarithmically transformed.

0.6 0.5 0.4

Residuals

0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −2.5

−1.5

0.5 −0.5 z-score

1.5

2.5

FIGURE 12.16 Normal probability plot for the data and the model shown in Figure 12.15.

TABLE 12.8 Parameter Estimates 95% Conﬁdence Intervals and Goodness of Fit for a Secondary Model for the Lag Time (Equation 12.23)

Estimate

p2

q2

m

Goodness of Fit

28.9 2.1

5.6 0.8

1.3 0.1

0.336

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controlling, following a zero-order reaction, and that this enzyme is reversibly denatured at very high and very low temperatures. The total amount of rate-controlling enzyme per cell is considered constant. Furthermore, the temperature dependence of the enzyme reaction as well as the inactivation reaction is supposed to follow an Arrhenius dependency (Equation 5.12). This results in the so-called Schoolﬁeld equation:

mmax

Ea A exp RT ¼ Eal Eah 1 þ Al exp þ Ah exp RT RT

(12:26)

A is the pre-exponential factor Ea the activation energy for the enzyme reaction, Al, Ah Eal, Eah, the pre-exponential factor and activation energy for the inactivation at low and high temperature, respectively As discussed in Chapter 7, this equation should be reparameterized because of the strong correlation between the parameters. Although this is not a problem as such, the problem with the Schoolﬁeld equation is that it contains many (six) parameters, and that the mechanistic basis is doubtful: it is not very likely that only one enzyme is rate controlling. This is reﬂected in the fact that the values derived for the parameters do in quite some cases not reﬂect biologically meaningful values. The Ratkowsky and Zwietering equations, although of an empirical nature, seem to be sounder from a modeling point of view. Secondary models for empirical parameters. As introduced above, some empirical models do not use the established microbiological parameters m, l, and As. For instance, in the shifted logistic Equation 12.16, the temperature dependence of the parameters YAs, k, and tc can be expressed as YAs (T) ¼ constant

(12:27)

k(T) ¼ k0 exp (ck T 0 )

(12:28)

tc (T) ¼ tc0 expðctc T0 Þ

(12:29)

These are purely empirical models that nevertheless can give a very good ﬁt, but they should, as with all empirical models, not be used outside the range for which they were derived. Figure 12.17 gives an example, obtained from ﬁtting the shifted logistic Equation 12.13 to growth curves of Pseudomonas species in ﬁsh. The resulting equations are just convenient ﬁts to the data obtained, without mechanistic meaning. Gamma concept and cardinal models. The so-called gamma concept relates so-called cardinal parameters (pH, T, aw) to each other on the assumption that they are not a function of the other parameters, so that they can be multiplied: gtotal ¼ g(T) g(aw ) g(pH)

(12:30)

The speciﬁc growth rate is related to temperature, pH, and water activity as follows: m ¼ mopt g(T)g(pH)g(aw )

(12:31)

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7

0.05 0.045

6

0.04 5

0.035 0.03

YA

s

k (h−1)

4 3

0.025 0.02 0.015

2

0.01 1

0.005

0 0 (A)

2

4 6 T (C)

8

0

10

0

2

4 6 T (C)

6

8

10

(B)

8

10

180 160 140 120 t c (h)

100 80 60 40 20 0

0

2

4

T (C)

(C)

FIGURE 12.17 Fits of empirical models to express temperature dependence of parameters of the shifted logistic Equation 12.13 to growth curves of Pseudomonas spp. on ﬁsh. YAs ¼ constant ¼ 5.86 (A), k ¼ 0.017 exp(0.125T 0 ) (B), tc ¼ 159.2exp(0.17T 0 ) (C). Dataset in Appendix 12.1, Table A.12.10.

in which g(T) ¼ g(pH) ¼

T Tmin Topt Tmin

2 (12:32)

(pH pHmin )(pHmax pH) (pHopt pHmin )(pHmax pHopt )

(12:33)

aw aw, min 1 aw, min

(12:34)

g(aw ) ¼

In this way, a correction is applied to the optimal growth rate if the pH, aw, or temperature are not optimal.

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Cardinal parameter models (CPMs) are also empirical secondary models and also assume that the inhibitory effect of factors such as temperature, pH, and water activity are multiplicative. CPM consist of a term (CMn) for each factor, while each term has a numerical value between 0 and 1 and acts as correction term to the optimal speciﬁc growth rate. Thus, when a term is unity, the growth rate is optimal for that factor. For optimal growth, all terms have to be unity. For CMn terms describing the effect of temperature, pH, water activity, a cardinal model would read: mmax ¼ mopt CM2 (T) CM1 (pH) CM2 (aw )

(12:35)

This equation can be expanded, if so desired, for any term CMn(X) that has an effect on the growth rate. Each term of CMn(X) is expressed as for X Xmin : CMn (X) ¼ 0 for Xmin < X < Xmax : CMn (X) ¼

(X Xmax )(X Xmin )n (Xopt Xmin )n1 (Xopt Xmin )(X Xopt ) (Xopt Xmax )((n 1)Xopt þ Xmin nX)

for X Xmax : CMn (X) ¼ 0

(12:36)

CPMs perform much like square root models discussed above. There is an extended literature on CPMs that will not be discussed here. Some references are given at the end of this chapter.

12.4

Nonisothermal Growth Modeling

The integrated modiﬁed logistic models, the modiﬁed Gompertz model, and the integrated Baranyi model are (semi)empirical models that are so-called time-explicit models because growth is an explicit function of time. This means that there are potential problems in describing time-varying changes in environmental conditions (notably temperature). Also, the modiﬁed gomperty model does not give fully correct values at t < l, as can be seen in Figure 12.6, as well as can be analytically derived from Equation 12.16. The problem becomes apparent when there is a sudden jump in temperature: the model also predicts a sudden jump in number of microorganisms (Figure 12.18). This is of course impossible: although growth rate will change, it cannot affect the number of microorganisms immediately. For this reason, there is a need for so-called dynamic growth models. Models that consist of differential equations such as the modiﬁed logistic model (Equation 12.4) and the Baranyi model, Equations 12.7 and 12.8 (which is actually based upon the logistic model) are capable to predict growth under dynamic conditions, such as varying temperatures. What is needed for this? First, the temperature proﬁle needs to be known, so that the temperature can be expressed as a function of time. Second, the parameters in the dynamic model must be made a function of temperature, and since temperature has been made a function of time, the parameters can be made time dependent also. As a next step, the differential equations can be integrated where the parameters are made time dependent as indicated. Using the Baranyi model for instance, the parameter mmax can be made temperature dependent using the square root model or the Zwietering model and using the known temperature–time dependency, mmax can be calculated at any time. Another way to describe population growth under varying temperature is due to Peleg and coworkers; they propose a method to estimate nonisothermal growth from isothermal growth curves. This is basically the same approach as discussed in Chapter 5 for the empirical modeling of chemical changes with variable temperature kinetics. Let us take the shifted logistic Equation 12.13 as an example. The momentary growth rate at constant temperature can be found by differentiating Equation 12.13 with respect to time:

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Kinetics of Microbial Growth

ln (N/N0)

12-21

T1 T0

Sudden temperature jump Time

FIGURE 12.18 Simulation showing the effect of a temperature jump (1.7 increase in mmax) on the Gompertz model (—) as compared to the growth curve at constant temperature (---).

dY(t) kYAs expðk(tc t)Þ ¼ dt T ð1 þ expðk(tc t ÞÞ2

(12:37)

This equation is valid at any temperature as long as this temperature is constant, but the values of the parameters change with temperature, so to make Equation 12.37 valid for every temperature we have to write: dY(t) k(T)YAs (T) expðk(T)(tc (T) t)Þ ¼ dt ð1 þ expðk(T)(tc (T) t ÞÞ2

(12:38)

As before, we need to make temperature a function of time from the known time–temperature relationship, and we need an expression that describes the dependence of the parameters as a function of temperature; for this we could use Equations 12.27 through 12.29. To relate time to the corresponding number of cells, we can reverse the original model (that describes the number of cells as a function of time) and calculate what Y(t) is at a certain time t* for any constant temperature. For the shifted logistic model this results in 1 expðk(T)tc (T)ÞðYAs (T) þ Y(t)ð1 þ expðk(T)tc (T)ÞÞÞ ln t* ¼ k(T) YAs (T) expðk(T)tc (T)Þ Y(t)ð1 þ expðk(T)tc (T)ÞÞ

(12:39)

So, we can now calculate the time that corresponds to the number of cells at a certain temperature. By making temperature a function of time and by combining Equations 12.38 and 12.39 we obtain the nonisothermal rate equation: dY(t) kðT(t)ÞYAs ðT(t)Þ expðkðT(t)Þðtc (T(t)) t*ÞÞ ¼ dt ð1 þ expðk(T(t))(tc (T(t)) t*)ÞÞ2

(12:40)

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Even though Equation 12.40 looks very awkward, it is a straightforward ordinary differential equation that can be solved numerically using appropriate software, but this can be done even in spreadsheets: see footnote on p. 12–3. It is basically the same approach as was discussed in Chapter 5 for variable temperature kinetics. The difference between Equations 12.40 and 12.37 is that Equation 12.37 is valid for any constant temperature, while Equation 12.40 is valid for a temperature that changes with time, i.e., nonisothermal conditions. Figure 12.19 and 12.20 gives an example using this approach. It concerns the growth of C. perfringens in ground meat during cooling after a heating process. Growth starts at temperatures below 458C. From isothermal experiments, the temperature dependence of the parameters of the shifted logistic equation was determined. The parameter YAs was virtually independent of temperature and was therefore held constant at a value of 5.8. The parameters k and tc were modeled with ad hoc empirical models (Figure 12.19). The results shown in Figure 12.20 are seen to give an excellent prediction of the nonisothermal dynamic growth; note that these are not model ﬁts but real predictions (i.e., not based on the same experiments for which the parameters were derived). Incidentally, this exercise can be done with any other equation than the shifted logistic equation, of course. The ﬁeld of predictive microbiology is rapidly developing, and it is not possible, also not very useful, to list all proposed equations. The ones that were given are reasonably well established, and most newly presented models are extensions or modiﬁcations of these.

12.5

Bayesian Modeling

1.6

16

1.4

14

1.2

12

1

10

0.8

8

0.6

6

0.4

4

0.2

2

0 20 (A)

tc (T ) (h)

k (T ) (h−1)

The Bayesian approach introduced in Chapter 7 can, of course, also be applied to microbial growth. As a ﬁrst example, we show a growth curve modeled using Markov Chain Monte Carlo (MCMC) integration as is possible with the program WINBUGS (introduced in Chapter 7). Figure 12.21 shows the code with the priors used. Figures 12.22, 12.23 and 12.24 show the results for ﬁtting the Gompertz model to growth data of Salmonellae while Table 12.9 shows the estimates for the parameters; the same data were used in Figure 12.6. Using the Bayesian approach just for simple growth models would not be a big advantage. However, the real advantage of this approach is that it allows tackling uncertainty and variability, which are

25

30

35 40 T (C)

45

0 20

50

(B)

25

30

35 T (C)

40

45

50

FIGURE 12.19 Empirical models describing temperature dependence of the shifted logistic equation parameters k(T) ¼ 0.27 þ 0.000855T2 (A) and tc ¼ 158 exp(0.09T) (B) for the case of isothermal growth of C. perfringens in ground ham. Dataset in Appendix 12.1, Table A.12.11.

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50

6

50

45

45 1

40 35

25 20

0.4

T (C)

30 0.6

35 30

3

25 20

2

15 10

0.2

40

4 log N/N0

0.8 log N/N0

5

15 10

1

5

5

0

0 0 (A)

1

2 3 Time (h)

4

T (C)

1.2

0 0

5 (B)

5

10

0 15

Time (h)

FIGURE 12.20 Fit of the shifted logistic model (Equation 12.40) to growth of C. perfringens in ground ham during cooling for two different cooling patterns. Dataset in Appendix 12.1, Table A.12.12.

important in predictive microbiology. Uncertainty refers to lack of knowledge on the values of model parameters (such as mmax), variability is due to biological variation in the behavior of the microorganism under study as well as in the food. The models discussed hitherto do not take into account uncertainty and variability explicitly. Bayesian hierarchical models on the other hand can take uncertainty and variability into account. The Bayesian approach was explained in Chapter 7, as well as a small introduction to hierarchical Bayesian models. Parameters in the Bayesian framework are considered as random

#WINBUGSmodel for bacterial growth model{ for (j in 1:N){ #likelihood Y[j]~dnorm(mu[j],tau); mu[j] 0, and the cumulative form is bW ! t F(t) ¼ exp aW

(13:7)

with aW and bW the two parameters of the distribution; aW is a scale parameter (a characteristic time) and bW is the so-called shape parameter for reasons that will become obvious below. In terms of a survival ratio S(t) ¼ N(t)=N0, the cumulative function is then

t S(t) ¼ exp aW

bW ! (13:8)

In microbiology, the logarithm of the survival ratio is commonly used: log S(t) ¼

bW 1 t 2:303 aW

(13:9)

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1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

S (t)

Frequency

13-6

0.5

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0 0 (A)

0.5

0.4

1 2 3 Time (arbitrary units)

4

0

1 2 3 Time (arbitrary units)

(B)

4

0 −0.5 −1

log S

−1.5 −2

−2.5 −3 −3.5 −4 (C)

0

1 2 Time (arbitrary units)

3

FIGURE 13.4 Simulations describing microbial inactivation using the Weibull distribution: density function (A), cumulative function (B), and a semilog plot (C) for aW ¼ 1 and bW ¼ 2.

The Weibull distribution corresponds to a concave upward survival curve if bW < 1 and concave downward if bW > 1. Figure 13.4 shows an example of the Weibull distribution. Interestingly, the Weibull distribution reduces to an exponential distribution for bW ¼ 1 and the cumulative distribution looks the same as the familiar ﬁrst-order equation: t S(t) ¼ exp aW

(13:10)

Although this exponential distribution has the same form as the ﬁrst-order Equation 13.3, the meaning of the parameters D and aW is different. D is the reciprocal of a ﬁrst-order rate constant, whereas aW represents the mean of the distribution describing the death times tc of the microbial population. It is worthwhile to explore the parameter aW a little bit further. At t ¼ aW, it represents a characteristic time at which the survival function log S(t) ¼ 0.434 (corresponding to S(t) ¼ exp(1)), regardless of the value of parameter bW. This becomes clear from Figure 13.5, where several theoretical cumulative functions are depicted. The parameter aW is called the hazard rate in reliability engineering. It is constant in the case of bW ¼ 1, increasing with time if bW < 1 and decreasing when bW > 1.

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13-7

0 −0.2 −0.4 −0.6 −0.8 log S

b W = 0.5

Characteristic time a W

b W = 1.0

−1 −1.2

b W = 2.0

−1.4 −1.6 −1.8 −2

0

0.5

1 1.5 2 Time ⫻ aW (arbitrary units)

2.5

3

FIGURE 13.5 Simulation describing microbial inactivation using the Weibull distribution for the shape parameter bW < 1, bW ¼ 1 and bW >1 and aW ¼ 1.

For the time needed to reduce the number of microorganisms by a factor 10 (analogous to the D-value) one could use the concept of reliable life tR used in reliability engineering, which is in fact the 90% percentile of the failure time distribution (Equation 13.11): tR ¼ aW (ln 0:1)1=bW ¼ aW (2:303)1=bW

(13:11)

These times are indicated by dashed lines in Figure 13.6 for various values of bW, and it shows its large dependence on bW. tR could play the same role as the parameter D, even though it has a different meaning. One could generalize Equation 13.11 into td ¼ aW ln (10d )1=bW

(13:12)

in which d is the number of decimal reductions. Thus for a 12D reduction, as is commonly used in sterilization td ¼ aW (25:328)1=bW . This equation shows that we need values for parameters aW and bW in order to do process calculations. These values need to be derived from experimental measurements, so this is what we have to discuss next. The Weibullian model. If we now move to practical applications, there is a complication with the numerical values of the Weibull parameters aW and bW if they are obtained by regression from experimental results. The problem lies in the scatter in experimental data and the log transformation commonly applied that changes the error distribution (Chapter 7). The parameters obtained experimentally from a plot of log S(t) depicted by Equation 13.10 may differ from the original parameters derived in Equation 13.9 when the untransformed survival ratio S(t) is used for estimation. This effect depends on the experimental errors (to be sure, in the absence of experimental errors there would be no difference). Therefore, it is better to introduce a ‘‘Weibullian’’ alternative, basically a power-law relation:

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Kinetic Modeling of Reactions in Foods

13-8 tR

0

tR

tR

−0.2 −0.4 −0.6

b W = 0.5

log S

−0.8 −1 −1.2

b W = 1.0

b W = 2.0

−1.4 −1.6 −1.8 −2

FIGURE 13.6

0

1

2 3 4 Time ⫻ aW (arbitrary units)

5

6

Simulated survival curves showing the effect of the Weibull parameters on the reliable life tR.

log S(t) ¼ b t n

(13:13)

Mathematically, this equation is similar to Equation 13.9 with b ¼ 1=2.303 3 (1=aW)b and n ¼ bW, but it serves to make a distinction in the parameters. The Weibullian model is quite ﬂexible and is capable to ﬁt microbial survival curves quite well. Figures 13.7 through 13.9 give some examples. It should be understood that the Weibull model is an empirical one in the sense that no attempts are made to link to mechanistic theories about microbial death. The only thing that is done is to give a statistical account of a failure time distribution. However, according to the author this is rather an 0 −0.5 −1

log S

−1.5 −2 −2.5 −3 −3.5 −4 −4.5

0

1

2

3

4 5 Time (min)

6

7

8

FIGURE 13.7 Fit of the Weibullian model to inactivation of Salmonella typhimurium with parameters b ¼ 0.037 and n ¼ 2.4. Dataset in Appendix 13.1, Table A.13.2.

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Kinetics of Inactivation of Microorganisms

13-9

0 −0.5 −1

log S

−1.5 −2

−2.5 −3 −3.5 −4

0

1

2 Time (min)

3

4

FIGURE 13.8 Fit of the Weibullian model to inactivation of Salmonella enteriditis in egg yolk with parameters b ¼ 2.6 and n ¼ 0.3. Dataset in Appendix 13.1, Table A.13.3. 0 −0.2 −0.4 −0.6 log S

−0.8 −1 −1.2 −1.4 −1.6 −1.8

0

10

20 Time (min)

30

40

FIGURE 13.9 Fit of the Weibullian model to inactivation of Saccharomyces cerevisiae with parameters b ¼ 0.046 and n ¼ 1.0. Dataset in Appendix 13.1, Table A.13.4.

advantage, because most likely there is not a single cause of death. Nevertheless, a link can be made with microbial inactivation as follows. If n < 1, it means that the remaining cells have less probability of dying, indicating that the remaining cells are the sturdy ones, or perhaps adapting to the stress. With n > 1 it means that the remaining cells become increasingly susceptible to heat; in other words this indicates that there is cumulative damage occurring making it increasingly difﬁcult for the cells to survive. With n ¼ 1 it means that the probability of dying does not depend on time, in other words each cell is equally susceptible no matter how long the treatment lasts. The implicit assumption of ﬁrst-order kinetics, when interpreted in a stochastic sense, is thus that there is no biological variation of the cells in response to stress, in other words, k or D do not depend on time. This seems unrealistic and it would explain why so few linear survival curves are actually found.

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In the language of reliability engineering, regression models take into account how the two parameters aW and bW depend on other conditions (covariates). Relevant for microorganisms are pH, temperature, water activity, pressure, ionic strength, etc. Experiments can be devised to determine the dependency of the Weibullian parameters on such conditions. It is usually assumed in reliability engineering that the shape parameter bW is independent of external conditions. We will discuss the effect of temperature below, but ﬁrst we will take a look at the behavior of spores. The conclusion is that the Weibullian model is very suitable to model nonlinear survival curves, and may be helpful to pinpoint relevant physiological effects caused by heating. As remarked before, there are certainly other distributions applicable than the Weibull model. The choice made here for the Weibull model is because it is a simple as well as a ﬂexible model that has proven to be useful in many applications.

13.3

Kinetics of Inactivation of Spores

Section 13.2 dealt with vegetative cells. It is also necessary to consider the inactivation of spores because they are known to be quite resistant (which is the very reason for the organism to form them), and because of that they can be a real nuisance if they are present in raw materials and foods. First-order inactivation kinetics. Like with vegetative cells, inactivation of spores is usually analyzed as if it were a ﬁrst-order reaction, even though more often than not a nonlinear survival curve is observed (see for instance Figure 13.10). One usually neglects the nonlinear part and takes the slope of the linear part as a measure for the rate constant (mostly expressed as D-values). Sometimes even an initial increase in numbers of spores is found when heating spore suspensions or foods containing spores, for instance, as shown in Figure 13.11. There is a perfectly logical explanation for this phenomenon, namely the presence of dormant spores that are not detected with plate counts. However, they may become recoverable by a heat shock and, if this happens, more spores are counted than in the case where no heat shock was given (i.e., at t ¼ 0). One cause for recovery is heat, other causes are high pressure, radiation, oxidizing and reducing agents, sonication. This recovery is usually called activation in the literature. It should be understood that dormant spores do not grow during the 0

0

−0.5

−1

−1 −2 log S (t)

log S (t)

−1.5 −2

−2.5

−3 −4

−3 −5

−3.5 −4 (A)

0

−6 100 200 300 400 500 600 700 0 (B) t (s)

1

2

3 t (min)

4

5

6

FIGURE 13.10 Inactivation of spores of B. licheniformis in concentrated milk ﬁtted with the Weibullian model, n ¼ 1.5, b ¼ 0.00021 (A) and of spores of C. botulinum ﬁtted with the Weibullian model, n ¼ 0.3, b ¼ 3.3 (B). Dataset in Appendix 13.1, Table A.13.5.

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Kinetics of Inactivation of Microorganisms

13-11

22 20

ln Sa/Sa0

18 16 14 12 10 8

0

100

200 300 Time (min)

400

500

FIGURE 13.11 Semilogarithmic inactivation plot of a suspension of spores of B. stearothermophilus heated at 1058C. The different symbols refer to replicate experiments. Dataset in Appendix 13.1, Table A.13.6.

experiment, they are present but not detected at t ¼ 0. With Sd the number of dormant spores, Sa the number of active spores, and Si the number of active spores, the simplest kinetic scheme is then in the framework of ﬁrst-order kinetics, ﬁrst proposed by Shull: Sd

kact

Sa

kinact

Si

SCHEME 13.1 Shull model: First-order kinetics scheme to describe inactivation of spores from dormant spores via activated to inactivated spores. Sd: number of dormant spores, Sa: number of active spores, Si: number of inactivated spores.

This is actually in the form of the consecutive reaction scheme A ! B ! C, as discussed in Chapter 4. The rate constant kact accounts for the rate of activation of dormant spores and kinact for the rate of inactivation of activated spores. This sequential kinetic model assumes that dormant spores are not inactivated; they ﬁrst have to be activated. It is not easy to determine the number of dormant spores because they do not grow on nutrient-agar plates. The difference between the total number of spores, determined by other methods of counting (microscopy, for instance), and active spores (those that grow on a nutrient-agar plate) gives the number of dormant spores, on the assumption that all counted particles are spores, that all spores are viable (i.e., no dead spores), and that there are no vegetative cells present (they would be counted as spores). In normal practice one therefore only determines the number of spores that grow on a plate, i.e., the number of activated spores. The fact that heat causes recovery (activation) complicates the subsequent inactivation process. For that reason, in laboratory situations, spores are sometimes ﬁrst activated by giving them a sublethal heat shock and then, after a while, they are subjected to a further heat treatment to determine inactivation. There are two objections to such a procedure. First, this does not correspond to the practice of heat treatment of food (no heat shock is given, and a considerable number of spores is supposed to be in a dormant state), and second, the time allowed between the heat shock and the further heat treatment may be crucial: it is not really known how long it takes for a spore suspension to completely transform from the dormant state into an activated

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state. So, it would be best to study inactivation kinetics using the spores as they would be present in the food, i.e., a mixture of dormant and active spores. Let us take Scheme 13.1 as a starting point of kinetic analysis, assuming that we can only determine the number of active spores Sa. The resulting equation is (Equation 4.48) Sa ¼ Sa0 exp (kinact t) þ Sd0

kact ðexp (kact t) exp (kinact t)Þ kinact kact

(13:14)

in which Sd0 is the initial number of dormant spores and Sa0 the initial number of active spores. Normalizing the number of spores we ﬁnd: Sa Sd0 kact ¼ exp (kinact t) þ ðexp (kact t) exp (kinact t)Þ Sa0 Sa0 kinact kact

(13:15)

and it is seen that the decrease in number of active spores is determined by the initial ratio of dormant to active spores and the two rate constants. Some simulations using this equation give interesting results (see Figure 13.12 in which the inactivation rate constant is varied and Figure 13.13, in which the ratio of

0.5

0.2

0.4

0.1 0 −0.1 ln Sa/Sa0

ln Sa/Sa0

0.3 0.2

−0.2

−0.3

0.1

−0.4

0

−0.5

−0.1 −0.2

−0.6 0

20

0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1

(C)

−0.7

100 (B)

0

20 40 60 80 Time (arbitrary values)

100

0 −0.5 ln Sa/Sa0

ln Sa/Sa0

(A)

40 60 80 Time (arbitrary values)

−1

−1.5 −2

0

20

40 60 80 Time (arbitrary values)

−2.5 100 0 (D)

20

40 60 80 Time (arbitrary values)

100

FIGURE 13.12 Simulated semilog plots for inactivation of spores according to Equation 13.16 with Sd0=Sa0 ¼ 10 and kact ¼ 0.01, kinact ¼ 0.05 (A), kinact ¼ 0.08 (B), kinact ¼ 0.098 (C), kinact ¼ 0.3 (D).

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13-13

0.5

0

0.4

− 0.1 − 0.2 − 0.3

0.2

ln Sa/Sa0

ln Sa/Sa0

0.3

− 0.4 − 0.5

0.1

− 0.6

0

− 0.7

− 0.1 − 0.2

(A)

− 0.8

0

20 40 60 80 Time (arbitrary values)

100

−0.9 0 (B)

20 40 60 80 Time (arbitrary values)

100

0

0

−0.5 −1 −1.5

−1

ln Sa/Sa0

ln Sa/Sa0

−0.5

−2

−2.5

−1.5

−3

−2

−3.5

−2.5

−4.5

−4

(C)

0

20 40 60 80 Time (arbitrary values)

100

0

(D)

20 40 60 80 Time (arbitrary values)

100

FIGURE 13.13 Simulated semilog plots for inactivation of spores according to Equation 13.16 with kact ¼ 0.01, kinact ¼ 0.05, and Sd0=Sa0 ¼ 10 (A), Sd0=Sa0 ¼ 5 (B), Sd0=Sa0 ¼ 1 (C), Sd0=Sa0 ¼ 0.1 (D).

dormant to active spores is varied). This ﬁrst-order model does actually result in nonlinear plots, depending on the values of the rate constants involved. In most cases reported in the literature, plots as in Figure 13.12B and C and Figure 13.13B are found. The important point is that the rate constant (or D-value) determined from the linear part of a logarithmic plot is actually the resultant of both rate constants kact and kinact and in some cases the slope could be signiﬁcantly inﬂuenced by the rate of activation rather than the rate of inactivation. The ﬁrst-order model given in Scheme 13.1 has been extended by Sapru to include also inactivation of dormant spores (Scheme 13.2): Si

kinact2

Sd

kact

Sa

kinact1

Si

SCHEME 13.2 Sapru model: First-order kinetics scheme to describe inactivation of spores, including the inactivation of dormant spores. Sd: number of dormant spores, Sa: number of active spores, Si: number of inactivated spores.

A possible problem with the Sapru model is that too many parameters are needed to describe inactivation while only Si can be measured; as a result they may be either inestimable or too imprecise to be useful.

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A further aspect to discuss is the applicability of ﬁrst-order kinetics. If the conclusion from Section 13.2 on inactivation of vegetative cells is plausible that ﬁrst-order kinetics is not really applicable, this same conclusion should hold for spore inactivation. We therefore need to consider other models as well. Alternative models for nonlinear spore survival curves. A logical extension of Section 13.2 on inactivation of vegetative cells would be to use the Weibull model again, but now also allowing for the phenomenon of ‘‘activation’’: log S(t) ¼ b1 t n1 b2 t n2

(13:16)

The ﬁrst part of the right-hand side refers to the activation part of the survival curve, and the second part to the inactivation part of the survival curve. If there is no activation shoulder, which implies that b1 ¼ 0, the model reduces to the Weibullian model (Equation 13.13). Another empirical model to handle activation shoulders was proposed by Peleg: log S(t) ¼

t f1 lnð1 þ exp (b3 t Þn3 g k1 þ k2 t

(13:17)

These empirical models are certainly not exclusive, others may perform equally well. It is perhaps instructive to compare the performance of the models discussed hitherto. A dataset was used that contained replicates and included activation shoulders and concerned the inactivation of Bacillus stearothermophilus spores. Four models were tested, the Shull and Sapru models displayed in Schemes 13.1 and 13.2, respectively, and the double Weibullian model and the Peleg model in Equations 13.16 and 13.17, respectively. Figure 13.14 shows the ﬁts obtained. Although the ﬁts do not differ that much by visual inspection, and they all seem to describe the experimental data quite well, there are big differences. The Shull and Sapru model suffer from very imprecise and sometimes indeterminate estimates, which is the reason that they coincide. This problem did not exist with the double Weibullian and the Peleg model. A model discrimination study with

2

1.5 1

1

0.5 0

0

−1

log S(t)

log S(t)

− 0.5

−1.5

−1 −2

−2 −3

−2.5 −3

−4

−3.5 −4 (A)

0

100

200

300 400 t (min)

500

−5

600 (B)

0

50

100 t (min)

150

200

FIGURE 13.14 Fits of four models to heat inactivation of spores of B. stearothermophilus at four temperatures. Peleg model (—), Shull model (- - -), Sapru model (. . . . ), double Weibullian model (— —) at 1058C (A), 1108C (B) Dataset in Appendix 13.1, Table A.13.7.

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13-15

2

2

1

1

0

0

−1

−1

−2

−2 log S(t)

log S(t)

Kinetics of Inactivation of Microorganisms

−3 −4

−3 −4

−5

−5

−6

−6

−7

−7

−8

0

10

20

(C)

FIGURE 13.14 (continued)

30 40 t (min)

50

60

70

−8

0

5

10

15

t (min)

(D)

1158C, (C), 1208C (D).

these data and models is shown in Tables 13.1 through 13.4. The criteria used have been discussed in Chapter 7; the goodness of ﬁt indicates the sampling probability of obtaining a higher ratio if the model is correct, the model having the lowest Akaike criterion number and the highest posterior probability performs best. The results clearly indicate that the double Weibullian and the Peleg model perform much better. The goodness of ﬁt is much less for the Sapru and Shull model at 1108C, while there is lack of ﬁt for all models at 1208C. As already mentioned, some parameter estimates were indeterminate with the Shull and Sapru model. All in all, it seems that the empirical models perform much better than the ones based on

TABLE 13.1 Goodness of Fit, Akaike Criterion (AIC, DAIC), and Logarithm of the Posterior Probability (log(PB)) for Four Models and Inactivation Data of Spores of B. stearothermophilus at 1058C DAIC

log(PB)

118.5

8.1

8.1

116.5

10.1

8.1

0.923

122.0

4.6

7.1

0.995

126.6

0

7.7

Model

Goodness of Fit

AIC

Shull

0.783

Sapru

0.783

Double Weibullian Peleg

TABLE 13.2 Goodness of Fit, Akaike Criterion (AIC, DAIC), and Logarithm of the Posterior Probability (log(PB)) for Four Models and Inactivation Data of Spores of B. stearothermophilus at 1108C Model

Goodness of Fit

AIC

DAIC

log(PB)

Shull

0.005

110.6

31.7

8.4

Sapru

0.005

108.6

33.7

8.4

Double Weibullian

0.941

139.5

2.9

4.4

Peleg

0.814

142.3

0

4.0

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13-16

TABLE 13.3 Goodness of Fit, Akaike Criterion (AIC, DAIC), and Logarithm of the Posterior Probability (log(PB)) for Four Models and Inactivation Data of Spores of B. stearothermophilus at 1158C Model

Goodness of Fit

AIC

DAIC

log(PB)

Shull

0.866

70.3

7.5

6.4

Sapru

0.866

68.3

9.5

6.4

Double Weibullian Peleg

1.000 0.999

77.3 77.8

0 0.5

5.6 5.6

TABLE 13.4 Goodness of Fit, Akaike Criterion (AIC, DAIC), and Logarithm of the Posterior Probability (log(PB)) for Four Models and Inactivation Data of Spores of B. stearothermophilus at 1208C Model

Goodness of Fit

AIC

DAIC

log(PB)

Shull

0.001

110.0

4.3

5.5

Sapru

0.001

108.0

6.3

5.5

Double Weibullian

0.005

114.3

0

4.8

Peleg

0.002

113.3

1.0

5.1

ﬁrst-order kinetics, at least for this dataset. However, if these models are to be used more datasets should be tested, and of course, also the temperature dependence should be studied. Temperature dependence is the topic of Section 13.4.

13.4

Temperature Dependence of Microbial Inactivation

Thermal death time (TDT) curves. The temperature dependence of microbial inactivation according to the classical ﬁrst-order model was already discussed in Chapter 5 when D- and Z-values were introduced. The analysis is based upon the assumption that the logarithm of the D-value varies linearly in a so-called TDT plot (see Equation 5.38 and Figure 5.14) and the Z-value can then be derived from the slope of the TDT plot. An example is given for the heat inactivation of Listeria monocytogenes suspended in a buffer (Figure 13.15A). As shown, the survival curve was more or less linear (one of the few examples to be found in the literature), so a D-value could be derived, and the D-values derived at four temperatures were used for a TDT plot (Figure 13.15B). The TDT curve could be regarded as linear, though there may also be a hint on nonlinearity but there are too few data points to check that. In any case, it is dangerous to extrapolate beyond the range (which is rather narrow: only 68C) for which the relation is derived. Most TDT curves published in the literature are based upon only three, sometimes four temperatures that are close together and then it is hard to discover nonlinearity. Moreover, the resulting parameter estimates are very imprecise due to the limited number of data, leading to very inaccurate predictions. A similar analysis for nonlinear survival curves by calculating a D-value makes no sense. However, one could investigate the behavior of the Weibullian parameters as a function of temperature; this is done below. D- and Z-values are the commonly used parameters in the literature to characterize the inactivation of microbes as a function of temperature. The inactivation of microbes in a process (lethality of a process) is commonly expressed in a so-called F0 value. This is an equivalent time at a reference temperature Tref that gives the same inactivation as the actual process:

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Kinetics of Inactivation of Microorganisms

13-17

0

0.6 0.4

−1

0.2 0 log D

log S(t)

−2 −3

−0.2 −0.4

−4

−0.6 −5 −6 (A)

−0.8 −1 0

1

2 3 t (min)

4

5

6

55

56

57

(B)

58 59 T ⬘ (⬚C)

60

61

62

63

FIGURE 13.15 Survival curve of L. monocytogenes at 588C in a buffer, pH 7.4 with a log-linear regression line (A) and a TDT curve for the same microorganism using the D-value calculated for four temperatures (B). Dataset in Appendix 13.1, Table A.13.8.

ðt T(t)Tref F0 ¼ 10 Z dt

(13:18)

0

The log reduction is related to F0 as F0 1 ¼ log S(t) ¼ Dref Dref

ðt 10

T(t)Tref Z

dt

(13:19)

0

where Dref is the D-value at the reference temperature. For vegetative cells the reference temperature is chosen usually around 708C (pasteurization range) and for spores 1218C (sterilization range). Sometimes, one uses ﬁrst-order rate constants and plots them according to the logarithmic form of the Arrhenius equation (Chapter 5). This author is of the opinion that the Arrhenius equation is not applicable at all to microbial inactivation. Arrhenius’ law was derived for simple elementary reactions, not for complicated events such as microbial death. Interpretation of activation energy in terms of J mol1 is impossible because one cannot refer to a speciﬁc elementary reaction, and it is hard to imagine a mole of bacteria anyway. Also, when the Arrhenius equation is applicable, it indicates that a reaction rate constant has a ﬁnite value both at low and high temperature and this makes sense for a chemical reaction: it will take place also at a low temperature, be it at a very low rate. With microbial inactivation, however, it becomes difﬁcult to imagine that inactivation takes place at a slow rate according to the same mechanism as the one at the higher temperature. On the contrary, at moderate temperature microorganisms grow and at temperatures below their minimum growth temperature they do not grow but they are usually not inactivated. Furthermore, inactivation of microbes takes place over a very narrow temperature interval and then it makes no sense to use the inverse of the absolute temperature, as is commonly done with the Arrhenius equation, nor does it makes sense to make a logarithmic transformation. So, we will not discuss the analysis of Arrhenius in microbial inactivation.

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13-18

One could argue that a Z-value does make sense, as long as it is not connected to activation energy (as in Equation 5.35). In other words, the Z-value could be used just as an empirical parameter to express temperature dependence of a rate parameter. However, the problem remains that if the derivation of the Z-value is based upon D-values that are derived from nonlinear plots, this Z-value has actually little value and, as stated before, there are not that many linear inactivation plots to be found. If we accept the Weibullian model (Equation 13.13) as the alternative model for microbial inactivation kinetics, the dependence of its two parameters b and n on temperature has to be studied. Literature analysis of microbial inactivation showed that the logarithm of the scale parameter b depended linearly on temperature, analogous to the classical D-value. However, the temperature dependence of the shape parameter n was not so clear. In only a few cases the shape parameter seemed to depend on temperature, in a linear way. In all other cases, no statistically signiﬁcant (linear) relation with temperature could be found. In most cases, the shape parameter n was larger than 1, and in fewer cases smaller than 1.

0

0

−0.5

−1

−1 −2

log S(t)

−1.5

−3

log S(t)

−2 −2.5

−4

−3 −5

−3.5 −4

0

100

200

(A)

300 400 t (min)

500

−6

600

0

500

−1.5 −2 log S(t)

log S(t)

400

−1

−3 −4

−2.5 −3 −3.5

−5

−4

−6

FIGURE 13.16 54.68C (D),

200 300 t (min)

−0.5

−2

(C)

100

0

−1

−7 0

0

(B)

−4.5 20

40

60 80 t (min)

100

120

−5

140 (D)

0

20

40 t (min)

60

80

Inactivation of E. coli K12 MG1655 suspended in growth medium at 49.98C (A), 528C (B), 548C (C), (continued )

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Kinetics of Inactivation of Microorganisms

13-19

0

0

−0.5

−1

−1

−2

−2

log S(t)

log S(t)

−1.5

−2.5 −3 −3.5

−4 −5

−4

−6

−4.5 −5

−3

0

10

20

30 t (min)

(E)

40

50

−7

60

0

5

10

15 t (min)

(F)

0

20

25

30

0 −1

−1

−2 −3 log S(t)

log S(t)

−2 −3

−4 −5

−4

−6 −5 −6 (G)

−7 0

2

4

t (min)

6

8

−8

10 (H)

0

1

2

t (min)

3

4

5

FIGURE 13.16 (continued) 558C (E), 56.68C (F), 58.68C (G), and 60.68C (H). The drawn lines are the ﬁts of the Weibullian model for a variable n and b, the broken line is for a ﬁxed n ¼ 1.6 (the average value for n), ﬁxed for all temperatures. Dataset in Appendix 13.1, Table A.13.9.

Only very rarely is the shape parameter n ¼ 1, indicating that the classical ﬁrst-order kinetics approach is indeed the exception rather than the rule. Most experimental results obtained so far point out that the n parameter is not, or only weakly, dependent on temperature. Figure 13.16 gives an example of nonlinear survival curves for an E. coli strain. The data were ﬁtted ﬁrst by allowing parameters n and b to vary. The variation of n with temperature is shown in Figure 13.17. It can be seen that parameter n does not have a constant value for each dataset, but on the other hand there does not seem to be a systematic variation with temperature; in this particular case the estimate for the data at 58.68C seems to be an outlier. Similar analyses from other datasets indicate the same behavior of the parameter n as a function of temperature, so to a ﬁrst approximation n can be assumed constant with varying temperature. For the data shown the average value of parameter n was 1.6. However, because of the variation, this adds uncertainty to subsequent ﬁts. To show the effect, the ﬁts with the ﬁxed value of n ¼ 1.6 are also shown in Figure 13.16.

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13-20

3

2.5

n (T⬘)

2

1.5

1

0.5

0 45

50

55 T⬘ (⬚C)

60

65

FIGURE 13.17 Dependence of the Weibullian parameter n on temperature for the ﬁts shown in Figure 13.16.

A ﬁxed value of n is needed if we want to study the effect of temperature on parameter b because the parameters b and n are correlated. By ﬁxing n at its average value 1.6 for this particular dataset, the dependence of b on temperature can be studied. A suitable model to capture the dependence of b on temperature T 0 is the log-logistic model: b(T 0 ) ¼ ln½1 þ expðk(T 0 Tc Þ

(13:20)

with k (8C1) and Tc (8C) as parameters. When T 0 > Tc, b(T 0 ) k(T 0 Tc). Another relation could be a power-law relation as depicted in the next equation: 0

b(T 0 ) ¼ 10aT þc

(13:21)

The ﬁt of these two models to the parameter estimates b for the datasets shown in Figure 13.16 is shown in Figure 13.18. For the data shown in Figure 13.18, the ﬁts by the two models are almost indistinguishable. This approach is an alternative for the TDT approach discussed above. To deal with nonisothermal situations the following analysis has been developed by Peleg and coworkers (see also Chapter 5 and Chapter 12 where this approach was applied). Nonisothermal conditions. If we take the Weibullian model (Equation 13.13) as our model, and we ﬁx its parameter n at a constant value (i.e., temperature independent) for the moment, the momentary rate at constant temperature can be calculated from this model as d logðS(t)Þ ¼ b(T) n t n1 dt T

(13:22)

It shows that the momentary rate varies with time. We need to know also how temperature varies with time as T(t). This information should be available from the experiment and the relation can be cast in any algebraic equation that ﬁts the experimental conditions. Then, we can make parameter b time dependent

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Kinetics of Inactivation of Microorganisms

13-21

0.7 0.6

b (T⬘)

0.5 0.4 0.3 0.2 0.1 0

45

50

55 T⬘ (⬚C)

60

65

FIGURE 13.18 Fit of the loglogistic model (Equation 13.20, solid line) and the power-law model (Equation 13.21, hyphenated line) to the Weibullian parameter b as a function of temperature for the ﬁts found to the dataset shown in Figure 13.16 while keeping the parameter n constant at its average value of n ¼ 1.6.

so that it can be written as b(T(t)). To know the survival ratio at any time t* the Weibullian Equation 13.13 can be inverted to ﬁnd: t* ¼

logðS(t)Þ bðT(t)Þ

1=n (13:23)

By combining these last two equations we ﬁnd d logðS(t)Þ logðS(t)Þ n1=n ¼ bðT(t)Þ n dt bðT(t)Þ

(13:24)

This equation can be numerically integrated if T(t) is known and the relation for b(T) has been established, for instance via Equations 13.20 or 13.21, or any other equation that would do the job. Thus, it allows calculating the survival ratio as a function of varying temperature in a relatively simple way. As an example, Figure 13.19 shows the inactivation of L. monocytogenes in a buffer at pH 4 at four temperatures, modeled via the Weibullian model. The shape parameter n was found to be relatively constant over this temperature range; the average value was n ¼ 1.265 and the results shown in Figure 13.19 are for this average value. From such data the temperature dependence of b can be derived. The temperature dependence of b(T) is shown in Figure 13.20 for two models, one is the log-logistic model of Equation 13.20, and the other the power-law relation shown in Equation 13.21. In this case, the log-logistic model appears not to be appropriate and the power-law model is performing well. So, the temperature dependence of parameter b(T) via the power-law model was used in Equation 13.24 to predict nonisothermal behavior of this bacterium. the results are shown in Figure 13.21 for a linear increasing temperature proﬁle.

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Kinetic Modeling of Reactions in Foods 0

0

−1

−1

−2

−2 log S (t)

log S (t)

13-22

−3

−3

−4

−4

−5

0

1

2

(A)

3 4 Time (min)

5

−5

6

0

0.5

1

(B)

0

1.5 2 Time (min)

2.5

3

0

−1

−1

−2 log S(t)

log S(t)

−2

−3

−3 −4

−4

−5 (C)

−5

0

0.5

1 Time (min)

1.5

−6 0 (D)

0.2

0.4 Time (min)

0.6

0.8

FIGURE 13.19 Survival curves of L. monocytogenes in a buffer of pH 4 at 548C (A), 568C (B), 588C (C), 608C (D). The curves are the ﬁts by the Weibullian model (Equation 13.13) for a ﬁxed shape parameter n ¼ 1.265 and varying parameter b. Dataset in Appendix 13.1, Table A.13.10.

Note that the predictions given in Figure 13.21 are real predictions: the data shown were obtained in independent experiments. The predictions look reasonable, though certainly not perfect. One possible reason for the discrepancy is the imprecision, but it is also not unthinkable that microorganisms respond differently to a variable temperature treatment than to a constant temperature treatment. As mentioned, this analysis can be done with any other model; one could also include temperature dependence of the parameter n if so desired. It is thus not necessary to work with activation energies to be able to do calculations for nonisothermal treatments.

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Kinetics of Inactivation of Microorganisms

13-23

18 16 14 12

b

10 8 6 4 2 0 40

45

50

55

60

65

T ⬘(⬚C)

FIGURE 13.20 Temperature dependence of the Weibullian parameter b for the inactivation of L. monocytogenes as shown in Figure 13.19 for the log-logistic model (hyphenated line, k ¼ 3.1, Tc ¼ 57.08C) and the power-law model (solid line, a ¼ 0.23, c ¼ 12.8).

0

−1

log S(t)

−2

−3

−4

−5

−6

0

10

20

30 40 Time (min)

50

60

FIGURE 13.21 Nonisothermal inactivation of L. monocytogenes in buffer at pH 4 for a linearly increasing temperature proﬁle from 308C with a rate of 0.58C min1 (&), 18C min1 (*), 28C min1 (~), and 58C min1 (*). The solid lines are the predictions by Equation 13.24, using the temperature dependence of b via the power-law model as indicated in Figure 13.20. Dataset in Appendix 13.1, Table A.13.11.

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Kinetic Modeling of Reactions in Foods

13-24

13.5

Food Matrix Effects

0

0

−1

−1

−2

−2

−3

−3 log S

log S

The kinetics of inactivation of microorganisms can be inﬂuenced strongly by the surroundings in which the microorganisms are. It is not really possible to predict the effects, sometimes the matrix appears to protect the microorganisms, and sometimes it is the other way around. Just to show the magnitudes of possible effects two examples are shown here. The ﬁrst example concerns the protective effect of the presence of fat on inactivation of a Salmonella species in meat (Figure 13.22). Table 13.5 shows the Weibullian parameter estimates for the data shown in Figure 13.22. Both the graphs and the parameter estimates show that the higher the fat content the longer the time needed for inactivation. In other words, fat appears to protect the microorganism. It is interesting to see that the shape parameter n increases with fat content, suggesting that the microorganism is better able to withstand the thermal stress when more fat is present. It is hard to think of a mechanism that could explain this, however. Another example is shown in Figure 13.23. This concerns the inactivation of spores of Bacillus subtilis in various media: 0.05 M phosphate solutions containing 3% and 5% NaCl, whole milk, soy sauce, and kayu (Japanese porridge). Table 13.6 shows the parameter estimates for Weibullian ﬁts to the data shown in Figure 13.23. It can be seen that in this case inactivation is faster in the food matrices shown as compared to a buffer system. Moreover, the shape factor is seen to vary as well, which may indicate that the inactivation mechanism depends on the food matrix. However, these few data do not allow ﬁrm conclusions to be drawn. The only thing that can be concluded is that matrix effects cannot be neglected, but unfortunately it is not possible to predict the magnitude or the direction of these effects. In any case, it shows that it can be dangerous to extrapolate from simple aqueous systems to real foods.

−4

−4

−5

−5

−6

−6

−7 (A)

0

1

2

3

Time (min)

4

−7

5 (B)

0

1

2

3

4

5

Time (min)

FIGURE 13.22 Effect of fat content on heat inactivation of Salmonellae species at 658C in beef (A) with 24% fat (^), 18% fat (~), 12% fat (*), 7% fat (&), and pork meat (B) with 28% fat (^), 24% fat (~), 10% fat (*), 4% fat (&). The drawn lines are ﬁts of the Weibullian model. Dataset in Appendix 13.1.

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Kinetics of Inactivation of Microorganisms

13-25

TABLE 13.5 Weibullian Parameter Estimates for the Inactivation of Salmonellae Species in Meat with Varying Fat Content (Figure 13.22) Matrix Beef 24% fat

b (min)

n

0.37

1.91

Beef 18% fat

0.60

1.57

Beef 12% fat Beef 7% fat

0.79 1.02

1.45 1.40

Pork 28% fat

0.21

2.26

Pork 24% fat

0.55

1.62

Pork 10% fat

0.67

1.51

Pork 4% fat

0.86

1.43

9 8 7

log N

6 5 4 3 2 1 0 0

10

20 30 Time (min)

40

50

FIGURE 13.23 Heat inactivation of B. subtilis spores in phosphate buffer with 3% NaCl (&) and 5% NaCl (^), whole milk (~), soy sauce (&), kayu (*). Dataset in Appendix 13.1, Table A.13.13.

TABLE 13.6 Parameter Estimates for the Weibullian Fits Shown in Figure 13.23 Matrix

b (min)

n

Phosphate buffer, 3% NaCl

0.02

1.37

Phosphate buffer, 5% NaCl

0.02

1.28

Whole milk

0.18

1.07

Soy sauce

0.06

1.62

Kayu

0.36

0.83

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13-26

13.6

Concluding Remarks

Kinetics of microbial inactivation has been investigated for almost a century now. For most of that century, ﬁrst-order kinetics has been applied, and not without success, because processed foods are actually quite safe. It is all the more remarkable that survival curves are not straight lines but rather are curved. The evidence is overwhelming that ﬁrst-order kinetics does not apply. This does not mean that it cannot be used in practice, but alternative models are widely available now and they seem to perform better in many cases from a statistical point of view. They also do more justice to the probabilistic character of inactivation phenomena. In the author’s view they deserve to be further investigated for practical applications. As a ﬁnal remark, it can be mentioned that the suggested models are applicable to other treatments than heat. Weibull models have already been tested quite extensively for inactivation due to pressure, pulsed electric ﬁelds, and chemical agents. We did not discuss this here, but the methods given can be applied directly. Some references are given in the Bibliography section.

Appendix 13.1

Datasets Used for Examples in This Chapter

TABLE A.13.1 Time 0

Survival Data for E. coli in Apple Cider (Figure 13.1) log N at 528C

log N at 558C

log N at 588C

9.65

9.65

9.65

1

9.10

2

8.75

3

8.35

4

7.12

5 6

5.45 3.77

10

9.2

8.05

15

8.9

6.78

20

8.4

6.25

25

7.95

4.75

30

7.5

35

6.8

45

5.8

Source: From Splittstoesser D.F., McLellan M.R., and Churey J.J. Heat resistance of Escherichia coli O157:H7 in apple juice. J Food Prot 59:226–229, 2000.

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Kinetics of Inactivation of Microorganisms TABLE A.13.2 (Figure 13.7) Time (min)

13-27

Inactivation of Salmonella typhimurium log S

0

0

0.47

0.041

0.984

0.124

1.476

0.145

1.968

0.373

2.527

0.373

2.997

0.539

3.556 4.048

0.622 1.119

4.607

1.513

5.099

1.575

6.128

2.632

6.597

3.565

7.089

3.876

Source: From Mackey B.M. and Derrick C.M. Elevation of the heat resistance of Salmonella typhimurium by sublethal heat shock. J Appl Bacteriol 61:389–393, 1986.

TABLE A.13.3 Inactivation of Salmonella enteriditis in Egg Yolk (Figure 13.8) Time (min) 0

log S 0

0.057

1

0.113 0.192

1.478 1.523

0.17

1.682

0.34

1.978

0.509

2.159

0.668

2.182

0.736

2.318

1.483

2.841

2.208 2.943

3.296 3.728

Source: From Michalski C.R., Brackett R.E., Hung Y.-C., and Ezeike G.O.I. Use of capillary tubes and plate heat exchanger to validate U.S. Department of Agriculture pasteurization protocols for elimination of Salmonella enteritidis from liquid egg products. J Food Prot 62:112– 117, 1999.

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13-28

TABLE A.13.4 Inactivation of Saccharomyces cerevisiae (Figure 13.9) Time (min) 0

log S 0

2.3

0.051

4.2

0.103

5.9

0.276

8.1

0.396

10.0

0.417

12.2

0.665

14.1 16.2

0.762 0.761

17.8

0.896

20.2

0.947

22.0

1.06

24.0

1.111

26.0

1.163

28.0

1.29

29.4 31.8

1.456 1.537

Source: From Van Uden N. and Vidal-Leiria M.M. Inactivation of S. cerevisiae in an aqueous solution of yeast nitrogen base and glucose. Arch Microbiol 108:293, 1976.

TABLE A.13.5 Inactivation of Spores of B. licheniformis in Concentrated Milk (Figure 13.10A) Time (s) 0

log N=N0 0

70

0.039

80

0.135

110

0.135

150 160

0.328 0.309

220

0.772

280

1.004

320

1.274

380

1.641

440

1.911

600

3.069

650

3.436

Source: From Behringer R. and Kessler H.G. Heat resistance of spores in skimmilk and skimmilk concentrates and determination of reaction kinetic parameters. Milchwissenschaft 46:488–492, 1991.

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13-29

TABLE A.13.6 Inactivation of Spores of C. botulinum (Figure 13.10B) Time (min)

log S 0.069

0 0

0.069

0.5

2.552

0.5

2.368

0.75

3.127

0.75

2.989

1

3.104

1 1.7

3.495 4

1.7

3.862

2.7

4.529

2.7

4.322

5

5.564

5

5.426

Source: From Anderson W.A., McClure P.J., Baird-Parker A.C., and Cole M.B. The application of a log-logistic model to describe the thermal inactivation of Clostridium botulinum 213B at temperatures below 121.18C. J Appl Bacteriol 80: 283–290, 1996.

TABLE A.13.7 Inactivation of Spores of B. stearothermophilus (Figures 13.11 and 13.14) Data for 1058C Time (min)

log S

0

0.145

0

0.140

0

0.055

1

0.282

1

0.112

1

0.191

2.5 2.5

0.404 0.191

2.5

0.305

5

0.594

5

0.421

5

0.446

10

0.594

10

0.305

10 15

0.506 0.700

15

0.552

15

0.565

25

0.828

25

0.627 (continued )

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Kinetic Modeling of Reactions in Foods

13-30

TABLE A.13.7 (continued) Inactivation of Spores of B. stearothermophilus (Figures 13.11 and 13.14) Data for 1058C Time (min)

log S

30

0.571

30

0.648

35

0.755

35 45

0.663 0.860

45

0.520

45

0.627

60

0.714

60

0.594

60

0.673

90

0.367

90 120

0.546 0.673

120

0.367

120

0.316

150

0.033

150

0.161

170

0.056

180

0.056

180 300

0.112 0.772

300

0.578

300

0.480

390

1.140

390

1.372

480

3.332

480

1.523

480

3.166

Data for 1108C Time (min)

log S

0

0.101

0

0.017

0

0.096

0.5

0.415

0.5 0.5

0.376 0.343

1

0.443

1

0.460

1

0.406

1.5

0.585

2

0.538

2

0.415

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13-31

TABLE A.13.7 (continued) Inactivation of Spores of B. stearothermophilus (Figures 13.11 and 13.14) Data for 1108C Time (min)

log S

2

0.682

4

0.832

4

0.545

4 6

0.832 0.735

6

0.592

6

0.832

10

0.832

10

0.667

10

0.794

15

0.832

15 15

0.656 0.095

20

0.832

20

0.740

20

0.832

30

0.702

30

0.509

30

0.573

40 40

0.434 0.151

40

0.309

60

0.151

60

0.071

60

0.032

90

0.402

90

0.885

90 120

0.566 1.308

120

2.247

120

1.471

180

3.991

180

4.690

180

4.512

Data for 1158C Time (min)

log S

0

0.101

0

0.017

0

0.095

0.25

0.565

0.25

0.460

0.25

0.406 (continued )

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TABLE A.13.7 (continued) Inactivation of Spores of B. stearothermophilus (Figures 13.11 and 13.14) Data for 1158C Time (min)

log S

0.5

0.702

0.5

0.616

0.5

0.692

0.75 1

0.634 0.794

1

0.753

1

0.650

1.5

0.899

2

0.832

2

0.867

2

0.794

2.5 3

0.753 0.794

3

0.867

3

0.983

5

0.899

5

0.716

10

0.452

10

0.271

15 15

0.017 0.206

15

0.832

20

0.366

20

0.816

30

1.414

30

2.323

45

4.101

45 45

3.168 5.017

60

6.946

Data for 1208C Time (min)

log S

0

0.101

0

0.017

0 0.08

0.095 0.386

0.08

0.524

0.08

0.284

0.17

0.753

0.17

0.592

0.17

0.434

0.33

0.744

0.33

0.832

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TABLE A.13.7 (continued) Inactivation of Spores of B. stearothermophilus (Figures 13.11 and 13.14) Data for 1208C Time (min)

log S

0.33

0.672

0.67

0.622

0.67

0.867

0.67 1

0.899 0.566

1

0.794

1

0.794

1.33

0.365

1.33

0.794

1.33

0.697

1.67

0.321

1.67 1.67

0.604 0.559

2

0.133

2

0.610

2

0.538

3

0.167

4

0.366

4

0.548

4 6

0.584 1.361

6

1.414

6

1.384

8

2.703

8

2.483

8

3.168

10

4.688

10 10

5.001 4.645

12

5.832

14

6.645

Source: From Sapru V. Mathematical modeling of bacterial spore population dynamics for design and validation of ultra high temperature sterilization processes. PhD thesis, University of Florida, Florida (1991).

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13-34 TABLE A.13.8 (Figure 13.15)

Survival Curve of L. monocytogenes

Data for Figure 13.15A Time (min)

log S

0

0 0.33 0.96

0.5 0.98 1.49

1.52

1.99

1.8

2.5

2.73

2.97

3.19

3.55

3.54

4.06

3.97

4.51 5.02

4.35 4.76

5.5

5.32

Data for Figure 13.15B T (8C)

log D

56

0.35

58

0.01

60 62

0.31 0.79

Source: From Hassani M., Alvarez I., Raso J., Condón S., and Pagán R. Comparing predicting models for heat inactivation of Listeria monocytogenes and Pseudomonas aeruginosa at different pH. Int J Food Microbiol 100: 213–222, 2005.

TABLE A.13.9 Inactivation of E. coli K12 MG1655 (Figure 13.16) 49.88C Time (min)

log S

0 50

0 0.02

66

0.07

85

0.07

100

0.17

125

0.25

150

0.3

165

0.32

185 200

0.55 0.54

220

0.64

235

0.79

250

0.74

265

0.77

270

1.02

290

1.12

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TABLE A.13.9 (continued) Inactivation of E. coli K12 MG1655 (Figure 13.16) 49.88C Time (min)

log S

300

1.24

325

1.42

350

1.59

370

1.84

390

1.79

410 430

2.17 2.47

450

2.29

475

2.54

490

2.52

510

2.99

570

3.69

528C Time (min)

log S

0

0

15

0.07

40

0.22

55

0.4

75

0.5

85 105

0.62 0.62

120

0.72

130

1.05

150

1.3

200

1.75

225

2.68

250

3.13

280 300

3.13 3.3

340

3.73

375

4.35

415

4.28

430

4.78

548C Time (min)

log S

0

0

3

0.15

10

0.33

25

0.53

32

0.75

44

1.5

45 50

1.83 2.36

60

2.51 (continued )

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13-36

TABLE A.13.9 (continued) Inactivation of E. coli K12 MG1655 (Figure 13.16) 548C Time (min)

log S

62

2.83

74

3.01

75

3.71

87

3.79

92

3.91

100 120

5.62 5.72

54.68C Time (min)

log S

0

0

9

0.13

11 18

0.05 0.35

25

0.73

30

1.46

35

1.74

40

2.39

45

2.49

50

3.07

55 60

3.15 3.35