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Progress in Reaction Kinetics and Mechanism. Vol. 30, pp. 3–113. 2005 1468-6783 # 2005 Science Reviews

REVIEW Thermodynamics and foundations of mass-action kinetics Miloslav Pekarˇ * Institute of Physical and Applied Chemistry, Faculty of Chemistry, Brno University of Technology, Purkynˇova 118, 612 00 Brno, Czech Republic. E-mail: [email protected]

Contents ABSTRACT 1.

INTRODUCTION

5

2.

CLASSICAL BACKGROUND

6

2.1. Reaction isotherm 2.2. Thermodynamic consistency of rate equations

6 9

3.

4. 5.

AFFINITY AND CHEMICAL KINETICS

13

3.1. 3.2. 3.3. 3.4. 3.5.

13 15 23 26 30

De Donder as originator Successors to De Donder Garﬁnkle’s original approach Critical slowing; linearity testing Summary

ACTIVITIES IN CHEMICAL KINETICS

31

CLASSICAL (LINEAR) IRREVERSIBLE THERMODYNAMICS

40

5.1. 5.2. 5.3. 5.4. 5.5. 5.6.

40 44 46 55 57 58

Fundamentals Tackling mass-action non-linearity and Onsager reciprocity Hungarian contribution I – Lengyel Onsager far from equilibrium Bro¨nsted re-discovered? Hungarian contribution II – Ola´h

6.

EXTENDED IRREVERSIBLE THERMODYNAMICS

62

7.

COMMON PROBLEMS IN CIT AND EIT APPROACHES

71

8.

RATIONAL OR CONTINUUM THERMODYNAMICS APPROACHES TO CHEMICAL KINETICS

74

8.1. Introduction 8.2. Bowen lays the foundation stone 8.3. Gurtin re-examines the classical theory 3

74 75 76

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Miloslav Pekarˇ

9. 10.

8.4. Treatments of more complex systems

81

8.5. Mu¨ller’s results

88

8.6. Samohy´l’s achievements

92

CHEMICAL POTENTIAL MODEL

105 107

CONCLUSIONS

ABSTRACT A critical overview is given of phenomenological thermodynamic approaches to reaction rate equations of the type based on the law of mass-action. The review covers treatments based on classical equilibrium and irreversible (linear) thermodynamics, extended irreversible, rational and continuum thermodynamics. Special attention is devoted to afﬁnity, the applications of activities in chemical kinetics and the importance of chemical potential. The review shows that chemical kinetics survives as the touchstone of these various thermodynamic theories. The traditional mass-action law is neither demonstrated nor proved and very often is only introduced post hoc into the framework of a particular thermodynamic theory, except for the case of rational thermodynamics. Most published ‘‘thermodynamic’’ kinetic equations are too complicated to ﬁnd application in practical kinetics and have merely theoretical value. Solely rational thermodynamics can provide, in the speciﬁc case of a ﬂuid reacting mixture, tractable rate equations which directly propose a possible reaction mechanism consistent with mass conservation and thermodynamics. It further shows that afﬁnity alone cannot determine the reaction rate and should be supplemented by a quantity provisionally called constitutive afﬁnity. Future research should focus on reaction rates in non-isotropic or non-homogeneous mixtures, the applicability of traditional (equilibrium) expressions relating chemical potential to activity in non-equilibrium states, and on using activities and activity coefﬁcients determined under equilibrium in non-equilibrium states. Prog React Kinet Mech 30:3-113 (c) 2004 Science Reviews

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Thermodynamics and mass-action chemical kinetics

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KEYWORDS: activated complex, activity, afﬁnity, chemical potential, continuum thermodynamics, equilibrium constant, extended irreversible thermodynamics, Guldberg – Waage law, ionic strength, irreversible thermodynamics, kinetic law, mass-action, Onsager reciprocity, rational thermodynamics, rate equation, reaction isotherm, reaction rate, strong equilibrium, weak equilibrium

1. INTRODUCTION The aim of this review is to give a critical overview of various thermodynamic approaches to the formulation of reaction rate equations, preferably of the mass-action law type. It aims to cover papers which directly derive kinetic equations from thermodynamic considerations or which try to obtain more general rate equations from the application of thermodynamic insights to common rate equations or which attempt to supply some established rate equation with proper thermodynamic rigour. ‘‘Kinetic equation’’ and ‘‘(reaction) rate equation’’ should be understood interchangeably as some equation relating chemical reaction rate and quantities, which should determine its value or as some function stating the dependence of the rate on particular (independent) variables. Brieﬂy, the goal is to give a review on thermodynamic derivations or proofs of the Guldberg – Waage kinetic law or of new rate equations applicable in experimental practice. It is just practical phenomenological kinetics which is the primary motivation of this review. Only phenomenological thermodynamic theories are covered, i.e. statistical or molecular approaches are not discussed. Also the large number of approaches which start directly with the mass-action rate equations and use them to study their properties or various properties of underlying systems are not considered. A short list of examples of work outside the scope of this review will make its coverage clearer: studies on mathematical structure and mathematical properties of mass-action type sets of equations [1 – 6], studies on properties of systems described by mass-action kinetics, e.g. their steady state multiplicities, their stability or dynamics [7 – 15], analyses of properties of solutions to (differential) equations embedding mass-action kinetics [16 – 20]. Nor is the detailed balancing included. This review should inform not only on the state-of-the-art of thermodynamic theory for mass-action kinetics but also on its origin. In some instances, the reference therefore goes back more than 100 years. Essentially, however, the period from about 1950 to the present day is covered. www.scilet.com

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Chemical kinetics and thermodynamics are usually considered as two independent disciplines describing reacting systems. Thermodynamics is said to state the conditions for the running and equilibrium of chemical reactions, while giving no information on how fast this all happens. The latter is the domain of kinetics. This review should further demonstrate that the relationships between thermodynamics and kinetics are much closer and that even from solely thermodynamic theories, some inferences on reaction rates can be obtained. Boyd [21] notes that, in contrast to thermodynamics, the kinetic description of a reaction system is less clear-cut. The value of an equilibrium constant is given unambiguously, together with the course of reaction, according to the sign of the Gibbs energy of reaction. On the contrary, it is often not clear whether a unique reaction velocity may be deﬁned, especially for multistep reaction mechanisms [21]. Another question concerns the circumstances under which the reaction rate may be expressed as the difference of two terms. This is very important because of frequent identiﬁcation of the two terms with forward and reverse rates, which balance at equilibrium. There is no speciﬁc thermodynamic reason why the observed reaction rate should be expressible as the difference of two terms [22]. The only observable is the net rate and the forward and backward rates have meaning only by interpretation. To conclude this introduction, a short note on symbolism should be made. The symbols used are a compromise between two extremes – an elaborate strictly uniﬁed nomenclature for this review or just to retain the differing symbols of the various original sources. In order to aid the interested reader, the speciﬁc original symbols of each paper referred to are used if possible, if these are not easily confused with one another. Universal variables like reaction rate, afﬁnity, concentration, activity etc. are given the common, usual symbols.

2. CLASSICAL BACKGROUND 2.1 Reaction isotherm A very lucid and ingenious discussion on the interrelationships between kinetics and thermodynamics from the standpoint of classical, reversible thermodynamics is given in Denbigh’s book [22], which remains even today one of the most lucid presentations of this topic. Denbigh asks following question: Which variables are determining the reaction rate? Is it the volume concentration of www.scilet.com

Thermodynamics and mass-action chemical kinetics

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each of the reacting species? Or is it some other concentration (e.g. molar fraction) or thermodynamic (chemical potential, activity) variable? These questions are not (sufﬁciently) answered by (classical) thermodynamic theory. Kinetic experience tells us that just the molar concentration is a very important variable, and that the rate can be expressed as the difference of two terms containing small powers of the molar concentrations. Denbigh further states that thermodynamics places only two requirements on the reaction rate: (1) a positive value of the rate in the direction of a decrease in Gibbs energy and (2) its zero value in the state of thermodynamic equilibrium. This requirement does not directly lead to the formulation of some explicit expression for the reaction rate. It can be used as a test for the ‘‘consistency’’ of proposed rate equation(s) with thermodynamics (see below) and as a restriction on the expression for the backward reaction rate if the expression for the forward rate has been formulated (as well as for the overall rate, usually as the difference of forward and backward rates). Before going into details let us make a small but very important digression. Many kinetic deductions, even in non-equilibrium thermodynamics, are in fact based on the well-known deﬁnitions of equilibrium thermodynamics. The principal relation is an equation, usually called the reaction isotherm. For a general chemical reaction 0¼

n X

i Ai

ð2:1Þ

i¼1

(i is the stoichiometric coefﬁcient, which is positive for products and negative for reactants) it is written as follows: DGr ¼ DGr þ RT ln

n Y

ai i :DGr þ RT ln Qr

ð2:2Þ

i¼1

where Qr is called the reaction quotient and DGr ¼ RT ln K, K is the equilibrium constant and ‘‘ ’’ denotes the standard state. The reaction isotherm was derived for systems at constant temperature and pressure starting from the Gibbs energy (G) considered to be a function of temperature, pressure and composition. In ideal systems, activities (ai ) may be substituted by concentrations. If the forward and backward reaction rates (r with respective arrow) are expressed according to the Guldberg – Waage law with orders equal to stoichiometric coefﬁcients, the reaction isotherm can be modiﬁed as follows: www.scilet.com

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Miloslav Pekarˇ

DGr ¼ RT ln K þ RT ln

n n ?/ ?/ Y Y ci i ¼ RT ln K þ RT ln ð k k y k k Þci i ¼ i¼1

i¼1 ? / / ?

RT ln K þ RT ln½ð k yk Þð r y r Þ

ð2:3Þ

? /

( k ; k are the rate constants in respective directions). Identifying the thermo? / dynamic with the kinetic ( k y k ) equilibrium constant, the ﬁnal equation results: ? /

DGr ¼ RT lnð r y r Þ

ð2:4Þ

It can be also rewritten introducing afﬁnity either by direct deﬁnition A ¼ DGr or in an alternative way through the chemical potential (m): A¼

n X

i mi ¼

i¼1

n X

ði mi þ i RT ln ai Þ ¼ DGr RT ln Qr

ð2:5Þ

i¼1

Two ﬂaws are hidden in this approach and often ignored. The ﬁrst one is direct identiﬁcation of activities with concentrations (in ideal systems). Activity is a dimensionless quantity and may be expressed as the product of activity coefﬁcient, which is in ideal systems equal to one, and the ratio of actual and standard state concentration. However, the Guldberg – Waage law contains actual concentrations, not related to the standard ones. The second ﬂaw is the identiﬁcation of kinetic and thermodynamic equilibrium constants, i.e. dimensional and dimensionless quantities, respectively. It should also be stressed that the use of stoichiometric coefﬁcients in place of reaction orders means that only elementary reactions are considered. From Eq. (2.4) other versions can be derived. The following relation is very popular: ?

/ ?

?

r ¼ r ð1 r y r Þ ¼ r ½1 expðAyRT Þ

ð2:6Þ

which can, close to equilibrium (AyRT 5 1), be linearized as follows: ?

expðAyRT Þ ¼ 1 ðAyRT Þy1 þ ðAyRT Þ2 y2 ) r % r AyRT

ð2:7Þ

A linear relationship between reaction rate and afﬁnity is thus obtained. As noted above, the reaction isotherm was originally born within equilibrium thermodynamics where it is used primarily to derive an expression for the equilibrium constant. Non-equilibrium applications of the reaction isotherm equation are plausible if the reaction Gibbs energy can be considered as a function of temperature, pressure, and composition only, or if the local www.scilet.com

Thermodynamics and mass-action chemical kinetics

9

equilibrium hypothesis is invoked and if the chemical potential dependence on composition can be expressed as indicated in Eq. (2.5). All these premises will be tackled several times throughout this review. 2.2 Thermodynamic consistency of rate equations Let us return to the ‘‘consistency’’ between thermodynamics and mass-action chemical kinetics. It has been already discussed by Boyd [21] with illustrative examples and therefore only the main points are reviewed here. ?

Gadsby et al. [23] claim, in fact, that for the forward ( r ) and backward / ( r ) reaction rates expressed by ?

?

/

r ¼ k ff ðci Þ;

/

r ¼ k fb ðci Þ

ð2:8Þ

where ci , i ¼ 1; . . . ; n, represent the concentrations of reacting species, to be consistent with the thermodynamic equilibrium condition (and constant), the ? / ratio of forward ( k ) and reverse (k ) rate constants must be equal to the equilibrium constant. Manes et al. [24] correct the conclusions of Gadsby et al. The rates for opposing reactions are formulated as ?

r ¼ ff ðci Þ;

/

r ¼ fb ðci Þ

ð2:9Þ

The only restrictions set by thermodynamics on functions f of the concentrations of reacting species ci are ? /

at equilibrium : r y r :ff yfb ¼ 1;

? /

r y r 41 when DGr 50

ð2:10Þ

In order to fulﬁl these conditions it is sufﬁcient to assume, for example, that " #z ? / Y ? / i ff yfb ¼ ð k y k Þ ci ; where k y k ¼ Kz ð2:11Þ i

where symbol ci again means the concentration of a particular specie and z is a positive constant. Examples of suitable (rational) functions f are given in the original paper. It should be stressed that the identiﬁcation of the kinetic with the (concentration-based) thermodynamic equilibrium constant (K) is assumed. The consistency condition (2.11) was generalised by Hollingsworth [25]. He also considers that the reaction rate is given by the forward and reverse reaction rate laws as in (2.9) but temperature is also included among the www.scilet.com

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Miloslav Pekarˇ

independent variables. The ratio of the forward and reverse rates (see the ﬁrst equation in (2.10)) is symbolized by f ðci ; T Þ. Two equilibrium conditions must be satisﬁed: Qr ¼ KðT Þ and f ¼ 1 ðequilibriumÞ

ð2:12Þ

A sufﬁcient condition for this is that f be expressible as a function of Qr such that f ðci ; T Þ ¼ FðQr ; T Þ and FðK; T Þ ¼ 1

ð2:13Þ

A necessary and sufﬁcient condition for Eqs (2.13) to hold could be that FðQ; T Þ be expressible as a function of Qr yKðT Þ such that FðQr ; T Þ ¼ FðQr yKÞ and Fð1Þ ¼ 1

ð2:14Þ

The condition given by Manes et al., see Eq. (2.11), is then considered as a special case: FðQr yKÞ ¼ ðQr yKÞz

ð2:15Þ

In a subsequent paper [26], Hollingsworth states that the conditions (2.14) are not necessary although sufﬁcient. He presents other sufﬁcient conditions: f ðci ; T ; uj Þ ¼ FðQr yK; uj Þ and Fð1; uj Þ ¼ 1

ð2:16Þ

where uj stands for a set of non-thermodynamic variables. Hollingsworth then shows that the necessary condition when f has continuous derivatives of all orders at Qr yK ¼ 1 is: it must be possible to express ( f 1) as a function which is divisible by the function (Qr yK 1) in the neighbourhood of Qr yK ¼ 1: f 1 ¼ ðQr yK 1ÞCðci ; T ; uj Þ

ð2:17Þ

It should be added that in his proof the invertibility of the function ðQr yKÞðci ; T Þ is tacitly supposed (not proved). An example of practical application of Hollingsworth’s approach is given by Boyd [21]. Blum and Luus [27] proved that condition (2.11)2 is not only sufﬁcient but also necessary providing the rate law is formulated as follows: m m ? Y / Y a0 a r ¼ k j ai i k j ai i i¼1

ð2:18Þ

i¼1

where j is some function of activities, ai , of reacting species, and ai and a 0i are coefﬁcients which may differ from the stoichiometric coefﬁcients. Equation (2.18) is some general law of mass-action inspired by the Bro¨nstedt’s work www.scilet.com

Thermodynamics and mass-action chemical kinetics

11 ?

/

(see below). Boyd reproduces it [21] in more general form with k j and k j 0 , introducing thus different coefﬁcients (phi’s) for the forward and backward directions. As stated by Denbigh [21,22], empiric experience allows one to set j ¼ j 0 . Coefﬁcient j, in fact, makes provision for the dependence upon ionic strength, etc. leaving the rate constants dependent only on temperature. At equilibrium, the following relation is valid: ? /

k yk ¼

n Y

ða 0 ai Þ

ai;eq

ð2:19Þ

i¼1

The proof [27] is based on the statement that both the equilibrium constant and the ratio of the rate constants are dependent only on temperature, which enables one to express the ratio as a function of the equilibrium constant (thus, the invertibility of one of the functions is tacitly introduced): ? /

k yk ¼ f ðKÞ

ð2:20Þ

As the equilibrium activities of all species except one may be selected arbitrarily, it is shown that function f inevitably has the form f ðKÞ ¼ Kz where z ¼ ða 0i ai Þyi ; i ¼ 1; . . . ; n

ð2:21Þ

Condition (2.11)2 was derived also by Van Rysselberghe [28] after introducing afﬁnity deﬁned using chemical potential, Eq. (2.5)1 and its dependence on activity, cf. Eq. (2.5)2, into the general mass-action law, Eq. (2.18). However, this law should be now formulated with stoichiometric coefﬁcients as exponents at activities, moreover, it was also supposed that only one reaction step is kinetically signiﬁcant and the overall afﬁnity is a g-multiple of the afﬁnity of this step. Under these conditions, z ¼ 1yg. In fact, this is another example of application of the reaction isotherm in the mass-action law. Boudart [29] joined equations (2.4) written for elementary steps of a reaction with Temkin’s theory of stationary reaction rates. The following equation for the ratio of overall reaction rates in both directions is thus obtained: ? /

r y r ¼ expðAysRT Þ

ð2:22Þ

where s is the average stoichiometric number and A the afﬁnity. Using again the reaction isotherm-based argument, another relation between the rate and equilibrium constants is obtained: www.scilet.com

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Miloslav Pekarˇ

? /

k yk ¼ K1ys

ð2:23Þ

All the consistency tests seek, from the mass-action law type rate equation, relations between the equilibrium constant and ratio of rate constants. A general ‘‘consistency’’ criterion, which does not refer to any particular rate equation, has been presented by Corio [30]. Function u is deﬁned u¼K

nr Y

ci i;reactant

i¼1

n Y

i ci;product

ð2:24Þ

i¼nr þ1

where nr symbolizes the number of reactants and ci represent the concentrations. The condition of thermodynamic equilibrium is written as u ¼ 0. On the other hand, the kinetic condition may be written as r ¼ 0. These two conditions can be interpreted as equations deﬁning two surfaces in a Euclidean space of dimension n þ R, where R is the number of reactions, which should touch at a single point only, as otherwise the equilibrium state would not be unique. Consequently, the surfaces have a common tangent plane, so that corresponding derivatives at the tangential point and equilibrium are proportional: ðqryqc1 Þyðquyqc1 Þ ¼ ðqryqc2 Þyðquyqc2 Þ ¼ ¼ ðqryqcn Þyðquyqcn Þ

ð2:25Þ

Using Eq. (2.24) these equations become: ðci yi Þðqryqci Þ ðciþ1 yiþ1 Þðqryqciþ1 Þ ¼ 0

ð2:26Þ

or, alternatively ci ðqryqci Þ ¼ li

ð2:27Þ

where l is a negative constant. Equations (2.26) or (2.27) represent the consistency condition to be fulﬁlled by any rate equation (expression for r) to be consistent with thermodynamics or, more precisely, with thermodynamic equilibrium. Corio also brieﬂy discusses a modiﬁcation for non-ideal systems, where the product of activity coefﬁcient and concentration should be used instead of concentration. It is also interesting to note that an equation similar to (2.24) was given already by Denbigh [22] as an example of a rate equation consistent with thermodynamics. Denbigh also states that the two thermodynamic requirements (see above) can be fulﬁlled by the rate equation www.scilet.com

Thermodynamics and mass-action chemical kinetics

r¼y

n X ði mi Þ

13

ð2:28Þ

i¼1

where y is some positive function of concentrations and mi are the chemical potentials. The disadvantage is that the reaction rate is not directly proportional to the volume concentrations. Eq. (2.28) is closely related to the afﬁnity approaches in chemical kinetics (see part 3). In summary, consistency tests do not provide a particular rate equation (law) but just test the consistency of some proposed rate equation with the condition of thermodynamic equilibrium where the overall reaction rate should vanish.

3. AFFINITY AND CHEMICAL KINETICS 3.1 De Donder as originator Afﬁnity was introduced by de Donder [31,32] in a rather awkward and nonrigorous fashion. As his original approach is nowadays only referred to and not discussed, let us review it here brieﬂy. Starting from the ﬁrst law of thermodynamics in the form dU ¼ dQ pdV and supposing that internal energy U (as well as volume V ) is a function of pressure (p), temperature (T ), and extent of * reaction (x), U ¼ Uðp; T ; xÞ, the following relation for the differential of heat (Q) was derived: dQ ¼ hT x dp þ Cpx dT rpT dx

ð3:1Þ

where *

*

*

*

*

*

hT x ¼ ðqUyqpÞT ;x þ pðqV yqpÞT ;x Cpx ¼ ðqUyqT Þp;x þ pðqV yqT Þp;x

ð3:2Þ

rpT ¼ ðqUyqxÞp;T þ pðqV yqxÞp;T De Donder also supposed that the second law of thermodynamics could be written (according to Clausius) as T dS dQ:dQ 0 0 and that entropy was a function of the same variables. Thus dQ 0 ¼ h 0T x dp þ C 0px dT r 0pT dx

ð3:3Þ

where www.scilet.com

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Miloslav Pekarˇ *

h 0T x ¼ T ðqS yqpÞT ;x hT x *

C 0px ¼ T ðqS yqT Þp;x Cpx

ð3:4Þ

*

r 0pT ¼ T ðqS yqxÞp;T þ rpT From Eq. (3.3) de Donder derived dQ 0 y dx ¼ h 0T x dpy dx þ C 0px dT y dx r 0pT

ð3:5Þ

Next he introduced the key hypothesis which is neither well substantiated nor supported: the derivative dQ 0 ydx has a constant value regardless of changes in p and T during the course of a reaction, which are dependent on x. There is no explicit motivation for this hypothesis, moreover, among the three independent variables there appears one which is ‘‘more independent’’ and governs the changes of the other two variables. From this hypothesis de Donder derived h 0T x ¼ 0

ð3:6Þ

C 0px ¼ 0 and deﬁned afﬁnity as A ¼ dQ 0 ydx: r 0xy ;

ð3:7Þ

where xy stands for the two (constant) independent variables other than the extent of reaction. The reason why de Donder’s afﬁnity often ‘‘works’’ lies probably in that it is applied under conditions where some quantities are constant, as indicated by Eq. (3.7) so the conditions (3.6) are superﬂuous. Further, afﬁnity can be related to the chemical potential which is also deﬁned by several alternative relations under conditions of constant various pairs of independent variables while not changing its value. For example, the afﬁnity of a reaction is simply given by the ﬁrst relation in (2.5). Expressing the total differential of the Gibbs energy as a * function of temperature, pressure and composition, G ¼ GðT ; p; ni Þ, using the extent of reaction as de Donder suggested, we obtain: X X mi dni ¼ i mi dx ¼ A dx ðconstant T and pÞ dG ¼ i

ð3:8Þ

i

As at constant temperature and pressure, heat is identical with the change of enthalpy (H), dQ 0 ¼ dG under these conditions and Eq. (3.7) is derived with no need for this strange hypothesis. www.scilet.com

Thermodynamics and mass-action chemical kinetics

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In fact, de Donder re-labelled some variables of classical thermodynamics and his main contribution consists of noticing that extent of reaction can be used as an independent variable instead of molar masses, concentrations and so on. However, the extent of reaction may not be useful in complex reactions. In this case, changes in molar amount of some or even all components may be caused by more than one reaction. The extent of reaction should be then deﬁned for every reaction step including only molar changes caused by the corresponding step. While this can be done in theory with no problem, it is useless in practice where molar changes caused by individual reactions cannot be always simply measured. As pointed by, e.g. Hollingsworth [33], it is often impossible to deﬁne some overall extent of reaction. Bowen has proved [34] that the extent of reaction cannot be used in reacting mixtures with diffusion unless the diffusion is so-called self-balancing [35]. As regards chemical kinetics, de Donder deduced from the second law and (3.8) 0 dQ 0 ydt ¼ AðdxydtÞ:Ar

ð3:9Þ

where r is the reaction rate.

3.2 Successors to De Donder Most applications of afﬁnity in chemical kinetics are, in fact, deductions based on the reaction isotherm outlined in Section 1. The ﬁrst work from this area is probably the paper by Prigogine et al. [36], which also refers to de Donder’s work. They started from the assumption that both reaction rate (r) and afﬁnity (A) depend on the same variables (xi ) and that the function for afﬁnity is invertible in at least one variable. Substituting this variable in the function for reaction rate, the following relationship results: r ¼ f ðx1 ; x2 ; . . . ; xl ; AÞ

ð3:10Þ

(variable xlþ1 was substituted). At equilibrium, both reaction rate and afﬁnity vanish. Expanding the function in (3.10), a close-to-equilibrium linear relationship is thus obtained: r ¼ ðqf yqAÞxi A

ð3:11Þ www.scilet.com

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Miloslav Pekarˇ

(so that the expansion was made keeping all xi constant!). Making use of manipulations with the Guldberg – Waage law and reaction isotherm (see part 2), this linear relation is illustrated by the linear relationships for the hydrogenation of benzene and dehydrogenation of cyclohexane. A subsequent paper by Manes et al. [24] derived the linear relationship in a somewhat more general fashion. The authors supposed that the reaction Gibbs energy (G) depends on some set of independent variables (aj ; j ¼ 1; 2; . . . ; m) and that the reaction rate depends on the same variables and some added, ‘‘nonthermodynamic’’ ones (bk ; k ¼ 1; 2; . . .). Using again the vanishing of the Gibbs energy and reaction rate at equilibrium simultaneously, they arrived at an equation valid sufﬁciently close to equilibrium: r ¼ xðaj ; bk Þ DG

ð3:12Þ

where the proportionality factor represents: xðaj ; bk Þ ¼ ½qryqðDGÞa2 ;a3 ;...;am ¼ ½qryqðDGÞa1 ;a3 ;...;am ¼ . . . ¼ ½qryqðDGÞa1 ;a2 ;...;am1 ð3:13Þ and depends on full sets of aj and bk . In the derivation, the implicit assumption on the invertibility of the reaction Gibbs energy function is hidden. Their thermodynamic approach gives no explicit relation for the proportionality factor. The authors also point that because x depends also on non-thermodynamic variables, Eq. (3.12) cannot be used to obtain absolute rates from thermodynamic data. How this could be achieved, when knowing the values of the additional variables, is not discussed. Another illustration of the application of the reaction isotherm and afﬁnity in chemical kinetics is given in the paper by Hall [37], which forms a part of the polemic between Haase and Hall mainly on kinetics in non-ideal systems and is therefore reviewed in part 4. Nebeker and Pings [38] tried to conﬁrm experimentally the linear relationship between afﬁnity and reaction rate. They measured the concentrations of components in a reacting mixture of NO, Cl2, NOCl, I2, and ICl. Two reactions were considered, viz.: 2 NO þ Cl2 ¼ 2 NOCl

ð3:14aÞ

2 NOCl þ I2 ¼ 2 NO þ 2 ICl

ð3:14bÞ www.scilet.com

Thermodynamics and mass-action chemical kinetics

17

Of course, afﬁnities were not measured but calculated from the reaction isotherm and concentration proﬁles. Rates of reactions (3.14a) and (3.14b) were taken as time derivatives of the chlorine and iodine concentrations. It was found that, for some portions of a run of the reacting system, the linear relationship is valid. In general, however, it was not veriﬁed as well as the socalled Onsager reciprocity relations, which are not discussed here. A linear relationship between reaction rate and afﬁnity near equilibrium was also derived by Gilkerson et al. [39] from the theory of absolute reaction rates. They identiﬁed the reaction Gibbs energy DGr ð:AÞ with DG6¼ r , i.e. the activation Gibbs energy, which might be questionable. Boudart shows in several papers more precisely the potential practical value of afﬁnity-containing equations in chemical kinetics. He distinguishes [40] between the de Donder inequality: Ar 0

ð3:15Þ

and de Donder equation: ? /

lnð r y r Þ ¼ AyRT

ð3:16Þ

Because Eq. (3.15) is valid for the overall reaction process, it may explain why some reaction steps may occur against the ‘‘thermodynamic direction’’ [41]. For instance, two reactions may occur simultaneously even when A1 r1 50

ð3:17Þ

providing that A1 r1 þ A2 r2 40

ð3:18Þ

It is said that reaction 1 is coupled to (driven by) the second one. Boudart shows [40] that this may be a useless idea, as the coupled reaction in many real cases does not proceed. Boudart argues that, in a reaction system consisting of a closed sequence of elementary reactions, at the steady state for each of the steps it is the case that: ?

/

r ¼ r i r i 40

ð3:19Þ

and from Eq. (3.15), which is valid for any step i with afﬁnity Ai , it follows that: Ai 40;

Ai ri 40

ð3:20Þ www.scilet.com

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Miloslav Pekarˇ

for all steps. It should be stressed that, in the case of more reactions than just one, there is no thermodynamic requirement (3.15) to be valid for any of these reactions separately. Only the sum of afﬁnity and rate products must be nonnegative as Eq. (3.18) illustrates. Inequalities (3.20)2 are just due to relations (3.19). But ‘‘kinetic coupling’’ may occur which can change substantially the steady-state concentrations of intermediates above their equilibrium value, if they are reactants, or below their equilibrium value if they are products. Particular examples are given in Boudart’s papers [40,42]. In a short report [43] Boudart et al. show, using the data by Prigogine et al. [36], that the linear relationship between reaction rate and afﬁnity remains valid also relatively far from equilibrium. Dumesic has published an analysis of the reaction scheme using ‘‘de Donder relations’’ [44]. It is claimed that the rate equation is derived from the reaction scheme in terms of these relations. In fact, all the results can be simply arrived at using traditional thermodynamics and kinetics. Further, the analysis is applicable only to stationary states and mechanisms in which the overall reaction is given as a sum of elementary reactions. The central quantity is the reversibility of the i-th step (elementary reaction) deﬁned by: ! Y ij zi ¼ expðAi yRT Þ ¼ aj yKi

ð3:21Þ

j

where Ai is the step afﬁnity, aj the activity of the j-th component and ij its stoichiometric coefﬁcient in the i-th step with the equilibrium constant Ki . Eq. (3.21) directly follows from the reaction isotherm. Reversibility was deﬁned by us (in ideal systems) the relative distance from equilibrium and shown to be useful to follow the evolution of reaction rates even in the non-steady state [45 – 47]. By the de Donder relation Eq. (2.6)2 is understood, and its exponential appears in (3.21). In fact, only (3.21)2 is used in the analysis and to calculate the reversibility. The rate equation is not derived but step rates are stated as massaction laws with activities instead of concentrations, Eqs (3.21) are used to eliminate the activities of intermediate species and analysis further continues within the idea of a rate-determining step. What could be done quite easily is complicated here by forcing deductions into the framework of the de Donder relation. For instance, it is ‘‘revealed’’ that the minimum number of kinetic parameters required to calculate the rate for three-step mechanism is equal to three, in contrast to expectation that ﬁve would be required, because all six step www.scilet.com

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rate constants are bounded by the total equilibrium constant. However, if it is realized that rate constants of each step are related by the kinetic equilibrium constant of the step, it immediately follows that only three kinetic parameters are necessary (and selectable independently). Reversibilities for each step are calculated from experimental data. Steps with close-to-one reversibility are (quasi-)equilibrated. If there is a step with reversibility far from a zero value, then this step is considered to be rate determining, and the overall reaction reversibility is equated to its reversibility whereas the other reversibilities are identiﬁed with unity. The overall rate is set equal to the rate-determining step rate. The whole procedure closely resembles the classical Langmuir – Hinshelwood – Hougen – Watson approach. This is felt also by the author as he states that his approach is advantageous because it provides the means to derive the overall reaction rate from the more general case where multiple steps are not in quasi-equilibrium. In fact, this means only that equilibrium constants of equilibrated steps, together with the overall equilibrium constant given as appropriate product of steps equilibrium constants, are used to eliminate intermediate activities. Let us illustrate this approach by the simple example of the three-step mechanism R 1 ¼ 2 I1 R2 þ I1 ¼ I2 I1 þ I2 ¼ P of the overall reaction R1 þ R2 ¼ P The rate of the ﬁrst step can be expressed as [44]: ?

r1 ¼ k 1 aR1 ð1 z1 Þ

ð3:22Þ

where z1 is given as follows from Eq. (3.21): z1 ¼ a2I1 yðK1 aR1 Þ

ð3:23Þ

If this step is rate-determining, then the overall rate (r) is equal to r1 . As the total reversibility (z) is given by www.scilet.com

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Miloslav Pekarˇ

z ¼ z1 z2 z3 ¼ aP yðKaR1 aR2 Þ

ð3:24Þ

and z2 , z3 are in this case equal to unity, it follows that ?

r ¼ k 1 aR1 ½1 aP yðKaR1 aR2 Þ

ð3:25Þ

This result can be derived by the usual procedure without reversibility or de Donder relations. Actually, in this example, the rate is given by: ?

/

r ¼ r1 ¼ k 1 aR1 k 1 a2I1

ð3:26Þ

From equilibrium constants of (quasi-)equilibrated steps 2 and 3: K2 ¼ aI2 yðaI1 aR2 Þ;

K3 ¼ aP yðaI1 aI2 Þ

ð3:27Þ

it can be easily derived: a2I1 ¼ aP yðaR2 K2 K3 Þ

ð3:28Þ

Introducing Eq. (3.28) into Eq. (3.26) and using the kinetic deﬁnition of equilibrium constant K1 and the relation K ¼ K1 K2 K3 , Eq. (3.25) is obtained. The very essence of Dumesic’s analysis can be reported in this way. Measure the values of equilibrium constants of elementary steps of interest or measure their rate constants and calculate equilibrium constants from them. Measure stationary concentrations (more rigorously, activities) and calculate reaction quotients from them. Compare all corresponding quotients and equilibrium constants to identify quasi-equilibrated steps. Use equilibrium constants of these steps to eliminate some (intermediates) concentrations. Set the overall rate to be equal to the rate of (some) non-equilibrated step. And make this analysis in terms of reversibilities and afﬁnities. There is nothing special to the thermodynamic analysis of chemical kinetics except comparing the actual stationary state of reacting system with its state of equilibrium. The principles of Dumesic’s analysis were combined by Fishtik and Datta [48] with their method of analysis and simpliﬁcation of reaction mechanisms, which is beyond the scope of this review. It should be only pointed that by the de Donder relations not only Eqs. (2.6)2 but also mass-action law expressions for forward reactions are understood in their paper. In principle, the relations are again used to eliminate the concentrations of intermediates. Afﬁnity is deﬁned in such a way that it directly accords with mass-action kinetics, viz. in concentrawww.scilet.com

Thermodynamics and mass-action chemical kinetics

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tions (more precisely, surface coverages and partial pressures) instead of activities. Timmermann [49] asserts that he obtained the general formula relating reaction rate and afﬁnity, and a general and rigorous statement of the thermodynamic restrictions on reaction rate is thus given. His proof is based only on the argument that the rate of increase of the extent of reaction has a unique value independent of the particular language used to describe the reaction and the afﬁnity. However, the key point of his proof is unclear. Timmermann deﬁnes the gross reaction rate (r) as the rate of increase of the extent of reaction (x): r ¼ dxydt ¼ dni yði dtÞ

ð3:29Þ

where ni is the amount of substance i in the whole system and i its stoichiometric coefﬁcient. Timmermann further states that the gross rate is generally not determined in a kinetic experiment. Instead, an intensive quantity is measured, which is the gross rate normalized to some extensive reference quantity. Two from several of Timmermann’s examples are reproduced here. The most common reference quantity is the volume of the system (V ) and the intensive reaction rate is then expressed as: rc ¼ ryV

ð3:30Þ

When the molality (m) reference basis is selected, we have: rm ¼ ryðn0 M0 Þ

ð3:31Þ

where n0 is the mole number of the solvent and M0 its molar mass. Clearly, r c V ¼ r m n0 M 0 Timmermann

ð3:32Þ

combines the classical mass-action rate equation ? P / P / where ¼ k c i ci i and r c ¼ k c j cj j (i runs through reactants, rc ¼ r c P j through products), with the classical deﬁnition of afﬁnity A ¼ k mk (k runs ?

/ r c,

then

? rc

k

through both reactants and products). Chemical potential (mk ) is expressed also traditionally, mk ¼ mok þ RT lnðgk ck yco Þ where ‘‘o ’’ denotes the standard state and gk is the activity coefﬁcient on the molarity scale. Timmermann ﬁnally arrives at the following expression: www.scilet.com

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Miloslav Pekarˇ

0

1

/

o

C k c Kg ðc Þ ? B C rc ¼ r c B @1 ? Q k expðAyRT ÞA k c gk

ð3:33Þ

k

where Kg is the thermodynamic equilibrium constant on the molarity scale and P ¼ k . He states that r cannot depend on the particular language used to describe the intensive reaction rate (i.e. on the referential quantity), consequently, the factor in Eq. (3.33) must be the same for every kinetic description, that is unity: /

k c Kgðco Þ ¼1 ? Q k c gkk

ð3:34Þ

k

This condition is acceptable as general at equilibrium with vanishing of both the gross rate and afﬁnity. Timmermann gives no explicit proof for its general validity (out of equilibrium) and his statement on the independence of the particular language is unclear as will be now shown. Consider his other example – molality scale. He derives the following alternative rate equation: 0

1

/

o

C k m Kj ðm Þ ? B C rm ¼ r m B @1 ? Q k expðAyRT ÞA k m jk

ð3:35Þ

k

where Kj is the thermodynamic equilibrium constant and jk the activity coefﬁcient on the molality scale this time. If Eqs (3.33) and (3.35) are substituted into Eq. (3.32) and if it is realized that Eq. (3.32) is valid also for forward or reverse rates, the following condition for ‘‘independence of particular language’’ is obtained: /

/

k c Kg ðco Þ k m Kj ðmo Þ ¼ ? Q ? Q k c gkk k m jkk k

ð3:36Þ

k

It is not clear why condition (3.36) is not sufﬁcient and why both fractions should be in addition equal to one everywhere. It seems that Timmermann’s condition (3.34) is unwarrantedly restrictive and his analysis questionable. www.scilet.com

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3.3 Garﬁnkle’s original approach Yet another approach to afﬁnity in relation to reaction kinetics was presented by Garﬁnkle. Actually, he takes the time derivative (symbolized by a dot) of the reaction isotherm written in terms of afﬁnity (A) instead of the Gibbs energy (and with concentrations approximating to activities) [50]: X _ ¼ RT A ð2i yci Þðdci yi dtÞ

ð3:37Þ

i

(i is the stoichiometric coefﬁcient and ci the concentration of the i-th component). According to Garﬁnkle, the term in the second parentheses is the reaction velocity r. After rearrangement, an equation relating reaction rate to _ ) is obtained: the afﬁnity decay rate (A _ yRT Þy r ¼ ðA

X 2i yci

ð3:38Þ

i

Because it is difﬁcult to obtain the afﬁnity decay rate directly, Garﬁnkle introduces an empirical relation between this quantity and the elapsed time of reaction (t): _ ¼ Ar ð1yt 1ytK Þ A

ð3:39Þ

where Ar and tK are parameters to be determined. The latter is called the mostprobable time to attain equilibrium and the meaning of both is discussed in the original papers, particularly ref. [51]. In practice, one must know the equilibrium constant of the reaction under study and the values of the reaction quotient at various reaction times. The latter is calculated from the measured concentration time proﬁles. From the reaction quotient and equilibrium constant, the afﬁnity is calculated and then a regression analysis devised by Garﬁnkle [51] is used to obtained the parameters of Eq. (3.39). Thus, the afﬁnity decay rate can be obtained and from it, using the concentrations of reacting species, the reaction rate at an appropriate instant in time can be calculated from Eq. (3.38). Garﬁnkle’s papers contain examples of afﬁnity or rate time proﬁles for many reactions and their comparison with conventional, mass-action rate equations. Garﬁnkle also shows [52,53] that for a (homogeneous) chemical reaction (in a closed isothermal system), there exists a unique natural path along which the rate of change in time of a thermodynamic function can be described. This, in fact, www.scilet.com

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Miloslav Pekarˇ

means that instead of reporting time proﬁles of concentrations (or, perhaps, reaction rate or afﬁnity), afﬁnity should be represented as a function of the following quantity: ln½ðtytK Þ expð1 tytK Þ, which appears in the integrated form of Eq. (3.39). Garﬁnkle shows that even for a reaction with ‘‘mechanistic differences’’, i.e. with different concentration time proﬁles (e.g. iodine atom recombination in different inert gases), it will have a unique natural path for afﬁnity. Garﬁnkle’s approach was criticized in details by Hjelmfelt et al. [54], Garﬁnkle responded in ref. [55]. We will not report here on this polemic and merely add some comments. First, it should be remembered that this method can be used only in closed isothermal systems where the reaction rate is directly given by the concentration time derivative. Second, it is limited only to the cases where the reaction rate is given by the time derivative of any reacting specie, i.e. where some overall reaction rate exists, to the stoichiometric systems. As Garﬁnkle states [55]: ‘‘The concentrations of reactants and products appearing in the stoichiometric equation that represents the overall chemical reaction under observation changes with elapsed time... The rate of change of these concentrations consistent with stoichiometric constratints is the reaction velocity...’’ As an example he gives the addition of iodine to styrene (St), I2 þ St ?IStI with a velocity deﬁned as r ¼ d½Stydt ¼ d½I2 ydt ¼ d½IStIydt

ð3:40Þ

where the square brackets symbolize concentrations. This deﬁnition supposes that product (IStI) appears immediately after the disappearing of reactants. This is generally not the case in reactions with a detailed mechanism [56], which is signiﬁcant for the concentration evolution of especially reaction intermediates. As an illustration, one of the simplest mechanisms can be used. Let us suppose that some general transformation A ?C goes through an intermediate B: A ?B ?C. From classical kinetics it follows that: dcA ydt ¼ k1 cA dcB ydt ¼ k1 cA k2 cB

ð3:41Þ

dcC ydt ¼ k2 cB where k1 is the rate constant of the step A ?B and k2 of the step B ?C. It is clear that the time derivatives are not in general equal, which is even more evident after inserting the analytical solutions: www.scilet.com

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dcA ydt ¼ k1 c0A expðk1 tÞ dcB ydt ¼ k1 c0A expðk1 tÞ k1 k2 c0A ½expðk1 tÞ expðk2 tÞyðk2 k1 Þ dcC ydt ¼

k1 k2 c0A ½expðk1 tÞ

ð3:42Þ

expðk2 tÞyðk2 k1 Þ

where 0 in the superscript denotes the initial concentration. So there is, in general, no simple single rate expression for the overall stoichiometric transformation A ?C and no identity dcA ydt ¼ dcC ydt. Only when k2 4 k1 can the last equation (3.42) be transformed practically to fulﬁl this identity. Equation (3.38) is not an expression of reaction rate as a function of afﬁnity decay rate but an expression of function of afﬁnity decay rate and concentrations, because they are also changing during the course of reaction and, in fact, determine the afﬁnity. Garﬁnkle presents an analysis of experimental data of many, essentially stoichiometric, reactions in terms of afﬁnity decay rate. He succeeded very well in ﬁtting experimental data translated into the reaction quotient by his Eq. (3.39). What is the value of this approach? Conventionally, concentrations are measured, and a kinetic-mechanistic model proposed and used to interpret the data. Rate expressions are obtained which can be used as rates of formation, e.g. in reactor balance equations to make its design possible. Afﬁnity decay methodology transforms concentrations to afﬁnity, the decay of which is ﬁtted by Eq. (3.39), and the decay rates may then be used to calculate reaction rate from Eq. (3.38). Garﬁnkle stresses that his approach gives correlations independent of reaction mechanism and, in contrast to the conventional description in terms of the time-dependency of the concentration of reacting components, it describes kinetic behaviour in terms of the time-dependency of a thermodynamic function. His approach could be viewed as an alternative of a data-ﬁtting procedure in closed isothermal systems with an unambiguously deﬁned and conﬁrmed overall reaction rate. Afﬁnity decay then describes the course of reaction not in terms of concentrations changing in time, i.e. in kinetic terms, but in terms of a thermodynamic quantity changing in time, i.e. in ‘‘energetic’’ terms. Although the kinetic details may be different even for very similar reactions (e.g. iodine atom recombination in different inert gases [52,53]), thermodynamic principles are general and really give identical decay curves for such reactions. The existence of a unique natural path is an interesting theoretical phenomenon and conﬁrmation of correctness of the reaction isotherm in stoichiometric systems. The natural path scales both the concentrations of reacting species and the elapsed reaction time. The former, through the afﬁnity www.scilet.com

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Miloslav Pekarˇ

embodying the reaction quotient and the equilibrium constant, which, in turn, contains equilibrium concentrations, the latter through the parameter tK , i.e. the most probable time of attaining equilibrium. As any chemical reaction proceeds from some initial concentrations and time to equilibrium concentrations and time, it may be expected that such ‘‘scaling to equilibrium’’ will work. 3.4 Critical slowing; linearity testing Afﬁnity- and reaction isotherm-based approaches have found some popularity in the interpretation of the slowing down of chemical reactions near some critical point, see e.g. refs [59 – 62]. Actually, the ‘‘linear’’ relationship (2.7) is used [59,60] for qualitative interpretations, not for quantitative evaluations. Recently, Kim and Baird [62] reported even a speeding up near the critical point. Several approximations are used, the nature of which is clearly seen from an inspiring older work by Meixner [63]. Meixner claims that the close-to-equilibrium reaction rate is expressed as dxydt and given by: dxydt ¼ eðT ; ; xÞAðT ; ; xÞ

ð3:43Þ

where x is the extent of reaction, e is the proportionality coefﬁcient dependent on temperature (T ), speciﬁc volume () and extent of reaction, and A is the afﬁnity determined by the same set of variables. First, Meixner states that the close-toequilibrium dependence on the extent of reaction in the functional expression for the coefﬁcient e in (3.43) can be abandoned by substituting its equilibrium value (xe ). Next, he expands the afﬁnity at constant temperature and speciﬁc volume up to the ﬁrst order: dxydt ¼ eðqAyqxÞT ; ½x xe ðT ; Þ

ð3:44Þ

Why the dependence on the extent of reaction is suppressed only in the ﬁrst function from (3.43), and why only the second one, afﬁnity, is expanded, is neither explained nor discussed. Coefﬁcient e in (3.44) is thus effectively a constant, which is stated, e.g. by Procaccia and Gitterman [60], as a fact at the outset. Kim and Baird [62] present a more correct derivation and expand, in fact, both functions in (3.43). In the end, however, they retain only the terms of ﬁrst order and arrive at Eq. (3.44) once more. From their procedure, the motivation for Meixner’s inconsequent treatment of functions can be clariﬁed a little. From Eq. (2.7) it is clear that coefﬁcient e is the forward reaction rate [62], which is www.scilet.com

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non-zero at equilibrium in contrast to the afﬁnity. Consequently, the ﬁrst term in the forward rate (or coefﬁcient e) expansion is non-zero whereas that in the afﬁnity expansion vanishes. What does an approximation like (3.44) using the equilibrium forward rate as a constant not-far-from-equilibrium mean in reality? From the more general Eqs (2.4) or (3.16), it is seen that within this approximation, the afﬁnity / at a given temperature is given by const RT ln r . All afﬁnity and, consequently, overall rate changes and evolution should be then governed by the reversed rate. This is also conﬁrmed by the expansion of (3.47) below. Even then it is rather arduous to accept that the backward rate changes markedly while the forward remains constant. Kim and Baird [62] claim even that the reaction they studied was essentially irreversible. From another point of view, the approximation used in (3.44) means a much slower approach (usually decrease) of the forward rate to its equilibrium value than afﬁnity decay to the equilibrium zero value. Rates of both decays are dictated by the values of the relevant concentrations. Decay of afﬁnity, anyway, corresponds to a decaying logarithm with the argument approaching to one, and it should be realized that whereas a logarithm is a ‘‘magnitude smoothing’’ function above one, at values very close to one it is a magnitude ampliﬁer. This elementary fact is illustrated by numbers given in Table 1, cf. also Eq. (3.16). Far from equilibrium, when the reaction rate in one direction, at least, is changing over several orders of magnitude, the afﬁnity decays by about only one order of magnitude. An afﬁnity decrease amounting to many orders of magnitude is not noticed before being very close to equilibrium when the rates in both directions are almost the same. Table 1 also models approximation (3.44) – if the forward reaction rate is considered to be constant, e.g. ﬁxed at its equilibrium value, than all changes of the ratio given in the ﬁrst column of the table are due to an increasing reverse rate on the approach to equilibrium. Consequently, when the reverse rate changes appreciably, the afﬁnity decreases (with extent of reaction) only slowly, whereas when the backward rate (and, consequently, the overall rate) almost attains its equilibrium value before the steep decay of afﬁnity starts. Perhaps Table 1 gives some answer to the question as to how far from equilibrium is too far [64]. On the other hand, should the numbers in the table mean that far from equilibrium, within a convenient time interval, the reaction rate could be approximated by ?

equation dxydt ¼ eðT ; ; xÞ const ½x x0 ðT ; Þ where e: r is not constant and the subscript ‘‘0’’ denotes some point within this interval? www.scilet.com

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Miloslav Pekarˇ Table 1

Decay of logarithm and its argument

!

!

ryr

lnð r y r Þ

1.0000000000E þ 10 1.0000000000E þ 09 1.0000000000E þ 08 1.0000000000E þ 07 1.0000000000E þ 06 1.0000000000E þ 05 1.0000000000E þ 04 1.0000000000E þ 03 1.0000000000E þ 02 1.0000000000E þ 01 1.1000000000E þ 00 1.0100000000E þ 00 1.0010000000E þ 00 1.0000001000E þ 00 1.0000000001E þ 00 1.0000000000E þ 00

23.03 20.72 18.42 16.12 13.82 11.51 9.210 6.908 4.605 2.303 9.531E-02 9.950E-03 9.995E-04 1.000E-07 1.000E-10 0.000

Our model calculations [45,47,65] demonstrated that (in ﬂow systems) the overall reaction rate can change appreciably even when the reaction is still very close to equilibrium (reaction quotient almost equal to one), its value can change abruptly just before reading equilibrium, or that both overall rate and afﬁnity may undergo steep changes close to equilibrium. In some cases the overall rate was even increasing at the same time as the ratio of reaction quotients and equilibrium constant approached to unity [66]. It should be also stressed that approximation (3.36) does not express the reaction rate as a function of afﬁnity partial derivative only but as a function of this derivative and extent of reaction. Linear approximations like (3.36) seem to be the result only of numerical trickiness in the logarithm and not consequences of some genuine thermodynamic principles. Experimental veriﬁcation of approximations involved in afﬁnity-rate deductions is still missing. Data by Prigogine et al. [36] show that the linear relationship between afﬁnity and reaction rate is valid also for values not fulﬁlling the inequality AyRT 5 1 (cf. Part 1.). The highest value of this ratio lying in the linear region is reported to be 2.3. Full revision of this paper is postponed to some future work, here only a short note is given. There must be some mathematical reason as it was the mathematical expansion of the exponential function, which enabled the disclosure of the linear relationship, cf. Eqs (2.6), (2.7), and not some ‘‘effort’’ of the reaction to keep linearity far from equilibrium. This is illustrated in Table 2. It is evident that the linear www.scilet.com

Thermodynamics and mass-action chemical kinetics

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Table 2 Comparison of exponential and the ﬁrst three terms of its series expansion x

0.01

0.1

1

2

expðxÞ 1x 1 x þ x2 y2 1 x þ x2 y2 x3 y6

0.99005 0.99000 0.99005 0.99005

0.90484 0.90000 0.90500 0.90483

0.36788 0.00000 0.50000 0.33333

0.13534 1.00000 1.00000 0.33333

approximation starting from an argument value equal to one, at least, is a nonsense. Let us analyze the reaction isotherm from the logarithmic side. If thermodynamic and kinetic equilibrium constants are identiﬁed, as necessary, Eq. (2.6) can be rewritten: ?

/

/

/

A ¼ RT ln K RT ln Q ¼ RT ln KyQ ¼ RT lnð r y r Þ ¼ RT ln½ðrþ r Þy r ¼ /

¼ RT lnðry r þ1Þ:RT lnðx þ 1Þ ¼ RT ðx x2 y2 þ x3 y3 x4 y4 þ Þ ð3:45Þ The expansion in Eq. (3.45) is valid only for 15x 1. From Eq. (3.45) it is better seen than from the last equality in (2.7) that the linear relationship between afﬁnity and rate is determined also by the rate in the reverse direction. The linear term in (3.45) can only be retained in the case when the ratio of the overall and reverse rates (x) is sufﬁciently small. In fact, Eq. (2.7) does not lead to a strict linear relationship unless the reverse rate is constant. Eq. (3.45) shows that the linear approximation may be acceptable regardless of the distance from equilibrium. For instance, if the overall rate has a formal value of 103, which is surely quite far from equilibrium, and the backward rate is 105, then the second order term gives less than 1% correction to the linear term. This short example is limited by the validity of the expansion used in Eq. (3.45) as stated above. In general, the logarithm can be expanded for all values of its argument (x40) in the following way: ln x ¼ 2ðy þ y3 y3 þ y5 y5 þ Þ; ?

where y ¼ ðx 1Þyðx þ 1Þ

ð3:46Þ

/

In our case x: r y r . From Eq. (3.46) then follows: h ? / i ? / ? / / A ¼ RT ln r y r ¼ RT 2 ð r y r 1Þyð r y r þ1 þ ¼ 2RTryðr þ 2 r Þ þ ð3:47Þ www.scilet.com

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Miloslav Pekarˇ

Thus, even the ﬁrst term is not linear in general. A linear relationship between afﬁnity and the overall rate can be obtained only if the ﬁrst term in approxima/

tion (3.47) is sufﬁcient and if r þ 2 r is constant. The latter condition can be ? / reformulated as r þ r ¼ const., which is easily imagined to be fulﬁlled in practice, because the forward rate is decreasing while the backward rate is increasing in the same time. 3.5 Summary The main problem of most afﬁnity-based approaches is that they are used for interpretation rather than for a theoretical explanation of experimental data. This is because afﬁnity usually cannot be measured. Concentrations (partial pressures, activities, etc.) are those quantities, which are measured by kineticists, and only from these quantities are afﬁnities calculated. The only exception is perhaps a reaction in a galvanic cell where the measured electromotive force (E) is directly related to afﬁnity through the well-known equation A ¼ zFE, where z is number of exchanged electrons and F is Faraday’s constant. Even in this case, if afﬁnity should be related to the reaction rate, concentrations (activities) within the cell should be utilised, i.e. the Nernst equation, which is a variant of the reaction isotherm. Thus in examples like that of Prigogine et al. [36], neither the afﬁnity nor reaction rate were directly and independently measured. Concentrations (composition) were determined and from them the rate and afﬁnity were computed. Afﬁnity-velocity linear tests are then no more than checking that concentrations behave in the manner predicted by the reaction isotherm. Equations (2.6) and (2.7) cannot be viewed as the function r ¼ f ðAÞ but as ? / ? a transformation of the function r ¼ f ð r ; r Þ to function r ¼ gð r ; AÞ using the reaction isotherm. Table 1 clearly illustrates that afﬁnity by itself is a problematic measure or determining quantity for reaction rate because it does not vary too much when the rate undergoes steep changes and vice versa. Afﬁnity or reaction Gibbs or free energy alone does not determine the reaction rate, or kinetic ‘‘driving force’’. Water synthesis from molecular oxygen and hydrogen is a notoriously well-known example – its (standard) reaction Gibbs energy amounts to several hundreds kJ but its reaction rate is negligible unless some external catalytic action is introduced. It follows from the reaction isotherm that any reaction mixture containing only reactants possesses in zero time an

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Thermodynamics and mass-action chemical kinetics

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inﬁnitely high afﬁnity but experimental evidence clearly shows that initial rates have ﬁnite and diverse values. Additional and very important information on the relation between afﬁnity and reaction rate is also provided by rational thermodynamics. For consistency, this is postponed to Section 7. 4. ACTIVITIES IN CHEMICAL KINETICS Rigorous thermodynamic treatments are given in activities. By contrast, kineticists prefer concentrations, and activities are rarely used. Proposals to replace concentrations in kinetic equations simply with activities appeared immediately after activities had been introduced by Lewis at the beginning of the 20th century. As expected, this substitution was being made particularly in ionic reactions where particle interactions are natural. Reviewing ionic reactions, salt effects etc., is beyond the scope of this review, because it can be found in many textbooks, e.g. refs [67, 68]. We will restrict ourselves here solely to the principal historical roots and modern work directly related to mass-action kinetics. Jones and Lewis [69] measured the rate of inversion of sucrose. Having estimated the unimolecular rate constant, they found its dependence on the initial concentrations of sugar and water. They measured also the activity of hydrogen ions using an electrochemical cell. Dividing the unimolecular constant by the hydrogen ion activity and water concentration, they obtained a constant value. In subsequent work, Moran and Lewis [70] also determined the activity of sucrose and water but the activity-based rate constants were not independent of the initial concentration of sucrose. The authors further developed a more elaborate approach including the effect of viscosity on the reaction rate. Livingston and Bray [71] studied the catalytic decomposition of hydrogen peroxide in a bromine-bromide solution. Substituting ion concentrations with activities (products of ion concentration and activity coefﬁcient) in the rate equation r ¼ kcH2 O2 cHþ cBr , they found a concentration-independent rate constant in most experiments, in contrast to the original rate equation. Later, Livingston reported [72] that the activity-based rate equation is valid only in solutions with an ionic strength less than unity. Scatchard [73, 74] carefully analyzed the issue arising from the sucrose inversion where discrepancies described in the above paragraphs, between theoretical and experimental proportionality of reaction rate and concentrawww.scilet.com

32

Miloslav Pekarˇ

tions, were found. He starts from the usual kinetic theory of the reaction in a perfect gas: the reaction rate is proportional to the concentration of each reacting species raised to the power which represents the number of molecules of that species which enter the reaction according to the stoichiometric equation. This gives the mass-action law rate equation, which, by analogy, is applied also in solutions. As thermodynamics formulates equilibrium constant in activities, and the ratio of the forward and reverse reaction rates must give the same equilibrium constant, it is much more logical, Scatchard argues [73], to express the reaction rate in activities. However, Scatchard is well aware of the dimensionality hitch, see Part 2. If the rate of sucrose inversion is accordingly formulated as: r ¼ kaaw abh acs

ð4:1Þ

where aw is the activity of water, ah the activity of hydrogen ion, and as the activity of sucrose (in Scatchard’s terminology, k is only a proportionality factor), Scatchard asks in what units should it be expressed, having to be measured by the number of molecules which react in unit time. The answer is not straightforward. First, the concept of a semi-ideal solution is introduced, which principal characteristic is that the activity of each component is proportional to the molar fraction of that component which actually exists in the solution. In fact, this means that thermodynamic environment is independent of concentration. Second, it is proposed to use the volume containing one mole as the ‘‘reaction volume’’ characteristic, which is the analogy of replacing ideal gas volume by the free volume to correct for non-idealities. Scatchard then concludes that the rate (r) is measured by the number of molecules transformed in unit time in the volume which contains one mole of total substance, i.e. r ¼ ðdxydtÞyC, where x is the transformed number in a litre and C is the total number of moles per litre. As an example, in the case of measuring the rate of sucrose inversion by its disappearance and considering that (reaction order) c ¼ 1, Eq. (4.1) is modiﬁed to: r ¼ ðdxydtÞyC ¼ kaaw abh ðC0s xÞyC or dxydt ¼ kaaw abh ðC0s xÞ

ð4:2Þ

(C0s is initial sucrose concentration). www.scilet.com

Thermodynamics and mass-action chemical kinetics

33

Although both Scatchard’s suppositions are rather operational and apparently formal, they are much better than simple replacement of (dimensional) concentrations with (non-dimensional) activities. The total concentration C has disappeared from Eq. (4.2) simply because only one of the three activities was substituted for the semi-dilute solution approximation. Had other activities also been replaced, C would be present. However, this was not important for Scatchard’s treatment as he could use measured activities of water and hydrogen ion. Just detailed considerations of water activity changes in sucrose solution enabled Scatchard to arrive ﬁnally to a k value independent of sucrose concentration [73]. Regardless of several assumptions, his work remains a representative example of a careful (practical) approach to activity-based kinetics. A different point of view was presented by Bro¨nsted [75] whose work has been here already mentioned several times. Bro¨nsted states that there exist many anomalies for ionic reactions in solutions in comparison to van’t Hoff’s kinetic law. He did not explicitly explain the anomalies nor give van’t Hoff’s law or any reference to it. Regarding van’t Hoff’s approach, from his original work [76] it is evident that his approach to kinetics is based on the work of Guldberg and Waage. van’t Hoff considers chemical equilibrium as the ﬁnal point of a chemical reaction described by the traditional thermodynamic equilibrium constant: Y Y K¼ ci i cj j ð4:3Þ products

reactants

from which he formulates the equilibrium condition: Y Y cj j ¼ ci i K reactants

ð4:4Þ

products

and on its basis he claims that the reaction rate should be proportional to the appropriate difference: r¼k

Y

cj j K reactants

Y

!

ci i products

ð4:5Þ

Bro¨nsted writes [75] that he is inspired by the ‘‘thermodynamic mass-action law’’ in which equilibrium activities appear instead of concentrations. By this law, the equilibrium constant expression (4.3) with activities should be understood. www.scilet.com

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Miloslav Pekarˇ

Therefore, also in kinetics, activities should replace concentrations. Bro¨nsted is less cautious than Scatchard but he is far from making only this simple substitution. He, in fact, recalls Marcellin’s ideas on the so-called critical or activated complex, which is some highly unstable intermediate assembled from reactants, which further decomposes to the products (or back to the reactants). It is a predecessor of the later transition state and is also referred to in pioneering work on transition state theory [77]. Bro¨nsted suggests that in the concentrationbased mass-action rate equations, corrections through the activity coefﬁcients not only of the reactants but also of the activated complex should be made. For instance, the rate equation r ¼ kcA cB

ð4:6Þ

should be replaced by the equation r ¼ kcA cB ð fA fB yfA ? B Þ

ð4:7Þ

where fi represents the activity coefﬁcient of, and A ? B denotes, the critical complex. Why should the rate be just inversely proportional to the activity coefﬁcient of the activated complex is explained by Bro¨nsted only by rather unclear physical reasoning, with no unambiguous proof being given. The inverse proportionality should make explicit, according to Bro¨nsted, that only those few reactant molecules possessing a sufﬁciently high activity to build up very unstable, i.e. a very ‘active’ activated complex. Thus, Bro¨nsted tried to formulate mathematically the decelerating effect of the necessity of existence of an activated complex with high ‘activity’. The two meanings of ‘activity’ are thus confused – that of high ‘reactivity’, which is rather vague, and that of the precisely-deﬁned thermodynamic quantity. The vagueness of Bro¨nsted’s reasoning prompted another Scandinavian, Bjerrum, who presented the whole matter more precisely two or three years later [78,79]. In fact, he made the same hypothesis as did formerly Arrhenius, and later Eyring and collaborators, in absolute reaction rate theory. Bjerrum supposed that Bro¨nsted’s activated complex is in equilibrium with the reactants, and that the reaction rate is directly proportional to its concentration. Expressing the activated complex concentration in terms of the thermodynamic equilibrium constant containing the products of concentration and activity coefﬁcient then resulted in a rate equation like Eq. (4.7). Bjerrum supported his argument with some ideas from kinetic-statistical theory. www.scilet.com

Thermodynamics and mass-action chemical kinetics

35

Using the same activity coefﬁcients for various ions with the same charge, i.e. coefﬁcients dependent only on the type of ion, Bro¨nsted further successfully applied his theory to many ionic reactions [75]. It is clear that Bro¨nsted’s treatment, exempliﬁed by Eq. (4.7), forms the basis of various non-ideal mass-action rate equations, e.g. (2.18), (3.22), (4.8), and forms the basis for treatment of the salt effect. Belton [80] applied activity-based kinetics in his study of the conversion of N-chloroacetanilide into p-chloroacetanilide by protons and chloride ions. He found little value in using activities, or, more precisely, the products of concentration and activity coefﬁcient both as a substitute in the normal massaction rate equation and in Bro¨nsted’s sense. Most activity-based approaches in modern kinetics stem from the reaction isotherm as explained in part 1. Thus, Haase [81], as stated in his paper abstract, gives a rigorous expression for the rate of a chemical reaction in a non-ideal system. In fact, he starts with an equation very similar to that discussed by Blum and Luus [27], see Eq. (2.18). The only difference is in the use of stoichiometric coefﬁcients (i ): ?

r ¼k l

m n Y Y / ai i k l ai i i¼1

ð4:8Þ

i¼mþ1

(a’s are activities) and considering only reactants or products in the ﬁrst or second term, respectively. Haase also refers to Bro¨nstedt’s work [75] as the origin of this equation. Haase requires that the general expression for the reaction rate must have a form which reduces to the classical rate expression for perfect gas mixtures and ideal dilute solutions and gives the correct formula for the equilibrium constant in any system. Using the ‘‘reaction isotherm-based’’ approach, described in part 1, he proves this to be valid for Eq. (4.8) and also derives the relationship between rate and reaction afﬁnity, see Eq. (2.6). Immediately after Haase’s paper, Hall’s contribution was published in the same journal [37] and a spirited discussion started between Haase and Hall. Hall [37] begins with the equation ?

/

r y r ¼ expðAyRT Þ

ð4:9Þ

and tries to show its validity for elementary reactions in non-ideal systems. To achieve this goal he uses traditional expressions for the dependence of chemical potential on concentration and the mass-action law in the usual, concentration www.scilet.com

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Miloslav Pekarˇ

form. The main point in his development is the rather strange hypothesis that the reaction is frozen for all but a very small fraction of the molecules present. This supposition might be perhaps accepted as a model of a non-ideal system in which intermolecular interactions deﬁnitely may affect the (‘‘frozen’’) ability of molecules to react. This hypothesis, with several additional physical premises, and not rigorous mathematical proofs, enable one to relate reaction rates and chemical potentials of (all) molecules present, leading thus to Eq. (4.9). The idea underlying all Hall’s premises and models is that, at constant temperature and pressure, reaction rates depend only on molecular environments. The main motivation of his rather incautious approach is an effort to avoid transition state theory, which is less readily applied to non-ideal systems. However, it is also not clear what is the advantage of Hall’s approach over the simple reaction isotherm-based derivation, except that he uses concentrations in the rate equation. To relate concentration-based kinetics with activity-based thermodynamics of non-ideal systems, he ﬁnally uses concentrations in expressions for chemical potential so the whole procedure loses its non-ideality status. In response to Haase’s paper [81], Hall claims [82] that Haase’s arguments lack rigour. Hall shows that Eq. (2.6) or (4.9) is not a logical consequence solely of Eq. (4.8) but may also be derived from its modiﬁed forms. Thus, Hall merely questions Haase’s derivation and does not add anything new to the kinetic-thermodynamic relationships. Haase rebuts [83] this criticism and shows by physical reasoning that Hall’s modiﬁcations reduce to Eq. (4.8), anyway. The following paper by Haase [84] generalizes his approach to any number of reactions. Hall responds to this several years later [85] and criticizes ﬁrst of all Haase’s reasoning in reference [83]. As well as this reasoning, the criticism is based upon physical argument and not mathematical proofs. In his ﬁnal response Haase published a mathematical proof that Hall’s more general form of Eq. (4.8), viz. m ?? Y

r ¼k l

//

ai i k l

i¼1

n Y

ai i

ð4:10Þ

i¼mþ1 ?

/

is superﬂuous because l ¼ l . Unfortunately, his proof lacks its claimed general validity as has been shown by Samohy´l (unpublished results) for the example of a gaseous reaction where it is not possible to choose the equilibrium pressure arbitrarily (one of the key points in Haase’s proof) when the temperature and composition are given, as can easily be checked by the interested reader. www.scilet.com

Thermodynamics and mass-action chemical kinetics

37

Haase notes, that l in Eq. (4.8) represents a function of temperature, pressure, and composition but gives no idea how this function can be obtained experimentally or theoretically to be useful in practice. Examples of practical applications of this equation are given by Baird [86]. In summary, Haase did not derive a ‘‘kinetic law’’ from thermodynamics. He was inspired by thermodynamics, used activities instead of concentrations, and the general form of the mass-action law, Eq. (4.8), directly. He did not tackle the question of whether there is also any other rate equation conforming to his postulates. Hall criticized the procedure, not this basis. Note that Hall derived Eq. (4.9) also using statistical thermodynamics [87]. Baird [86] claims that the generalized law of mass-action (4.8) is consistent with transition state theory. He considers the example of the simple reaction 1 ½1 þ 2 ½2 ?½6¼ ?3 ½3 þ 4 ½4

ð4:11Þ

In transition state theory, the reactants are considered to be in equilibrium with the transition state ([6¼]). The true thermodynamic equilibrium constant is then given by ?

K¼ a6¼ yða11 a22 Þ

ð4:12Þ

The reaction rate is proportional to the concentration of transition state, ? ? r ¼ c6¼ . Expressing activity as the product of activity coefﬁcient (g) and relative concentration, i.e. the ratio of the actual and the standard concentration (co ), the reaction rate in the forward direction is as follows: ?

??

?

r ¼ K c a12 yg6¼ : k a11 a22 yg6¼

ð4:13Þ

By the principle of microscopic reversibility, the reaction must proceed in the reverse direction via the same transition state [86]. Therefore the products are also in equilibrium with the same transition state: /

K¼ a6¼ yða33 a44 Þ

ð4:14Þ

and by analogy: /

//

/

r ¼ K c a33 a44 yg6¼ : k a33 a44 yg6¼ www.scilet.com

ð4:15Þ

38

Miloslav Pekarˇ

By subtracting the forward and reverse reaction rates, Eq. (4.8) is obtained with l ¼ 1yg6¼ . However, from the supposed equilibria, it also follows that the reactants are in equilibrium with the products:

?

/

a33 a44 yða11 a22 Þ ¼K y K¼ ðequilibriumÞ constant

ð4:16Þ

The entire analysis could thus be valid only for equilibrium where the overall rate is zero! Introducing Eq. (4.16) into the generalized rate equation (4.8), we obtain:

?

/?

/

r ¼ a11 a22 ð k k K y KÞyg= :ka11 a22 yg=

ð4:17Þ ?

/?

/

This generally gives non-zero equilibrium rate unless k ¼ k K y K, which leads to ? / ¼ . Otherwise, Eq. (4.17) would give the very strange result that the overall rate of a reversible reaction is independent of the concentrations of products, i.e. of the reverse direction. Thus, transition state theory does not prove in this way the generalized mass-action law (4.8). Obstacles could be overcome perhaps by considering different transition states [88] in both directions with concentrations given by: ?

/

c ? ¼K c a11 a22 yg ? ; =

c / ¼K c a33 a44 yg /

=

=

=

ð4:18Þ

The ﬁnal result is: ?

/

r ¼ k a11 a22 yg ? k a33 a44 yg / =

ð4:19Þ

=

which is, in fact, Hall’s general mass-action law (4.10). The same result can be obtained considering different activity coefﬁcients, i.e. different activities of a common transition state in the forward and reverse directions. Both different transition states and different activities sound rather strange and illustrate the problems which are encountered when applying transition state theory to reactions occurring simultaneously in both directions out of equilibrium. Considering different transition states in different directions of the same reaction may violate microscopic reversibility. It might be therefore supplemented by the hypothesis that the transition states are different in non-equilibrium states only, and become identical when equilibrium is attained. www.scilet.com

Thermodynamics and mass-action chemical kinetics

39

Activities were introduced into the mass-action kinetic equation also by Ola´h [89] using his ‘‘thermokinetic’’ theory. This theory is analyzed in Part 5 below. Now it is sufﬁcient to state that it is in fact an ordinary afﬁnity-based approach. As afﬁnities are directly related to chemical potentials, see (2.5)1 and cf. Ola´h’s Eq. (5.88), which in turn are, by deﬁnition, related to activities, nothing fundamentally new is added. Eckert and Boudart [90] successfully described gas phase kinetics using a fugacities-based mass-action rate equation of the Bro¨nstedt type in contrast to the traditional concentration-based treatment. Mason [91], however, demonstrated using the same data set that the activity-based rate coefﬁcient shows a much stronger pressure dependence than the concentration-based coefﬁcient. Activity-based kinetic equations have also started to become popular in enzyme kinetics. Van Tol et al. [92] probably pioneered this approach to circumvent problems with solvent effects on reaction rates, substrate – solvent interactions in nonaqueous enzymology, or with the substrate concentration in biphasic systems. Their study of lipase-catalyzed ester hydrolysis in biphasic systems with various solvents did not give fully satisfactory results. Experimental data obtained in isooctane could be well ﬁtted to the activity-based equation whereas for the other solvents the ﬁt was poor. The latter was attributed to unrealistic premises employed in modelling (equal binding of the solvents to the active site, no solvent effect on the mechanism, equal activity coefﬁcients of the enzyme species in the catalytic cycle, and others). Activity coefﬁcients were calculated from UNIFAC or determined from equilibrium solubility or partitioning. From subsequent papers, let us mention only that by Sandoval et al. [93] who used activities in the traditional equations of enzyme kinetics, i.e. in the initial rate expression originally derived from the mass-action law. The authors simply replaced concentrations with activities and used UNIFAC group contribution methodology to compute the activity coefﬁcients. From experiments made in one solvent, kinetic parameters, free of solvent effect, were determined. They were used to predict the reaction rate in other solvents using, of course, the activity coefﬁcient computed for the respective solvent. From a comparison of predictions with measured data, it seems that this approach works in most systems. Van Tol et al. [94] summarize that when organic solvents do not interfere with the binding process nor with the catalytic mechanism of enzyme-catalyzed reactions, the contribution of substrate-solvent interactions to enzyme kinetics www.scilet.com

40

Miloslav Pekarˇ

can be accounted for by just replacing substrate concentrations in the kinetic equations by thermodynamic activities. Only the afﬁnity parameters (substrate afﬁnity, speciﬁcity constant) are affected by this transformation and corrected parameters and the maximal rate should be equal for all media. Experimental data show, however, that although the kinetic performance of each enzyme in the solvents became much more similar after correction, differences still remain. They are caused mainly by incomplete shielding of the bound substrate from the solvent, the non-constancy of the activity coefﬁcient of the enzyme species in the catalytic cycle, and by solvent competition with substrate for binding to the active site. Published data on activity-based mass-action kinetics generally give no decisive conclusion. The idea, already formulated in Hougen-Watson’s classic monograph [95], that mass-action law should be generally formulated in activities and not in concentrations does not have general validity. It seems that ion (salt) effects mostly cannot be included by simply using activities in place of concentrations whereas solvent effects usually can be. In any case, introducing activity coefﬁcients into the mass-action rate equation is identical to considering a concentration-dependent rate ‘‘constant’’.

5. CLASSICAL (LINEAR) IRREVERSIBLE THERMODYNAMICS 5.1 Fundamentals Haase’s book [96] gives probably the most comprehensive explanation of the basis of the classical or linear irreversible thermodynamic (CIT) approach to chemical kinetics, compared to other books in this ﬁeld. Haase, in the part of his book devoted to homogeneous systems, presents an attempt to combine well-known kinetic ‘‘laws’’ with the phenomenological or ﬂux-force laws. This is a typical effort of CIT. As the driving ‘‘force’’ for chemical reaction, or chemically reacting systems in general, the afﬁnity (A) is selected. The phenomenological law for the reaction rate (ri ), the ‘‘ﬂux’’, may be written, close to equilibrium, in linear form ri ¼

R X aij Aj ;

i ¼ 1; 2; . . . ; R

ð5:1Þ

j¼1

where R is the total number of independent reactions and aij are the phenomenological coefﬁcients. The law of mass-action is used in the form www.scilet.com

Thermodynamics and mass-action chemical kinetics

! Y Y Y mi 0 ni ki ri ¼ ki cm k i cn ¼ oi 1 li ck ; m

n

41

ð5:2Þ

k

where o i ¼ ki

Y

cmmi ; li ¼ ki 0 yki

ð5:3Þ

m

and m goes through all reactants, n through all products and k through both these kinds of constituents; ’s are the stoichiometric coefﬁcients and c’s are the concentrations. Using the deﬁnition of afﬁnity in terms of chemical potential and the classical relationship between chemical potential and concentration, the following equation is obtained: ri ¼ oi ½1 li Ki ðco Þi expðAi yRT Þ

ð5:4Þ

where Ki is the equilibrium constant, co the standard concentration, and P i ¼ k ki . It can be shown that the multiplicative factor at the exponential is equal to one. Close to equilibrium (jAi yRT j 5 1; index ‘‘eq’’), the exponential may be expanded in a series retaining only the ﬁrst member. The linear phenomenological relation is ﬁnally obtained: ri ¼ ðoeq i yRT ÞAi

ð5:5Þ

(note that oi , once more, was not expanded in contrast to the afﬁnity), which is a special case of Eq. (5.1). Haase is very careful to identify kinetic and thermodynamic equilibrium constants and also to ignore standard concentrations when substituting concentrations for activities, cf. Eq. (5.4). In the ﬂow-through systems, where the spatial distribution of variables must also be taken into account, the following expression for the local entropy production (u) is derived: C:T u ¼ JQ XQ þ

X k

J;k Xk þ

X r

or Ar þ

3 X 3 X

Pij Xij 0

ð5:6Þ

i¼1 j¼1

where XQ ¼ ð1yT ÞgradT

ð5:7Þ

Xk ¼ Kk ðgradmk ÞT

ð5:8Þ www.scilet.com

42

Miloslav Pekarˇ

Xij ð1y2Þðqi yqzj þ qj yqzi Þ;

i; j ¼ 1; 2; 3

ð5:9Þ

and JQ is the density of heat ﬂow, J;k are the densities of diffusion ﬂows, or is the rate and Ar the afﬁnity of the r-th reaction, Pij ¼ Pji are the frictional forces, Kk the external molar force acting on the k-th component, mk its chemical potential and i is the component of the barycentric rate vector v, z’s are spatial coordinates. The gradient in Eq. (5.8) is taken at constant temperature. The so-called dissipative function C is interpreted with the aid of the concepts of ﬂuxes and forces, viz. ‘‘phenomenological relationships’’ among them are sought. To this end, forces are considered to be independent, and ﬂuxes dependent, variables. Their mutual functional connections are formulated with a rather intuitive use of linear isotropic function representation, here called Curie’s principle. First, the tension term is excluded from the considerations, giving nothing new to chemical kinetics, as the relevant phenomenological relation is again postulated to be: X or ¼ ars As

ð5:10Þ

s

where index s also refers to all reactions and a’s are the proportionality (phenomenological) coefﬁcients. Second, the tension term is taken into account but together with only the chemical rate term from the dissipative function (5.6) and not also with the other two, transport terms: C¼

X r

or Ar þ

3 X 3 X Pij Xij 0

ð5:11Þ

i¼1 j¼1

This is vaguely substantiated by Curie’s principle and should probably be understood as the fact that the vectorial linear isotropic function depends only on vectors whereas the tensorial function may depend also on scalars. Consequently, the reaction rate is written (better speaking, represented) as: X or ¼ ars As L r div v ð5:12Þ s

(L

r

is the proportionality, phenomenological, coefﬁcient) because div v is

considered to be the trace of tensor X given by Eq. (5.9). Eq. (5.12) is claimed to be a generalization of Eq. (5.10). It can be interpreted as a warning that the chemical reaction rate may be affected by viscous processes. The functional www.scilet.com

Thermodynamics and mass-action chemical kinetics

43

dependence, Eq. (5.12), of the former on the latter is more a matter of interpretation than of exact proof. Very recently, Cukrowski and Kolbus [97] published another paper utilising ﬂux and force. They found a new, ‘superior’ force, which enables one to use linear ﬂux-force over a wider range (i.e. farther from equilibrium) than usually expected. The new force is deﬁned as the difference between the reactive absolute activities of reactants and products. The absolute activity (of component i, li ) was introduced into thermodynamics by Fowler and Guggenheim [98] as the exponential of chemical potential: li ¼ expðmi yRT Þ

ð5:13Þ

The reactive absolute activity is deﬁned by Cukrowski and Kolbus using the ‘‘reactive chemical potential’’ mre i , which they deﬁne as the difference between the chemical potential and its equilibrium (‘‘eq’’) value: eq mre i ¼ mi mi

ð5:14Þ

The reactive absolute activity of component i is then deﬁned as follows: re lre i ¼ expðmi yRT Þ

ð5:15Þ

and the reactive absolute activity of reactants (sufﬁx R) or products (sufﬁx P) as products: lre R ¼

Y reactants

lre i ;

lre P ¼

Y

lre j

ð5:16Þ

products

re The new force is then X ¼ lre R lP . Cukrowski and Kolbus [97] then present

several examples of model reactions, which were analyzed using the new ‘force’. Their approach is in fact another exercise in combining the traditional massaction law and the traditional expression for chemical potential as a function of concentration similar to those reviewed in Parts 2.1, 3.1, or 3.2. What is important and new in this approach is the stress on equilibrium and its use as a referential state. It will be seen later that the equilibrium also has similar importance in rational thermodynamic theories. A similar approach, but with no such stress on equilibrium, was presented by Parmon [99,100].

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Miloslav Pekarˇ

5.2 Tackling mass-action non-linearity and Onsager reciprocity Anderson and Boyd [101] extended the ﬂux-force approach to the nonlinear area and focused on the so-called Onsager’s reciprocity relations, another popular theme of CIT. They suppose that the reaction rate (ﬂux) J depends on the appropriate force X in the following way: J ¼ L X þ MX2

ð5:17Þ

where L and M are again proportionality, phenomenological, coefﬁcients. Further they take over the general rate equation: r ¼ kb

m Y

m Y 0 ðai Þa i

ðai Þai k 0 b

i¼1

ð5:18Þ

i¼1

where a’s are the activities and a’s are the reaction orders and not stoichiometric coefﬁcients. Combination of this equation and the condition of a vanishing rate in equilibrium, together with the deﬁnition of afﬁnity (A) by Eq. (2.5)1 and with common expression for the relationship between chemical potential and activity, gives the ﬁnal rate equation: "

# m Y ai r ¼ kb ða1 Þ 6½1 expðnAyRT Þ

ð5:19Þ

i¼1

where n ¼ ðai 0 ai Þyi

ð5:20Þ

Factors in Eq. (5.19) are expanded in Taylor series and the form of Eq. (5.17) is thus obtained as follows: "

# m Y eq ai r ¼ kb ðai Þ 6½nðAyRT Þ þ nðd1 ny2ÞðAyRT Þ2 þ

ð5:21Þ

i¼1

where "

# m Y eq ai d1 ¼ q ðai yai Þ yqðAyRT ÞA¼0 i¼1

and index ‘‘eq’’ refers to equilibrium. www.scilet.com

ð5:22Þ

Thermodynamics and mass-action chemical kinetics

45

The authors then assume that Onsager’s reciprocity relations hold only in the linear range. However: ‘‘Even in the linear regime...the coefﬁcients (in Eq. (5.21)) are functions of the equilibrium state of the system. This implies that phenomenological coefﬁcients, measured in one reaction mixture, cannot be applied directly to another.’’ The authors therefore conclude that the language of nonlinear thermodynamics is not suitable for chemical kinetics. The authors agree that there is no ‘‘thermodynamic’’ substitute for the established, mostly empirical, rate equations and Eq. (5.21) serves only as a connection between this tradition and the CIT approach. It could be that the phenomenological coefﬁcients measured and general values obtained, perhaps Eq. (5.17) might substitute the traditional rate equation, with the phenomenological coefﬁcients playing the role of rate constants. Bataille et al. [102] also dealt with Onsager’s relations. They state that linear ﬂux-force relations are not adequate for reaction kinetics and try to discover an extension of Onsager’s reciprocal relations into the non-linear domain. They start with the generalized rate equation for reaction a (Ja ) in a rather unusual form but close to Eq. (5.18): Ja ¼ k a

m m Y Y þ ð fi Þi kþ ð fi Þi a i¼1

ð5:23Þ

i¼1

þ where fi denote fugacities and i , i are the (positive) stoichiometric coefﬁcients

for the backward and forward directions, respectively. Three things are then to be checked: (a)

the J ’s can be expressed in terms of the A’s (afﬁnities) and the thermostatic state variables,

(b)

the entropy production is non-negative and vanishes only when all of the A’s vanish,

(c)

approximation of the J ’s by linear functions of the A’s over a sufﬁciently small neighborhood of the equilibrium values Aa gives Ja &

R X L

ab Ab

b¼1

with satisfaction of the Onsager reciprocity relations L

www.scilet.com

ab

¼L

ba .

46

Miloslav Pekarˇ

Whereas the last two conditions are quite simply satisﬁed, the ﬁrst one requires more elaborate treatment. Using afﬁnity deﬁnition (2.5)1 and introdu cing forward and backward afﬁnities such that Aa ¼ Aþ a Aa , Eq. (5.23) is modiﬁed:

Ja ¼ ka expðA a yRT Þ½1 expðAa yRT Þ

ð5:24Þ

From Eq. (5.24) it is clear that the ﬁrst condition transforms to the question whether the backward afﬁnity can be expressed in terms of the A’s and the thermostatic state variables. This leads to the standard task of linear algebra, viz. ﬁnding (general) solutions of the system of equations m X ai mi ¼ Aa ;

a ¼ 1; . . . ; R

ð5:25Þ

i¼1

As there are more unknowns (mi ) than given (Aa ) quantities, the number of solutions is inﬁnite. Anyway, the ‘‘unknowns’’ may be expressed from the system with the aid of afﬁnities or other unknowns, which are themselves functions of state variables. Consequently, condition (a) is conﬁrmed. After introducing the general solution into the rate equation (5.24), it is immediately found that Onsager’s relations ðqJa yqAb Þp;T ;bl ¼ ðqJb yqAa Þp;T ;bl

ð5:26Þ

(p is pressure and bl denotes set of parameters in the general solution of the system of algebraic equations) are not generally valid in the non-linear domain. The authors point out that Edelen’s generalized dissipation potential [103] is still applicable in this domain and its symmetry relations hold as well. 5.3 Hungarian contribution I – Lengyel Several papers have appeared from the Hungarian school based on Gyarmati’s ‘‘integral principle of thermodynamics.’’ Gyarmati’s approach is, in principle, a certain reformulation of the irreversible thermodynamic approach into the terms of variational principles. Its application to chemically reacting systems in general is described in Sa´ndor’s papers [104,105]. The ﬁrst contribution dealing with kinetics in more detail is probably the paper by Lengyel and Gyarmati [106,107]. It is interesting to cite the authors’ motivation: ‘‘This consistency (between kinetics and thermodynamics) is both theoretically and practically important also from the aspect of reaction kinetics. www.scilet.com

Thermodynamics and mass-action chemical kinetics

47

If we can show this consistency, then the whole phenomenological theory of chemical reactions will become a special, but organic, branch of non-equilibrium thermodynamics in the same way as the theory of chemical equilibria has become a special chapter of thermostatics as a result of Gibbs’ work. From the practical point of view the description in non-equilibrium thermodynamics not only offers an alternative description of chemical reactions but can complete the Guldberg – Waage theory. We think that reaction kinetics describe only the concentrations as a function of time but the reaction heats involved in the reaction, i.e. energetics, are not included in the description. In non-equilibrium thermodynamics this inclusion is quite natural; moreover, if the equivalence of both theories can be assumed, then stationary states, the stability and evolution of open kinetic systems, may become objects of exact studies...To illustrate...let us assume that we could show the consistency of nonlinear thermodynamics and the nonlinear theory of chemical kinetics. In this case, instead of the Guldberg – Waage form of the kinetic equations, the consistent differential equations of the nonlinear thermodynamic theory have to be solved.’’ Although the authors present basic thermodynamic equations, including Gyarmati’s principle, in the introduction, the procedure adopted is standard, close to that given in the papers described above. The authors write the Guldberg – Waage law, in this case with molar fractions. Further, the expression of chemical potential in term of molar fraction in ideal systems and the traditional deﬁnition of afﬁnity are used and combined with the Guldberg – Waage law to arrive at the general equation Jr ¼ Jer jr ðA1 ; . . . ; AR Þ

ð5:27Þ

for the rate (Jr ) of reaction r (which are R in total), called the non-linear phenomenological equation. The novelty in this general equation is the particular representation in a)

the components’ deviations from equilibrium

Dni ¼ ni nei ;

i ¼ 1; . . . ; m

ð5:28Þ

(n’s are mole numbers and ‘‘e’’ refers to equilibrium) which can be introduced into the Guldberg – Waage law through rewriting it into the form:

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48

Miloslav Pekarˇ

Jr ¼ k 0r

Y

ðnei 0 ynÞi 0 ;r

Y

i0

ðni 0 ynei 0 Þi 0 ;r k00r

i0

Y Y ðnei00 ynÞi00 ;r ðni00 ynei00 Þi00 ;r i00

ð5:29Þ

i00

giving " Jr ¼

Jer

Y Y ð1 þ Dni 0 ynei 0 Þi 0 ;r ð1 þ Dni00 ynei00 Þi00 ;r i0

# ð5:20Þ

i00

where the prime denotes the forward, and double prime the backward, reaction direction and ’s are the stoichiometric coefﬁcients, n the total mole number and Y Y Jer ¼ k 0r ðnei 0 ynÞi 0 ;r ¼ k00r ðnei00 ynÞi00 ;r ð5:31Þ i0

b)

i00

the reaction’s deviations from equilibrium

Dni ¼

R X i;r Dxr

ð5:32Þ

r¼1

(xr is extent of reaction r and D has the same meaning as in Eq. (5.28)) which, after introducing into Eq. (5.30), give: " # R R Y X Y X e e i 0 ;r e i00 ;r Jr ¼ Jr ð1 þ i 0 ;r Dxr yni 0 Þ ð1 þ i00 ;r Dxr yni00 Þ i0

c)

i00

r¼1

ð5:33Þ

r¼1

and introducing the ‘‘absolute afﬁnities’’ "

Lr ¼

Y i0

# " # R R . Y X X e i 0 ;r e i00 ;r ð1 þ i 0 ;r Dxr yni 0 Þ ð1 þ i00 ;r Dxr yni00 Þ i00

r¼1

ð5:34Þ

r¼1

which are related to common afﬁnities (Ar ) by Lr ¼ expðAr yRT Þ. Eq. (5.34) can be considered to be a system of algebraic equations which can be solved for the deviations of the extent of reaction from equilibrium: Dxr ¼ fr ðL1 ; . . . ; LR Þ or

Dxr ¼ jr ðA1 ; . . . ; AR Þ

ð5:35Þ

Substituting (5.35) into (5.33), the general Eq. (5.27) is obtained. Using the absolute afﬁnities, ‘‘general non-linear constitutive equations between reaction rates and afﬁnities’’ are derived in the form Jr ¼ Jer ðLr 1Þu00r ðA1 ; . . . ; AR Þ

ð5:36Þ

where www.scilet.com

Thermodynamics and mass-action chemical kinetics

u00r

¼

Y

R X 1þ i00 ;r Dxr ynei00

i00

49

!i00 ;r ð5:37Þ

r¼1

and is a function of all afﬁnities as indicated in Eq. (5.36). Onsager’s relations in the linear approximation close to equilibrium are then proved for both stoichiometrically independent and dependent reaction systems. The authors then present several examples in which they also test the ‘‘Rysselberghe generalized reciprocity relations’’. These were postulated by Rysselberghe [108,109] to be valid for the non-linear equations Jk ¼ L kk Ak þ L kl Al þ L kkk A2k þ L kkl Ak Al þ L kll A2l

ð5:38Þ

in the form qJk yqAl ¼ qJl yqAk

ð5:39Þ

originally set forth by Pe´ne´loux [110,111]. As this attempt was unsuccessful, the authors conclude that Rysselberghe’s relations are inconsistent with classical chemical kinetics. The authors speculate that the cause lies in the improper choice of thermodynamic forces and some new parameter should be sought instead of afﬁnities. They also stress that, for example, Gyarmati never identiﬁed the thermodynamic forces in chemical kinetics with afﬁnities. As he also never gave any speciﬁcation of these general forces to kinetics, ﬁnding the right forces remains an unresolved task. Summarizing, classical kinetic or thermodynamic quantities and relations are combined and subjected to the interpretation within the ﬂux-force framework. As this works in the linear domain only, new, superior ‘forces’ should be found, without asking whether the ﬂux-force approach is correct, necessary or of any practical use, at all. The proclaimed practical aim is not demonstrated even in the linear domain where the ﬂux-force interpretation is satisfactory. The practical value of Eq. (5.34) is questionable especially with regard to the fact that each extent of reaction contains only those moles which have reacted just in that reaction. The ‘right’ forces are claimed to be found in subsequent papers by Lengyel [112,113], the ﬁrst one being, in fact, a shortened version of the second. Moreover, it is stated that the mass-action law was deduced from Gyarmati’s governing principle of dissipative processes. This principle reformulates the results of CIT in terms of variational principle. Locally, it asserts that the density (o) of the so-called Onsager – Machlup function www.scilet.com

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Miloslav Pekarˇ

o¼scj

ð5:40Þ

keeps its extremum value at any point in the system. In other words, its variation (do) is always and everywhere vanishing: do ¼ ds dc dj ¼ 0

ð5:41Þ

In the equations, s is the density of the local entropy production rate, c and j are the so-called dissipation functions (in the forms of densities) or potentials. Function c is said to depend on all (independent) forces whereas j depends on all ﬂuxes. Consequently, the former is sometimes called the force potential, the latter the ﬂux potential. The dissipation functions are selected by Lengyel to be " ! !# Q Q S X X X 00 0 c ¼ 2C ¼ 2 Rlt exp gt X yR þ exp gt X yR t¼1

¼1

ð5:42Þ

¼1

and j ¼ 2F ¼ 2

S h? X ? / / ? / i R J t lnð J t ylt Þ þ J t lnðJ t ylt Þ ð J t þ J t Þ

ð5:43Þ

t¼1

In these equations S is the total number of reactions and Q the number of independent reactions. Parameter lt comes from nothing more than the massaction law written in the form ?

Jt ¼ k t

Y N0 / Y 00 ðci Þ i;t k t ðci ÞNi;t i

ð5:44Þ

i

(c’s are concentrations) and transformed to the form ? / Jt ¼ J t J t ¼ lt expðXt 0 yRÞ expðX00t yRÞ ;

t ¼ 1; . . . ; S

ð5:45Þ

The prime or left-to-right arrow, and double prime or right-to-left arrow, represent forward and reverse reaction directions respectively, N’s are the orders (not stoichiometric coefﬁcients). Coefﬁcient gt results from the relations between the dependent and independent (marked by an asterisk) reactions: J* ¼

S X gt Jt ;

¼ 1; . . . ; Q

ð5:46Þ

t¼1

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Thermodynamics and mass-action chemical kinetics

51

Quantities X’s are related to afﬁnities: A00t ¼ TX00t ;

At 0 ¼ TXt 0 ;

t ¼ 1; . . . ; S

ð5:47Þ

(all stoichiometric coefﬁcients are considered positive). Introducing variations of the dissipation functions (5.42) and (5.43), together with the variation of s (which will be discussed later) into Eq. (5.41), the following equation is obtained: " Q X

J* 2

S X

u

u¼1

þ

" Q X

gtu lt exp

t¼1

J* þ 2

Q X

!# 0

gt X yR

¼1 S X

u

gtu lt exp

t¼1

u¼1

Q X

dXu 0 !#

gt X00 yR

dX00u

¼1

S h S h i ? i / X X ? / þ Xt 2R lnð J t ylt Þ d J t þ Xt 2R lnðJ t ylt Þ dJ t ¼ 0 t¼1

ð5:48Þ

t¼1

As all the varied variables are mutually independent, it follows that the expressions in brackets vanish. Combining the two equal-to-zero equations for Ju* , relation (5.49) follows: J*u ¼

S X t¼1

" gtu lt exp

Q X ¼1

! 0

gt X yR exp

Q X

!# gt X00 yR

;

u ¼ 1; . . . ; Q

¼1

ð5:49Þ which is said to be nothing more than the Guldberg – Waage mass-action law. However, the whole deduction suffers from several deﬁciencies. The forms of dissipation functions (5.42) and (5.43) are not proved but stated. They are given in such a way to obtain immediately (5.49) after introducing them into (5.41). This is no deduction but a tautology. Of course, Eqs (5.42) and (5.43) do have physical motivation. It stems from the well-known Eq. (5.44), which was transformed to Eq. (5.45) using the classical relation for the chemical potential in ideal systems. It should be pointed out also that the postulate of expressing the rate as a difference between the forward and backward rate was introduced. Only stoichiometrically independent reactions are considered. They are selected from the whole reaction set by means of relations (5.42) and their afﬁnities are used to express forward and backward rates of independent reactions, e.g. (cf. also Eq. (5.45)): www.scilet.com

52

Miloslav Pekarˇ

?

J u* ¼

S X

?

gtu J t ¼

t¼1

S X

gtu lt exp

t¼1

Q X

! gt X 0 yR

;

u ¼ 1; . . . ; Q

ð5:50Þ

¼1

Now, the inspiration for the ﬂux potential is clear. The origin of the force potential is more unclear. From the inversion of expressions for both forward and backward rates in Eq. (5.45), the author ﬁnds relations: ?

Xt 0 ¼ R lnð J t ylt Þ;

/

X00t ¼ R lnðJ t ylt Þ;

t ¼ 1; . . . ; S

ð5:51Þ

which, as he states, satisfy the reciprocal relations. Relations of Eq. (5.51) appear in ﬂux potential (5.43). In this case no attempt is made to use only independent reactions or their afﬁnities and no explanation is given as to why the minus sign in the second expression in (5.51) is not retained in Eq. (5.43). The tautology is even deeper. The author starts from the mass-action law, either in the form (5.44) or (5.45), to discover it again after several lines of manipulating with it. To recover the desired result, multiplication by 2 is necessary not only in (5.42) and (5.43) but also in the entropy production density. This is achieved in a particularly intriguing manner. The author states that the local entropy density (s) is the function of some set of ‘‘independent extensive state’’ variables x1 ; . . . ; xi ; . . . ; xf

ð5:52Þ

The partial time derivative of this function is given by qsyqt ¼

f X

Gi qxi yqt

ð5:53Þ

i¼1

and may be used in the general entropy balance equation of CIT: qsyqt þ divJs ¼ ss

ð5:54Þ

in which Js is the entropy ﬂux density and ss its source density. Similar balance equations are supposed to be valid also for the independent variables (5.52). Combining all balances, the following expression for the entropy production rate is found: ss ¼

f X i¼1

Ji grad Gi þ

f X

Gi si

ð5:55Þ

i¼1

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Thermodynamics and mass-action chemical kinetics

53

where Ji are current densities and si source densities of variables (5.52) in their balances like (5.54). The author claims that the density of entropy production can be written in the form f X

ss ¼

Ji grad Gi þ

S X

i¼1

?

/

ðXt 0 þ X00t Þð J t J t Þ

ð5:56Þ

t¼1

because of Gi ¼ mi yT A 0t ¼

X

ð5:57Þ

0i;t mi ;

A00t ¼

X

i

si ¼

00i;t mi

ð5:58Þ

i

S X

ð00i;t 0i;t ÞJt

ð5:59Þ

t¼1

where m’s are chemical potentials and ’s (positive) stoichiometric coefﬁcients. Note that the postulate xi :ci was introduced. The part of the entropy production rate due to the chemical reactions (the second term in (5.56)) can be expressed using only the independent reactions: S X

?

/

Xt ð J t J t Þ ¼

t¼1

Q X

Ju* ðXt 0 þ J00t Þ

ð5:60Þ

u¼1

In the variation condition (5.48), both versions from Eq. (5.60) are summed forming 2s! So, in fact, 2s is used in the Onsager – Machlup function (5.40) instead of s. Further, there is an obscurity with the independent variables. Initially, it is stated that entropy density is a function of variables (5.52), which are also subjected to the balance equations like (5.54). Later, they are identiﬁed with the volume concentrations, see above. However, it is then declared that there are two complete sets of independent variables, viz. X1 0 ; . . . Xu 0 ; . . . ; XQ 0 ;

X001 ; . . . X00u ; . . . ; X00Q

ð5:61Þ

and ?

?

?

J 1; . . . ; J t; . . . ; J S;

/

/

/

J 1; . . . ; J t; . . . ; J S www.scilet.com

ð5:62Þ

54

Miloslav Pekarˇ

which can be used alternatively to express the part of entropy production caused by the chemical reactions, see (5.60). No explanation for this transformation is given. It should be probably understood as a sudden return to the ﬂux-force area. These new sets are used in the variation condition, however, not alternatively, but all-at-once with no explanation, again. The author also claims that whereas overall reaction rates of all elementary reactions may be dependent, this is not true for the backward and forward rates, which are completely independent. It was clearly demonstrated by Bowen [34] that the (linear) dependence of reaction rates is the result of the permanence of atoms and not of the way the rate is expressed. Consider two reactions with one common reactant. Kinetic experience tells that the rate of one reaction may affect the forward rate of the other through its inﬂuence on the concentration of the common reactant. This can be supported also by formal argument. Let us suppose that among S reactions, the ﬁrst R of them are independent. This means that rate of any other reaction may be obtained as a linear combination of the rates of the independent reactions: Jk ¼

R X

lki Ji ;

k4R

ð5:63Þ

i¼1

It is further supposed that every rate is given as the difference between the ?

/

forward and backward rates, Ji ¼ J i J i . Thus ?

/

J k Jk ¼

R X

?

/

lki ð J i J i Þ;

k4R

ð5:64Þ

i¼1

This equation can be modiﬁed as ?

/

J k Jk ¼

R X

?

lki J i

i¼1

R X

/

lki J i ;

k4R

ð5:65Þ

i¼1

Stating ?

Jk ¼

R X i¼1

?

/

lki J i ; J k ¼

R X

/

lki J i ;

k4R

ð5:66Þ

i¼1

it is seen that some forward or backward rates can be expressed as a linear combination of the other forward or backward rates, respectively. It follows that all forward or backward rates are not independent. www.scilet.com

Thermodynamics and mass-action chemical kinetics

55

The author considers as the most essential point of the work ‘‘the identiﬁcation of the thermodynamic forces with the collection of the independent forward and backward afﬁnities, instead of the full afﬁnities (divided by the temperature)’’. However, it is not further explained why these should be the true forces when it is the difference between the chemical potentials (which are closely related to afﬁnities) of reactants and products which drives the chemical reaction. No ideas as to how to measure reaction afﬁnities (or rates) separately in the forward and backward directions are given. Perhaps some electrochemical cell under special conditions can serve this purpose. Lengyel’s approach is given once more in his next paper [114]. It is a review on the relationships between chemical kinetics and thermodynamics, which were resolved, at last, by the author. The same procedure is presented as in ref. [113], only here the rates are expressed using the extent of reaction. Thus, the results should be applicable only to closed systems with no diffusion (or with the so-called self-balanced diffusion [35] only). To conclude, ignoring the ﬂaws described, the well-known kinetic law was introduced into a certain thermodynamic formalism to re-derive it in a rather different form.

5.4 Onsager far from equilibrium Shiner [115] tries to prove the Onsager reciprocity or symmetry relations for chemical kinetics not only in the vicinity of equilibrium but also far from it. In fact, he starts again from the Guldberg – Waage law in its general form with activities (a’s): f x_ r :_ni;r yðbi;r i;r Þ ¼ krf

Y i

f

ðai Þi;r kbr

Y

b

ðai Þi;r

ð5:67Þ

i

ðxr is the extent of the r-th reaction, dot means the time derivative and f and b the forward and backward directions, respectively, ’s are the positive stoichiometric coefﬁcients), introduces common relations between activity and chemical potential, between the equilibrium constant and ratio of the rate constants in the forward and backward directions, and uses the common deﬁnition of afﬁnity (Ar ) through chemical potentials, cf. Eq. (2.5)1. This results in the trivial relation for afﬁnity www.scilet.com

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Miloslav Pekarˇ

" # " # . Y b f Y i;rf b Ar ¼ kB T ln kr ðai Þ kr ðai Þ i;r i i

ð5:68Þ

(kB is the Boltzmann constant) which is interpreted as ‘‘the driving force of a dissipative process ¼ product of the resistance and the ﬂow of the process’’: A r ¼ R r xr

ð5:69Þ

Using the resistances (Rr ), Shiner succeeds in proving symmetry relations in the following sense. Supposing that the thermodynamic force Xi and its conjugate ﬂow Ji are connected by the general linear relation Xi ¼

X Rij Jj

ð5:70Þ

j

the symmetry means Rij ¼ Rji

ð5:71Þ

or qXi yqJj ¼ qXj yqJi

ð5:72Þ

Shiner shows that (5.71) is valid also for stationary states far from equilibrium or, to put it better, ﬁnds from the kinetic and thermodynamic equations and relations used, the right form of X, R, J for (5.71) to be valid. Of course, in this case the condition cannot be reformulated into the differential form (5.72). Shiner therefore states that for stationary states far from equilibrium only algebraic symmetry (5.71), in contrast to the differential symmetry (5.72) found close to equilibrium, is valid. Differential symmetry is thus less stringent. The ﬁnal form of resistances (not reproduced here) contains rather complicated combinations of stoichiometric coefﬁcients and (‘‘Guldberg – Waage’’) expressions for forward and backward rates, resulting, in fact, from the material balance. The proof includes one essential point. It is supposed that chemical potentials of some species are controlled from some ‘‘external source’’ and are held at constant values equal to the values of the source (index ‘‘ex’’): mk ¼ mex k for some k

ð5:73Þ www.scilet.com

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This artiﬁcial-looking presumption should probably represent the continuous inand outﬂow in the stationary system. It is not proved nor is its reality discussed. It may refer to the equality of activities (or concentrations) inside the stationary system and in incoming and outcoming ﬂows, which would mean the nonreacting system! Anyway, the result should be considered to be valid only for those stationary systems, in which Eq. (5.73) is valid. Shiner’s analysis is another example of using the well-known kinetic equation in a certain thermodynamic formalism. The usefulness of such derived symmetry relations for practical kinetics is unclear. 5.5 Bro¨nsted re-discovered? An example of an approach based on very unclear justiﬁcation is Grigin’s paper [116]. His aim is to derive a Bro¨nsted-type relationship between rate and equilibrium constant from the generalized thermodynamic force for chemical reactions. The starting point is again the classical rate equation, and the reaction rate is considered to be also the thermodynamic ﬂow. Therefore, another, now thermodynamic, equation should be found for it. It is stated that this equation will be in the form of product I ¼ Rf of the two functions. The ﬁrst one (R) depends on the concentrations, the second one ( f ) on the difference between the chemical potentials of reactants and products (Dm). The author very vaguely and inadequately explains (not proves) why this is possible even when the chemical potential is a function of concentrations. It does not even prevent him from introducing the dependence of the difference between chemical potentials on the concentrations into the supposed form of the second function: f ðDmÞ ¼ expðb1 DmyT Þ expðb2 DmyT Þ

ð5:74Þ

(b1 and b2 are proportionality coefﬁcients); this equation is another premise. The resulting power-law equation immediately leads to the form of the ﬁrst function (for the model reaction A þ B ¼ C): a

R ¼ lðcA cB Þa1 cC2

ð5:75Þ

(l and a’s are again proportionality coefﬁcients). For some unexplained reason, the powers of cA and cB are not independent but equal. Comparing the powers of the corresponding concentrations (or their products) in both terms (separately!) of the initial Guldberg – Waage and resulting ‘‘thermodynamic’’ rate www.scilet.com

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Miloslav Pekarˇ

equations, it is revealed that the powers are expressible through some common parameter a. Finally the following equation is obtained: ð5:76Þ IðDmÞ ¼ l cA cB expbð1 aÞDm yT c cC expðaDm yT Þ (this is the correct form of the misprinted Eq. (12) in ref. [116]) where ‘‘ ’’ indicates the standard state). This enables the formulation of the ‘‘connection between the kinetic and thermodynamic reaction parameters’’: ?

p ¼ l expbð1 aÞDm yT c;

/

p ¼ l expðaDm yT Þ

ð5:77Þ ?

/

So, now the chemical potential appears directly in the rate constant p or p (despite the author’s nomenclature naming it as the probability instead of the

rate constant)! Knowing the still intriguing parameters l, a it would be possible to calculate the values of rate constants!! A Bro¨nsted-type relationship then easily follows: /

p ¼ lKa

ð5:78Þ

where K is the equilibrium constant. It is evident from this short report that the whole deduction is depreciated by several ad hoc, non-justiﬁed presumptions or steps. 5.6 Hungarian contribution II – Ola´h Ola´h has developed ‘‘thermokinetics’’ which is claimed to be a general kinetic theory of physico-chemical phenomena [117 – 122]. It is based on some general features of equations of motion. Thermokinetics is not, in fact, a new or deductive theory but another (re)interpretation of well-known concepts and approaches. We will brieﬂy review its main points without evaluation and concentrate on its application in chemical kinetics. Ola´h calls classical (equilibrium) thermodynamics thermostatics and by (non-equilibrium) thermodynamics, he understands classical irreversible thermodynamics [117,121]. While the former works with extensive state properties or their densities and suitable potentials, the latter introduces ﬂuxes and forces. Thermokinetics adds so-called partial ﬂuxes in contrast to the net ﬂuxes of CIT. Partial ﬂuxes mean that every process is treated separately in the forward and reverse directions and the ﬂuxes are considered as the most valuable contributions of thermostatics. Probably the most important equations are the so-called ‘‘constitutive relations of thermokinetics’’, i.e. relations between partial ﬂuxes www.scilet.com

Thermodynamics and mass-action chemical kinetics

59

and (classical) potentials. The functional relationships are not clearly stated but in any case are supposed to be invertible, thus partial ﬂuxes may be functions of potentials or vice versa. Because of the principle of microreversibility, opposite partial ﬂuxes of the same process should be equal at equilibrium. From this statement it is ‘‘proved’’ (no discussion of this proof is given here) that both ﬂuxes have the same canonical functional form [117]. As noted, thermokinetics was inspired by physics where in the case of conservative ﬁelds the force (X) may be given by appropriate potential (F) [121]: X ¼ m grad F

ð5:79Þ

where m is the proportionality factor or the ‘‘charge’’. The force acts only on the movement of objects carrying the charge. The differential of the potential is related to the force by: m dF ¼ X dr

ð5:80Þ

where r is the positional vector. In the case of gravitational ﬁeld and potential, it follows from Newton’s law that m dF ¼ mv dv ¼ mv2 d ln v:m2 d ln where v2 ¼ ð21 ; 22 ; 23 Þ. If the potential acting on the object is the resultant of two potentials (e.g. gravitational and electrostatic), Ola´h supposes that it is given by m dF ¼ m2 d ln ¼ Q1 dF1 þ Q2 dF2

ð5:81Þ

In thermodynamics, the following potentials are considered: 1yT ; PyT ; mi yT (P is the pressure, m’s are the chemical potentials). Charges (Qi ) are not derived or deduced from the theory but rather selected in such a way that some wellknown relation describing a particular process is obtained. It is stated (neither proved nor derived) that charges relate motion with forces by relations analogous to Eq. (5.81): X R d ln jz ¼ Qzk dFk

ð5:82Þ

k

where R is the universal gas constant and jz is the partial ﬂux of the z-th process. Ola´h states [117] that analysis of known rate equations (here lies the source of the interpretative nature of his approach) shows that partial ﬂuxes can be written as products of factors depending on various potentials. For instance, mass ﬂuxes www.scilet.com

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Miloslav Pekarˇ

( j) depend on chemical and thermal potentials: j ¼ j0 jm ðmyT Þjq ð1yT Þ

ð5:83Þ

(parentheses just show the independent variables) where the thermal factor jq is given by jq ¼ T n expðE* yRT Þ

ð5:84Þ

which is equivalent to the thermal charge according to Eq. (5.78) in the form: Qq ¼ E* þ nRT

ð5:85Þ

(n is the exponent from (5.84), and not the number of moles). Equation (5.82) is used in ref. [118] to derive reaction rate equations. It is stated without proof (only by comparison with the mass-action law) that, in the case of chemical reaction, the partial ﬂuxes are the forward and backward reaction rates with stoichiometric coefﬁcients being the appropriate charges [120]. However, in ref. [118] it is further stated that, due to the stoichiometric constraints, the charge changes to one and the potential to quantity Fr (instead of mi yT ), which is given below. An additional partial ﬂux is the ﬂux of the transported energy with charge given by Eq. (5.85) where E* is now the activation energy (subscript ‘‘act’’). Introducing the charges and potentials into Eq. (5.82), we obtain: R d ln jr ¼ dFr þ ðEact þ nRT Þdð1yT Þ

ð5:86Þ

and after integration: jr ¼ j0r T n expðEact yRT Þ expðFr yRÞ

ð5:87Þ

Finally, after subtracting expressions (5.87) for the forward and backward P reactions and substitution from the statement Fr ¼ k;r mk yT , the following k equation is obtained: " ! !# X ðþÞ X ðÞ 0 n Jr ¼ jr T expðEact yRT Þ exp k;r mk yRT exp k;r mk yRT ð5:88Þ k

k

ðÞ where ðþÞ k;r ; k;r are the stoichiometric coefﬁcients of the k-th component in the r-

th reaction in the forward ‘‘( þ )’’ and backward ‘‘( )’’ directions, respectively. Another version of potentials is given a few paragraphs later. It originates from the dependence of chemical potential on concentration www.scilet.com

Thermodynamics and mass-action chemical kinetics

mk ¼ mk þ mEk þ RT ln ck

61

ð5:89Þ

(‘‘E’’ means excess) and the Gibbs – Duhem equation d ðmk þ mEk ÞyT ¼ Uk dð1yT Þ þ V k dðPyT Þ

ð5:90Þ

and becomes at constant PyT (or constant V in gases): ðþÞ

dFðþÞ ¼ Ur dð1yT Þ R d ln r

Y ðþÞ ckk;r

ð5:91Þ

k

ðÞ

dFðÞ ¼ Ur dð1yT Þ R d ln r

Y ðÞ ckk;r

ð5:92Þ

k

By inserting (5.91) and (5.92) into Eq. (5.86), the mass-action equation can be derived, e.g.: Y ðþÞ Y ðþÞ jðþÞ ¼ jr0 T n exp ðEact U ðþÞ ckk;r ¼ kðþÞ ckk;r r r ÞyRT r k

ð5:83Þ

k

Once more, the potentials (5.91), (5.92) are neither derived nor proved but selected to obtain the desired results. Ola´h also claims [118] that potentials and partial ﬂuxes can be derived from the entropy and entropy dissipation function, respectively. The potentials are given by the partial derivatives of entropy (S) with respect to all densities X dS ¼ Fi dci ; Fi ¼ qSyqci ð5:94Þ i

i.e. by the Gibbs equation. The entropy dissipation function is introduced in ref. [119] as X ji Fi ð5:95Þ DS ¼ i

dDS ¼

X i

ji dFi þ

X Fi dji :dDFS dDjS

ð5:96Þ

i

Because the entropy dissipation function is deﬁned using partial ﬂuxes, the derivation of them from this equation is tautological. We can only ﬁnd the identity www.scilet.com

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ji ¼ qDFS yqFi

ð5:97Þ

and also Fi ¼ qDjS yqji

ð5:98Þ

To conclude, this approach is another combination of the mass-action law, reaction isotherm and chemical potential outlined in Part 1. No fundamentally new deductions or proofs were achieved, only old facts were given new interpretations.

6. EXTENDED IRREVERSIBLE THERMODYNAMICS Extended irreversible thermodynamics (EIT) continues its role in the ﬂux-forces arena. Its principal contribution lies in the extension of the (classical) set of independent variables (speciﬁc internal energy, speciﬁc volume, mass fractions) by some of the so-called ﬂuxes (e.g. heat ﬂux or stress tensor). Thus, applications of EIT to chemical kinetics is usually nothing more than the introduction of some chemical ﬂux among independent variables. Perhaps the ﬁrst contribution of EIT to the analysis of thermodynamicskinetics relationships was the paper by Garcı´ a-Colı´ n and de la Selva [123]. They suppose that there exists some function (Z), in fact the non-equilibrium entropy, of the following variables: Z ¼ Zðe; ; ci ; J; Jd ; pÞ

ð6:1Þ

where e is the speciﬁc internal energy, is the speciﬁc volume, ci the mass fraction of component i (i ¼ 1, 2 in ref. [123] for simplicity), J is the chemical ﬂux and Jd the diffusive ﬂux of one of the species, and p is the trace of the viscous tensor. Partial derivatives occurring in the total differential of this function are either expressed by relations resembling relations of classical reversible thermodynamics or modelled by relations suitable for further developments: qZyqe ¼ Y1 ;

qZyq ¼ PY1 ;

qZyqJ ¼ ar Y1 ;

qZyqci ¼ Mi Y1

qZyqJd ¼ ad Y1 ;

qZyqp ¼ ap Y1

ð6:2Þ ð6:3Þ

where Y represents the non-equilibrium temperature, P is the non-equilibrium pressure, Mi the molar mass, a’s are the proportionality coefﬁcients. www.scilet.com

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The chemical ﬂux is not explicitly stated but from the symbol used it is clear that the ﬂux is actually the reaction rate. Unfortunately, this is deﬁned using the time derivatives of the mass fractions, which is in the modelled system with diffusion either improper or, at least, not easily applicable to experiment. The partial derivatives are then approximated by their expansion in some of the independent variables, e.g.: Mi Y1 ¼ mi T 1 þ bqðMi Y1 ÞyqJcJ þ bqðMi Y1 ÞyqJd cJd þ bqðMi Y1 Þyqpcpþ þ ð1y2!Þ q2 ðMi Y1 ÞyqJ2 J2 þ ð6:4Þ where T is the (equilibrium) temperature and mi the (classical) chemical potential. The expansions are nothing more than following transformation of functions: Z ¼ Zðe; ; ci ; J; Jd ; pÞ?qZyqx ¼ f ðJ; Jd ; pÞ þ Cx where x represents any variable from the set fe; ; ci ; J; Jd ; pg and Cx the relevant classical term. This transformation is substantiated by stating that for the classical case, i.e. for the disappearance of extending, ﬂux variables, classical expressions like qZyqe ¼ T 1 should be obtained. Why this equation cannot be arrived at by disappearing corresponding partial derivatives in full functional representation is not explained. Moreover, functions f ðJ; Jd ; pÞ look like a McLaurin series expansion and Cx is the equilibrium expression for the appropriate partial derivative of entropy. Thus, the approximation of partial derivatives is an expansion around equilibrium. It is therefore not clear where the partial derivatives in this expansion should be evaluated, as at equilibrium they should vanish, i.e. be equal to zero. Consequently, Eqs (6.2) – (6.4) should be considered only as a speciﬁc model and the whole analysis is valid only for systems complying with this model. Which real systems or materials correspond to the model is not discussed in the original paper. Another particular model in this work is the expression for the entropy ﬂux, which is constructed just as the sum of the diffusion ﬂux, the only one vectorial independent variable, multiplied successively by some of the scalar independent variables and a term which should again probably resemble some classical term: JZ ¼ Y1 ðM1 M2 ÞJd þ b01 JJd þ b02 pJd þ www.scilet.com

ð6:5Þ

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Miloslav Pekarˇ

(b’s are proportionality coefﬁcients). This should be viewed only as a speciﬁc example of representation of the linear isotropic vectorial function. The tacit construction of models continues: the next model is the representation of the entropy source (s) as a nonlinear isotropic scalar function: s ¼ Jd Xd þ JXr þ pX

ð6:6Þ

where the ‘‘generalized forces’’ are deﬁned as Xd ¼ m20 Jd þ m21 JJd þ

ð6:7aÞ

Xr ¼ mr1 J þ mr2 J2d þ mr3 J2 þ mr4 J3 þ mr5 p

ð6:7bÞ

X ¼ m1 p þ m2 J þ

ð6:7cÞ

Then, a preliminary result is derived – an equation for evolution of the chemical ﬂux, i.e. an equation with the material derivative of J (J_ ). This equation is solved, or approximated to successively higher orders in J, by some strange procedure referring to the stationary state. The ﬁnal result, Eq. (6.8) below – ‘‘general phenomenological relation between the rate of the reaction and the chemical afﬁnity’’ – is a mere summation of several of these approximations and is not proved for consistency with the initial expression for J_ . This general phenomenological relation expresses reaction rate as a function of powers of afﬁnity: J¼

ðrAymr1T Þðmr3 r2A2 Þyðm3r1 T 2 Þð1y2Þ 1 ðqar1 yqc1 Þþ2 ðqar1 yqc2 Þ ðr2A2 Þyðm3r1 T 3 Þþ þ ð2m2r3 ymr1 Þ mr4 ðr3 A3 Þyðm3r1 T 3 Þ ð6:8Þ here, r is the density, A the afﬁnity, i is the product of the stoichiometric coefﬁcient and molar mass of the component i. The afﬁnity is introduced through the classical deﬁnition (2.5)1, which is also used in CIT, supposing the same concentration dependence. There is no extended approach. More peculiar is the way that led to the power law of Eq. (6.8). This was not due to the speciﬁc claims of EIT but just due to the models introduced and used in an unusual manner: 1.

powers of J , which are the causes of later powers of A in the ‘‘general phenomenological relation’’, are introduced due to the model (6.7b),

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2.

afﬁnity is originally introduced, in fact, just as a ﬁrst – classical, equilibrium (!) – member of a series approximation (6.4),

3.

non-equilibrium corrections, introduced into this approximation are ignored in the above-mentioned stationary state analysis,

4.

powers of J , which should be the other non-equilibrium corrections in model (6.7b), are systematically expressed in this analysis in powers of afﬁnity despite its status given under 2 above. There are other unclear points. During the constructing of the models, it

is several times stated, but never proved, that the models should reduce to the standard form of CIT. Rate (chemical ﬂux) is ﬁnally expressed as a function of afﬁnity, which itself is a function of chemical potential, which itself is a function of concentration. Although the chemical ﬂux, or reaction rate, is included among the independent variables, the dependence of the rate on other variable(s), usually afﬁnity, is sought. The next work of the same authors gives only moderate progress. The motivation for the EIT approach is stated in ref. [124]: ‘‘For many years linear irreversible thermodynamics has been the only theory available to account for the empirical kinetic mass-action law (KMAL) as a ﬂux-force relation between the chemical rate J and the afﬁnity A namely, J & ½expðAyRT Þ 1. In spite of the fact that such a relation is a nonlinear one, it has been shown that at least for the reaction B þ C , D þ E, and using a kinetic theory model, the entropy source JAyT is consistent with it. Therefore the rate J is interpreted as a thermodynamic ﬂux of the same footing as the heat ﬂux, the diffusion ﬂux, and the stress tensor. Thus, KMAL has been viewed as a constitutive relation, analogous to Fourier’s heat equation, Fick’s equation for diffusion, and the Newton – Navier equation for the transmission of momentum. On the other hand, the coupling between the chemical rate with its generating forces, namely A, and the divergence of the hydrodynamic velocity satisfy Onsager’s reciprocity theorem in the linear approximation only.’’ A paper entitled simply ‘‘Consistency of the Kinetic Mass Action Law with Thermodynamics’’ [125] starts with the function Z ¼ Zðe; ; ci ; q; J; Ji Þ

ð6:9Þ

thus, instead of the stress tensor, the heat ﬂux q is considered; e is the internal energy density, the other symbols have the same meaning as in Eq. (6.1), but the diffusion ﬂuxes (Ji ) are considered for each component i separately. Again, some www.scilet.com

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speciﬁc models of representations of functions of various tensorial orders are introduced. As usual in EIT, the total derivative of the function is written and the partial derivatives with respect to the ‘‘ﬂux variables’’, viz. q; J; Ji are approximated by expansion around the local equilibrium where these variables should, of course vanish. No true Taylor expansion is used, as the ‘‘classical variables’’ (e; ; ci ) are not included, and do not expand anywhere. Rather, a combination of the isotropic function representation and a Taylor series is used, at least for qZyqJ: qZyqJ ¼ T 1 o0 J

ð6:10Þ

(o0 is the proportionality coefﬁcient). This is the ﬁrst model used. The EIT postulate of the entropy balance equation, viz.: r dZydt þ divJZ ¼ sZ

ð6:11Þ

is a necessary intermediate step calling for at least two expressions – for entropy ﬂux (JZ ) and source (sZ ). The entropy ﬂux is represented as an isotropic vectorial function, which is immediately speciﬁed by the following equation X X JZ ¼ ð1yT Þq þ b0 q ðmi yTMi ÞJi þ bi0 Ji ð6:12Þ i

i

(b’s are the proportionality coefﬁcients, which are, in turn, functions of all scalar invariants) again claiming, not proving, that this form is reducible to the normal entropy ﬂux of CIT where ‘‘ﬂux variables’’ can be ignored. This is the second model which is then combined with mass, energy, and entropy balances and time derivative of Z, coming from the ﬁrst model (6.10). An expression for entropy production then results as follows: sZ ¼ "

# X q grad T 1 þ grad b0 þ ða0 yT ÞðdqydtÞ þ ðg0i yT ÞðdJi ydtÞ þ i

"

# X X Ji grad ðmi yTMi Þ þ grad bi0 þ ða0i yT ÞðdqydtÞ þ ðgij yT ÞðdJj ydtÞ þ þ i

j

X X ð1yT ÞJi Fi þ b0 div q þ bi0 div Ji þ AJyT þ o0 JT 1 ðdJydtÞ ð6:13Þ þ i

i

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where a’s and g’s are the coefﬁcients from equations analogous to Eq. (6.10) but for the partial derivative with respect to q and Ji , respectively; mi is the (classical) chemical potential, Fi the external force on i per unit mass and A is the (classical) afﬁnity. It is claimed that entropy production is a scalar function of deﬁning scalar variables. Therefore, it can be represented as an isotropic function. The representation is again rather speciﬁc: X sZ ¼ P 0 þ X q q þ Xi Ji ð6:14aÞ i

X q ¼ x0 q þ

X

xi Ji

ð6:14bÞ

lij Jj

ð6:14cÞ

i

X i ¼ li q þ

X j

and can be considered as the third model. Symbol P 0 represents a function of all scalar invariants. The two expressions for entropy production, (6.13) and (6.14a), should be consistent. But the consistency is not straightforward and explicit. Therefore, further models have to be invoked. In contrast to the authors’ contention, chemical ﬂux does appear as a multiplicative factor in one of the two equations for entropy production, cf. Eq. (6.13). This appearance is not sufﬁcient and must be supported by extracting the chemical ﬂux from (only some!) scalar coefﬁcients in functional representations. Several additional models can therefore be constructed: b0 ¼ b 00 J;

bi0 ¼ b 0i0 J;

P 0 ¼ JPðJ; . . .Þ

ð6:15Þ

where P in the last equation is again a function of all scalar invariants including J. At last, the desired relation – an equation for the time derivative of chemical ﬂux or the general mass-action law – is obtained: X ðo0 yT ÞðdJy dtÞ ¼ AyT b 00 div q b 0i0 div Ji þ P ð6:16Þ i

Afﬁnity was introduced again due to the reminder of the classical term in the representation of entropy ﬂux. The evolution equation enables, after introducing further models or simpliﬁcations, discussion in the terms ‘‘chemical ﬂux is forced www.scilet.com

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by the afﬁnity’’. It should be stressed that an explicit dependence on afﬁnity is found only for the time derivative of the chemical ﬂux J (which is, perhaps, the reaction rate); the direct relationship between chemical ﬂux (reaction rate) and afﬁnity is obtainable only for the stationary state. To summarize – there are too many models with vague relations to real systems or materials, and too general ﬁnal equations, not containing the main quantity measured by kineticists, namely concentration. Of course, in Eq. (6.16) concentration is hidden in afﬁnity, however, this is not an equation for the reaction rate itself, but for its time derivative. Lebon et al. [126] write that they adopt a position intermediate between classical theory and EIT. Their work was competently criticized by Garcı´ a-Colı´ n [127] to say nothing about its limitation to homogeneous (non-diffusing) mixtures. Instead of the reaction rate, the authors use the degree of advancement (x) deﬁned as x_ ¼ c_ i yi

ð6:17Þ

(ci is the mass fraction of i-th component and i its stoichiometric coefﬁcient). This means that only closed systems are considered. As an independent variable, however, the following difference is used: x ¼ x xe

ð6:18Þ

where xe denotes the equilibrium value. It is assumed that the time evolution of the new variable is given by x_ ¼ jðT ; p; xÞ

ð6:19Þ

(T is temperature, p pressure). It is further postulated that function j is expressed as follows: jðT ; p; xÞ ¼ xcðT ; p; xÞ

ð6:20Þ

During further development, no special irreversible thermodynamical approach is used. Only combinations and manipulations with the postulate, integrated form (6.17), and relations well-known from reversible thermodynamics, are used, viz. the deﬁnition of the relation of chemical potential to the component activity, the deﬁnition of afﬁnity (2.5)1, and the expressing of equilibrium constants by standard chemical potentials. www.scilet.com

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The ﬁnal result is the following equation x_ ¼ oðT ; p; xÞ½1 expðAyRT Þ

ð6:21Þ

containing another function (o) which includes also the function c. Eq. (6.21) is again an expression for the time derivative of a certain reaction rate quantity (degree of advancement), which is claimed to be the standard law of massaction. No explicit rate equation or function was derived. Function c remains undetermined and Garcı´ a-Colı´ n [127] showed how it can be related to phenomenological coefﬁcients of CIT. In summary: no irreversible thermodynamics is utilised and the only new feature is postulate (6.20), which is used several times and coupled with the wellknown relations of classical thermodynamics. This work, as well as other irreversible thermodynamics approaches (e.g. [125]), were brieﬂy criticized by Ross and Garcı´ a-Colı´ n [128]. However, no new ideas were presented, just some reservoirs for reactants and products are introduced with no clear distinction between the reaction Gibbs free energy and the Gibbs free energy of the whole system. The critique of EIT approaches is based on the initial task of EIT – to describe fast processes by introducing new, extending, variables: ‘‘For most reactions, especially in liquids, reaction times are long compared to other relaxation times (vibrations, etc.)... For such cases the condition of local equilibrium holds well; the thermodynamic variables including the progress variable are on the same time scale and there is no need for an extended thermodynamics. That need may arise when the reaction time is more comparable to other relaxation times and the predicted rate coefﬁcients become time-dependent, which expresses the effect of the relaxation of the fast(er) variables of, say, vibrational relaxation, compared to the slow(er) chemical rate.’’ Thus, there is usually no need for incorporating some ‘‘chemical ﬂux’’ among the independent variables. An interesting note is given in the conclusion: ‘‘Furthermore, the identiﬁcation of a generating function (Z above) with an entropy has not yet been justiﬁed.’’ The last contribution from EIT was due to Fort et al. [129] who try to ﬁnd new developments from the same starting point. First, they would like to ascertain whether EIT methodology gives entropy as a sum of its equilibrium value and some non-equilibrium correction also for chemically reacting systems. Second, they support this ﬁnding with deductions from the kinetic theory of www.scilet.com

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gases. As statistical theories are beyond the scope of this review, we will focus mainly on the EIT part. Speciﬁc entropy (s) is, again, considered to be a function of ‘‘classical’’ variables and one extending, which is, of course, the reaction rate J: s ¼ sðu; ; ci ; JÞ; i ¼ 1; 2; . . . ; n

ð6:22Þ

(u is the total speciﬁc internal energy, the speciﬁc volume and ci the mass fraction) and its total differential is constructed; partial derivatives with respect to the classical variables are expressed with the aid of the generalized temperature (Y), pressure (P), and chemical potentials (Zi ) as dictated by the EIT standard procedure. Then, the ﬁrst new postulate or, more appropriately, model is introduced: ðqsyqJÞu;;ci ¼ ðayT ÞJ

ð6:23Þ

where a is some coefﬁcient depending only on the classical variables. This model is substantiated by the traditional claim that the generalized entropy ðsÞ must reduce to the classical one at equilibrium where the rate is zero, which is not proved. Consequently, the generalized local Gibbs equation can be formulated: X ds ¼ ð1yYÞ du þ ðPyYÞ d ðZi yYÞ dci ðayT ÞJ dJ ð6:24Þ i

However, the evolution equation for speciﬁc entropy is restricted to depend on the reaction rate only, which is explained by considering only non-equilibrium processes in an incompressible ﬂuid in the absence of heat and diffusion effects, speciﬁcally: r dsy dt ¼ J½AyT ðaryT Þ dJy dt

ð6:25Þ

In the (second) postulate (6.25), the classical equilibrium deﬁnition of afﬁnity (A) through the classical chemical potentials (not Zi ’s!), cf. Eq. (2.5)1, was applied together with the mass fraction balance, namely dci y dt ¼ ði yrÞJ, where i is the stoichiometric coefﬁcient and r the density. The evolution equation (6.25) is compared with the general law of entropy balance of EIT, see Eq. (7.3) below, and the entropy source (more precisely, the rate of entropy production per unit volume) is then expressed as the right hand side of Eq. (6.25). It should be noted that in contrast to the other EIT approaches, which considered the reaction rate as a part of the entropy ﬂux, here the rate is included in the entropy source! www.scilet.com

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As the second law of thermodynamics calls for a positive value for entropy production, the simplest way to assure this is to have it in only the second power of the reaction rate. Therefore, a third model is proposed: AyT ðarÞyT dJy dt ¼ bJ

ð6:26Þ

This model is particularly convenient for EIT as it is analogous to the Maxwell – Cattaneo equations, which were successfully explained within the EIT approach. With this model, the generalized Gibbs equation (6.24) may be written as X ds ¼ ð1yT Þ du þ ðpyT Þ d ðmi yT Þ dci ðtyrlÞJ dJ ð6:27Þ i

where l ¼ 1yb and t ¼ arlyT , with no explanation as to why the generalized variables were substituted by their classical (equilibrium?) analogues, i.e. temperature (T ), pressure (p) and chemical potential (mi ). Its integrated form is simply expressed as sðu; ; ci ; JÞ ¼ sðu; ; ci Þ ðty2rlÞJ2

ð6:28Þ

And this is all for chemical kinetics. Entropy was, ﬁnally, expressed as its equilibrium value and non-equilibrium correction, which is second order in the reaction rate. The kinetic theory part of this work derives a similar expression for entropy and even the usual proportionality of the rate to the afﬁnity. Interestingly, in the conclusion the authors write: ‘‘...the reaction rate is not a ﬂux in the usual sense because it does not appear as a true ﬂux in the balance equations of mass fractions; instead, it appears as a source term there.’’ Further. ‘‘In spite of this, we have shown how chemical reactions can be included in the much broader framework of EIT.’’ This means that ﬂuxes, forces, sources are interpretations according to some particular motivation and not the results of rigorous deﬁnitions or proofs.

7. COMMON PROBLEMS IN CIT AND EIT APPROACHES Balance equations, well-known from other branches of physics or chemistry, are used in both approaches, of course. They have the general form [130]: ð

qðrbÞ dV ¼ V ðtÞ qt

ð SðtÞ

Jb ? n dS þ

ð

sb dV

V ðtÞ

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ð7:1Þ

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Miloslav Pekarˇ

Here Jb , which may be a vector or tensor, represents the quantity ﬂowing per unit area and unit time (t) through the boundary S of some volume V , and sb is the rate of production or destruction per unit volume of the quantity, which has the speciﬁc value b, r is the mass density and n is the positive unit normal vector to the boundary. Both the volume and its boundary are in general changing in time. The local form of the balance under pertinent conditions is as follows: qðrbÞyqt ¼ div Jb þ sb

ð7:2Þ

It is further supposed that the same balance is valid also for the speciﬁc entropy: qðrsÞyqt ¼ div ðJs þ rs vÞ þ ss

ð7:3Þ

where v is the barycentric velocity. Entropy ﬂux Js is deﬁned by: ð e Js n dS d Sydt ¼

ð7:4Þ

SðtÞ

where superscript ‘‘e’’ refers to the entropy (S) exchanged with the surroundings (balances (7.2) and (7.3) are rather inconsistent). That every approach considering reaction rate as a part of entropy ﬂux is dubious. The reaction rate is certainly not limited to something which only ﬂows through the boundary. The same problem is with the ‘‘chemical ﬂux’’, widely used by LI and EI thermodynamicists, in fact the reaction rate. Considering the reaction (rate) to be some ﬂux is unacceptable. As this is usually no more than a linguistic denomination, it may present no serious problem in the results of particular deductions. Jou et al. [131] state: ‘‘. . . decomposition into thermodynamic ﬂuxes and forces is arbitrary to a certain extent. . .one could permute the deﬁnitions of ﬂuxes and forces.’’ Why use this arbitrary concept, at all? The ﬂux-force concept is a somewhat conﬁning trap of CIT or EIT. Jou et al. [131] write when introducing ﬂux-force conception into balances, particularly into the entropy balance: ‘‘. . . the cause is provided by the driving thermodynamic force, which elicits the effect manifested through the conjugated ﬂux.’’, whereas several pages further on [132]: ‘‘In order to obtain evolution equations for the ﬂuxes compatible with the positiveness of the entropy source, we express the forces as functions of the ﬂuxes.’’ The logic of the cause – effect is suddenly abandoned. Whereas in many other cases it is really only an arbitrariness of improper nomenclature, despite of which realistic equations can be obtained, in the chemically reacting systems this effort of forcing the wellwww.scilet.com

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known equations into the ﬂux-force framework is much less productive. Instead of searching for relationships between ﬂuxes and forces a priori selected to be the reaction rate and afﬁnity, respectively, a description of the actual problems should be sought – what are the correct independent variables in chemically reacting systems. Denbigh in the early 1950’s even argued [133] that the rate of a chemical reaction is not proportional to a thermodynamic force in general, and is primarily determined by the concentrations and not by the difference of chemical potentials. Chemical ﬂux, which is in fact the reaction rate, is systematically introduced among the independent variables. Because concentrations (in the general meaning, not just molar concentrations) are also and traditionally independent variables, it follows that the reaction rate is not dependent on the concentration of the reacting species! This is in strong contradiction with experience and theory of chemical kinetics. A particular example is found in ref. [129]. Requiring Eq. (6.24) to be an exact differential, the following condition, among others, should be fulﬁlled: qðZi yYÞyqJ ¼ q½ðayT ÞJyqci

ð7:5Þ

This condition is written in ref. [129] as qðZi yYÞyqJ ¼ JqðayT Þyqci

ð7:6Þ

i.e. the reaction rate is independent of concentrations! Fortunately, Eq. (7.6) is of no further use in the development given in ref. [129]. The CIT and EIT approaches also seeks evolution equations for perhaps every variable, i.e. also for the chemical ﬂux or reaction rate. Thus, equations for the (material) time derivative of this ﬂux, i.e. of reaction rate, are being derived, sometimes including also powers of the ﬂux. This is unusual in common chemical kinetics where (rate) equations for the time evolution of concentrations are the desired object of study. Further, substantiation of the various postulates for non-equilibrium or extending variables often calls for the necessity of their reduction to classical equations (containing classical variables only) at equilibrium. It seems then that classical variables, concentrations among them, have no effect in the nonequilibrium regime. This is certainly not true for the non-equilibrium reaction rate. www.scilet.com

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The Guldberg – Waage law is still empirical. Instead of implementing it into the theoretical ﬂux-force framework, it might be desirable to derive it, or something analogous to it, from the theoretical thermodynamic framework. To this point, EIT is of much more beneﬁt than CIT. As Bowen has clearly stated in the beginning of his development of thermodynamic mixture theory [134]: ‘‘In a general theory of mixtures . . . it should be possible to prescribe, as independent ﬁelds, . . . certain parameters characterizing the state of any chemical reactions taking place.’’

8. RATIONAL OR CONTINUUM THERMODYNAMICS APPROACHES TO CHEMICAL KINETICS 8.1 Introduction Under this heading, all papers, more or less using balance equations, entropy inequality, and constitutive equations in the continuum approach, are collected. Rational thermodynamics provides an elaborated rigorous framework for the description of various continuum systems. Because of its axiomatic basis and strong mathematical language, it is rather difﬁcult to give here a brief yet exhaustive review. Therefore we will focus only on the most important works in regard to chemical kinetics. A very general overview of basic balance laws (equations), which lie at the heart of all rational thermodynamics treatment for a reacting continuum, was given by Kelly in the 1960’s [135]. The reaction rate, of course, enters into the mass balance but usually in a very general way. Most rational thermodynamics contributions thus have a primarily theoretical value. The integral form of material balance for component a reads [135]: ð ð d ra d ¼ rba d ð8:1Þ dt Va Va where ra is the mass density of constituent a, r the mass density of the reacting P continuum, r ¼ a ra , and rba is the volume supply of mass of species a due to chemical reaction. The differential form of the material balance (8.1) is obtained under certain continuity suppositions as: qra yqt þ div ðra va Þ ¼ rba

ð8:2Þ

where va is the velocity of component a. www.scilet.com

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The reaction rate is ‘‘hidden’’ in the mass supply rba and perhaps virtually no publication gives its explicit functional form or, using the terminology of rational thermodynamics, constitutive equation. 8.2 Bowen lays the foundation stone Bowen [136] treats the model of a non-isothermal reacting mixture with elastic deformations and viscous stresses but without diffusion. Instead of reaction rate or mass supply, he works with the extent of reaction deﬁned using his algebraicstoichiometric formalism [34], which is not identical with the deﬁnition known from (irreversible) classical thermodynamics but closely related to it. It is deﬁned as a vector (j), i.e. for each component of the reacting mixture as the difference between its actual and referential (initial) molar fraction. It is thus neither deﬁned for, nor related to, a particular reaction. It summarizes the effects of all reactions, which may inﬂuence the concentration of a particular component. However, it can be related to reaction rates through the algebraic-stoichiometric formalism [34]; this is not used in ref. [136]. Moreover, Bowen introduces extent of reaction among the independent variables, so this seems to be analogous to the CIT and EIT practices criticized in Part 6. However, concentrations are not explicitly included in the set of independent variables (which can be seen on the right hand side of Eq. (8.3)), the extent of reaction is not a reaction-rate quantity in this case and, in fact, one of the constitutive equations, viz. j_ ¼ wðu; g; F; F_ ; jÞ

ð8:3Þ

resembles traditional rate equations setting the reaction rate (concentration time variation) to be some function of concentrations themselves. In Eq. (8.3), w stands for some function, u is the temperature, g its gradient, F is the gradient of the deformation and the dot means (material) time derivative. Unfortunately, no explicit, practically convenient expression of function w is derived. Bowen arrives, however, at interesting conclusions regarding the relationships between afﬁnity and reaction rate (at equilibrium). Afﬁnity (A) is deﬁned by him as follows: A ¼ Aðu; g; F; F_ ; jÞ ¼ ðqcyqjÞðu; g; F; F_ ; jÞ

ð8:4Þ

and the state of equilibrium (denoted ‘‘ þ ’’) by: g ¼ 0;

F_ ¼ 0;

wðuþ ; 0; Fþ ; 0; jþ Þ ¼ 0 www.scilet.com

ð8:5Þ

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Miloslav Pekarˇ

Function c means free energy as a function of the same variables as the extent of reaction in Eq. (8.3), which is expressed by the last parentheses in Eq. (8.4). Bowen shows that, at equilibrium, the afﬁnity is restricted but it is not necessarily zero. It must vanish if qwyqj is at equilibrium non-zero (regular); this special equilibrium state is called by Bowen ‘classical equilibrium’ and, without additional assumptions, classical equilibrium and Bowen’s equilibrium states are generally different. Further, Bowen tests what the vanishing of afﬁnity implies about the reaction vector and proves that in this state this vector need not be zero. It must vanish (only) if qAyqj is regular in the state deﬁned by (8.5)1,2. It is thus seen that more complicated material systems are richer in equilibrium states and therefore LIT (EIT) approaches are restricted in their generality. The next Bowen paper [137] treats a reacting mixture of elastic materials with diffusion. The extent of reaction is abandoned, as it is not in general deﬁnable in systems with diffusion [34,35]. Further, concentrations (precisely, densities or weight concentrations) are among the independent variables and the same form of constitutive equation as for other dependent variables is formulated for the mass supply (closely related to the reaction rate): ba ¼ f ðy; g; Fb ; Gb ; vb ; gb ; db Þ

ð8:6Þ

where ba ¼ jdet Fa j _ ca

ð8:7Þ

The repeated symbols have the same meaning as above, Fb is the deformation gradient and Gb the second deformation gradient for the constituent b, vb its velocity, gb ¼ rb jdet Fb j, db is the certain gradient of density of component b (rb ) and _ ca is the mass supply for constituent a. Subscripts a and b apply to all constituents of the mixture which are n in total. Thus, in this case, the mixture is not considered and balanced as a whole, as a single-continuum, but as a mixture of continua of individual components. Thus reaction rate is, in fact, one of the proposed dependent variables because it is determined by _ ca , which in traditional chemical kinetics is called the production rate of component a. No further simpliﬁcation of its functional form is obtained. Equations (8.6) and (8.3) are typical examples of (only) results of most rational thermodynamics works related to kinetics. Even for a reacting www.scilet.com

Thermodynamics and mass-action chemical kinetics

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mixture of two ideal gases with different temperatures, the mass production rate remains as a general function of densities, temperatures and their gradients, gas velocities, and deformation tensors [138]. The consequences of Bowen’s model (8.6) to afﬁnity-rate relationships are similar to the preceding case. Now, the afﬁnity of reaction p is deﬁned as: Ap ¼

n X ma jdet Fa jPpa qCyqga ¼ Ap ðy; Fb ; gb ; db Þ

ð8:8Þ

a¼1

where ma is the molecular weight of constituent a, C is the (inert part of) free energy density, Ppa is the stoichiometric coefﬁcient related to the mass supply through [34] _ ca yma

¼

R X

Ppa jp

ð8:9Þ

p¼1

where R is the number of independent reactions and jp their rates. Although now the afﬁnity is deﬁned as the afﬁnity of a reaction, it is still in principle identical to the deﬁnition in the previous paper and has not the same meaning as its usual, classical counterpart. Using ca ¼ ra yr

ð8:10Þ

and ga ¼ ra jdet Fa j

ð8:11Þ

deﬁnition (8.8) can be transformed to deﬁnition (8.4) for non-diffusing mixtures where concentration changes determined solely by the reaction and extent of reaction can be used: qCyqga ¼ ðqCyqca Þðqca yqga Þ ¼ ðqCyqca Þð1yrjdet Fa jÞ qCyqxa :ðqCyqca Þma

ð8:12Þ

Bowen shows that for the state of weak equilibrium (identical to equilibrium from ref. [136]) deﬁned as þ jp ðyþ ; 0; Fþ b ; 0; 0; gb ; 0Þ ¼ 0;

p ¼ 1; . . . ; R

ð8:13Þ

T afﬁnity need not be zero in general but it must vanish if qjþ p yqga , a ¼ 1; . . . ; n, is regular. This state, where both (8.13) and www.scilet.com

78 þ Ap ðyþ ; Fþ b ; gb ; 0Þ ¼ 0;

Miloslav Pekarˇ

p ¼ 1; . . . ; R

ð8:14Þ

are valid, is called strong equilibrium [35]. Conversely, in an equilibrium state where (8.14) is fulﬁlled, the reaction T þ rate need not vanish but must be zero (only) if qAp yqga is regular. In other T T þ words, if both q jp yqga and qAp yqga are regular at (yþ ; 0; Fþ b ; 0; 0; gb ; 0), þ þ þ þ þ p then jp ðy ; 0; Fþ b ; 0; 0; gb ; 0Þ ¼ 0 if and only if A ðy ; Fb ; gb ; 0Þ ¼ 0 for p ¼ 1; . . . ; R. Interestingly, Bowen asserts that there is no natural relationship between chemical potential and chemical afﬁnities. This contradiction with traditional irreversible thermodynamics originates probably in his deﬁnition of the chemical potential (tensor) Ka : Ka ¼ ca I TTa yra

ð8:15Þ

where ca is the constituent’s partial free energy density and Ta is the partial stress on the a-th constituent. Besides its inherent (third order) tensorial character (in contrast to scalar afﬁnity), this quantity is deﬁned using free energy directly and not only its (concentration) derivative as traditionally. Deﬁnition (8.15) derives from the classical relation between the Gibbs (G) and Helmholtz (F) energies: G ¼ H T S ¼ U þ PV T S ¼ F þ PV

ð8:16Þ

together with the classical equivalence between chemical potential and partial molar Gibbs energy. Deﬁnition (8.15) can be then understood as a generalization of (8.16) to materials, which can support stress. As noted by Nunziato and Walsh [139], this deﬁnition views the chemical potential as reﬂecting any change in the free energy due to the deformation of a constituent, and not due just to the changes in its mass. 8.3 Gurtin re-examines the classical theory Gurtin and Vargas [140] tried to reinvigorate the classical theory of reacting ﬂuid mixtures using an appropriate constitutive model. The mixture is treated as a single body: balance laws for momentum, energy, and entropy are postulated for the mixture as a whole. Their effort was successful using the following independent variables in constitutive equations: speciﬁc volume (), temperature (y), concentrations (in the above mentioned sense; collected in vector c), and www.scilet.com

Thermodynamics and mass-action chemical kinetics

79

gradients of all these quantities. Among others, they proved that in this mixture, stress reduces to a pressure, that free energy, pressure, and entropy depend only on speciﬁc volume, temperature, and concentrations, and are related by the classical relations (e.g. Maxwell or Gibbs relations), and that there exists a chemical potential given as the derivative of the free energy with respect to the corresponding concentration. Regarding our interest in chemical kinetics, the reaction rate, or, more precisely, the mass supply ma (of any constituent a) is considered to be one of the dependent variables possessing thus the standard constitutive equation: ma ¼ _ ma ð; y; c; grad ; grad y; grad cÞ: _ ma ðLÞ;

a ¼ 1; . . . ; n

ð8:17Þ

Unfortunately, the rational thermodynamic procedure again gives no further simpliﬁcation of this general equation. Thus, even with this simple mixture, otherwise obeying classical thermodynamic relations, the reaction rate is not in general a function only of temperature and concentrations as supposed by traditional mass-action kinetics, but also a function of, for example, their gradients. Gurtin and Vargas’ results have consequences also for afﬁnity-based approaches. Expressing the mass supply through the rates of R independent reactions (Jr ): ma ¼

R X ar Jr

ð8:18Þ

r¼1

(ar divided by the molecular mass of constituent a is proportional to the corresponding stoichiometric coefﬁcient) and deﬁning afﬁnity using the chemical potential ma : X ar ma ð8:19Þ Ar ¼ a

the product of mass supply and chemical potential occurring in the ﬁnal entropic inequality can be written as ma ma ¼

R X Jr Ar

ð8:20Þ

r¼1

and the entropic inequality reads: X Jr A r 0 ha grad ma þ ð1yyÞq grad y þ www.scilet.com

ð8:21Þ

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Miloslav Pekarˇ

Here, ha is the relative mass ﬂux of constituent a (diffusive ﬂux) and q is the heat ﬂux. Thus, only, for example, in the non-diffusing mixture with no temperature gradients, the (‘‘de Donder’s’’) relation R X Jr Ar 0

ð8:22Þ

r¼1

is valid (and for independent reactions only). Gurtin and Vargas formulate results for the equilibrium state and its neighbourhood with unusual mathematical care. The equilibrium state is deﬁned as a homogeneous state Lþ , i.e. a state with grad ¼ grad T ¼ grad ca ¼ 0, with the property that ma ðLþ Þ ¼ 0;

_

a ¼ 1; . . . ; n

ð8:23Þ

It is proved that þ 2 ma mþ a ¼ OðjL L j Þ;

ma ma ¼ OðjL Lþ j2 Þ

ð8:24Þ

thus, within the terms of OðjL Lþ j2 Þ, the mass supply and the chemical potential (called the reduced chemical potential by Gurtin and Vargas) are ‘‘orthogonal’’. Equations (8.24) may be viewed as a certain generalization of the equilibrium (equality) version of (3.12). Strong equilibrium is deﬁned by þ 3 ma ðLÞmþ a ¼ OðjL L j Þ

_

ð8:25Þ

as jL Lþ j ?0 in addition to (8.23). These notions of strong equilibrium and equilibrium are somewhat weaker than those given above, yet their consequences regarding the simultaneous vanishing of reaction rate and afﬁnity are similar. Gurtin and Vargas prove that for a strong equilibrium, mass supply depends only on the chemical potential (as jL Lþ j ?0): ma ¼

n X þ 2 tab ðmb mþ b Þ þ OðjL L j Þ

ð8:26Þ

b¼1

or, if the scalar coefﬁcients tab fulﬁl tab ¼ tba , ma ¼

n X tab mb þ OðjL Lþ j2 Þ

ð8:27Þ

b¼1

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Thermodynamics and mass-action chemical kinetics

81

þ 2 because tab mþ a ¼ 0. Thus, within the terms of OðjL L j Þ, the reaction rate could be expressed directly through the chemical potentials, or, if the ideal

relation between the chemical potential and concentration is inserted, through the concentrations. Equations (8.26) or (8.27) show that the mass supply of any component is in general determined by the chemical potential of all the other constituents. Further, if a linear relationship between reaction rate and afﬁnity around strong equilibrium is assumed: Jr ¼

R X þ 2 L rs ðAs Aþ s Þ þ OðjL L j Þ

ð8:28Þ

s¼1

then R X þ 2 Jr Aþ r ¼ OðjL L j Þ

ð8:29Þ

r¼1

Equation (8.29) is the ﬁnal result restricting simultaneously the values of reaction rate and (equilibrium) afﬁnity. It should be stressed that this general approach gives no restrictions on the equilibrium value of the afﬁnity. Although the equilibrium criteria are deﬁned with minimum (mathematical) restrictions, they are probably too weak to be used and tested in practice. On the other hand, statements on relationships between reaction rate and afﬁnity reviewed in Section 2 may have very limited validity even if they are deduced for the vicinity of equilibrium only. ‘‘Kinetic’’ results obtained by Gurtin and Vargas are very close to those presented by Gurtin [141] for a mixture of inviscid (zero or near-zero viscosity) ﬂuids treated more generally, i.e. considering each constituent as a single body (although the chemical potential should be deﬁned in a slightly different way). 8.4 Treatments of more complex systems Nunziato and Walsh [139] apply continuum theory to (ideal) multiphase mixtures. In fact, they do not explicitly consider individual phases and multiphasicity is included (among others) by considering the volume fraction of each constituent as an independent kinematical quantity. To this end, an additional balance equation of forces is added. The volume fraction ja is related to the usual continuum mass density ra (called by Nunziato and Walsh partial density) of constituent a through the actual or local density (called by Nunziato and www.scilet.com

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Miloslav Pekarˇ

Walsh simply density) ra by the relation: ja ¼ ðma yVtot Þyðma yVa Þ:ra yra

ð8:30Þ

where Vtot is the volume of the total continuum and Va the volume occupied by the constituent a. To imply phase separation, the thermal and mechanical response of the a-th constituent depends only upon the properties of that component. However, the remaining responses, including mass supply, are functions of interactions with all components. Also in this model, the constitutive equation for the mass supply c# a of constituent a does not simplify during further development and remains in the very general form: _ b ; ub ; yb ; Fb ; grad yb ; cb Þ:Fa ðLÞ; c# a ¼ Fa ðjb ; grad jb ; j

a; b ¼ 1; . . . ; n ð8:31Þ

Here the dot means the material time derivative, ub is the diffusion velocity of component b, yb its temperature and Fb its deformation gradient, cb stands for the component mass concentration. The development of entropic inequality naturally leads to the following deﬁnition of chemical potential: ma ¼ qðca ca Þyqca

ð8:32Þ

where ca is the (Helmholtz) free energy of constituent a. Nunziato and Walsh distinguish between ‘‘unsaturated’’ and ‘‘saturated’’ mixtures. In general, the sum of volume fractions is restricted by: 05:

X a 1

ð8:33Þ

a

and saturated mixtures are those where ¼ 1. Nunziato and Walsh deﬁne thermochemical equilibrium (denoted again by ‘‘ þ ’’) in an unsaturated mixture by þ þ ðc# a Þ :Fa ðL Þ ¼ 0;

a ¼ 1; . . . ; n

ð8:34Þ

and þ þ þ þ Lþ ¼ ðjþ b ; grad jb ; 0; 0; yb ; Fb ; 0; cb Þ;

b ¼ 1; . . . ; n

Among other consequences, the following relation is found: www.scilet.com

ð8:35Þ

Thermodynamics and mass-action chemical kinetics n X

þ mþ b ðqFb yqca Þ ¼ 0;

a ¼ 1; . . . ; n

83

ð8:36Þ

b¼1

which means that all the chemical potentials need not be equal at equilibrium. This led the authors to designate the equilibrium deﬁned by Eqs (8.34) and (8.35) as ‘weak’ equilibrium in contrast to the ‘strong’ equilibrium, which was deﬁned by adding the following conditions to Eqs (8.34) and (8.35): þ mþ a ¼ mb ; a 6¼ b;

a; b ¼ 1; . . . ; n

ð8:37Þ

It should be pointed out that the chemical potentials used by Nunziato and Walsh differ from those familiar in classical reversible thermodynamics, as these authors do not explicitly consider individual phases with their own potentials. Moreover, the classical condition of phase equilibrium does not follow (8.37) but states the equality of chemical potentials in all phases for every component. It is not clear how the strong equilibrium condition (8.37) is related to the classical phase equilibrium condition. As a consequence of strong equilibrium in both saturated and unsaturated mixtures, this inequality is also found: ðqFa yqca Þþ 0;

a ¼ 1; . . . ; n

ð8:38Þ

Referring to (8.34) this means that the production rate (rate of formation) of a component has, at strong equilibrium, its maximum value with respect to the concentration of this specie, providing the strong inequality is valid in Eq. (8.38). If the afﬁnity of a reaction is deﬁned as X Ar ¼ Ma Pra ma ð8:39Þ a

where Ma is the molecular weight of constituent a and Pra its stoichiometric coefﬁcient in the (independent) reaction r, then the strong equilibrium conditions guarantee its disappearance. Drumheller and Bedford presented an extension of their variational approach [142] to reacting immiscible mixtures [143]. Their methodology was criticized by Passman, Nunziato and Walsh [144]; the critical remarks as well as Drumheller’s responses [145] are not repeated here but some others related to the scope of this review are added. To incorporate immiscibility, Drumheller and Bedford also introduce local density ra and volume fraction ja , as Nunziato and Walsh (see above), and suppose that the constitutive equation for the free energy www.scilet.com

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Miloslav Pekarˇ

of a constituent depends only on the variables pertinent to that constituent alone. The remaining dependent variables are assumed to be functions of variables of all constituents. Among independent variables, the reaction rates of constituents (more precisely, the rates of variation in time constituents, i.e. their material derivatives) are also included. Thus, the authors do not formulate a constitutive equation for the reaction rate or the reaction rate equation. Because (local) densities are also included among the independent variables, the same questions and problems arise regarding the dependence of reaction rate on concentrations, which were discussed in Section 6. Moreover, also the time dependence of the local densities (material derivatives) appear in the set of independent variables. However, the reaction rates, densities and their changes with time are not mutually independent but related through the mass balance equation. This paradox can perhaps be resolved by noting that volume fraction, which also appears in the mass balance after substituting densities by partial densities, is not included among the independent variables. But then it cannot vary independently (of component density) as stated by Drumheller and Bedford (for compressible constituents). A much more acceptable and lucid version of the application of continuum theory to immiscible mixtures was elaborated subsequently by Drumheller [145]. The developments are much more relevant to the principles of immiscible mixture theory than to chemical kinetics themselves, and are not described here. Drumheller considers for simplicity a mixture of a (porous) solid and gas. The constitutive equations are again controlled by the immiscibility postulate (alternatively, the principle of phase separation): the constitutive equations of any constituent depend only upon the variables of this constituent, while the mixture variables (production terms) depend upon all variables. Mass production rates are included among the dependent variables. As Drumheller takes into account only one direct reaction between the gas (G) and solid (S), i.e. _ cG

¼ _ cS , only one constitutive rate equation is necessary, viz.:

_ cS

E E _ E ; G ; d ; r ; T ; G ; v ; v Þ:f ðL Þ ¼ fM ðFS ; AES ; TS ; F_ S ; A S S S G G S G M M G

ð8:40Þ

E

where _ c is the mass production rate, F the elastic part of true deformation gradient, AE the elastic part of distention gradient, T the temperature and G its gradient, d is the damage (represents damage to the solid matrix) and v is the material velocity; symbols S and G refer to the solid and gas constituents, www.scilet.com

Thermodynamics and mass-action chemical kinetics

85

respectively and symbol M stresses the whole mixture. This very general relation is again not simpliﬁed by subsequent development. Using a special constitutive postulate for the energy change of the solid component, and concentrating on only sufﬁcient conditions to fulﬁl the entropy inequality, one of these conditions may be written: _ cS ðGS yTS GG yTG Þ 0

ð8:41Þ

Here, G represents the (equilibrium) Gibbs energy. Condition (8.41) can be realized by the following constitutive postulate for the reaction rate: _ cS

¼ ðGS yTS GG yTG ÞFðLM Þ

ð8:42Þ

where FðLM Þ is an arbitrary positive scalar function of LM . Experience with detonations of granular explosives suggests that in this case the function might be proportional to the pressure. Even under these special conditions, rate equation (8.42) remains too general to be directly applied in practical kinetics. Thomas [146] considers an ‘‘open mass’’ continuum. His approach was motivated by the requirement to describe the evolution of complex material systems composed of constituents in varying physical states. The body of the mixture is taken to correspond to the body of one of the constituents (symbolized by r), preferably solid. This reference constituent deﬁnes the mixture. One of the consequences of this model is the non-constant mass of the whole mixture. While this can be acceptable and understandable, it seems that problems in balances arise. The total mass balance of the mixture is written in integral or local forms as: dr dt

ð

ð r dV ¼

Vr

3 X

dr rydt þ r Vr

! qir yqxi dV ¼

! 3 nþ1 X X qJi yqxi þ r ca _ ca dV

ð Vr

i¼1

i¼1

a¼1

ð8:43Þ or nþ1 X ca _ ca

qryqt þ div ðrvr Þ þ div J ¼ r

ð8:44Þ

a¼1

respectively. Symbol dr stresses the material derivative following the motion of the reference constituent body with volume Vr , r is the mixture density, vector vr the velocity of the reference constituent, vector J is the total diffusion mass ﬂux, ca is the mass fraction of constituent a and _ ca its mass supply rate per unit mass. www.scilet.com

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It is assumed that _ cnþ1 : _ cr ¼ 0, so the total number of constituents is equal to n þ 1. The right-hand-sides of Eqs (8.43) and (8.44) are zero in the usual ‘‘closed mass’’ systems. The mass supply rate S is considered to be

S¼

nþ1 X ca _ ca

ð8:45Þ

a¼1

and is not in general equal to zero. The non-vanishing ﬁrst term on the right hand side of Eq. (8.43) is perhaps easily acceptable. However, it is not clear how the total mass supply (8.45) accords with simple mass conservation in chemical reactions, which is valid, of course, if we take into account the indestructibility of atoms [34]. This difﬁculty is illustrated in Thomas’s chapter 5, part V [146] where for the closed systems he arrives at div J ¼ rS

ð8:46Þ

and simply states that this cannot be physically enforced in reacting and diffusing mixtures and (only?) non-reacting (S ¼ 0) and diluted (i.e. J & 0) mixtures can be modelled as closed. Clearly, there is some confusion in mass exchange within a real mixture and in the reference-constituent-model for the mixture. It is not clear how in reality the non-reference constituents enter the volume occupied by the reference one (Vr ), react within it but not with the reference constituent, while keeping a non-zero mass supply (8.45). There are other ﬂaws. The independent variables are not clearly stated. At the beginning, the mixture density, species concentrations, diffusional mass ﬂuxes of all constituents, and velocity of the reference constituent are stated to be selected independent variables. Later, in discussion of inequality of entropy, a new and larger set of independent variables is suddenly deﬁned. And yet another set is used on the page before when expressing the derivative of free energy. Some relations, which are in continuum theories proved on the basis of particular constitutive equations and the general validity of entropic inequality, are here postulated and then used in discussion of the consequences of entropy inequality. In this paper the methodologies of rational and classical irreversible thermodynamics are inconsistently combined. Ignoring all the controversial points, let us look at the implications for reaction kinetics and rates. The following relationships for the mass supplies (i.e. equivalents of reaction rates) result in: www.scilet.com

Thermodynamics and mass-action chemical kinetics _

rca _ ca ðX; vÞ ¼ qFðX; vÞyqAa Ua ðX; vÞ;

87

a ¼ 1; . . . ; n

ð8:47Þ

where parentheses only show independent variables. Here F is the so-called generalized dissipation potential, which is determined by the entropy production sS or by the vectors of thermodynamic ﬂuxes JT and forces X as follows: ð1

ð1 sS ðlX; vÞdlyl ¼

FðX; vÞ ¼ 0

X JT ðlX; vÞdl

ð8:48Þ

0

v is the second part of the set of independent variables; details on independent variables can be found in the original source. Quantities Aa are given by the following equation: Aa ¼ ma yT

n X _ cb mb yT v2 y2T þ CyT ;

a ¼ 1; . . . ; n

ð8:49Þ

b¼1 _

_

where ma is the reduced chemical potential deﬁned by ma ¼ qCyqca , C being the _

free energy. Quantity Ua represents the components of the so-called nondissipative thermodynamic ﬂux (for details see again the original source). As the author states in the conclusion, to apply such a complex equation, models for the generalized dissipation potential and non-dissipative ﬂux must be constructed, as well as for the free energy. Thus, within one material model (of open mass continuum), a couple of other (material) models (of classical irreversible thermodynamics) are necessary. The practical value of equations like (8.47) plus (8.48) and (8.49) in chemical kinetics remains remote. Jabbour and Bhattacharya [147] made some compromise between rate equations that are too general and the direct use of simple mass-action kinetics. They proposed a continuum theory for the chemical vapour deposition of thin solid ﬁlms. The rate of reaction in the gas ﬂow is expressed by the mass-action difference of the forward and reverse rates with mass fractions in place of concentrations and with rate constants dependent on temperature and speciﬁc volume. Similarly, the rate of surface reactions is expressed with densities (per unit area) instead of concentrations and rate constants dependent on temperature, surface orientation (described by the unit normal vector) and on chemical composition (!) of the ﬁlm surface. Either rates are introduced a priori, as constitutive postulates. The authors test their consistency with thermodynamics. From the entropic inequality, it follows as sufﬁcient that all rate constants are non-negative. www.scilet.com

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8.5 Mu¨ller’s results Mu¨ller developed a linear approach [148] and went a step further beyond his and Shih-Liu’s preceding work [149]. Mu¨ller considers a mixture of inviscid ﬂuids and discusses only its linear case, i.e. constitutive equations are linear representations of isotropic functions. Mu¨ller operates with ‘‘reaction rate densities’’ Lp (p is the index of independent reactions) which are related to the mass supplies ta (called by him production densities of mass) of components a by: ta ¼

R X

gpa ma Lp

ð8:50Þ

p¼1

where gpa is the stoichiometric coefﬁcient of constituent a in reaction p and ma is the molecular mass of constituent a. The original constitutive equations of a mixture of inviscid ﬂuids [148,149] Lp ¼ Lp ðra ; grad ra ; va ; T ; grad T Þ

ð8:51Þ

are simpliﬁed in the linear approximation to [148,149] Lp ¼ Lp ðra ; T Þ

ð8:52Þ

In fact, Eq. (8.52) a priori postulates what had been proved long before by Samohy´l for his somewhat more general mixture model. Using the deﬁnition of equilibrium as a process of uniform temperature, with the same speed of all constituents and no chemical reactions, it is shown that entropy production vanishes and assumes its minimum at equilibrium (denoted by ‘‘ þ ’’). From this, the equilibrium mass-action relation follows [148,149]: n X

gpa Ma mþ a ¼0

ð8:53Þ

a¼1

where ma is the ‘‘intrinsic’’ chemical potential of constituent a deﬁned in the course of applying entropic inequality to simplify constitutive equations (for details see refs [148,149]). Equation (8.53) provides some constraints on the equilibrium densities of the constituents, leaving only n R of them independent. Due to constitutive equations (8.52), the rate densities should vanish after insertion of equilibrium densities (and temperature) into them. Capitalizing upon (8.53), the constitutive equations are then written as [148] www.scilet.com

Thermodynamics and mass-action chemical kinetics

Lp ¼

R X

L pq ðr1 ; . . . ; rnR ; T Þ

q¼1

n X

gqa Ma ma

89

ð8:54Þ

a¼1

Mu¨ller [148] also presents a simple example of its application – a binary mixture of atomic and molecular oxygen between two inﬁnite parallel plates with several assumptions (e.g. a stationary and one-dimensional process at constant total pressure). Only one reaction, viz. 12 O2 ¼ O is considered and only the linear vicinity of equilibrium is treated. Thus, just one mass balance of components is sufﬁcient, e.g. of atomic oxygen numbered as 1: 1 rþ 1 q yqx1 ¼ m1 L

ð8:55Þ

Here ‘‘ þ ’’ means the equilibrium value, x1 is the coordinate in that one dimension in which the process occurs and 1 is the shortened notation of the (one-dimensional) velocity of constituent 1. In fact, example balances are written in a rather unusual way. The values of some quantities are taken at equilibrium whereas others are not considered to be ﬁxed and probably allowed to evolve not far from equilibrium. The former are called coefﬁcients, see, e.g. density in Eq. (8.55) and appear in balances as multipliers at partial derivatives of the latter. This is what should be understood under the ‘‘linearization about an equilibrium’’ in this example. Thus, when temperature appears in balances as a multiplicative factor, it takes the equilibrium value, whereas when it appears in the partial derivative, it is a general (ﬁeld) variable. The only exception is chemical potential which appears also in the product due to Eq. (8.54) with a non-equilibrium value, see Eq. (8.56) below, because otherwise the whole example would be nulliﬁed (zero reaction rate). Instead of linearization close to equilibrium, the whole procedure should be simply called a model. Introducing constitutive equation (8.54) into Eq. (8.55) we obtain: 2 1 þ rþ 1 q yqx1 L ðm1 Þ ðm1 m2 Þ ¼ 0

ð8:56Þ

(molecular oxygen takes number 2). This balance equation is solved together with the others giving the result pﬃﬃﬃﬃﬃﬃﬃ m1 m2 ¼ b sinh ABx1 ð8:57Þ which could be, in principle, used to compute the reaction rate from Eq. (8.54). Symbols b, A and B stand for rather complicated expressions which can be found in the original reference [148]. However, to compute a reaction rate from www.scilet.com

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Eq. (8.54), the (equilibrium) value of coefﬁcient L þ is also needed. As stated by Mu¨ller [148], the only way to get it is by calculation from statistical mechanics. This simple example clearly illustrates the common problems of many theoretical treatments – too many approximations and simpliﬁcations lead to too complicated expressions, which are difﬁcult to use in practice. Mu¨ller’s later treatment goes deeper and develops so-called rational extended thermodynamics including relativistic effects. The principle of relativity was added to the entropy principle (split into the entropy inequality and into stability, i.e. convexity conditions) as one of the universal principles to be satisﬁed by constitutive or ﬁeld equations. Relativity demands form invariance of the equations under Galilean or Lorentzian transformations, depending on whether the theory is non-relativistic or relativistic. Rational extended thermodynamics lies between extended irreversible and rational thermodynamics. It combines adding ﬂux variables, etc. with constitutive equations restricted by entropic inequality and the kinetic theory (of gases). The main substance of its contribution to the kinetics of chemically reacting systems is given in the paper by Kremer and Mu¨ller [150] which is reproduced in a shortened version in the book by Mu¨ller and Ruggeri [151]. A simpler, non-relativistic derivation is given in the paper by Kremer and Mu¨ller [152]. In fact, the results are given only for a non-diffusing binary mixture (of ideal gases). The authors claim that in a reacting mixture, the rest mass is not conserved and there is a mass-defect M given by M¼

X ga ma

ð8:58Þ

a

where ga is the stoichiometric coefﬁcient and ma is the molecular mass of constituent a. The mass-defect determines the heat of reaction which is Mc2 , where c should be the speed of light. The reacting mixture is exempliﬁed by a binary, non-diffusing mixture described by 15 ﬁelds: mixture fugacity (a), chemical afﬁnity (A), temperature (T ), mixture velocity vector (U), stress deviator tensor (t), dynamic pressure (p; non-equilibrium part of the pressure) and heat ﬂux vector (q). Mixture fugacity is deﬁned using equilibrium (!; symbol þ ‘‘ þ ’’) values of the components’ fugacities as a ¼ g1 m1 aþ 1 ¼ g2 m2 a2 and P afﬁnity by A ¼ ga ma aa . The relevant balance equations and constitutive a equations for, among others, reaction rate density (L) are written as www.scilet.com

Thermodynamics and mass-action chemical kinetics

div

X a

ma na U ¼

X

ga m a L

91

ð8:59Þ

a

(na represents the component’s number density) and _

L ¼ Lða; A; T ; U; t; p; qÞ

ð8:60Þ

respectively, where the reaction rate density is deﬁned as in Eq. (8.50) (only one reaction is considered). The complete derivation of further results is not given in the original papers because of the lengthy cumbersome algebra. It is therefore not easy to trace the origin of the varied and complex equations found in them. Nevertheless, some remarks can be made. The principle of relativity requires that, among others, the reaction rate density is an isotropic function of its variables and attention is then restricted to linear functions, i.e. the reaction rate density is a linear function of only the chemical afﬁnity and dynamic pressure. The latter is in fact a ‘‘linearization of the linear isotropic function.’’ After transformation of variables, and the requirement of the general validity of entropic inequality, the latter is simpliﬁed to X l ga m a L l I 0

ð8:61Þ

a

The lambdas (l and elements of the tensor l) belong to the set of Lagrange multipliers introduced as a result of incorporating balance equations into the entropic inequality. Tensor I (of the 3rd order) is called the ﬂux production and it is the divergence of the ﬂux tensor, which is of the 4th order. The next step is to ﬁnd the physical meaning of the formal Lagrange multipliers, i.e. determining them in terms of physical quantities. This can be done (simply) only for nearequilibrium processes. Because at equilibrium, the productions L and I must vanish, it is concluded from (8.61) that l and l also vanish. It is not clear how this conclusion is arrived at, because the reaction rate disappears at equilibrium by deﬁnition, and not due to the vanishing of some Lagrange multipliers. The relevant equations or terms are then linearized near equilibrium. For the reaction rate density, the following equation is ﬁnally derived þ L ¼ ð1yDÞðMc2 ykT Þ ðnþ ð8:62Þ 1 þ n2 Þ div U ð1ykT Þ div q where D stands for www.scilet.com

92

D¼

Miloslav Pekarˇ

2 2

2 Mc Mc þ3ðg1 þ g2 Þ kT kT ð3y4Þg1 g2

2 m2 ðnþ Þ2 þ m22 ðnþ m21 þ m22 2Þ 2 1 1 þ 5 6 þ m1 m2 m 1 m 2 nþ 1 n2

! ð8:63Þ

k being the Boltzmann constant. An alternative expression can be derived from the linear constitutive representation of the reaction rate density mentioned above and approximating the dynamic pressure p by the Maxwellian iteration of the kinetic theory of gases. The result is as follows: þ A p L ¼ ð2lp y3DBp1 ÞðMc2 ykT Þ2 ðnþ 1 þ n2 Þ div U þ ðlA lp B1 yB1 ÞA

ð8:64Þ

where lp and lA are coefﬁcients of the linear representation of the reaction rate density and Bp1 and BA 1 are coefﬁcients from similar linear representation of the ﬂux production tensor I. The resulting Equations (8.62) and (8.64) are really complicated considering that they are only linear, near-equilibrium approximations for nondiffusing, two-component ideal gas mixture with only one reaction, i.e. with some isomerizations at most. Their practical value is perhaps low not only due to this complexity but also due to the following consequences. Equation (8.62) states that, with no expansion or heating, there is no chemical reaction. This must sound rather strange to the ordinary chemical kineticist. Should it mean that in the constant volume adiabatic autoclave there is no possible chemical reaction? More serious is the absence of the very essence of chemical kinetics – the dependence of the reaction rate on the concentrations of reacting species. The only parameters involved are the molar masses and equilibrium particle densities, which are constants under given conditions. There is no change of rate with change of concentrations. Certainly, this is a very serious drawback to the near-equilibrium approximation. Equation (8.64) is simply an attempt to recover a linear phenomenological relation of LIT between reaction rate and afﬁnity with still undetermined coefﬁcients. 8.6 Samohy´l’s achievements Perhaps the most signiﬁcant and successful step towards relating mass-action kinetics and thermodynamics was made by Samohy´l [153 – 155]. Using the standard rational thermodynamics procedure, Samohy´l proved [154 – 156] www.scilet.com

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that, in a mixture of ﬂuids with ‘‘linear transport properties’’, reaction rates (collected in a vector J) are functions of only temperature and the densities of the mixture components (collected in a vector r): J ¼ JðT ; rÞ

ð8:65Þ

The elements of the vector J are the rates of individual reactions, J ¼ ðJ1 ; J2 ; . . . ; Jr Þ. Alternatively, this function can be expressed with (molar) concentrations (c) as independent variables instead of densities: J ¼ JðT ; cÞ

ð8:66Þ

Function (8.66) is approximated by a polynomial of degree M [153,157,158]: J¼

Z X

knb

b¼1

n Y

caba ;

a¼1

n X

nba M

ð8:67Þ

a¼1

This approximation can be viewed as an application of Weierstrass’ theorem on the polynomial approximation of continuous functions. Vector knb ¼ ðk1nb ; k2nb ; . . . ; krnb Þ of polynomial coefﬁcients is to be interpreted as a vector of rate constants and vector nb ¼ ðb1 ; b2 ; . . . ; bn Þ as a vector of reaction orders with respect to components 1; 2; . . . ; n. The number of terms in the polynomial is equal to Z¼

M X ðn þ k 1Þ! k¼0

ð8:68Þ

n!ðn 1Þ!

The fundamental point in further development in this methodology is, in fact, applying the condition of consistency of the kinetic and thermodynamic descriptions of equilibrium. From the kinetic viewpoint J ¼ 0 ðequilibriumÞ

ð8:69Þ

From the thermodynamic viewpoint, equilibrium concentrations must accord with the value of equilibrium constant Kp of the p-th reaction: Kp ¼

n Y pa cPa ;

p ¼ 1; 2; . . . ; n h ðequilibrium concentrationsÞ

a¼1

www.scilet.com

ð8:70Þ

94

Miloslav Pekarˇ

Here, it should be underlined that only independent reactions are considered, which number n h, where h is the rank of a certain matrix describing the atomic composition of the mixture components (see below). Ppa stands for the elements of the stoichiometric matrix, i.e. for the stoichiometric coefﬁcient of component a in reaction p. The consistency condition sets restrictions on the polynomial (8.67). The correct, consistent form of the polynomial is found by the standard rational thermodynamics procedure. First, the independent reactions are selected. Second, their equilibrium constants are used to express some concentrations as functions of the remaining ones. These expressions are substituted into the polynomial (8.67). The modiﬁed polynomial should be zero for arbitrary equilibrium values of the remaining concentrations. The arbitrariness demands further modiﬁcation of the polynomial – some coefﬁcients (rate constants) vanish, some others are related through the equilibrium constants. As the values of the coefﬁcients and equilibrium constants depend only on temperature, these results are valid also out of equilibrium and can be put into the initial (i.e. non-vanishing) polynomial (8.67). The ﬁnal rate equation results, having the same general form for all (independent) reactions. Details on the procedure, and examples of rate equations, can be found elsewhere [153,159]. For the purpose of this review it should be sufﬁcient to note the principal features of this method. These features result when interpreting the polynomial rate equations within the framework of traditional mass-action kinetics; that is, interpreting the polynomial terms as corresponding to individual steps of the reaction mechanism and (the difference of) their forward and reverse rates. Thus, polynomial coefﬁcients are considered to be rate constants. The principal features of Samohy´l’s method are as follows: Rate equations of the mass-action type are directly derived from thermodynamic considerations. The rate of any step may be affected also by species, which do not directly take

part in this step or, in other words, by the rate of (some) other steps; similarly, the effect of an inert additive on the reaction rate can be included naturally. The reaction mechanism, relevant for the description of kinetics, ‘‘automatically’’ follows from the rate equation, contrary to conventional procedure.

The complexity of the mechanism can be changed by the number of components, which consequently determines the number of independent reactions, or by the degree of the approximating polynomial. www.scilet.com

Thermodynamics and mass-action chemical kinetics

95

The resulting rate equations contain only integral-number reaction orders with respect to any component. The procedure starts with independent reactions only, however, the resulting rate equations, in the sense of above interpretation, may also contain other

reactions relevant for the description of the reaction kinetics. The resulting ‘‘kinetic’’ mechanism may be thus different from, and more complex, than the initial, let us say ‘‘thermodynamic’’ one. The procedure does not preclude (‘‘thermodynamically’’) dependent reactions to be kinetically important – reaction steps, which are not among the independent ones may occur in the rate equation. Rational thermodynamics naturally differentiates thermodynamic from kinetic independence. Kinetic equilibrium criteria are fulﬁlled with a more general equation than usual Guldberg – Waage, considering only true thermodynamic equilibrium constants, with no need for any ‘‘kinetic’’ equilibrium constant, given as the ratio of rate constants in the forward and reverse reaction directions. The method can be used to design a suitable mechanism for the reaction under study which, fully and consistently with (irreversible) thermodynamics,

describes the reaction kinetics. Samohy´l’s methodology can be viewed as a purely phenomenological theory of the mass-action law of chemical kinetics giving generally rate equations of type (2.18) or (4.1). It can be also considered as a modern and more general version of van’t Hoff’s equation (4.5). Let us consider just one example – the oxidation of carbon monoxide, CO þ 12 O2 ¼ CO2 , in the presence of some catalyst S. There are supposed to be six components (CO, O2, CO2, S, OS, COS) and, consequently, three independent reactions are possible. There were selected as 1.

O2 þ 2 S ¼ 2 OS

2.

CO þ S ¼ COS

3.

OS þ COS ¼ 2 S þ CO2

The approximation by the second degree polynomial leads to the following the rate equation:

1 1 J ¼ k100100 ðcCO cS K1 2 cCOS Þ þ k100010 ðcCO cOS K2 K3 cCO2 cS Þ 1 þ k010001 ðcO2 cCOS K1 1 K3 cCO2 cOS Þ

www.scilet.com

ð8:71Þ

96

Miloslav Pekarˇ

Thus, the following three reactions ‘‘appear’’ in this approximation necessary for the description of the kinetics of catalytic oxidation of CO: CO þ S ¼ COS CO þ OS ¼ CO2 þ S O2 þ COS ¼ CO2 þ OS As has been stated already, this method was derived for a mixture of ﬂuids with linear transport properties. This model should be plausible for most homogeneous chemically reacting systems but is in general inapplicable to heterogeneous mixtures or for complex (e.g. viscoelastic) ﬂuids. However, conventional phenomenological kinetics of such systems are inspired by equations used in homogeneous cases, anyway, and uses them directly. In other words, the massaction rate equations are frequently used in any system regardless of their phase or rheological state. More general studies indicate that, in more complex systems, the reaction rate can be a function of various deformation (kinematics) variables not usual in chemical kinetics (see also above). To stay within the scope of Samohy´l’s work, let us refer to his treatment of general reacting mixtures [160]. In it, reaction rates remain functions of deformation gradients, second deformation gradients, densities, density gradients, diffusion velocities, and velocity gradients of all components and of temperature and its gradient. Restricting discussion solely to mixtures of ﬂuids brings just moderate simpliﬁcation [161] – reaction rates are functions of densities, density gradients, diffusion velocities, and velocity gradients of all components and of temperature and its gradient. Samohy´l’s results also shed more light on the afﬁnity-reaction rate relation. For the reader’s convenience, it is now necessary to review the basic ﬁndings of linear algebra in chemical stoichiometry [34]. The most essential point is the material balance of the mass of the reacting mixture. If ra indicates the mass source of component a, representing the mass of component a produced or consumed by chemical reactions in unit time and volume, then the usual (molar) reaction rate of component a (Ja ) is deﬁned by: Ja ¼ ra yMa

ð8:72Þ www.scilet.com

Thermodynamics and mass-action chemical kinetics

97

where Ma is the molar mass (weight). The balance of total mass can be written as [35,162,163]: n X

ra ¼ 0

ð8:73Þ

a¼1

Utilising the last two equations, the material balance can be rewritten as: n X

Ja Ma ¼ 0

ð8:74Þ

a¼1

As Eq. (8.74) resembles the relation for the scalar product of two vectors, let us deﬁne an abstract n-dimensional vector space of mixture components U with base ea and reciprocal base ea . From the individual molar masses and reaction rates respectively, vectors M and J are constructed as follows: M¼

n X

M a ; ea ;

J¼

a¼1

n X

Ja ea

ð8:75Þ

a¼1

The material balance (8.74) can be rewritten as M?J ¼ 0. It can be proved [34] that space U can be decomposed into two complementary orthogonal subspaces V and W; the vector of molar weights lies in subspace W and that of reaction rates in subspace V: U ¼ V+W;

V ? W;

M P W;

JPV

ð8:76Þ

Subspace V is called the reaction subspace. The molar weights of components are determined by their composition and atomic weights: Ma ¼

z X

Msa Tsa

ð8:77Þ

s¼1

where z is the number of atoms, Msa is the atomic weight of atom s and Tsa indicates the number of atoms s in component a. Atoms are assumed to be indestructible and are thus conserved in chemical reactions: n X

Tsa Ja ¼ 0;

s ¼ 1; 2; . . . ; z

ð8:78Þ

a¼1

However, not all relations in (8.78) may be independent. Their maximum number is given by the rank of matrix k Tsa k which will be denoted by h. Keeping only linearly independent relations, Eq. (8.78) is transformed into: www.scilet.com

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Miloslav Pekarˇ

n X Ssa Ja ¼ 0;

s ¼ 1; 2; . . . ; h

ð8:79Þ

a¼1

where the h6n matrix k Ssa k with rank h was constructed by elimination of linearly dependent relations from (8.78), i.e. rows (columns) from matrix k Tsa k. It can be proved [34] that vectors fs ¼

n X

Ssa ea ;

s ¼ 1; 2; . . . ; h

ð8:80Þ

a¼1

form a basis in the subspace W. Let us also select the basis for the subspace V and denote the (reciprocal) base vectors as dp , p ¼ 1; 2; . . . ; n h. As these vectors also belong to the (original) space U, they can be expressed using its basis: dp ¼

n X

Ppa ea ;

p ¼ 1; 2; . . . ; n h

ð8:81Þ

a¼1

Matrix k Ppa k of dimension ðn hÞ6n is a matrix of the stoichiometric coefﬁcients of components a in the independent reaction p. Thus there are n h independent reactions and only they are needed for the kinetic and thermodynamic description. As vectors f s and dp lie in orthogonal (and complementary) subspaces, their scalar product is zero: f s ? dp ¼

n X Ssa Ppa ¼ 0

or

ð8:82aÞ

a¼1

k Ppa k 6 k Ssa kT ¼k 0 k

ð8:82bÞ

Equations (8.82) relate the stoichiometric matrix with matrix k Ssa k describing the atomic composition of the constituents by linearly independent relations. Samohy´l deﬁnes chemical potential as follows: ga ¼ qðrf Þyqra

ð8:83Þ

where r is the mixture density, ra represents the density of component a, and f states for the speciﬁc free energy ( f ) as a function of relevant independent variables (component densities being among them), i.e. f ¼ f (independent variables). The molar chemical potential is then given by ma ¼ ga Ma

ð8:84Þ www.scilet.com

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As chemical potentials are inseparably associated with the components, they should and easily can be introduced into the component space U by deﬁning the vector of molar chemical potential in this space: m¼

n X

ma ea

ð8:85Þ

a¼1

Because the component space is decomposable to subspaces V and W, so is the vector m: m ¼ A þ B;

m P U;

A P V;

BPW

ð8:86Þ

Equation (8.86) expresses the molar chemical potential vector as a function of A and B. Projection A into the reaction subspace can be expressed using its base vectors: A¼

nh X

Ap dp

ð8:87Þ

p¼1

The relation of coordinates Ap to molar chemical potentials can be found using linear algebra and the above deﬁnitions: A p ¼ m ? dp ¼

n X

ma Ppa ;

p ¼ 1; 2; . . . ; n h

ð8:88Þ

a¼1

The ﬁnal expression in Eq. (8.88) is the classical deﬁnition of afﬁnity of the p-th reaction with a reverse sign. Therefore, vector A is called the (chemical) afﬁnity vector and Ap the (chemical) afﬁnity of reaction p. Similar equations can be written for the projection B, lying in the subspace W: B¼

h h X X Bs f s ¼ Bs f s s¼1

Bs ¼ m?f s ¼

ð8:89Þ

s¼1 n X

ma Ssa

ð8:90Þ

a¼1

Now we are ready to ﬁnd the relation between reaction rate and afﬁnity. Samohy´l has proved Eq. (8.78) using rational thermodynamics and a model of ﬂuids with linear transport properties. Supposing the invertibility of another www.scilet.com

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Miloslav Pekarˇ

proved function, viz. ga ¼ ga ðT ; rÞ, i.e. assuming the regularity of the matrix k qga yqra k, which was proved, for example, for stable mixtures [164,165], and then using Eqs (8.84), (8.85), densities can be substituted by chemical potentials and function (8.65) transformed to J ¼ JðT ; mÞ

ð8:91Þ

Introducing decomposition (8.86), the following function is obtained: J ¼ JðT ; A; BÞ

ð8:92Þ

Equation (8.92) states that, in general, there is no direct, unambiguous functional dependence of reaction rate on afﬁnity, as noted also by Bataille et al. [102] (cf. Part 5), even in the case of a simple model of linear ﬂuids. Besides afﬁnities, vector B also appears in such a function. The reason as to why the chemical potentials in Eq. (8.91) cannot be directly substituted only by afﬁnities lies in the deﬁnition of afﬁnity itself, Eq. (8.88). The latter equation does not, in general, provide an unambiguous expression for molar chemical potential ma as a function of afﬁnities; see the example below – the ﬁrst equality in (8.98a). This conclusion was arrived mathematically rigorously and not by making ad hoc and a priori statements on the functional dependence. The following example will aid better understanding. In the reaction mixture of NO2 and N2O4 there is only one independent reaction possible [154,155]. Let us select it as N2 O4 ¼ 2 NO2 and index it by 1. Using Samohy´l’s method of polynomial approximation with a second degree polynomial, the following rate equation is obtained [154,155]: J1 ¼ k01 c2 k01 K1 c21

ð8:93Þ

(i.e. virtually the usual mass-action expression) where NO2 is numbered as 1 and N2O4 as 2 and K is the equilibrium constant of the selected independent reaction. Unfortunately, an explicit particular expression for the equality in (8.86) is not known and no useful explicit relation (8.92) can be found from (8.91). Instead we are to use some model equation relating chemical potential and concentrations. The ﬁrst choice we have available is the common relation for ideal mixtures: ma ¼ ma ðT Þ þ RT ln ca

ð8:94Þ www.scilet.com

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where the standard chemical potential is a function of temperature only and concentrations are, in reality, divided by the standard value 1 mol dm 3. In fact, this model has already been used in Eq. (8.93) where the equilibrium constant was deﬁned generally by: RT ln Kp ¼

n X

ma Ppa

ð8:95Þ

a¼1

and transformed by (8.94) into: Kp ¼

n Y Ppa ðcþ aÞ

ð8:96Þ

a¼1

where symbol ‘‘ þ ’’ again indicates the equilibrium value. Using Eqs (8.94) and (8.95), chemical potential can be introduced into the rate equation (8.93) giving: J1 ¼ k01 expðm2 yRT Þ expðm2 yRT Þ expð2m1 yRT Þ ð8:97Þ which is a function of the form (8.91). Equations (8.88), (8.90), (8.89), and (8.93) give: A1 ¼ 2m1 m2 ;

B1 ¼ ðm1 þ 2m2 Þy5

ð8:98aÞ

m1 ¼ ð2y5ÞA1 þ B1 ;

m2 ¼ ð1y5ÞA1 þ 2B1

ð8:98bÞ

Combining (8.98b) and (8.97), the ﬁnal rate equation follows: J1 ¼ k01 expðm2 yRT Þ expð2B1 yRT Þ expð4A1 y5RT Þ expðA1 yRT Þ 1

ð8:99Þ

This is the rate equation of type (8.92), i.e. a reaction rate expressed as a function of afﬁnity and not of afﬁnity and concentration as in the case of equations like (2.4) or (2.6). Note that at reaction equilibrium, deﬁned as A1 ¼ 0, the reaction rate really vanishes, and note that rational afﬁnities are deﬁned with the opposite sign compared to the classical deﬁnition. The ﬁrst equality in (8.98a) clearly illustrates that there is no inversion function mi ¼ mi ðA1 Þ. Expression (8.99) is more complex than the usual classical analogue (2.4) or (2.6) and also contains, as emphasised above, quantity B. Let us compare the rational derivation with that based on the classical reaction isotherm. Using Eqs (8.88), (8.94), and (8.95), we obtain for the present example: A1 ¼ 2m1 m2 ¼ 2m1 m2 þ RT lnðc21 yc2 Þ ¼ RT ln K þ RT lnðc21 yc2 Þ ¼ RT lnðc21 yKc2 Þ www.scilet.com

ð8:100Þ

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Miloslav Pekarˇ

Because there is no presumption as to the reverse rate constant and the relation between the ratio of rate constants and equilibrium constants in rational theory, there is no equation like (2.4) or (2.6). Of course, we can easily derive, combining Eqs (8.93) and (8.100), the rate equation J1 ¼ k01 c2 ½1 expðA1 yRT Þ but this is not an equation of type (8.92). The equilibrium implications of the rational rate equation (8.93) are as follows: 1 þ 2 þ 1 þ 2 0 ¼ k01 cþ 2 k01 K ðc1 Þ ) 0 ¼ c2 K ðc1 Þ

ð8:101Þ

and only the relation for equilibrium constant (8.96) is thus recovered. The ﬂaws of the (classical) reaction isotherm-afﬁnity-based procedures in chemical kinetics emphasised in Part 1 are conﬁrmed once more. Classical afﬁnity approaches are limited only within the model of the reverse rate constant and the identity between the thermodynamic equilibrium constant and the ratio of the forward and reverse rate constants. Rational thermodynamics offers a different, more general alternative - there is no need for any assumption about reverse rate constants, instead equilibrium constants are used in rate equations directly, and consistency between thermodynamic and kinetics is achieved straightforwardly simply through this thermodynamic quantity with no additional presumptions on a ‘‘kinetic equilibrium constant’’. Eq. (8.99) is a fully equivalent version of Eq. (8.93), of course, within the model used for chemical potential (8.94) and equilibrium constant (8.95). However, it is clear that the latter equation is much more useful for practical kinetics and evaluation of kinetic data. The algebraic essence of the conservation of mass and atoms conceals another interesting result which is ignored in equilibrium and other nonequilibrium theories. Because the reaction rate vector J is located in the reaction subspace V, whereas vector B is in its orthogonal counterpart W, the scalar product m?J is identically equal to A?J. Thus, vector B is ‘‘invisible’’ in the product m?J and also therefore in the modiﬁed form of the entropy inequality: A?J ¼

nh X

Ap Jp 0

ð8:102Þ

p¼1

(again, note that rational afﬁnities are deﬁned with the opposite sign in comparison to the traditional deﬁnition) which is often the starting point for deductions on afﬁnity-rate relationships. Flux-forces approaches of linear or extended irreversible thermodynamics usually postulate independent variables, and the idea of the afﬁnity as the variable (force) determining the reaction rate www.scilet.com

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(ﬂux) comes from entropy inequality of the type of Eq. (8.102). Consequently, vector B is overlooked as the other determining quantity in LIT (EIT). What is the meaning of quantity B? Eq.(8.90) indicates that it is also closely related to the molar chemical potential. It is located in the same subspace as the molar weights of the components where, on the other hand, reaction rates are not located. Thus, whereas afﬁnities relate the chemical potentials of individual components to the reactions (and their stoichiometry), in which the components take part, vector B relates the chemical potentials to the composition of the components. Matrix k Ssa k appearing in Eq. (8.90) followed from the atomic composition of the individual components, see text following Eq. (8.92). This matrix represents the atomic composition in linearly independent relations, i.e. using independent (pseudo)atomic substances. For instance, in the example discussed of the mixture with nitrogen oxides, the composition matrix k Tsa k reads (s ¼ 1 for nitrogen and s ¼ 2 for oxygen): k Tsa k¼

1 2 2 4

ð8:103Þ

and its rank is 1 and only one (pseudo)atomic substance exists and is necessary for the description of the mixture. Thus, the matrix k Ssa k can be selected as k Ssa k¼ ½1 2

ð8:104Þ

and represents the composition of nitrogen oxides in terms of the pseudoatomic substance NO2. Then B1 ¼ m1 þ 2m2 (and also B1 ¼ 5 B1 , cf. Eq. (8.98a)); recall that m1 is the chemical potential of NO2 and m2 of N2O4. Another selection of the matrix k Ssa k, viz. k Ssa k¼ ½2 4 gives the composition in terms of the pseudoatomic substance N1=2O and has perhaps only theoretical signiﬁcance. Whereas the afﬁnity combines chemical potentials according to the stoichiometric coefﬁcients of reactions in which individual components (may) take part, quantity B combines them according to the numbers of atomic or pseudoatomic substances which are contained in the individual components. The number of afﬁnities is equal to the number of (independent) reactions, cf. Eq. (8.101), the number of components of vector B, is equal to the number of (pseudo)atomic substances, cf. Eq. (8.102). Afﬁnity can be therefore viewed as the afﬁnity or potential of a particular specie in a particular reaction, whereas quantity B represents the afﬁnity or potential of (pseudo)atomic substances www.scilet.com

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themselves, which are in the number necessary and sufﬁcient for the thermodynamic description. Let us add two more examples in which there are no pseudo- but only real atomic substances, i.e. in which h ¼ z. The ﬁrst one is the simple combination D þ E ¼ DE, where n ¼ 3, h ¼ z ¼ 2; with numbering of atoms as 1 ¼ D, 2 ¼ E, and constituents as 1 ¼ D, 2 ¼ E, 3 ¼ DE matrices read:

1 0 1 k Tsa k¼ : k Ssa k ð8:105Þ 0 1 1 The components of B are then given by: B1 ¼ m1 þ m3 :mD þ mDE ;

B2 ¼ m2 þ m3 :mE þ mDE

ð8:106Þ

and we can consider B1 :BD ; B2 :BE . The second example refers to the abovementioned mixture involved in carbon monoxide oxidation. Here, n ¼ 6, h ¼ z ¼ 3, the atoms are numbered: 1 ¼ C, 2 ¼ O, 3 ¼ S, components: 1 ¼ CO, 2 ¼ O2, 3 ¼ CO2, 2 1 0 k Tsa k¼ 4 1 2 0 0

4 ¼ S, 5 ¼ OS, 6 ¼ COS, and the matrices are selected as: 3 1 0 0 1 2 0 1 1 5: k Ssa k ð8:107Þ 0 1 1 1

The components of B are then determined by: B1 ¼ m1 þ m3 þ m6 :mCO þ mCO2 þ mCOS B2 ¼ m1 þ 2m2 þ 2m3 þ m5 þ m6 :mCO þ 2mO2 þ 2mCO2 þ mOS þ mCOS B3 ¼ m4 þ m5 þ m6 :mS þ mOS þ mCOS

ð8:108Þ

and, in fact, B1 :BC , B2 :BO , B3 :BS . Equations (8.106) or (8.108) combine the constituents’ chemical potentials with respect to the individual atomic substances according to the number of particular atomic substances in all constituents. Vector B refers to the atomic constitution of the components in a similar or, so to speak, reciprocal manner as does vector M. The latter expresses the molecular weight as the respective sum of atomic weights, the former is the respective sum of chemical potentials of all constituents which contain the relevant atomic substance. Within this framework, what counts are not individual chemical potentials but their combinations, partly according to the individual reactions and partly according to the content of the (pseudo)atomic substances in the individual constituents. This is the essence of the decomposition in (8.99). Eq. (8.102) states that only the ﬁrst www.scilet.com

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part is signiﬁcant and acting in non-equilibrium effects described by the entropy inequality. Coordinates Bs of quantity B can be perhaps called the constitutive afﬁnity and B itself the constitutive afﬁnity vector; consequently, the molar chemical potential vector is decomposed into the chemical and constitutive afﬁnity vectors, see Eq. (8.99). To conclude, nearly all classical afﬁnity approaches to rate equations are valid only within the model of identity between thermodynamic and kinetic equilibrium constants, which is (at least one) necessary condition for deducing equations like (2.4), (2.6) or (2.7). This need not mean that such a model is incorrect, inappropriate, and useless. However, Eq. (8.92) and also other rational thermodynamic results given above, cast serious doubt upon nonequilibrium attempts to ﬁnd the reaction rate as a function of (only chemical) afﬁnity. In his typically cautious procedure, Haase [96] was very careful and avoided this model, see Eqs (5.2), (5.4), still persisting in afﬁnity as the quantity determining the reaction rate, cf. Eq. (5.1).

9. CHEMICAL POTENTIAL MODEL This short section concerns almost all reviewed thermodynamic treatments. It refers to expressing the dependence of chemical potential (m) on concentration or, generally, on activity (a) for the i-th component: mi ¼ m þ RT ln ai , cf. Eq. (2.5). This expression is commonly introduced as a ‘‘well-known’’, ‘‘common’’, ‘‘usual’’ or even as a deﬁnition of chemical potential and it looks as though its origin has been forgotten. Let us recall it. Chemical potential is, in equilibrium thermodynamics, deﬁned as, among others, the partial derivative of the Gibbs energy (G) at constant temperature (T ) and pressure (P): m ¼ ðqGyqnÞT ;P

ð9:1Þ

where n is the molar quantity. Considering volume work only and reversible, equilibrium processes only, the classical ﬁrst and second laws of thermodynamics can be combined giving, for example, the following equation for the differential of Gibbs energy: dG ¼ S dT þ V dP

ð9:2Þ

(S is the entropy, V is the volume) from which it follows that Gibbs energy is a function of temperature and pressure and also: www.scilet.com

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V ¼ ðqGyqPÞT

ð9:3Þ

From Eq. (9.3) we have: Vm ¼ ðqmyqPÞT

ð9:4Þ

Integration of Eq. (9.4) from P to P with the ideal gas state equation results in: mðT ; PÞ ¼ mðT ; P Þ þ RT lnðPyP Þ

ð9:5Þ

This procedure can be easily extended to mixtures, which introduces no essential change to our discussion. Equation (9.5) is the basis for all expressions for the dependence of chemical potential on concentrations, fugacities or activities, and serves to deﬁne ideal gas, ideal or real mixtures. It is extended to the liquid state by means of vapour-liquid phase equilibrium. Thus, even the expression mi ¼ m þ RT ln ai should be viewed as a model, and not as a deﬁnition of chemical potential. It can, at most, be considered as a deﬁnition of some material system. This model comes from, and depends on, the ideas of equilibrium thermodynamics. Its validity in non-equilibrium theories and systems should be carefully tested. All theories using it have their validity restricted to the range of validity of this model. Of course, the validity of this model in equilibrium systems has been conﬁrmed exhaustively. Even the very deﬁnition (9.1) should be re-checked for non-equilibrium states. It is applicable only when it is proved that the Gibbs energy is a function of temperature, pressure and composition only. The chemical potential model is also included in Samohy´l’s methodology reviewed in the preceding section. Samohy´l deﬁned chemical potential by Eq. (8.83) where function f is not a function of pressure. Samohy´l proved that, in the simple model of linear ﬂuids, the free (and Gibbs) energy is a function of temperature and component densities only. Introducing classical (thermodynamic) pressure, even into this simple material model, is not straightforward [166,167]. The main cause is rooted in that the true (measured) pressure is generally not equal to the thermodynamic pressure P except at equilibrium. Independently of this model, Samohy´l has proved that reaction rate can be expressed as a function of chemical potentials (and temperature) only, at least in a stable mixture of ﬂuids with linear transport properties. As mentioned in the preceding section, matrix k qga yqra k (a is the constituent index) is regular in this www.scilet.com

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mixture when it is stable and enables one to substitute (molar) chemical potentials, cf. Eq. (8.84), in place of densities into the functional dependence of reaction rate (8.65). It should be stressed that the dependence of reaction rate on chemical potential was thus proved using the reversibility of the relevant function and not by stating this relationship a priori or by stating that free (Gibbs) energy is the driving force for reaction and its rate. Within rational thermodynamics there is no proof that reaction rate is a function of free energy (and temperature, at least) even in the simple case of linear ﬂuids, i.e. no proof exists of the invertibility of function f ðT ; rÞ or f a ðT ; rÞ. Probably, these functions are not invertible, because partial free energy ( fa ) is not ‘‘mixture invariant’’ [166 – 168] in contrast to chemical potential. 10. CONCLUSIONS Chemical kinetics seems to survive as the touchstone of all various thermodynamic theories. Thermodynamic approaches to the mass-action law hitherto achieved are still not very satisfactory. Speciﬁcally, the traditional mass-action law – an expression for the reaction rate as the difference between ‘‘forward’’ and ‘‘reverse’’ rates – is not discovered, recovered, proved. Very often, this law is only introduced in some way into the framework of a particular thermodynamic theory. The notable exception (perhaps the only one) is Samohy´l’s work on rational thermodynamics of ﬂuids with linear transport properties, although it is still more a matter of mathematical approximation and qualitative interpretation of approximating polynomials. Many (irreversible) deductions rely on afﬁnity as a device to relate reaction rate and thermodynamics. However, we have clearly shown that afﬁnity is not well suited for this purpose, particularly as a variable which should determine the value of a rate of reaction. Its real practical value consists, eventually, in determining how far the actual concentrations are from their equilibrium values. Most ‘‘thermodynamic’’ kinetic equations have only theoretical value. They are too complicated to be useful for practical kinetics and contain parameters whose values are difﬁcult to ﬁnd. Interestingly,

all

non-equilibrium

thermodynamic

approaches

to

chemical kinetics demonstrate the power of equilibrium. Even in Samohy´l’s approach, just equilibrium constants and the universality of equilibrium enabled arrival at ﬁnal, mass-action-type rate equations. www.scilet.com

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Samohy´l’s rational thermodynamic procedure also implies a possible change in current paradigms in the reaction mechanism-reaction rate arena. Traditionally, based on (kinetic) experiments, a reaction mechanism is proposed and from it reaction rates are formulated. Samohy´l’s methodology leads to the opposite approach. Knowing the species involved in the reacting mixture, (polynomial) rate equations are derived and, by interpreting them within the common mass-action law framework, an appropriate mechanism is found. Indeed, usually we do not measure mechanisms and even not just the reaction rates but we do measure concentrations of the determined reacting species. Samohy´l’s way thus looks more natural. Modern thermodynamic treatments show that chemistry is not just some simple generalization of mechanics or dynamics. It seems even that there is still not full agreement which variables are pertinent for a full and correct description of chemically reacting systems or, in the narrower sense, reaction rates occurring in these systems generally, i.e. not only in isotropic or homogeneous mixtures. Resolving this task calls not only for rigorous theoretical considerations but also for detailed experimental data on reaction kinetics in spatially inhomogeneous systems where mechanical (hydrodynamic, deformation) variables may also inﬂuence rates of chemical reactions. Future work is necessary to clarify also the role of ‘‘constitutive afﬁnity’’ (vector B from Part 8.) in chemical potential and reaction rate, the applicability of the traditional (equilibrium) expression relating chemical potential and activity in non-equilibrium states as well as using activity and activity coefﬁcients determined at equilibrium in non-equilibrium states. To conclude, from all the approaches discussed, only rational thermodynamics was able to come really close to the traditions of chemical kinetics and to derive the classical kinetic mass-action law. The other theories simply only implement it into their frameworks. ACKNOWLEDGEMENT I thank all authors of every work reviewed here, despite the existence of some inevitable criticism. Their ideas enabled me to better understand the role and signiﬁcance of thermodynamics in chemical kinetics. Particularly I am indebted to Ivan Samohy´l for many valuable discussions, and not only on his rational thermodynamics method. This work was partially supported by the Ministry of Education of the Czech Republic, project. No. MSM 0021630501. www.scilet.com

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Miloslav Pekarˇ S. Lengyel, J. Chem. Phys., 1988, 88, 1617. S. Lengyel, Z. phys. Chem., 1989, 270, 577. S. Lengyel, Comput. Math. Appl., 1989, 17, 443. J. S. Shiner, J. Chem. Phys., 1987, 87, 1089. A. P. Grigin, Izv. Akad. nauk. Ser. khim., 1996, 12, 2898. K. Ola´h, Acta Chim. Hung., 1988, 125, 117. K. Ola´h and J. Bo´diss, Hung. J. Ind. Chem., 1988, 16, 39. K. Ola´h, H. Farkas and J. Bo´diss, Period. Polytech. Chem. Eng., 1989, 33, 125. K. Ola´h and J. Bo´diss, React. Kinet. Catal. Lett., 1989, 39, 163. K. Ola´h, J. Bo´diss and H. Farkas, Acta Chim. Hung., 1990, 127, 783. K. Ola´h, Period. Polytech. Ser. Chem. Eng., 1998, 42, 21. L. S. Garcı´ a-Colı´ n and S. M. T. de la Selva, J. Non-Equilib. Thermodyn., 1983, 8, 277. S. M. T. de la Selva and L. S. Garcı´ a-Colı´ n, J. Chem. Phys., 1986, 85, 2140. L. S. Garcı´ a-Colı´ n, S. M. T. de la Selva and E. Pin˜a, J. Phys. Chem., 1986, 90, 953. G. Lebon, M. Torrisi and A. Valenti, J. Phys. Chem., 1987, 91, 5103. L. S. Garcı´ a-Colı´ n, J. Phys. Chem., 1988, 92, 3017. J. Ross and L. S. Garcı´ a-Colı´ n, J. Phys. Chem., 1989, 93, 2091. J. Fort, J. Casas-Va´zquez and V. Me´ndez, J. Phys. Chem. B, 1999, 103, 860. D. Jou, J. Casas-Va´zquez and G. Lebon, Extended Irreversible Thermodynamics, Springer, Berlin, Heidelberg, 1996; ch. 1. D. Jou, J. Casas-Va´zquez and G. Lebon, Extended Irreversible Thermodynamics, Springer, Berlin, Heidelberg, 1996; ch. 1, p. 18. D. Jou, J. Casas-Va´zquez and G. Lebon, Extended Irreversible Thermodynamics, Springer, Berlin, Heidelberg, 1996; ch. 1, p. 46. K. G. Denbigh, Trans. Faraday Soc., 1952, 48, 389. R. M. Bowen, Arch. Rational Mech. Anal., 1967, 24, 370. P. D. Kelly, Int. J. Eng. Sci., 1964, 2, 129. M. Bowen, J. Chem. Phys., 1968, 49, 1625. M. Bowen, Arch. Rational. Mech. Anal., 1969, 34, 97. N. T. Dunwoody and I. Mu¨ller, Arch. Rational Mech. Anal., 1968, 29, 344. J. W. Nunziato and E. K. Walsh, Arch. Rational Mech. Anal., 1980, 73, 285. M. E. Gurtin and A. S. Vargas, Arch. Rational Mech. Anal., 1971, 43, 179. M. E. Gurtin, Arch. Rational Mech. Anal., 1971, 43, 198. A. Bedford and D. S. Drumheller, Arch. Rational Mech. Anal., 1978, 68, 37. D. S. Drumheller and A. Bedford, Arch. Rational Mech. Anal., 1980, 73, 257. C. Truesdell, Rational Thermodynamics, McGraw-Hill, New York, 1984; lect. 6, app. 5C. D. S. Drumheller, Int. J. Eng. Sci., 2000, 38, 347. J. P. Thomas, Int. J. Eng. Sci., 1991, 29, 1451. M. E. Jabbour and K. Bhattacharya, J. Elasticity, 2003, 73, 13. I. Mu¨ller, Physica D, 1986, 20, 35. C. Truesdell, Rational Thermodynamics, McGraw-Hill, New York, 1984; lect. 6, app. 5B.

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Thermodynamics and mass-action chemical kinetics [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168]

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G. M. Kremer and I. Mu¨ller, Ann. Inst. Henri Poincare´, 1998, 69, 309. I. Mu¨ller and T. Ruggeri, Rational Extended Thermodynamics, Springer, NewYork, 1998; ch. 7. G. M. Kremer and I. Mu¨ller, Acta Mechanica, 1999, 132, 37. I. Samohy´l and A. Malijevsky´, Collect. Czech. Chem. Commun., 1976, 41, 2131. I. Samohy´l, Rational thermodynamics of chemically reacting mixtures (in Czech); Academia, Praha, 1982; ch. 43. I. Samohy´l, Thermodynamics of Irreversible Processes in Fluid Mixtures, Teubner, Leipzig, 1987; ch. 22. I. Samohy´l, Collection Czechoslov. Chem. Commun., 1975, 40, 3409. I. Samohy´l, Rational thermodynamics of chemically reacting mixtures (in Czech); Academia, Praha, 1982; ch. 49. I. Samohy´l, Thermodynamics of Irreversible Processes in Fluid Mixtures, Teubner, Leipzig, 1987; ch. 26. M. Pekarˇ , Chem. Eng. Sci., 2004, 59, 4103. I. Samohy´l, Arch. Rational Mech. Anal., 1999, 147, 1. I. Samohy´l, Int. J. Non-Lin. Mech., 1997, 32, 241. I. Samohy´l, Rational thermodynamics of chemically reacting mixtures (in Czech); Academia, Praha, 1982; ch. 27. I. Samohy´l, Thermodynamics of Irreversible Processes in Fluid Mixtures, Teubner, Leipzig, 1987; ch. 19. I. Samohy´l, Rational thermodynamics of chemically reacting mixtures (in Czech); Academia, Praha, 1982; ch. 48. I. Samohy´l, Thermodynamics of Irreversible Processes in Fluid Mixtures, Teubner, Leipzig, 1987; ch. 24. I. Samohy´l, Rational thermodynamics of chemically reacting mixtures (in Czech); Academia, Praha, 1982; ch. 45. I. Samohy´l, Thermodynamics of Irreversible Processes in Fluid Mixtures, Teubner, Leipzig, 1987; ch. 23. I. Samohy´l and M. Sˇilhavy´, Arch. Rational Mech. Anal., 1990, 109, 299.

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REVIEW Thermodynamics and foundations of mass-action kinetics Miloslav Pekarˇ * Institute of Physical and Applied Chemistry, Faculty of Chemistry, Brno University of Technology, Purkynˇova 118, 612 00 Brno, Czech Republic. E-mail: [email protected]

Contents ABSTRACT 1.

INTRODUCTION

5

2.

CLASSICAL BACKGROUND

6

2.1. Reaction isotherm 2.2. Thermodynamic consistency of rate equations

6 9

3.

4. 5.

AFFINITY AND CHEMICAL KINETICS

13

3.1. 3.2. 3.3. 3.4. 3.5.

13 15 23 26 30

De Donder as originator Successors to De Donder Garﬁnkle’s original approach Critical slowing; linearity testing Summary

ACTIVITIES IN CHEMICAL KINETICS

31

CLASSICAL (LINEAR) IRREVERSIBLE THERMODYNAMICS

40

5.1. 5.2. 5.3. 5.4. 5.5. 5.6.

40 44 46 55 57 58

Fundamentals Tackling mass-action non-linearity and Onsager reciprocity Hungarian contribution I – Lengyel Onsager far from equilibrium Bro¨nsted re-discovered? Hungarian contribution II – Ola´h

6.

EXTENDED IRREVERSIBLE THERMODYNAMICS

62

7.

COMMON PROBLEMS IN CIT AND EIT APPROACHES

71

8.

RATIONAL OR CONTINUUM THERMODYNAMICS APPROACHES TO CHEMICAL KINETICS

74

8.1. Introduction 8.2. Bowen lays the foundation stone 8.3. Gurtin re-examines the classical theory 3

74 75 76

4

Miloslav Pekarˇ

9. 10.

8.4. Treatments of more complex systems

81

8.5. Mu¨ller’s results

88

8.6. Samohy´l’s achievements

92

CHEMICAL POTENTIAL MODEL

105 107

CONCLUSIONS

ABSTRACT A critical overview is given of phenomenological thermodynamic approaches to reaction rate equations of the type based on the law of mass-action. The review covers treatments based on classical equilibrium and irreversible (linear) thermodynamics, extended irreversible, rational and continuum thermodynamics. Special attention is devoted to afﬁnity, the applications of activities in chemical kinetics and the importance of chemical potential. The review shows that chemical kinetics survives as the touchstone of these various thermodynamic theories. The traditional mass-action law is neither demonstrated nor proved and very often is only introduced post hoc into the framework of a particular thermodynamic theory, except for the case of rational thermodynamics. Most published ‘‘thermodynamic’’ kinetic equations are too complicated to ﬁnd application in practical kinetics and have merely theoretical value. Solely rational thermodynamics can provide, in the speciﬁc case of a ﬂuid reacting mixture, tractable rate equations which directly propose a possible reaction mechanism consistent with mass conservation and thermodynamics. It further shows that afﬁnity alone cannot determine the reaction rate and should be supplemented by a quantity provisionally called constitutive afﬁnity. Future research should focus on reaction rates in non-isotropic or non-homogeneous mixtures, the applicability of traditional (equilibrium) expressions relating chemical potential to activity in non-equilibrium states, and on using activities and activity coefﬁcients determined under equilibrium in non-equilibrium states. Prog React Kinet Mech 30:3-113 (c) 2004 Science Reviews

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Thermodynamics and mass-action chemical kinetics

5

KEYWORDS: activated complex, activity, afﬁnity, chemical potential, continuum thermodynamics, equilibrium constant, extended irreversible thermodynamics, Guldberg – Waage law, ionic strength, irreversible thermodynamics, kinetic law, mass-action, Onsager reciprocity, rational thermodynamics, rate equation, reaction isotherm, reaction rate, strong equilibrium, weak equilibrium

1. INTRODUCTION The aim of this review is to give a critical overview of various thermodynamic approaches to the formulation of reaction rate equations, preferably of the mass-action law type. It aims to cover papers which directly derive kinetic equations from thermodynamic considerations or which try to obtain more general rate equations from the application of thermodynamic insights to common rate equations or which attempt to supply some established rate equation with proper thermodynamic rigour. ‘‘Kinetic equation’’ and ‘‘(reaction) rate equation’’ should be understood interchangeably as some equation relating chemical reaction rate and quantities, which should determine its value or as some function stating the dependence of the rate on particular (independent) variables. Brieﬂy, the goal is to give a review on thermodynamic derivations or proofs of the Guldberg – Waage kinetic law or of new rate equations applicable in experimental practice. It is just practical phenomenological kinetics which is the primary motivation of this review. Only phenomenological thermodynamic theories are covered, i.e. statistical or molecular approaches are not discussed. Also the large number of approaches which start directly with the mass-action rate equations and use them to study their properties or various properties of underlying systems are not considered. A short list of examples of work outside the scope of this review will make its coverage clearer: studies on mathematical structure and mathematical properties of mass-action type sets of equations [1 – 6], studies on properties of systems described by mass-action kinetics, e.g. their steady state multiplicities, their stability or dynamics [7 – 15], analyses of properties of solutions to (differential) equations embedding mass-action kinetics [16 – 20]. Nor is the detailed balancing included. This review should inform not only on the state-of-the-art of thermodynamic theory for mass-action kinetics but also on its origin. In some instances, the reference therefore goes back more than 100 years. Essentially, however, the period from about 1950 to the present day is covered. www.scilet.com

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Miloslav Pekarˇ

Chemical kinetics and thermodynamics are usually considered as two independent disciplines describing reacting systems. Thermodynamics is said to state the conditions for the running and equilibrium of chemical reactions, while giving no information on how fast this all happens. The latter is the domain of kinetics. This review should further demonstrate that the relationships between thermodynamics and kinetics are much closer and that even from solely thermodynamic theories, some inferences on reaction rates can be obtained. Boyd [21] notes that, in contrast to thermodynamics, the kinetic description of a reaction system is less clear-cut. The value of an equilibrium constant is given unambiguously, together with the course of reaction, according to the sign of the Gibbs energy of reaction. On the contrary, it is often not clear whether a unique reaction velocity may be deﬁned, especially for multistep reaction mechanisms [21]. Another question concerns the circumstances under which the reaction rate may be expressed as the difference of two terms. This is very important because of frequent identiﬁcation of the two terms with forward and reverse rates, which balance at equilibrium. There is no speciﬁc thermodynamic reason why the observed reaction rate should be expressible as the difference of two terms [22]. The only observable is the net rate and the forward and backward rates have meaning only by interpretation. To conclude this introduction, a short note on symbolism should be made. The symbols used are a compromise between two extremes – an elaborate strictly uniﬁed nomenclature for this review or just to retain the differing symbols of the various original sources. In order to aid the interested reader, the speciﬁc original symbols of each paper referred to are used if possible, if these are not easily confused with one another. Universal variables like reaction rate, afﬁnity, concentration, activity etc. are given the common, usual symbols.

2. CLASSICAL BACKGROUND 2.1 Reaction isotherm A very lucid and ingenious discussion on the interrelationships between kinetics and thermodynamics from the standpoint of classical, reversible thermodynamics is given in Denbigh’s book [22], which remains even today one of the most lucid presentations of this topic. Denbigh asks following question: Which variables are determining the reaction rate? Is it the volume concentration of www.scilet.com

Thermodynamics and mass-action chemical kinetics

7

each of the reacting species? Or is it some other concentration (e.g. molar fraction) or thermodynamic (chemical potential, activity) variable? These questions are not (sufﬁciently) answered by (classical) thermodynamic theory. Kinetic experience tells us that just the molar concentration is a very important variable, and that the rate can be expressed as the difference of two terms containing small powers of the molar concentrations. Denbigh further states that thermodynamics places only two requirements on the reaction rate: (1) a positive value of the rate in the direction of a decrease in Gibbs energy and (2) its zero value in the state of thermodynamic equilibrium. This requirement does not directly lead to the formulation of some explicit expression for the reaction rate. It can be used as a test for the ‘‘consistency’’ of proposed rate equation(s) with thermodynamics (see below) and as a restriction on the expression for the backward reaction rate if the expression for the forward rate has been formulated (as well as for the overall rate, usually as the difference of forward and backward rates). Before going into details let us make a small but very important digression. Many kinetic deductions, even in non-equilibrium thermodynamics, are in fact based on the well-known deﬁnitions of equilibrium thermodynamics. The principal relation is an equation, usually called the reaction isotherm. For a general chemical reaction 0¼

n X

i Ai

ð2:1Þ

i¼1

(i is the stoichiometric coefﬁcient, which is positive for products and negative for reactants) it is written as follows: DGr ¼ DGr þ RT ln

n Y

ai i :DGr þ RT ln Qr

ð2:2Þ

i¼1

where Qr is called the reaction quotient and DGr ¼ RT ln K, K is the equilibrium constant and ‘‘ ’’ denotes the standard state. The reaction isotherm was derived for systems at constant temperature and pressure starting from the Gibbs energy (G) considered to be a function of temperature, pressure and composition. In ideal systems, activities (ai ) may be substituted by concentrations. If the forward and backward reaction rates (r with respective arrow) are expressed according to the Guldberg – Waage law with orders equal to stoichiometric coefﬁcients, the reaction isotherm can be modiﬁed as follows: www.scilet.com

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Miloslav Pekarˇ

DGr ¼ RT ln K þ RT ln

n n ?/ ?/ Y Y ci i ¼ RT ln K þ RT ln ð k k y k k Þci i ¼ i¼1

i¼1 ? / / ?

RT ln K þ RT ln½ð k yk Þð r y r Þ

ð2:3Þ

? /

( k ; k are the rate constants in respective directions). Identifying the thermo? / dynamic with the kinetic ( k y k ) equilibrium constant, the ﬁnal equation results: ? /

DGr ¼ RT lnð r y r Þ

ð2:4Þ

It can be also rewritten introducing afﬁnity either by direct deﬁnition A ¼ DGr or in an alternative way through the chemical potential (m): A¼

n X

i mi ¼

i¼1

n X

ði mi þ i RT ln ai Þ ¼ DGr RT ln Qr

ð2:5Þ

i¼1

Two ﬂaws are hidden in this approach and often ignored. The ﬁrst one is direct identiﬁcation of activities with concentrations (in ideal systems). Activity is a dimensionless quantity and may be expressed as the product of activity coefﬁcient, which is in ideal systems equal to one, and the ratio of actual and standard state concentration. However, the Guldberg – Waage law contains actual concentrations, not related to the standard ones. The second ﬂaw is the identiﬁcation of kinetic and thermodynamic equilibrium constants, i.e. dimensional and dimensionless quantities, respectively. It should also be stressed that the use of stoichiometric coefﬁcients in place of reaction orders means that only elementary reactions are considered. From Eq. (2.4) other versions can be derived. The following relation is very popular: ?

/ ?

?

r ¼ r ð1 r y r Þ ¼ r ½1 expðAyRT Þ

ð2:6Þ

which can, close to equilibrium (AyRT 5 1), be linearized as follows: ?

expðAyRT Þ ¼ 1 ðAyRT Þy1 þ ðAyRT Þ2 y2 ) r % r AyRT

ð2:7Þ

A linear relationship between reaction rate and afﬁnity is thus obtained. As noted above, the reaction isotherm was originally born within equilibrium thermodynamics where it is used primarily to derive an expression for the equilibrium constant. Non-equilibrium applications of the reaction isotherm equation are plausible if the reaction Gibbs energy can be considered as a function of temperature, pressure, and composition only, or if the local www.scilet.com

Thermodynamics and mass-action chemical kinetics

9

equilibrium hypothesis is invoked and if the chemical potential dependence on composition can be expressed as indicated in Eq. (2.5). All these premises will be tackled several times throughout this review. 2.2 Thermodynamic consistency of rate equations Let us return to the ‘‘consistency’’ between thermodynamics and mass-action chemical kinetics. It has been already discussed by Boyd [21] with illustrative examples and therefore only the main points are reviewed here. ?

Gadsby et al. [23] claim, in fact, that for the forward ( r ) and backward / ( r ) reaction rates expressed by ?

?

/

r ¼ k ff ðci Þ;

/

r ¼ k fb ðci Þ

ð2:8Þ

where ci , i ¼ 1; . . . ; n, represent the concentrations of reacting species, to be consistent with the thermodynamic equilibrium condition (and constant), the ? / ratio of forward ( k ) and reverse (k ) rate constants must be equal to the equilibrium constant. Manes et al. [24] correct the conclusions of Gadsby et al. The rates for opposing reactions are formulated as ?

r ¼ ff ðci Þ;

/

r ¼ fb ðci Þ

ð2:9Þ

The only restrictions set by thermodynamics on functions f of the concentrations of reacting species ci are ? /

at equilibrium : r y r :ff yfb ¼ 1;

? /

r y r 41 when DGr 50

ð2:10Þ

In order to fulﬁl these conditions it is sufﬁcient to assume, for example, that " #z ? / Y ? / i ff yfb ¼ ð k y k Þ ci ; where k y k ¼ Kz ð2:11Þ i

where symbol ci again means the concentration of a particular specie and z is a positive constant. Examples of suitable (rational) functions f are given in the original paper. It should be stressed that the identiﬁcation of the kinetic with the (concentration-based) thermodynamic equilibrium constant (K) is assumed. The consistency condition (2.11) was generalised by Hollingsworth [25]. He also considers that the reaction rate is given by the forward and reverse reaction rate laws as in (2.9) but temperature is also included among the www.scilet.com

10

Miloslav Pekarˇ

independent variables. The ratio of the forward and reverse rates (see the ﬁrst equation in (2.10)) is symbolized by f ðci ; T Þ. Two equilibrium conditions must be satisﬁed: Qr ¼ KðT Þ and f ¼ 1 ðequilibriumÞ

ð2:12Þ

A sufﬁcient condition for this is that f be expressible as a function of Qr such that f ðci ; T Þ ¼ FðQr ; T Þ and FðK; T Þ ¼ 1

ð2:13Þ

A necessary and sufﬁcient condition for Eqs (2.13) to hold could be that FðQ; T Þ be expressible as a function of Qr yKðT Þ such that FðQr ; T Þ ¼ FðQr yKÞ and Fð1Þ ¼ 1

ð2:14Þ

The condition given by Manes et al., see Eq. (2.11), is then considered as a special case: FðQr yKÞ ¼ ðQr yKÞz

ð2:15Þ

In a subsequent paper [26], Hollingsworth states that the conditions (2.14) are not necessary although sufﬁcient. He presents other sufﬁcient conditions: f ðci ; T ; uj Þ ¼ FðQr yK; uj Þ and Fð1; uj Þ ¼ 1

ð2:16Þ

where uj stands for a set of non-thermodynamic variables. Hollingsworth then shows that the necessary condition when f has continuous derivatives of all orders at Qr yK ¼ 1 is: it must be possible to express ( f 1) as a function which is divisible by the function (Qr yK 1) in the neighbourhood of Qr yK ¼ 1: f 1 ¼ ðQr yK 1ÞCðci ; T ; uj Þ

ð2:17Þ

It should be added that in his proof the invertibility of the function ðQr yKÞðci ; T Þ is tacitly supposed (not proved). An example of practical application of Hollingsworth’s approach is given by Boyd [21]. Blum and Luus [27] proved that condition (2.11)2 is not only sufﬁcient but also necessary providing the rate law is formulated as follows: m m ? Y / Y a0 a r ¼ k j ai i k j ai i i¼1

ð2:18Þ

i¼1

where j is some function of activities, ai , of reacting species, and ai and a 0i are coefﬁcients which may differ from the stoichiometric coefﬁcients. Equation (2.18) is some general law of mass-action inspired by the Bro¨nstedt’s work www.scilet.com

Thermodynamics and mass-action chemical kinetics

11 ?

/

(see below). Boyd reproduces it [21] in more general form with k j and k j 0 , introducing thus different coefﬁcients (phi’s) for the forward and backward directions. As stated by Denbigh [21,22], empiric experience allows one to set j ¼ j 0 . Coefﬁcient j, in fact, makes provision for the dependence upon ionic strength, etc. leaving the rate constants dependent only on temperature. At equilibrium, the following relation is valid: ? /

k yk ¼

n Y

ða 0 ai Þ

ai;eq

ð2:19Þ

i¼1

The proof [27] is based on the statement that both the equilibrium constant and the ratio of the rate constants are dependent only on temperature, which enables one to express the ratio as a function of the equilibrium constant (thus, the invertibility of one of the functions is tacitly introduced): ? /

k yk ¼ f ðKÞ

ð2:20Þ

As the equilibrium activities of all species except one may be selected arbitrarily, it is shown that function f inevitably has the form f ðKÞ ¼ Kz where z ¼ ða 0i ai Þyi ; i ¼ 1; . . . ; n

ð2:21Þ

Condition (2.11)2 was derived also by Van Rysselberghe [28] after introducing afﬁnity deﬁned using chemical potential, Eq. (2.5)1 and its dependence on activity, cf. Eq. (2.5)2, into the general mass-action law, Eq. (2.18). However, this law should be now formulated with stoichiometric coefﬁcients as exponents at activities, moreover, it was also supposed that only one reaction step is kinetically signiﬁcant and the overall afﬁnity is a g-multiple of the afﬁnity of this step. Under these conditions, z ¼ 1yg. In fact, this is another example of application of the reaction isotherm in the mass-action law. Boudart [29] joined equations (2.4) written for elementary steps of a reaction with Temkin’s theory of stationary reaction rates. The following equation for the ratio of overall reaction rates in both directions is thus obtained: ? /

r y r ¼ expðAysRT Þ

ð2:22Þ

where s is the average stoichiometric number and A the afﬁnity. Using again the reaction isotherm-based argument, another relation between the rate and equilibrium constants is obtained: www.scilet.com

12

Miloslav Pekarˇ

? /

k yk ¼ K1ys

ð2:23Þ

All the consistency tests seek, from the mass-action law type rate equation, relations between the equilibrium constant and ratio of rate constants. A general ‘‘consistency’’ criterion, which does not refer to any particular rate equation, has been presented by Corio [30]. Function u is deﬁned u¼K

nr Y

ci i;reactant

i¼1

n Y

i ci;product

ð2:24Þ

i¼nr þ1

where nr symbolizes the number of reactants and ci represent the concentrations. The condition of thermodynamic equilibrium is written as u ¼ 0. On the other hand, the kinetic condition may be written as r ¼ 0. These two conditions can be interpreted as equations deﬁning two surfaces in a Euclidean space of dimension n þ R, where R is the number of reactions, which should touch at a single point only, as otherwise the equilibrium state would not be unique. Consequently, the surfaces have a common tangent plane, so that corresponding derivatives at the tangential point and equilibrium are proportional: ðqryqc1 Þyðquyqc1 Þ ¼ ðqryqc2 Þyðquyqc2 Þ ¼ ¼ ðqryqcn Þyðquyqcn Þ

ð2:25Þ

Using Eq. (2.24) these equations become: ðci yi Þðqryqci Þ ðciþ1 yiþ1 Þðqryqciþ1 Þ ¼ 0

ð2:26Þ

or, alternatively ci ðqryqci Þ ¼ li

ð2:27Þ

where l is a negative constant. Equations (2.26) or (2.27) represent the consistency condition to be fulﬁlled by any rate equation (expression for r) to be consistent with thermodynamics or, more precisely, with thermodynamic equilibrium. Corio also brieﬂy discusses a modiﬁcation for non-ideal systems, where the product of activity coefﬁcient and concentration should be used instead of concentration. It is also interesting to note that an equation similar to (2.24) was given already by Denbigh [22] as an example of a rate equation consistent with thermodynamics. Denbigh also states that the two thermodynamic requirements (see above) can be fulﬁlled by the rate equation www.scilet.com

Thermodynamics and mass-action chemical kinetics

r¼y

n X ði mi Þ

13

ð2:28Þ

i¼1

where y is some positive function of concentrations and mi are the chemical potentials. The disadvantage is that the reaction rate is not directly proportional to the volume concentrations. Eq. (2.28) is closely related to the afﬁnity approaches in chemical kinetics (see part 3). In summary, consistency tests do not provide a particular rate equation (law) but just test the consistency of some proposed rate equation with the condition of thermodynamic equilibrium where the overall reaction rate should vanish.

3. AFFINITY AND CHEMICAL KINETICS 3.1 De Donder as originator Afﬁnity was introduced by de Donder [31,32] in a rather awkward and nonrigorous fashion. As his original approach is nowadays only referred to and not discussed, let us review it here brieﬂy. Starting from the ﬁrst law of thermodynamics in the form dU ¼ dQ pdV and supposing that internal energy U (as well as volume V ) is a function of pressure (p), temperature (T ), and extent of * reaction (x), U ¼ Uðp; T ; xÞ, the following relation for the differential of heat (Q) was derived: dQ ¼ hT x dp þ Cpx dT rpT dx

ð3:1Þ

where *

*

*

*

*

*

hT x ¼ ðqUyqpÞT ;x þ pðqV yqpÞT ;x Cpx ¼ ðqUyqT Þp;x þ pðqV yqT Þp;x

ð3:2Þ

rpT ¼ ðqUyqxÞp;T þ pðqV yqxÞp;T De Donder also supposed that the second law of thermodynamics could be written (according to Clausius) as T dS dQ:dQ 0 0 and that entropy was a function of the same variables. Thus dQ 0 ¼ h 0T x dp þ C 0px dT r 0pT dx

ð3:3Þ

where www.scilet.com

14

Miloslav Pekarˇ *

h 0T x ¼ T ðqS yqpÞT ;x hT x *

C 0px ¼ T ðqS yqT Þp;x Cpx

ð3:4Þ

*

r 0pT ¼ T ðqS yqxÞp;T þ rpT From Eq. (3.3) de Donder derived dQ 0 y dx ¼ h 0T x dpy dx þ C 0px dT y dx r 0pT

ð3:5Þ

Next he introduced the key hypothesis which is neither well substantiated nor supported: the derivative dQ 0 ydx has a constant value regardless of changes in p and T during the course of a reaction, which are dependent on x. There is no explicit motivation for this hypothesis, moreover, among the three independent variables there appears one which is ‘‘more independent’’ and governs the changes of the other two variables. From this hypothesis de Donder derived h 0T x ¼ 0

ð3:6Þ

C 0px ¼ 0 and deﬁned afﬁnity as A ¼ dQ 0 ydx: r 0xy ;

ð3:7Þ

where xy stands for the two (constant) independent variables other than the extent of reaction. The reason why de Donder’s afﬁnity often ‘‘works’’ lies probably in that it is applied under conditions where some quantities are constant, as indicated by Eq. (3.7) so the conditions (3.6) are superﬂuous. Further, afﬁnity can be related to the chemical potential which is also deﬁned by several alternative relations under conditions of constant various pairs of independent variables while not changing its value. For example, the afﬁnity of a reaction is simply given by the ﬁrst relation in (2.5). Expressing the total differential of the Gibbs energy as a * function of temperature, pressure and composition, G ¼ GðT ; p; ni Þ, using the extent of reaction as de Donder suggested, we obtain: X X mi dni ¼ i mi dx ¼ A dx ðconstant T and pÞ dG ¼ i

ð3:8Þ

i

As at constant temperature and pressure, heat is identical with the change of enthalpy (H), dQ 0 ¼ dG under these conditions and Eq. (3.7) is derived with no need for this strange hypothesis. www.scilet.com

Thermodynamics and mass-action chemical kinetics

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In fact, de Donder re-labelled some variables of classical thermodynamics and his main contribution consists of noticing that extent of reaction can be used as an independent variable instead of molar masses, concentrations and so on. However, the extent of reaction may not be useful in complex reactions. In this case, changes in molar amount of some or even all components may be caused by more than one reaction. The extent of reaction should be then deﬁned for every reaction step including only molar changes caused by the corresponding step. While this can be done in theory with no problem, it is useless in practice where molar changes caused by individual reactions cannot be always simply measured. As pointed by, e.g. Hollingsworth [33], it is often impossible to deﬁne some overall extent of reaction. Bowen has proved [34] that the extent of reaction cannot be used in reacting mixtures with diffusion unless the diffusion is so-called self-balancing [35]. As regards chemical kinetics, de Donder deduced from the second law and (3.8) 0 dQ 0 ydt ¼ AðdxydtÞ:Ar

ð3:9Þ

where r is the reaction rate.

3.2 Successors to De Donder Most applications of afﬁnity in chemical kinetics are, in fact, deductions based on the reaction isotherm outlined in Section 1. The ﬁrst work from this area is probably the paper by Prigogine et al. [36], which also refers to de Donder’s work. They started from the assumption that both reaction rate (r) and afﬁnity (A) depend on the same variables (xi ) and that the function for afﬁnity is invertible in at least one variable. Substituting this variable in the function for reaction rate, the following relationship results: r ¼ f ðx1 ; x2 ; . . . ; xl ; AÞ

ð3:10Þ

(variable xlþ1 was substituted). At equilibrium, both reaction rate and afﬁnity vanish. Expanding the function in (3.10), a close-to-equilibrium linear relationship is thus obtained: r ¼ ðqf yqAÞxi A

ð3:11Þ www.scilet.com

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Miloslav Pekarˇ

(so that the expansion was made keeping all xi constant!). Making use of manipulations with the Guldberg – Waage law and reaction isotherm (see part 2), this linear relation is illustrated by the linear relationships for the hydrogenation of benzene and dehydrogenation of cyclohexane. A subsequent paper by Manes et al. [24] derived the linear relationship in a somewhat more general fashion. The authors supposed that the reaction Gibbs energy (G) depends on some set of independent variables (aj ; j ¼ 1; 2; . . . ; m) and that the reaction rate depends on the same variables and some added, ‘‘nonthermodynamic’’ ones (bk ; k ¼ 1; 2; . . .). Using again the vanishing of the Gibbs energy and reaction rate at equilibrium simultaneously, they arrived at an equation valid sufﬁciently close to equilibrium: r ¼ xðaj ; bk Þ DG

ð3:12Þ

where the proportionality factor represents: xðaj ; bk Þ ¼ ½qryqðDGÞa2 ;a3 ;...;am ¼ ½qryqðDGÞa1 ;a3 ;...;am ¼ . . . ¼ ½qryqðDGÞa1 ;a2 ;...;am1 ð3:13Þ and depends on full sets of aj and bk . In the derivation, the implicit assumption on the invertibility of the reaction Gibbs energy function is hidden. Their thermodynamic approach gives no explicit relation for the proportionality factor. The authors also point that because x depends also on non-thermodynamic variables, Eq. (3.12) cannot be used to obtain absolute rates from thermodynamic data. How this could be achieved, when knowing the values of the additional variables, is not discussed. Another illustration of the application of the reaction isotherm and afﬁnity in chemical kinetics is given in the paper by Hall [37], which forms a part of the polemic between Haase and Hall mainly on kinetics in non-ideal systems and is therefore reviewed in part 4. Nebeker and Pings [38] tried to conﬁrm experimentally the linear relationship between afﬁnity and reaction rate. They measured the concentrations of components in a reacting mixture of NO, Cl2, NOCl, I2, and ICl. Two reactions were considered, viz.: 2 NO þ Cl2 ¼ 2 NOCl

ð3:14aÞ

2 NOCl þ I2 ¼ 2 NO þ 2 ICl

ð3:14bÞ www.scilet.com

Thermodynamics and mass-action chemical kinetics

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Of course, afﬁnities were not measured but calculated from the reaction isotherm and concentration proﬁles. Rates of reactions (3.14a) and (3.14b) were taken as time derivatives of the chlorine and iodine concentrations. It was found that, for some portions of a run of the reacting system, the linear relationship is valid. In general, however, it was not veriﬁed as well as the socalled Onsager reciprocity relations, which are not discussed here. A linear relationship between reaction rate and afﬁnity near equilibrium was also derived by Gilkerson et al. [39] from the theory of absolute reaction rates. They identiﬁed the reaction Gibbs energy DGr ð:AÞ with DG6¼ r , i.e. the activation Gibbs energy, which might be questionable. Boudart shows in several papers more precisely the potential practical value of afﬁnity-containing equations in chemical kinetics. He distinguishes [40] between the de Donder inequality: Ar 0

ð3:15Þ

and de Donder equation: ? /

lnð r y r Þ ¼ AyRT

ð3:16Þ

Because Eq. (3.15) is valid for the overall reaction process, it may explain why some reaction steps may occur against the ‘‘thermodynamic direction’’ [41]. For instance, two reactions may occur simultaneously even when A1 r1 50

ð3:17Þ

providing that A1 r1 þ A2 r2 40

ð3:18Þ

It is said that reaction 1 is coupled to (driven by) the second one. Boudart shows [40] that this may be a useless idea, as the coupled reaction in many real cases does not proceed. Boudart argues that, in a reaction system consisting of a closed sequence of elementary reactions, at the steady state for each of the steps it is the case that: ?

/

r ¼ r i r i 40

ð3:19Þ

and from Eq. (3.15), which is valid for any step i with afﬁnity Ai , it follows that: Ai 40;

Ai ri 40

ð3:20Þ www.scilet.com

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Miloslav Pekarˇ

for all steps. It should be stressed that, in the case of more reactions than just one, there is no thermodynamic requirement (3.15) to be valid for any of these reactions separately. Only the sum of afﬁnity and rate products must be nonnegative as Eq. (3.18) illustrates. Inequalities (3.20)2 are just due to relations (3.19). But ‘‘kinetic coupling’’ may occur which can change substantially the steady-state concentrations of intermediates above their equilibrium value, if they are reactants, or below their equilibrium value if they are products. Particular examples are given in Boudart’s papers [40,42]. In a short report [43] Boudart et al. show, using the data by Prigogine et al. [36], that the linear relationship between reaction rate and afﬁnity remains valid also relatively far from equilibrium. Dumesic has published an analysis of the reaction scheme using ‘‘de Donder relations’’ [44]. It is claimed that the rate equation is derived from the reaction scheme in terms of these relations. In fact, all the results can be simply arrived at using traditional thermodynamics and kinetics. Further, the analysis is applicable only to stationary states and mechanisms in which the overall reaction is given as a sum of elementary reactions. The central quantity is the reversibility of the i-th step (elementary reaction) deﬁned by: ! Y ij zi ¼ expðAi yRT Þ ¼ aj yKi

ð3:21Þ

j

where Ai is the step afﬁnity, aj the activity of the j-th component and ij its stoichiometric coefﬁcient in the i-th step with the equilibrium constant Ki . Eq. (3.21) directly follows from the reaction isotherm. Reversibility was deﬁned by us (in ideal systems) the relative distance from equilibrium and shown to be useful to follow the evolution of reaction rates even in the non-steady state [45 – 47]. By the de Donder relation Eq. (2.6)2 is understood, and its exponential appears in (3.21). In fact, only (3.21)2 is used in the analysis and to calculate the reversibility. The rate equation is not derived but step rates are stated as massaction laws with activities instead of concentrations, Eqs (3.21) are used to eliminate the activities of intermediate species and analysis further continues within the idea of a rate-determining step. What could be done quite easily is complicated here by forcing deductions into the framework of the de Donder relation. For instance, it is ‘‘revealed’’ that the minimum number of kinetic parameters required to calculate the rate for three-step mechanism is equal to three, in contrast to expectation that ﬁve would be required, because all six step www.scilet.com

Thermodynamics and mass-action chemical kinetics

19

rate constants are bounded by the total equilibrium constant. However, if it is realized that rate constants of each step are related by the kinetic equilibrium constant of the step, it immediately follows that only three kinetic parameters are necessary (and selectable independently). Reversibilities for each step are calculated from experimental data. Steps with close-to-one reversibility are (quasi-)equilibrated. If there is a step with reversibility far from a zero value, then this step is considered to be rate determining, and the overall reaction reversibility is equated to its reversibility whereas the other reversibilities are identiﬁed with unity. The overall rate is set equal to the rate-determining step rate. The whole procedure closely resembles the classical Langmuir – Hinshelwood – Hougen – Watson approach. This is felt also by the author as he states that his approach is advantageous because it provides the means to derive the overall reaction rate from the more general case where multiple steps are not in quasi-equilibrium. In fact, this means only that equilibrium constants of equilibrated steps, together with the overall equilibrium constant given as appropriate product of steps equilibrium constants, are used to eliminate intermediate activities. Let us illustrate this approach by the simple example of the three-step mechanism R 1 ¼ 2 I1 R2 þ I1 ¼ I2 I1 þ I2 ¼ P of the overall reaction R1 þ R2 ¼ P The rate of the ﬁrst step can be expressed as [44]: ?

r1 ¼ k 1 aR1 ð1 z1 Þ

ð3:22Þ

where z1 is given as follows from Eq. (3.21): z1 ¼ a2I1 yðK1 aR1 Þ

ð3:23Þ

If this step is rate-determining, then the overall rate (r) is equal to r1 . As the total reversibility (z) is given by www.scilet.com

20

Miloslav Pekarˇ

z ¼ z1 z2 z3 ¼ aP yðKaR1 aR2 Þ

ð3:24Þ

and z2 , z3 are in this case equal to unity, it follows that ?

r ¼ k 1 aR1 ½1 aP yðKaR1 aR2 Þ

ð3:25Þ

This result can be derived by the usual procedure without reversibility or de Donder relations. Actually, in this example, the rate is given by: ?

/

r ¼ r1 ¼ k 1 aR1 k 1 a2I1

ð3:26Þ

From equilibrium constants of (quasi-)equilibrated steps 2 and 3: K2 ¼ aI2 yðaI1 aR2 Þ;

K3 ¼ aP yðaI1 aI2 Þ

ð3:27Þ

it can be easily derived: a2I1 ¼ aP yðaR2 K2 K3 Þ

ð3:28Þ

Introducing Eq. (3.28) into Eq. (3.26) and using the kinetic deﬁnition of equilibrium constant K1 and the relation K ¼ K1 K2 K3 , Eq. (3.25) is obtained. The very essence of Dumesic’s analysis can be reported in this way. Measure the values of equilibrium constants of elementary steps of interest or measure their rate constants and calculate equilibrium constants from them. Measure stationary concentrations (more rigorously, activities) and calculate reaction quotients from them. Compare all corresponding quotients and equilibrium constants to identify quasi-equilibrated steps. Use equilibrium constants of these steps to eliminate some (intermediates) concentrations. Set the overall rate to be equal to the rate of (some) non-equilibrated step. And make this analysis in terms of reversibilities and afﬁnities. There is nothing special to the thermodynamic analysis of chemical kinetics except comparing the actual stationary state of reacting system with its state of equilibrium. The principles of Dumesic’s analysis were combined by Fishtik and Datta [48] with their method of analysis and simpliﬁcation of reaction mechanisms, which is beyond the scope of this review. It should be only pointed that by the de Donder relations not only Eqs. (2.6)2 but also mass-action law expressions for forward reactions are understood in their paper. In principle, the relations are again used to eliminate the concentrations of intermediates. Afﬁnity is deﬁned in such a way that it directly accords with mass-action kinetics, viz. in concentrawww.scilet.com

Thermodynamics and mass-action chemical kinetics

21

tions (more precisely, surface coverages and partial pressures) instead of activities. Timmermann [49] asserts that he obtained the general formula relating reaction rate and afﬁnity, and a general and rigorous statement of the thermodynamic restrictions on reaction rate is thus given. His proof is based only on the argument that the rate of increase of the extent of reaction has a unique value independent of the particular language used to describe the reaction and the afﬁnity. However, the key point of his proof is unclear. Timmermann deﬁnes the gross reaction rate (r) as the rate of increase of the extent of reaction (x): r ¼ dxydt ¼ dni yði dtÞ

ð3:29Þ

where ni is the amount of substance i in the whole system and i its stoichiometric coefﬁcient. Timmermann further states that the gross rate is generally not determined in a kinetic experiment. Instead, an intensive quantity is measured, which is the gross rate normalized to some extensive reference quantity. Two from several of Timmermann’s examples are reproduced here. The most common reference quantity is the volume of the system (V ) and the intensive reaction rate is then expressed as: rc ¼ ryV

ð3:30Þ

When the molality (m) reference basis is selected, we have: rm ¼ ryðn0 M0 Þ

ð3:31Þ

where n0 is the mole number of the solvent and M0 its molar mass. Clearly, r c V ¼ r m n0 M 0 Timmermann

ð3:32Þ

combines the classical mass-action rate equation ? P / P / where ¼ k c i ci i and r c ¼ k c j cj j (i runs through reactants, rc ¼ r c P j through products), with the classical deﬁnition of afﬁnity A ¼ k mk (k runs ?

/ r c,

then

? rc

k

through both reactants and products). Chemical potential (mk ) is expressed also traditionally, mk ¼ mok þ RT lnðgk ck yco Þ where ‘‘o ’’ denotes the standard state and gk is the activity coefﬁcient on the molarity scale. Timmermann ﬁnally arrives at the following expression: www.scilet.com

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Miloslav Pekarˇ

0

1

/

o

C k c Kg ðc Þ ? B C rc ¼ r c B @1 ? Q k expðAyRT ÞA k c gk

ð3:33Þ

k

where Kg is the thermodynamic equilibrium constant on the molarity scale and P ¼ k . He states that r cannot depend on the particular language used to describe the intensive reaction rate (i.e. on the referential quantity), consequently, the factor in Eq. (3.33) must be the same for every kinetic description, that is unity: /

k c Kgðco Þ ¼1 ? Q k c gkk

ð3:34Þ

k

This condition is acceptable as general at equilibrium with vanishing of both the gross rate and afﬁnity. Timmermann gives no explicit proof for its general validity (out of equilibrium) and his statement on the independence of the particular language is unclear as will be now shown. Consider his other example – molality scale. He derives the following alternative rate equation: 0

1

/

o

C k m Kj ðm Þ ? B C rm ¼ r m B @1 ? Q k expðAyRT ÞA k m jk

ð3:35Þ

k

where Kj is the thermodynamic equilibrium constant and jk the activity coefﬁcient on the molality scale this time. If Eqs (3.33) and (3.35) are substituted into Eq. (3.32) and if it is realized that Eq. (3.32) is valid also for forward or reverse rates, the following condition for ‘‘independence of particular language’’ is obtained: /

/

k c Kg ðco Þ k m Kj ðmo Þ ¼ ? Q ? Q k c gkk k m jkk k

ð3:36Þ

k

It is not clear why condition (3.36) is not sufﬁcient and why both fractions should be in addition equal to one everywhere. It seems that Timmermann’s condition (3.34) is unwarrantedly restrictive and his analysis questionable. www.scilet.com

Thermodynamics and mass-action chemical kinetics

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3.3 Garﬁnkle’s original approach Yet another approach to afﬁnity in relation to reaction kinetics was presented by Garﬁnkle. Actually, he takes the time derivative (symbolized by a dot) of the reaction isotherm written in terms of afﬁnity (A) instead of the Gibbs energy (and with concentrations approximating to activities) [50]: X _ ¼ RT A ð2i yci Þðdci yi dtÞ

ð3:37Þ

i

(i is the stoichiometric coefﬁcient and ci the concentration of the i-th component). According to Garﬁnkle, the term in the second parentheses is the reaction velocity r. After rearrangement, an equation relating reaction rate to _ ) is obtained: the afﬁnity decay rate (A _ yRT Þy r ¼ ðA

X 2i yci

ð3:38Þ

i

Because it is difﬁcult to obtain the afﬁnity decay rate directly, Garﬁnkle introduces an empirical relation between this quantity and the elapsed time of reaction (t): _ ¼ Ar ð1yt 1ytK Þ A

ð3:39Þ

where Ar and tK are parameters to be determined. The latter is called the mostprobable time to attain equilibrium and the meaning of both is discussed in the original papers, particularly ref. [51]. In practice, one must know the equilibrium constant of the reaction under study and the values of the reaction quotient at various reaction times. The latter is calculated from the measured concentration time proﬁles. From the reaction quotient and equilibrium constant, the afﬁnity is calculated and then a regression analysis devised by Garﬁnkle [51] is used to obtained the parameters of Eq. (3.39). Thus, the afﬁnity decay rate can be obtained and from it, using the concentrations of reacting species, the reaction rate at an appropriate instant in time can be calculated from Eq. (3.38). Garﬁnkle’s papers contain examples of afﬁnity or rate time proﬁles for many reactions and their comparison with conventional, mass-action rate equations. Garﬁnkle also shows [52,53] that for a (homogeneous) chemical reaction (in a closed isothermal system), there exists a unique natural path along which the rate of change in time of a thermodynamic function can be described. This, in fact, www.scilet.com

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Miloslav Pekarˇ

means that instead of reporting time proﬁles of concentrations (or, perhaps, reaction rate or afﬁnity), afﬁnity should be represented as a function of the following quantity: ln½ðtytK Þ expð1 tytK Þ, which appears in the integrated form of Eq. (3.39). Garﬁnkle shows that even for a reaction with ‘‘mechanistic differences’’, i.e. with different concentration time proﬁles (e.g. iodine atom recombination in different inert gases), it will have a unique natural path for afﬁnity. Garﬁnkle’s approach was criticized in details by Hjelmfelt et al. [54], Garﬁnkle responded in ref. [55]. We will not report here on this polemic and merely add some comments. First, it should be remembered that this method can be used only in closed isothermal systems where the reaction rate is directly given by the concentration time derivative. Second, it is limited only to the cases where the reaction rate is given by the time derivative of any reacting specie, i.e. where some overall reaction rate exists, to the stoichiometric systems. As Garﬁnkle states [55]: ‘‘The concentrations of reactants and products appearing in the stoichiometric equation that represents the overall chemical reaction under observation changes with elapsed time... The rate of change of these concentrations consistent with stoichiometric constratints is the reaction velocity...’’ As an example he gives the addition of iodine to styrene (St), I2 þ St ?IStI with a velocity deﬁned as r ¼ d½Stydt ¼ d½I2 ydt ¼ d½IStIydt

ð3:40Þ

where the square brackets symbolize concentrations. This deﬁnition supposes that product (IStI) appears immediately after the disappearing of reactants. This is generally not the case in reactions with a detailed mechanism [56], which is signiﬁcant for the concentration evolution of especially reaction intermediates. As an illustration, one of the simplest mechanisms can be used. Let us suppose that some general transformation A ?C goes through an intermediate B: A ?B ?C. From classical kinetics it follows that: dcA ydt ¼ k1 cA dcB ydt ¼ k1 cA k2 cB

ð3:41Þ

dcC ydt ¼ k2 cB where k1 is the rate constant of the step A ?B and k2 of the step B ?C. It is clear that the time derivatives are not in general equal, which is even more evident after inserting the analytical solutions: www.scilet.com

Thermodynamics and mass-action chemical kinetics

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dcA ydt ¼ k1 c0A expðk1 tÞ dcB ydt ¼ k1 c0A expðk1 tÞ k1 k2 c0A ½expðk1 tÞ expðk2 tÞyðk2 k1 Þ dcC ydt ¼

k1 k2 c0A ½expðk1 tÞ

ð3:42Þ

expðk2 tÞyðk2 k1 Þ

where 0 in the superscript denotes the initial concentration. So there is, in general, no simple single rate expression for the overall stoichiometric transformation A ?C and no identity dcA ydt ¼ dcC ydt. Only when k2 4 k1 can the last equation (3.42) be transformed practically to fulﬁl this identity. Equation (3.38) is not an expression of reaction rate as a function of afﬁnity decay rate but an expression of function of afﬁnity decay rate and concentrations, because they are also changing during the course of reaction and, in fact, determine the afﬁnity. Garﬁnkle presents an analysis of experimental data of many, essentially stoichiometric, reactions in terms of afﬁnity decay rate. He succeeded very well in ﬁtting experimental data translated into the reaction quotient by his Eq. (3.39). What is the value of this approach? Conventionally, concentrations are measured, and a kinetic-mechanistic model proposed and used to interpret the data. Rate expressions are obtained which can be used as rates of formation, e.g. in reactor balance equations to make its design possible. Afﬁnity decay methodology transforms concentrations to afﬁnity, the decay of which is ﬁtted by Eq. (3.39), and the decay rates may then be used to calculate reaction rate from Eq. (3.38). Garﬁnkle stresses that his approach gives correlations independent of reaction mechanism and, in contrast to the conventional description in terms of the time-dependency of the concentration of reacting components, it describes kinetic behaviour in terms of the time-dependency of a thermodynamic function. His approach could be viewed as an alternative of a data-ﬁtting procedure in closed isothermal systems with an unambiguously deﬁned and conﬁrmed overall reaction rate. Afﬁnity decay then describes the course of reaction not in terms of concentrations changing in time, i.e. in kinetic terms, but in terms of a thermodynamic quantity changing in time, i.e. in ‘‘energetic’’ terms. Although the kinetic details may be different even for very similar reactions (e.g. iodine atom recombination in different inert gases [52,53]), thermodynamic principles are general and really give identical decay curves for such reactions. The existence of a unique natural path is an interesting theoretical phenomenon and conﬁrmation of correctness of the reaction isotherm in stoichiometric systems. The natural path scales both the concentrations of reacting species and the elapsed reaction time. The former, through the afﬁnity www.scilet.com

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Miloslav Pekarˇ

embodying the reaction quotient and the equilibrium constant, which, in turn, contains equilibrium concentrations, the latter through the parameter tK , i.e. the most probable time of attaining equilibrium. As any chemical reaction proceeds from some initial concentrations and time to equilibrium concentrations and time, it may be expected that such ‘‘scaling to equilibrium’’ will work. 3.4 Critical slowing; linearity testing Afﬁnity- and reaction isotherm-based approaches have found some popularity in the interpretation of the slowing down of chemical reactions near some critical point, see e.g. refs [59 – 62]. Actually, the ‘‘linear’’ relationship (2.7) is used [59,60] for qualitative interpretations, not for quantitative evaluations. Recently, Kim and Baird [62] reported even a speeding up near the critical point. Several approximations are used, the nature of which is clearly seen from an inspiring older work by Meixner [63]. Meixner claims that the close-to-equilibrium reaction rate is expressed as dxydt and given by: dxydt ¼ eðT ; ; xÞAðT ; ; xÞ

ð3:43Þ

where x is the extent of reaction, e is the proportionality coefﬁcient dependent on temperature (T ), speciﬁc volume () and extent of reaction, and A is the afﬁnity determined by the same set of variables. First, Meixner states that the close-toequilibrium dependence on the extent of reaction in the functional expression for the coefﬁcient e in (3.43) can be abandoned by substituting its equilibrium value (xe ). Next, he expands the afﬁnity at constant temperature and speciﬁc volume up to the ﬁrst order: dxydt ¼ eðqAyqxÞT ; ½x xe ðT ; Þ

ð3:44Þ

Why the dependence on the extent of reaction is suppressed only in the ﬁrst function from (3.43), and why only the second one, afﬁnity, is expanded, is neither explained nor discussed. Coefﬁcient e in (3.44) is thus effectively a constant, which is stated, e.g. by Procaccia and Gitterman [60], as a fact at the outset. Kim and Baird [62] present a more correct derivation and expand, in fact, both functions in (3.43). In the end, however, they retain only the terms of ﬁrst order and arrive at Eq. (3.44) once more. From their procedure, the motivation for Meixner’s inconsequent treatment of functions can be clariﬁed a little. From Eq. (2.7) it is clear that coefﬁcient e is the forward reaction rate [62], which is www.scilet.com

Thermodynamics and mass-action chemical kinetics

27

non-zero at equilibrium in contrast to the afﬁnity. Consequently, the ﬁrst term in the forward rate (or coefﬁcient e) expansion is non-zero whereas that in the afﬁnity expansion vanishes. What does an approximation like (3.44) using the equilibrium forward rate as a constant not-far-from-equilibrium mean in reality? From the more general Eqs (2.4) or (3.16), it is seen that within this approximation, the afﬁnity / at a given temperature is given by const RT ln r . All afﬁnity and, consequently, overall rate changes and evolution should be then governed by the reversed rate. This is also conﬁrmed by the expansion of (3.47) below. Even then it is rather arduous to accept that the backward rate changes markedly while the forward remains constant. Kim and Baird [62] claim even that the reaction they studied was essentially irreversible. From another point of view, the approximation used in (3.44) means a much slower approach (usually decrease) of the forward rate to its equilibrium value than afﬁnity decay to the equilibrium zero value. Rates of both decays are dictated by the values of the relevant concentrations. Decay of afﬁnity, anyway, corresponds to a decaying logarithm with the argument approaching to one, and it should be realized that whereas a logarithm is a ‘‘magnitude smoothing’’ function above one, at values very close to one it is a magnitude ampliﬁer. This elementary fact is illustrated by numbers given in Table 1, cf. also Eq. (3.16). Far from equilibrium, when the reaction rate in one direction, at least, is changing over several orders of magnitude, the afﬁnity decays by about only one order of magnitude. An afﬁnity decrease amounting to many orders of magnitude is not noticed before being very close to equilibrium when the rates in both directions are almost the same. Table 1 also models approximation (3.44) – if the forward reaction rate is considered to be constant, e.g. ﬁxed at its equilibrium value, than all changes of the ratio given in the ﬁrst column of the table are due to an increasing reverse rate on the approach to equilibrium. Consequently, when the reverse rate changes appreciably, the afﬁnity decreases (with extent of reaction) only slowly, whereas when the backward rate (and, consequently, the overall rate) almost attains its equilibrium value before the steep decay of afﬁnity starts. Perhaps Table 1 gives some answer to the question as to how far from equilibrium is too far [64]. On the other hand, should the numbers in the table mean that far from equilibrium, within a convenient time interval, the reaction rate could be approximated by ?

equation dxydt ¼ eðT ; ; xÞ const ½x x0 ðT ; Þ where e: r is not constant and the subscript ‘‘0’’ denotes some point within this interval? www.scilet.com

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Miloslav Pekarˇ Table 1

Decay of logarithm and its argument

!

!

ryr

lnð r y r Þ

1.0000000000E þ 10 1.0000000000E þ 09 1.0000000000E þ 08 1.0000000000E þ 07 1.0000000000E þ 06 1.0000000000E þ 05 1.0000000000E þ 04 1.0000000000E þ 03 1.0000000000E þ 02 1.0000000000E þ 01 1.1000000000E þ 00 1.0100000000E þ 00 1.0010000000E þ 00 1.0000001000E þ 00 1.0000000001E þ 00 1.0000000000E þ 00

23.03 20.72 18.42 16.12 13.82 11.51 9.210 6.908 4.605 2.303 9.531E-02 9.950E-03 9.995E-04 1.000E-07 1.000E-10 0.000

Our model calculations [45,47,65] demonstrated that (in ﬂow systems) the overall reaction rate can change appreciably even when the reaction is still very close to equilibrium (reaction quotient almost equal to one), its value can change abruptly just before reading equilibrium, or that both overall rate and afﬁnity may undergo steep changes close to equilibrium. In some cases the overall rate was even increasing at the same time as the ratio of reaction quotients and equilibrium constant approached to unity [66]. It should be also stressed that approximation (3.36) does not express the reaction rate as a function of afﬁnity partial derivative only but as a function of this derivative and extent of reaction. Linear approximations like (3.36) seem to be the result only of numerical trickiness in the logarithm and not consequences of some genuine thermodynamic principles. Experimental veriﬁcation of approximations involved in afﬁnity-rate deductions is still missing. Data by Prigogine et al. [36] show that the linear relationship between afﬁnity and reaction rate is valid also for values not fulﬁlling the inequality AyRT 5 1 (cf. Part 1.). The highest value of this ratio lying in the linear region is reported to be 2.3. Full revision of this paper is postponed to some future work, here only a short note is given. There must be some mathematical reason as it was the mathematical expansion of the exponential function, which enabled the disclosure of the linear relationship, cf. Eqs (2.6), (2.7), and not some ‘‘effort’’ of the reaction to keep linearity far from equilibrium. This is illustrated in Table 2. It is evident that the linear www.scilet.com

Thermodynamics and mass-action chemical kinetics

29

Table 2 Comparison of exponential and the ﬁrst three terms of its series expansion x

0.01

0.1

1

2

expðxÞ 1x 1 x þ x2 y2 1 x þ x2 y2 x3 y6

0.99005 0.99000 0.99005 0.99005

0.90484 0.90000 0.90500 0.90483

0.36788 0.00000 0.50000 0.33333

0.13534 1.00000 1.00000 0.33333

approximation starting from an argument value equal to one, at least, is a nonsense. Let us analyze the reaction isotherm from the logarithmic side. If thermodynamic and kinetic equilibrium constants are identiﬁed, as necessary, Eq. (2.6) can be rewritten: ?

/

/

/

A ¼ RT ln K RT ln Q ¼ RT ln KyQ ¼ RT lnð r y r Þ ¼ RT ln½ðrþ r Þy r ¼ /

¼ RT lnðry r þ1Þ:RT lnðx þ 1Þ ¼ RT ðx x2 y2 þ x3 y3 x4 y4 þ Þ ð3:45Þ The expansion in Eq. (3.45) is valid only for 15x 1. From Eq. (3.45) it is better seen than from the last equality in (2.7) that the linear relationship between afﬁnity and rate is determined also by the rate in the reverse direction. The linear term in (3.45) can only be retained in the case when the ratio of the overall and reverse rates (x) is sufﬁciently small. In fact, Eq. (2.7) does not lead to a strict linear relationship unless the reverse rate is constant. Eq. (3.45) shows that the linear approximation may be acceptable regardless of the distance from equilibrium. For instance, if the overall rate has a formal value of 103, which is surely quite far from equilibrium, and the backward rate is 105, then the second order term gives less than 1% correction to the linear term. This short example is limited by the validity of the expansion used in Eq. (3.45) as stated above. In general, the logarithm can be expanded for all values of its argument (x40) in the following way: ln x ¼ 2ðy þ y3 y3 þ y5 y5 þ Þ; ?

where y ¼ ðx 1Þyðx þ 1Þ

ð3:46Þ

/

In our case x: r y r . From Eq. (3.46) then follows: h ? / i ? / ? / / A ¼ RT ln r y r ¼ RT 2 ð r y r 1Þyð r y r þ1 þ ¼ 2RTryðr þ 2 r Þ þ ð3:47Þ www.scilet.com

30

Miloslav Pekarˇ

Thus, even the ﬁrst term is not linear in general. A linear relationship between afﬁnity and the overall rate can be obtained only if the ﬁrst term in approxima/

tion (3.47) is sufﬁcient and if r þ 2 r is constant. The latter condition can be ? / reformulated as r þ r ¼ const., which is easily imagined to be fulﬁlled in practice, because the forward rate is decreasing while the backward rate is increasing in the same time. 3.5 Summary The main problem of most afﬁnity-based approaches is that they are used for interpretation rather than for a theoretical explanation of experimental data. This is because afﬁnity usually cannot be measured. Concentrations (partial pressures, activities, etc.) are those quantities, which are measured by kineticists, and only from these quantities are afﬁnities calculated. The only exception is perhaps a reaction in a galvanic cell where the measured electromotive force (E) is directly related to afﬁnity through the well-known equation A ¼ zFE, where z is number of exchanged electrons and F is Faraday’s constant. Even in this case, if afﬁnity should be related to the reaction rate, concentrations (activities) within the cell should be utilised, i.e. the Nernst equation, which is a variant of the reaction isotherm. Thus in examples like that of Prigogine et al. [36], neither the afﬁnity nor reaction rate were directly and independently measured. Concentrations (composition) were determined and from them the rate and afﬁnity were computed. Afﬁnity-velocity linear tests are then no more than checking that concentrations behave in the manner predicted by the reaction isotherm. Equations (2.6) and (2.7) cannot be viewed as the function r ¼ f ðAÞ but as ? / ? a transformation of the function r ¼ f ð r ; r Þ to function r ¼ gð r ; AÞ using the reaction isotherm. Table 1 clearly illustrates that afﬁnity by itself is a problematic measure or determining quantity for reaction rate because it does not vary too much when the rate undergoes steep changes and vice versa. Afﬁnity or reaction Gibbs or free energy alone does not determine the reaction rate, or kinetic ‘‘driving force’’. Water synthesis from molecular oxygen and hydrogen is a notoriously well-known example – its (standard) reaction Gibbs energy amounts to several hundreds kJ but its reaction rate is negligible unless some external catalytic action is introduced. It follows from the reaction isotherm that any reaction mixture containing only reactants possesses in zero time an

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Thermodynamics and mass-action chemical kinetics

31

inﬁnitely high afﬁnity but experimental evidence clearly shows that initial rates have ﬁnite and diverse values. Additional and very important information on the relation between afﬁnity and reaction rate is also provided by rational thermodynamics. For consistency, this is postponed to Section 7. 4. ACTIVITIES IN CHEMICAL KINETICS Rigorous thermodynamic treatments are given in activities. By contrast, kineticists prefer concentrations, and activities are rarely used. Proposals to replace concentrations in kinetic equations simply with activities appeared immediately after activities had been introduced by Lewis at the beginning of the 20th century. As expected, this substitution was being made particularly in ionic reactions where particle interactions are natural. Reviewing ionic reactions, salt effects etc., is beyond the scope of this review, because it can be found in many textbooks, e.g. refs [67, 68]. We will restrict ourselves here solely to the principal historical roots and modern work directly related to mass-action kinetics. Jones and Lewis [69] measured the rate of inversion of sucrose. Having estimated the unimolecular rate constant, they found its dependence on the initial concentrations of sugar and water. They measured also the activity of hydrogen ions using an electrochemical cell. Dividing the unimolecular constant by the hydrogen ion activity and water concentration, they obtained a constant value. In subsequent work, Moran and Lewis [70] also determined the activity of sucrose and water but the activity-based rate constants were not independent of the initial concentration of sucrose. The authors further developed a more elaborate approach including the effect of viscosity on the reaction rate. Livingston and Bray [71] studied the catalytic decomposition of hydrogen peroxide in a bromine-bromide solution. Substituting ion concentrations with activities (products of ion concentration and activity coefﬁcient) in the rate equation r ¼ kcH2 O2 cHþ cBr , they found a concentration-independent rate constant in most experiments, in contrast to the original rate equation. Later, Livingston reported [72] that the activity-based rate equation is valid only in solutions with an ionic strength less than unity. Scatchard [73, 74] carefully analyzed the issue arising from the sucrose inversion where discrepancies described in the above paragraphs, between theoretical and experimental proportionality of reaction rate and concentrawww.scilet.com

32

Miloslav Pekarˇ

tions, were found. He starts from the usual kinetic theory of the reaction in a perfect gas: the reaction rate is proportional to the concentration of each reacting species raised to the power which represents the number of molecules of that species which enter the reaction according to the stoichiometric equation. This gives the mass-action law rate equation, which, by analogy, is applied also in solutions. As thermodynamics formulates equilibrium constant in activities, and the ratio of the forward and reverse reaction rates must give the same equilibrium constant, it is much more logical, Scatchard argues [73], to express the reaction rate in activities. However, Scatchard is well aware of the dimensionality hitch, see Part 2. If the rate of sucrose inversion is accordingly formulated as: r ¼ kaaw abh acs

ð4:1Þ

where aw is the activity of water, ah the activity of hydrogen ion, and as the activity of sucrose (in Scatchard’s terminology, k is only a proportionality factor), Scatchard asks in what units should it be expressed, having to be measured by the number of molecules which react in unit time. The answer is not straightforward. First, the concept of a semi-ideal solution is introduced, which principal characteristic is that the activity of each component is proportional to the molar fraction of that component which actually exists in the solution. In fact, this means that thermodynamic environment is independent of concentration. Second, it is proposed to use the volume containing one mole as the ‘‘reaction volume’’ characteristic, which is the analogy of replacing ideal gas volume by the free volume to correct for non-idealities. Scatchard then concludes that the rate (r) is measured by the number of molecules transformed in unit time in the volume which contains one mole of total substance, i.e. r ¼ ðdxydtÞyC, where x is the transformed number in a litre and C is the total number of moles per litre. As an example, in the case of measuring the rate of sucrose inversion by its disappearance and considering that (reaction order) c ¼ 1, Eq. (4.1) is modiﬁed to: r ¼ ðdxydtÞyC ¼ kaaw abh ðC0s xÞyC or dxydt ¼ kaaw abh ðC0s xÞ

ð4:2Þ

(C0s is initial sucrose concentration). www.scilet.com

Thermodynamics and mass-action chemical kinetics

33

Although both Scatchard’s suppositions are rather operational and apparently formal, they are much better than simple replacement of (dimensional) concentrations with (non-dimensional) activities. The total concentration C has disappeared from Eq. (4.2) simply because only one of the three activities was substituted for the semi-dilute solution approximation. Had other activities also been replaced, C would be present. However, this was not important for Scatchard’s treatment as he could use measured activities of water and hydrogen ion. Just detailed considerations of water activity changes in sucrose solution enabled Scatchard to arrive ﬁnally to a k value independent of sucrose concentration [73]. Regardless of several assumptions, his work remains a representative example of a careful (practical) approach to activity-based kinetics. A different point of view was presented by Bro¨nsted [75] whose work has been here already mentioned several times. Bro¨nsted states that there exist many anomalies for ionic reactions in solutions in comparison to van’t Hoff’s kinetic law. He did not explicitly explain the anomalies nor give van’t Hoff’s law or any reference to it. Regarding van’t Hoff’s approach, from his original work [76] it is evident that his approach to kinetics is based on the work of Guldberg and Waage. van’t Hoff considers chemical equilibrium as the ﬁnal point of a chemical reaction described by the traditional thermodynamic equilibrium constant: Y Y K¼ ci i cj j ð4:3Þ products

reactants

from which he formulates the equilibrium condition: Y Y cj j ¼ ci i K reactants

ð4:4Þ

products

and on its basis he claims that the reaction rate should be proportional to the appropriate difference: r¼k

Y

cj j K reactants

Y

!

ci i products

ð4:5Þ

Bro¨nsted writes [75] that he is inspired by the ‘‘thermodynamic mass-action law’’ in which equilibrium activities appear instead of concentrations. By this law, the equilibrium constant expression (4.3) with activities should be understood. www.scilet.com

34

Miloslav Pekarˇ

Therefore, also in kinetics, activities should replace concentrations. Bro¨nsted is less cautious than Scatchard but he is far from making only this simple substitution. He, in fact, recalls Marcellin’s ideas on the so-called critical or activated complex, which is some highly unstable intermediate assembled from reactants, which further decomposes to the products (or back to the reactants). It is a predecessor of the later transition state and is also referred to in pioneering work on transition state theory [77]. Bro¨nsted suggests that in the concentrationbased mass-action rate equations, corrections through the activity coefﬁcients not only of the reactants but also of the activated complex should be made. For instance, the rate equation r ¼ kcA cB

ð4:6Þ

should be replaced by the equation r ¼ kcA cB ð fA fB yfA ? B Þ

ð4:7Þ

where fi represents the activity coefﬁcient of, and A ? B denotes, the critical complex. Why should the rate be just inversely proportional to the activity coefﬁcient of the activated complex is explained by Bro¨nsted only by rather unclear physical reasoning, with no unambiguous proof being given. The inverse proportionality should make explicit, according to Bro¨nsted, that only those few reactant molecules possessing a sufﬁciently high activity to build up very unstable, i.e. a very ‘active’ activated complex. Thus, Bro¨nsted tried to formulate mathematically the decelerating effect of the necessity of existence of an activated complex with high ‘activity’. The two meanings of ‘activity’ are thus confused – that of high ‘reactivity’, which is rather vague, and that of the precisely-deﬁned thermodynamic quantity. The vagueness of Bro¨nsted’s reasoning prompted another Scandinavian, Bjerrum, who presented the whole matter more precisely two or three years later [78,79]. In fact, he made the same hypothesis as did formerly Arrhenius, and later Eyring and collaborators, in absolute reaction rate theory. Bjerrum supposed that Bro¨nsted’s activated complex is in equilibrium with the reactants, and that the reaction rate is directly proportional to its concentration. Expressing the activated complex concentration in terms of the thermodynamic equilibrium constant containing the products of concentration and activity coefﬁcient then resulted in a rate equation like Eq. (4.7). Bjerrum supported his argument with some ideas from kinetic-statistical theory. www.scilet.com

Thermodynamics and mass-action chemical kinetics

35

Using the same activity coefﬁcients for various ions with the same charge, i.e. coefﬁcients dependent only on the type of ion, Bro¨nsted further successfully applied his theory to many ionic reactions [75]. It is clear that Bro¨nsted’s treatment, exempliﬁed by Eq. (4.7), forms the basis of various non-ideal mass-action rate equations, e.g. (2.18), (3.22), (4.8), and forms the basis for treatment of the salt effect. Belton [80] applied activity-based kinetics in his study of the conversion of N-chloroacetanilide into p-chloroacetanilide by protons and chloride ions. He found little value in using activities, or, more precisely, the products of concentration and activity coefﬁcient both as a substitute in the normal massaction rate equation and in Bro¨nsted’s sense. Most activity-based approaches in modern kinetics stem from the reaction isotherm as explained in part 1. Thus, Haase [81], as stated in his paper abstract, gives a rigorous expression for the rate of a chemical reaction in a non-ideal system. In fact, he starts with an equation very similar to that discussed by Blum and Luus [27], see Eq. (2.18). The only difference is in the use of stoichiometric coefﬁcients (i ): ?

r ¼k l

m n Y Y / ai i k l ai i i¼1

ð4:8Þ

i¼mþ1

(a’s are activities) and considering only reactants or products in the ﬁrst or second term, respectively. Haase also refers to Bro¨nstedt’s work [75] as the origin of this equation. Haase requires that the general expression for the reaction rate must have a form which reduces to the classical rate expression for perfect gas mixtures and ideal dilute solutions and gives the correct formula for the equilibrium constant in any system. Using the ‘‘reaction isotherm-based’’ approach, described in part 1, he proves this to be valid for Eq. (4.8) and also derives the relationship between rate and reaction afﬁnity, see Eq. (2.6). Immediately after Haase’s paper, Hall’s contribution was published in the same journal [37] and a spirited discussion started between Haase and Hall. Hall [37] begins with the equation ?

/

r y r ¼ expðAyRT Þ

ð4:9Þ

and tries to show its validity for elementary reactions in non-ideal systems. To achieve this goal he uses traditional expressions for the dependence of chemical potential on concentration and the mass-action law in the usual, concentration www.scilet.com

36

Miloslav Pekarˇ

form. The main point in his development is the rather strange hypothesis that the reaction is frozen for all but a very small fraction of the molecules present. This supposition might be perhaps accepted as a model of a non-ideal system in which intermolecular interactions deﬁnitely may affect the (‘‘frozen’’) ability of molecules to react. This hypothesis, with several additional physical premises, and not rigorous mathematical proofs, enable one to relate reaction rates and chemical potentials of (all) molecules present, leading thus to Eq. (4.9). The idea underlying all Hall’s premises and models is that, at constant temperature and pressure, reaction rates depend only on molecular environments. The main motivation of his rather incautious approach is an effort to avoid transition state theory, which is less readily applied to non-ideal systems. However, it is also not clear what is the advantage of Hall’s approach over the simple reaction isotherm-based derivation, except that he uses concentrations in the rate equation. To relate concentration-based kinetics with activity-based thermodynamics of non-ideal systems, he ﬁnally uses concentrations in expressions for chemical potential so the whole procedure loses its non-ideality status. In response to Haase’s paper [81], Hall claims [82] that Haase’s arguments lack rigour. Hall shows that Eq. (2.6) or (4.9) is not a logical consequence solely of Eq. (4.8) but may also be derived from its modiﬁed forms. Thus, Hall merely questions Haase’s derivation and does not add anything new to the kinetic-thermodynamic relationships. Haase rebuts [83] this criticism and shows by physical reasoning that Hall’s modiﬁcations reduce to Eq. (4.8), anyway. The following paper by Haase [84] generalizes his approach to any number of reactions. Hall responds to this several years later [85] and criticizes ﬁrst of all Haase’s reasoning in reference [83]. As well as this reasoning, the criticism is based upon physical argument and not mathematical proofs. In his ﬁnal response Haase published a mathematical proof that Hall’s more general form of Eq. (4.8), viz. m ?? Y

r ¼k l

//

ai i k l

i¼1

n Y

ai i

ð4:10Þ

i¼mþ1 ?

/

is superﬂuous because l ¼ l . Unfortunately, his proof lacks its claimed general validity as has been shown by Samohy´l (unpublished results) for the example of a gaseous reaction where it is not possible to choose the equilibrium pressure arbitrarily (one of the key points in Haase’s proof) when the temperature and composition are given, as can easily be checked by the interested reader. www.scilet.com

Thermodynamics and mass-action chemical kinetics

37

Haase notes, that l in Eq. (4.8) represents a function of temperature, pressure, and composition but gives no idea how this function can be obtained experimentally or theoretically to be useful in practice. Examples of practical applications of this equation are given by Baird [86]. In summary, Haase did not derive a ‘‘kinetic law’’ from thermodynamics. He was inspired by thermodynamics, used activities instead of concentrations, and the general form of the mass-action law, Eq. (4.8), directly. He did not tackle the question of whether there is also any other rate equation conforming to his postulates. Hall criticized the procedure, not this basis. Note that Hall derived Eq. (4.9) also using statistical thermodynamics [87]. Baird [86] claims that the generalized law of mass-action (4.8) is consistent with transition state theory. He considers the example of the simple reaction 1 ½1 þ 2 ½2 ?½6¼ ?3 ½3 þ 4 ½4

ð4:11Þ

In transition state theory, the reactants are considered to be in equilibrium with the transition state ([6¼]). The true thermodynamic equilibrium constant is then given by ?

K¼ a6¼ yða11 a22 Þ

ð4:12Þ

The reaction rate is proportional to the concentration of transition state, ? ? r ¼ c6¼ . Expressing activity as the product of activity coefﬁcient (g) and relative concentration, i.e. the ratio of the actual and the standard concentration (co ), the reaction rate in the forward direction is as follows: ?

??

?

r ¼ K c a12 yg6¼ : k a11 a22 yg6¼

ð4:13Þ

By the principle of microscopic reversibility, the reaction must proceed in the reverse direction via the same transition state [86]. Therefore the products are also in equilibrium with the same transition state: /

K¼ a6¼ yða33 a44 Þ

ð4:14Þ

and by analogy: /

//

/

r ¼ K c a33 a44 yg6¼ : k a33 a44 yg6¼ www.scilet.com

ð4:15Þ

38

Miloslav Pekarˇ

By subtracting the forward and reverse reaction rates, Eq. (4.8) is obtained with l ¼ 1yg6¼ . However, from the supposed equilibria, it also follows that the reactants are in equilibrium with the products:

?

/

a33 a44 yða11 a22 Þ ¼K y K¼ ðequilibriumÞ constant

ð4:16Þ

The entire analysis could thus be valid only for equilibrium where the overall rate is zero! Introducing Eq. (4.16) into the generalized rate equation (4.8), we obtain:

?

/?

/

r ¼ a11 a22 ð k k K y KÞyg= :ka11 a22 yg=

ð4:17Þ ?

/?

/

This generally gives non-zero equilibrium rate unless k ¼ k K y K, which leads to ? / ¼ . Otherwise, Eq. (4.17) would give the very strange result that the overall rate of a reversible reaction is independent of the concentrations of products, i.e. of the reverse direction. Thus, transition state theory does not prove in this way the generalized mass-action law (4.8). Obstacles could be overcome perhaps by considering different transition states [88] in both directions with concentrations given by: ?

/

c ? ¼K c a11 a22 yg ? ; =

c / ¼K c a33 a44 yg /

=

=

=

ð4:18Þ

The ﬁnal result is: ?

/

r ¼ k a11 a22 yg ? k a33 a44 yg / =

ð4:19Þ

=

which is, in fact, Hall’s general mass-action law (4.10). The same result can be obtained considering different activity coefﬁcients, i.e. different activities of a common transition state in the forward and reverse directions. Both different transition states and different activities sound rather strange and illustrate the problems which are encountered when applying transition state theory to reactions occurring simultaneously in both directions out of equilibrium. Considering different transition states in different directions of the same reaction may violate microscopic reversibility. It might be therefore supplemented by the hypothesis that the transition states are different in non-equilibrium states only, and become identical when equilibrium is attained. www.scilet.com

Thermodynamics and mass-action chemical kinetics

39

Activities were introduced into the mass-action kinetic equation also by Ola´h [89] using his ‘‘thermokinetic’’ theory. This theory is analyzed in Part 5 below. Now it is sufﬁcient to state that it is in fact an ordinary afﬁnity-based approach. As afﬁnities are directly related to chemical potentials, see (2.5)1 and cf. Ola´h’s Eq. (5.88), which in turn are, by deﬁnition, related to activities, nothing fundamentally new is added. Eckert and Boudart [90] successfully described gas phase kinetics using a fugacities-based mass-action rate equation of the Bro¨nstedt type in contrast to the traditional concentration-based treatment. Mason [91], however, demonstrated using the same data set that the activity-based rate coefﬁcient shows a much stronger pressure dependence than the concentration-based coefﬁcient. Activity-based kinetic equations have also started to become popular in enzyme kinetics. Van Tol et al. [92] probably pioneered this approach to circumvent problems with solvent effects on reaction rates, substrate – solvent interactions in nonaqueous enzymology, or with the substrate concentration in biphasic systems. Their study of lipase-catalyzed ester hydrolysis in biphasic systems with various solvents did not give fully satisfactory results. Experimental data obtained in isooctane could be well ﬁtted to the activity-based equation whereas for the other solvents the ﬁt was poor. The latter was attributed to unrealistic premises employed in modelling (equal binding of the solvents to the active site, no solvent effect on the mechanism, equal activity coefﬁcients of the enzyme species in the catalytic cycle, and others). Activity coefﬁcients were calculated from UNIFAC or determined from equilibrium solubility or partitioning. From subsequent papers, let us mention only that by Sandoval et al. [93] who used activities in the traditional equations of enzyme kinetics, i.e. in the initial rate expression originally derived from the mass-action law. The authors simply replaced concentrations with activities and used UNIFAC group contribution methodology to compute the activity coefﬁcients. From experiments made in one solvent, kinetic parameters, free of solvent effect, were determined. They were used to predict the reaction rate in other solvents using, of course, the activity coefﬁcient computed for the respective solvent. From a comparison of predictions with measured data, it seems that this approach works in most systems. Van Tol et al. [94] summarize that when organic solvents do not interfere with the binding process nor with the catalytic mechanism of enzyme-catalyzed reactions, the contribution of substrate-solvent interactions to enzyme kinetics www.scilet.com

40

Miloslav Pekarˇ

can be accounted for by just replacing substrate concentrations in the kinetic equations by thermodynamic activities. Only the afﬁnity parameters (substrate afﬁnity, speciﬁcity constant) are affected by this transformation and corrected parameters and the maximal rate should be equal for all media. Experimental data show, however, that although the kinetic performance of each enzyme in the solvents became much more similar after correction, differences still remain. They are caused mainly by incomplete shielding of the bound substrate from the solvent, the non-constancy of the activity coefﬁcient of the enzyme species in the catalytic cycle, and by solvent competition with substrate for binding to the active site. Published data on activity-based mass-action kinetics generally give no decisive conclusion. The idea, already formulated in Hougen-Watson’s classic monograph [95], that mass-action law should be generally formulated in activities and not in concentrations does not have general validity. It seems that ion (salt) effects mostly cannot be included by simply using activities in place of concentrations whereas solvent effects usually can be. In any case, introducing activity coefﬁcients into the mass-action rate equation is identical to considering a concentration-dependent rate ‘‘constant’’.

5. CLASSICAL (LINEAR) IRREVERSIBLE THERMODYNAMICS 5.1 Fundamentals Haase’s book [96] gives probably the most comprehensive explanation of the basis of the classical or linear irreversible thermodynamic (CIT) approach to chemical kinetics, compared to other books in this ﬁeld. Haase, in the part of his book devoted to homogeneous systems, presents an attempt to combine well-known kinetic ‘‘laws’’ with the phenomenological or ﬂux-force laws. This is a typical effort of CIT. As the driving ‘‘force’’ for chemical reaction, or chemically reacting systems in general, the afﬁnity (A) is selected. The phenomenological law for the reaction rate (ri ), the ‘‘ﬂux’’, may be written, close to equilibrium, in linear form ri ¼

R X aij Aj ;

i ¼ 1; 2; . . . ; R

ð5:1Þ

j¼1

where R is the total number of independent reactions and aij are the phenomenological coefﬁcients. The law of mass-action is used in the form www.scilet.com

Thermodynamics and mass-action chemical kinetics

! Y Y Y mi 0 ni ki ri ¼ ki cm k i cn ¼ oi 1 li ck ; m

n

41

ð5:2Þ

k

where o i ¼ ki

Y

cmmi ; li ¼ ki 0 yki

ð5:3Þ

m

and m goes through all reactants, n through all products and k through both these kinds of constituents; ’s are the stoichiometric coefﬁcients and c’s are the concentrations. Using the deﬁnition of afﬁnity in terms of chemical potential and the classical relationship between chemical potential and concentration, the following equation is obtained: ri ¼ oi ½1 li Ki ðco Þi expðAi yRT Þ

ð5:4Þ

where Ki is the equilibrium constant, co the standard concentration, and P i ¼ k ki . It can be shown that the multiplicative factor at the exponential is equal to one. Close to equilibrium (jAi yRT j 5 1; index ‘‘eq’’), the exponential may be expanded in a series retaining only the ﬁrst member. The linear phenomenological relation is ﬁnally obtained: ri ¼ ðoeq i yRT ÞAi

ð5:5Þ

(note that oi , once more, was not expanded in contrast to the afﬁnity), which is a special case of Eq. (5.1). Haase is very careful to identify kinetic and thermodynamic equilibrium constants and also to ignore standard concentrations when substituting concentrations for activities, cf. Eq. (5.4). In the ﬂow-through systems, where the spatial distribution of variables must also be taken into account, the following expression for the local entropy production (u) is derived: C:T u ¼ JQ XQ þ

X k

J;k Xk þ

X r

or Ar þ

3 X 3 X

Pij Xij 0

ð5:6Þ

i¼1 j¼1

where XQ ¼ ð1yT ÞgradT

ð5:7Þ

Xk ¼ Kk ðgradmk ÞT

ð5:8Þ www.scilet.com

42

Miloslav Pekarˇ

Xij ð1y2Þðqi yqzj þ qj yqzi Þ;

i; j ¼ 1; 2; 3

ð5:9Þ

and JQ is the density of heat ﬂow, J;k are the densities of diffusion ﬂows, or is the rate and Ar the afﬁnity of the r-th reaction, Pij ¼ Pji are the frictional forces, Kk the external molar force acting on the k-th component, mk its chemical potential and i is the component of the barycentric rate vector v, z’s are spatial coordinates. The gradient in Eq. (5.8) is taken at constant temperature. The so-called dissipative function C is interpreted with the aid of the concepts of ﬂuxes and forces, viz. ‘‘phenomenological relationships’’ among them are sought. To this end, forces are considered to be independent, and ﬂuxes dependent, variables. Their mutual functional connections are formulated with a rather intuitive use of linear isotropic function representation, here called Curie’s principle. First, the tension term is excluded from the considerations, giving nothing new to chemical kinetics, as the relevant phenomenological relation is again postulated to be: X or ¼ ars As

ð5:10Þ

s

where index s also refers to all reactions and a’s are the proportionality (phenomenological) coefﬁcients. Second, the tension term is taken into account but together with only the chemical rate term from the dissipative function (5.6) and not also with the other two, transport terms: C¼

X r

or Ar þ

3 X 3 X Pij Xij 0

ð5:11Þ

i¼1 j¼1

This is vaguely substantiated by Curie’s principle and should probably be understood as the fact that the vectorial linear isotropic function depends only on vectors whereas the tensorial function may depend also on scalars. Consequently, the reaction rate is written (better speaking, represented) as: X or ¼ ars As L r div v ð5:12Þ s

(L

r

is the proportionality, phenomenological, coefﬁcient) because div v is

considered to be the trace of tensor X given by Eq. (5.9). Eq. (5.12) is claimed to be a generalization of Eq. (5.10). It can be interpreted as a warning that the chemical reaction rate may be affected by viscous processes. The functional www.scilet.com

Thermodynamics and mass-action chemical kinetics

43

dependence, Eq. (5.12), of the former on the latter is more a matter of interpretation than of exact proof. Very recently, Cukrowski and Kolbus [97] published another paper utilising ﬂux and force. They found a new, ‘superior’ force, which enables one to use linear ﬂux-force over a wider range (i.e. farther from equilibrium) than usually expected. The new force is deﬁned as the difference between the reactive absolute activities of reactants and products. The absolute activity (of component i, li ) was introduced into thermodynamics by Fowler and Guggenheim [98] as the exponential of chemical potential: li ¼ expðmi yRT Þ

ð5:13Þ

The reactive absolute activity is deﬁned by Cukrowski and Kolbus using the ‘‘reactive chemical potential’’ mre i , which they deﬁne as the difference between the chemical potential and its equilibrium (‘‘eq’’) value: eq mre i ¼ mi mi

ð5:14Þ

The reactive absolute activity of component i is then deﬁned as follows: re lre i ¼ expðmi yRT Þ

ð5:15Þ

and the reactive absolute activity of reactants (sufﬁx R) or products (sufﬁx P) as products: lre R ¼

Y reactants

lre i ;

lre P ¼

Y

lre j

ð5:16Þ

products

re The new force is then X ¼ lre R lP . Cukrowski and Kolbus [97] then present

several examples of model reactions, which were analyzed using the new ‘force’. Their approach is in fact another exercise in combining the traditional massaction law and the traditional expression for chemical potential as a function of concentration similar to those reviewed in Parts 2.1, 3.1, or 3.2. What is important and new in this approach is the stress on equilibrium and its use as a referential state. It will be seen later that the equilibrium also has similar importance in rational thermodynamic theories. A similar approach, but with no such stress on equilibrium, was presented by Parmon [99,100].

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Miloslav Pekarˇ

5.2 Tackling mass-action non-linearity and Onsager reciprocity Anderson and Boyd [101] extended the ﬂux-force approach to the nonlinear area and focused on the so-called Onsager’s reciprocity relations, another popular theme of CIT. They suppose that the reaction rate (ﬂux) J depends on the appropriate force X in the following way: J ¼ L X þ MX2

ð5:17Þ

where L and M are again proportionality, phenomenological, coefﬁcients. Further they take over the general rate equation: r ¼ kb

m Y

m Y 0 ðai Þa i

ðai Þai k 0 b

i¼1

ð5:18Þ

i¼1

where a’s are the activities and a’s are the reaction orders and not stoichiometric coefﬁcients. Combination of this equation and the condition of a vanishing rate in equilibrium, together with the deﬁnition of afﬁnity (A) by Eq. (2.5)1 and with common expression for the relationship between chemical potential and activity, gives the ﬁnal rate equation: "

# m Y ai r ¼ kb ða1 Þ 6½1 expðnAyRT Þ

ð5:19Þ

i¼1

where n ¼ ðai 0 ai Þyi

ð5:20Þ

Factors in Eq. (5.19) are expanded in Taylor series and the form of Eq. (5.17) is thus obtained as follows: "

# m Y eq ai r ¼ kb ðai Þ 6½nðAyRT Þ þ nðd1 ny2ÞðAyRT Þ2 þ

ð5:21Þ

i¼1

where "

# m Y eq ai d1 ¼ q ðai yai Þ yqðAyRT ÞA¼0 i¼1

and index ‘‘eq’’ refers to equilibrium. www.scilet.com

ð5:22Þ

Thermodynamics and mass-action chemical kinetics

45

The authors then assume that Onsager’s reciprocity relations hold only in the linear range. However: ‘‘Even in the linear regime...the coefﬁcients (in Eq. (5.21)) are functions of the equilibrium state of the system. This implies that phenomenological coefﬁcients, measured in one reaction mixture, cannot be applied directly to another.’’ The authors therefore conclude that the language of nonlinear thermodynamics is not suitable for chemical kinetics. The authors agree that there is no ‘‘thermodynamic’’ substitute for the established, mostly empirical, rate equations and Eq. (5.21) serves only as a connection between this tradition and the CIT approach. It could be that the phenomenological coefﬁcients measured and general values obtained, perhaps Eq. (5.17) might substitute the traditional rate equation, with the phenomenological coefﬁcients playing the role of rate constants. Bataille et al. [102] also dealt with Onsager’s relations. They state that linear ﬂux-force relations are not adequate for reaction kinetics and try to discover an extension of Onsager’s reciprocal relations into the non-linear domain. They start with the generalized rate equation for reaction a (Ja ) in a rather unusual form but close to Eq. (5.18): Ja ¼ k a

m m Y Y þ ð fi Þi kþ ð fi Þi a i¼1

ð5:23Þ

i¼1

þ where fi denote fugacities and i , i are the (positive) stoichiometric coefﬁcients

for the backward and forward directions, respectively. Three things are then to be checked: (a)

the J ’s can be expressed in terms of the A’s (afﬁnities) and the thermostatic state variables,

(b)

the entropy production is non-negative and vanishes only when all of the A’s vanish,

(c)

approximation of the J ’s by linear functions of the A’s over a sufﬁciently small neighborhood of the equilibrium values Aa gives Ja &

R X L

ab Ab

b¼1

with satisfaction of the Onsager reciprocity relations L

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ab

¼L

ba .

46

Miloslav Pekarˇ

Whereas the last two conditions are quite simply satisﬁed, the ﬁrst one requires more elaborate treatment. Using afﬁnity deﬁnition (2.5)1 and introdu cing forward and backward afﬁnities such that Aa ¼ Aþ a Aa , Eq. (5.23) is modiﬁed:

Ja ¼ ka expðA a yRT Þ½1 expðAa yRT Þ

ð5:24Þ

From Eq. (5.24) it is clear that the ﬁrst condition transforms to the question whether the backward afﬁnity can be expressed in terms of the A’s and the thermostatic state variables. This leads to the standard task of linear algebra, viz. ﬁnding (general) solutions of the system of equations m X ai mi ¼ Aa ;

a ¼ 1; . . . ; R

ð5:25Þ

i¼1

As there are more unknowns (mi ) than given (Aa ) quantities, the number of solutions is inﬁnite. Anyway, the ‘‘unknowns’’ may be expressed from the system with the aid of afﬁnities or other unknowns, which are themselves functions of state variables. Consequently, condition (a) is conﬁrmed. After introducing the general solution into the rate equation (5.24), it is immediately found that Onsager’s relations ðqJa yqAb Þp;T ;bl ¼ ðqJb yqAa Þp;T ;bl

ð5:26Þ

(p is pressure and bl denotes set of parameters in the general solution of the system of algebraic equations) are not generally valid in the non-linear domain. The authors point out that Edelen’s generalized dissipation potential [103] is still applicable in this domain and its symmetry relations hold as well. 5.3 Hungarian contribution I – Lengyel Several papers have appeared from the Hungarian school based on Gyarmati’s ‘‘integral principle of thermodynamics.’’ Gyarmati’s approach is, in principle, a certain reformulation of the irreversible thermodynamic approach into the terms of variational principles. Its application to chemically reacting systems in general is described in Sa´ndor’s papers [104,105]. The ﬁrst contribution dealing with kinetics in more detail is probably the paper by Lengyel and Gyarmati [106,107]. It is interesting to cite the authors’ motivation: ‘‘This consistency (between kinetics and thermodynamics) is both theoretically and practically important also from the aspect of reaction kinetics. www.scilet.com

Thermodynamics and mass-action chemical kinetics

47

If we can show this consistency, then the whole phenomenological theory of chemical reactions will become a special, but organic, branch of non-equilibrium thermodynamics in the same way as the theory of chemical equilibria has become a special chapter of thermostatics as a result of Gibbs’ work. From the practical point of view the description in non-equilibrium thermodynamics not only offers an alternative description of chemical reactions but can complete the Guldberg – Waage theory. We think that reaction kinetics describe only the concentrations as a function of time but the reaction heats involved in the reaction, i.e. energetics, are not included in the description. In non-equilibrium thermodynamics this inclusion is quite natural; moreover, if the equivalence of both theories can be assumed, then stationary states, the stability and evolution of open kinetic systems, may become objects of exact studies...To illustrate...let us assume that we could show the consistency of nonlinear thermodynamics and the nonlinear theory of chemical kinetics. In this case, instead of the Guldberg – Waage form of the kinetic equations, the consistent differential equations of the nonlinear thermodynamic theory have to be solved.’’ Although the authors present basic thermodynamic equations, including Gyarmati’s principle, in the introduction, the procedure adopted is standard, close to that given in the papers described above. The authors write the Guldberg – Waage law, in this case with molar fractions. Further, the expression of chemical potential in term of molar fraction in ideal systems and the traditional deﬁnition of afﬁnity are used and combined with the Guldberg – Waage law to arrive at the general equation Jr ¼ Jer jr ðA1 ; . . . ; AR Þ

ð5:27Þ

for the rate (Jr ) of reaction r (which are R in total), called the non-linear phenomenological equation. The novelty in this general equation is the particular representation in a)

the components’ deviations from equilibrium

Dni ¼ ni nei ;

i ¼ 1; . . . ; m

ð5:28Þ

(n’s are mole numbers and ‘‘e’’ refers to equilibrium) which can be introduced into the Guldberg – Waage law through rewriting it into the form:

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48

Miloslav Pekarˇ

Jr ¼ k 0r

Y

ðnei 0 ynÞi 0 ;r

Y

i0

ðni 0 ynei 0 Þi 0 ;r k00r

i0

Y Y ðnei00 ynÞi00 ;r ðni00 ynei00 Þi00 ;r i00

ð5:29Þ

i00

giving " Jr ¼

Jer

Y Y ð1 þ Dni 0 ynei 0 Þi 0 ;r ð1 þ Dni00 ynei00 Þi00 ;r i0

# ð5:20Þ

i00

where the prime denotes the forward, and double prime the backward, reaction direction and ’s are the stoichiometric coefﬁcients, n the total mole number and Y Y Jer ¼ k 0r ðnei 0 ynÞi 0 ;r ¼ k00r ðnei00 ynÞi00 ;r ð5:31Þ i0

b)

i00

the reaction’s deviations from equilibrium

Dni ¼

R X i;r Dxr

ð5:32Þ

r¼1

(xr is extent of reaction r and D has the same meaning as in Eq. (5.28)) which, after introducing into Eq. (5.30), give: " # R R Y X Y X e e i 0 ;r e i00 ;r Jr ¼ Jr ð1 þ i 0 ;r Dxr yni 0 Þ ð1 þ i00 ;r Dxr yni00 Þ i0

c)

i00

r¼1

ð5:33Þ

r¼1

and introducing the ‘‘absolute afﬁnities’’ "

Lr ¼

Y i0

# " # R R . Y X X e i 0 ;r e i00 ;r ð1 þ i 0 ;r Dxr yni 0 Þ ð1 þ i00 ;r Dxr yni00 Þ i00

r¼1

ð5:34Þ

r¼1

which are related to common afﬁnities (Ar ) by Lr ¼ expðAr yRT Þ. Eq. (5.34) can be considered to be a system of algebraic equations which can be solved for the deviations of the extent of reaction from equilibrium: Dxr ¼ fr ðL1 ; . . . ; LR Þ or

Dxr ¼ jr ðA1 ; . . . ; AR Þ

ð5:35Þ

Substituting (5.35) into (5.33), the general Eq. (5.27) is obtained. Using the absolute afﬁnities, ‘‘general non-linear constitutive equations between reaction rates and afﬁnities’’ are derived in the form Jr ¼ Jer ðLr 1Þu00r ðA1 ; . . . ; AR Þ

ð5:36Þ

where www.scilet.com

Thermodynamics and mass-action chemical kinetics

u00r

¼

Y

R X 1þ i00 ;r Dxr ynei00

i00

49

!i00 ;r ð5:37Þ

r¼1

and is a function of all afﬁnities as indicated in Eq. (5.36). Onsager’s relations in the linear approximation close to equilibrium are then proved for both stoichiometrically independent and dependent reaction systems. The authors then present several examples in which they also test the ‘‘Rysselberghe generalized reciprocity relations’’. These were postulated by Rysselberghe [108,109] to be valid for the non-linear equations Jk ¼ L kk Ak þ L kl Al þ L kkk A2k þ L kkl Ak Al þ L kll A2l

ð5:38Þ

in the form qJk yqAl ¼ qJl yqAk

ð5:39Þ

originally set forth by Pe´ne´loux [110,111]. As this attempt was unsuccessful, the authors conclude that Rysselberghe’s relations are inconsistent with classical chemical kinetics. The authors speculate that the cause lies in the improper choice of thermodynamic forces and some new parameter should be sought instead of afﬁnities. They also stress that, for example, Gyarmati never identiﬁed the thermodynamic forces in chemical kinetics with afﬁnities. As he also never gave any speciﬁcation of these general forces to kinetics, ﬁnding the right forces remains an unresolved task. Summarizing, classical kinetic or thermodynamic quantities and relations are combined and subjected to the interpretation within the ﬂux-force framework. As this works in the linear domain only, new, superior ‘forces’ should be found, without asking whether the ﬂux-force approach is correct, necessary or of any practical use, at all. The proclaimed practical aim is not demonstrated even in the linear domain where the ﬂux-force interpretation is satisfactory. The practical value of Eq. (5.34) is questionable especially with regard to the fact that each extent of reaction contains only those moles which have reacted just in that reaction. The ‘right’ forces are claimed to be found in subsequent papers by Lengyel [112,113], the ﬁrst one being, in fact, a shortened version of the second. Moreover, it is stated that the mass-action law was deduced from Gyarmati’s governing principle of dissipative processes. This principle reformulates the results of CIT in terms of variational principle. Locally, it asserts that the density (o) of the so-called Onsager – Machlup function www.scilet.com

50

Miloslav Pekarˇ

o¼scj

ð5:40Þ

keeps its extremum value at any point in the system. In other words, its variation (do) is always and everywhere vanishing: do ¼ ds dc dj ¼ 0

ð5:41Þ

In the equations, s is the density of the local entropy production rate, c and j are the so-called dissipation functions (in the forms of densities) or potentials. Function c is said to depend on all (independent) forces whereas j depends on all ﬂuxes. Consequently, the former is sometimes called the force potential, the latter the ﬂux potential. The dissipation functions are selected by Lengyel to be " ! !# Q Q S X X X 00 0 c ¼ 2C ¼ 2 Rlt exp gt X yR þ exp gt X yR t¼1

¼1

ð5:42Þ

¼1

and j ¼ 2F ¼ 2

S h? X ? / / ? / i R J t lnð J t ylt Þ þ J t lnðJ t ylt Þ ð J t þ J t Þ

ð5:43Þ

t¼1

In these equations S is the total number of reactions and Q the number of independent reactions. Parameter lt comes from nothing more than the massaction law written in the form ?

Jt ¼ k t

Y N0 / Y 00 ðci Þ i;t k t ðci ÞNi;t i

ð5:44Þ

i

(c’s are concentrations) and transformed to the form ? / Jt ¼ J t J t ¼ lt expðXt 0 yRÞ expðX00t yRÞ ;

t ¼ 1; . . . ; S

ð5:45Þ

The prime or left-to-right arrow, and double prime or right-to-left arrow, represent forward and reverse reaction directions respectively, N’s are the orders (not stoichiometric coefﬁcients). Coefﬁcient gt results from the relations between the dependent and independent (marked by an asterisk) reactions: J* ¼

S X gt Jt ;

¼ 1; . . . ; Q

ð5:46Þ

t¼1

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Thermodynamics and mass-action chemical kinetics

51

Quantities X’s are related to afﬁnities: A00t ¼ TX00t ;

At 0 ¼ TXt 0 ;

t ¼ 1; . . . ; S

ð5:47Þ

(all stoichiometric coefﬁcients are considered positive). Introducing variations of the dissipation functions (5.42) and (5.43), together with the variation of s (which will be discussed later) into Eq. (5.41), the following equation is obtained: " Q X

J* 2

S X

u

u¼1

þ

" Q X

gtu lt exp

t¼1

J* þ 2

Q X

!# 0

gt X yR

¼1 S X

u

gtu lt exp

t¼1

u¼1

Q X

dXu 0 !#

gt X00 yR

dX00u

¼1

S h S h i ? i / X X ? / þ Xt 2R lnð J t ylt Þ d J t þ Xt 2R lnðJ t ylt Þ dJ t ¼ 0 t¼1

ð5:48Þ

t¼1

As all the varied variables are mutually independent, it follows that the expressions in brackets vanish. Combining the two equal-to-zero equations for Ju* , relation (5.49) follows: J*u ¼

S X t¼1

" gtu lt exp

Q X ¼1

! 0

gt X yR exp

Q X

!# gt X00 yR

;

u ¼ 1; . . . ; Q

¼1

ð5:49Þ which is said to be nothing more than the Guldberg – Waage mass-action law. However, the whole deduction suffers from several deﬁciencies. The forms of dissipation functions (5.42) and (5.43) are not proved but stated. They are given in such a way to obtain immediately (5.49) after introducing them into (5.41). This is no deduction but a tautology. Of course, Eqs (5.42) and (5.43) do have physical motivation. It stems from the well-known Eq. (5.44), which was transformed to Eq. (5.45) using the classical relation for the chemical potential in ideal systems. It should be pointed out also that the postulate of expressing the rate as a difference between the forward and backward rate was introduced. Only stoichiometrically independent reactions are considered. They are selected from the whole reaction set by means of relations (5.42) and their afﬁnities are used to express forward and backward rates of independent reactions, e.g. (cf. also Eq. (5.45)): www.scilet.com

52

Miloslav Pekarˇ

?

J u* ¼

S X

?

gtu J t ¼

t¼1

S X

gtu lt exp

t¼1

Q X

! gt X 0 yR

;

u ¼ 1; . . . ; Q

ð5:50Þ

¼1

Now, the inspiration for the ﬂux potential is clear. The origin of the force potential is more unclear. From the inversion of expressions for both forward and backward rates in Eq. (5.45), the author ﬁnds relations: ?

Xt 0 ¼ R lnð J t ylt Þ;

/

X00t ¼ R lnðJ t ylt Þ;

t ¼ 1; . . . ; S

ð5:51Þ

which, as he states, satisfy the reciprocal relations. Relations of Eq. (5.51) appear in ﬂux potential (5.43). In this case no attempt is made to use only independent reactions or their afﬁnities and no explanation is given as to why the minus sign in the second expression in (5.51) is not retained in Eq. (5.43). The tautology is even deeper. The author starts from the mass-action law, either in the form (5.44) or (5.45), to discover it again after several lines of manipulating with it. To recover the desired result, multiplication by 2 is necessary not only in (5.42) and (5.43) but also in the entropy production density. This is achieved in a particularly intriguing manner. The author states that the local entropy density (s) is the function of some set of ‘‘independent extensive state’’ variables x1 ; . . . ; xi ; . . . ; xf

ð5:52Þ

The partial time derivative of this function is given by qsyqt ¼

f X

Gi qxi yqt

ð5:53Þ

i¼1

and may be used in the general entropy balance equation of CIT: qsyqt þ divJs ¼ ss

ð5:54Þ

in which Js is the entropy ﬂux density and ss its source density. Similar balance equations are supposed to be valid also for the independent variables (5.52). Combining all balances, the following expression for the entropy production rate is found: ss ¼

f X i¼1

Ji grad Gi þ

f X

Gi si

ð5:55Þ

i¼1

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Thermodynamics and mass-action chemical kinetics

53

where Ji are current densities and si source densities of variables (5.52) in their balances like (5.54). The author claims that the density of entropy production can be written in the form f X

ss ¼

Ji grad Gi þ

S X

i¼1

?

/

ðXt 0 þ X00t Þð J t J t Þ

ð5:56Þ

t¼1

because of Gi ¼ mi yT A 0t ¼

X

ð5:57Þ

0i;t mi ;

A00t ¼

X

i

si ¼

00i;t mi

ð5:58Þ

i

S X

ð00i;t 0i;t ÞJt

ð5:59Þ

t¼1

where m’s are chemical potentials and ’s (positive) stoichiometric coefﬁcients. Note that the postulate xi :ci was introduced. The part of the entropy production rate due to the chemical reactions (the second term in (5.56)) can be expressed using only the independent reactions: S X

?

/

Xt ð J t J t Þ ¼

t¼1

Q X

Ju* ðXt 0 þ J00t Þ

ð5:60Þ

u¼1

In the variation condition (5.48), both versions from Eq. (5.60) are summed forming 2s! So, in fact, 2s is used in the Onsager – Machlup function (5.40) instead of s. Further, there is an obscurity with the independent variables. Initially, it is stated that entropy density is a function of variables (5.52), which are also subjected to the balance equations like (5.54). Later, they are identiﬁed with the volume concentrations, see above. However, it is then declared that there are two complete sets of independent variables, viz. X1 0 ; . . . Xu 0 ; . . . ; XQ 0 ;

X001 ; . . . X00u ; . . . ; X00Q

ð5:61Þ

and ?

?

?

J 1; . . . ; J t; . . . ; J S;

/

/

/

J 1; . . . ; J t; . . . ; J S www.scilet.com

ð5:62Þ

54

Miloslav Pekarˇ

which can be used alternatively to express the part of entropy production caused by the chemical reactions, see (5.60). No explanation for this transformation is given. It should be probably understood as a sudden return to the ﬂux-force area. These new sets are used in the variation condition, however, not alternatively, but all-at-once with no explanation, again. The author also claims that whereas overall reaction rates of all elementary reactions may be dependent, this is not true for the backward and forward rates, which are completely independent. It was clearly demonstrated by Bowen [34] that the (linear) dependence of reaction rates is the result of the permanence of atoms and not of the way the rate is expressed. Consider two reactions with one common reactant. Kinetic experience tells that the rate of one reaction may affect the forward rate of the other through its inﬂuence on the concentration of the common reactant. This can be supported also by formal argument. Let us suppose that among S reactions, the ﬁrst R of them are independent. This means that rate of any other reaction may be obtained as a linear combination of the rates of the independent reactions: Jk ¼

R X

lki Ji ;

k4R

ð5:63Þ

i¼1

It is further supposed that every rate is given as the difference between the ?

/

forward and backward rates, Ji ¼ J i J i . Thus ?

/

J k Jk ¼

R X

?

/

lki ð J i J i Þ;

k4R

ð5:64Þ

i¼1

This equation can be modiﬁed as ?

/

J k Jk ¼

R X

?

lki J i

i¼1

R X

/

lki J i ;

k4R

ð5:65Þ

i¼1

Stating ?

Jk ¼

R X i¼1

?

/

lki J i ; J k ¼

R X

/

lki J i ;

k4R

ð5:66Þ

i¼1

it is seen that some forward or backward rates can be expressed as a linear combination of the other forward or backward rates, respectively. It follows that all forward or backward rates are not independent. www.scilet.com

Thermodynamics and mass-action chemical kinetics

55

The author considers as the most essential point of the work ‘‘the identiﬁcation of the thermodynamic forces with the collection of the independent forward and backward afﬁnities, instead of the full afﬁnities (divided by the temperature)’’. However, it is not further explained why these should be the true forces when it is the difference between the chemical potentials (which are closely related to afﬁnities) of reactants and products which drives the chemical reaction. No ideas as to how to measure reaction afﬁnities (or rates) separately in the forward and backward directions are given. Perhaps some electrochemical cell under special conditions can serve this purpose. Lengyel’s approach is given once more in his next paper [114]. It is a review on the relationships between chemical kinetics and thermodynamics, which were resolved, at last, by the author. The same procedure is presented as in ref. [113], only here the rates are expressed using the extent of reaction. Thus, the results should be applicable only to closed systems with no diffusion (or with the so-called self-balanced diffusion [35] only). To conclude, ignoring the ﬂaws described, the well-known kinetic law was introduced into a certain thermodynamic formalism to re-derive it in a rather different form.

5.4 Onsager far from equilibrium Shiner [115] tries to prove the Onsager reciprocity or symmetry relations for chemical kinetics not only in the vicinity of equilibrium but also far from it. In fact, he starts again from the Guldberg – Waage law in its general form with activities (a’s): f x_ r :_ni;r yðbi;r i;r Þ ¼ krf

Y i

f

ðai Þi;r kbr

Y

b

ðai Þi;r

ð5:67Þ

i

ðxr is the extent of the r-th reaction, dot means the time derivative and f and b the forward and backward directions, respectively, ’s are the positive stoichiometric coefﬁcients), introduces common relations between activity and chemical potential, between the equilibrium constant and ratio of the rate constants in the forward and backward directions, and uses the common deﬁnition of afﬁnity (Ar ) through chemical potentials, cf. Eq. (2.5)1. This results in the trivial relation for afﬁnity www.scilet.com

56

Miloslav Pekarˇ

" # " # . Y b f Y i;rf b Ar ¼ kB T ln kr ðai Þ kr ðai Þ i;r i i

ð5:68Þ

(kB is the Boltzmann constant) which is interpreted as ‘‘the driving force of a dissipative process ¼ product of the resistance and the ﬂow of the process’’: A r ¼ R r xr

ð5:69Þ

Using the resistances (Rr ), Shiner succeeds in proving symmetry relations in the following sense. Supposing that the thermodynamic force Xi and its conjugate ﬂow Ji are connected by the general linear relation Xi ¼

X Rij Jj

ð5:70Þ

j

the symmetry means Rij ¼ Rji

ð5:71Þ

or qXi yqJj ¼ qXj yqJi

ð5:72Þ

Shiner shows that (5.71) is valid also for stationary states far from equilibrium or, to put it better, ﬁnds from the kinetic and thermodynamic equations and relations used, the right form of X, R, J for (5.71) to be valid. Of course, in this case the condition cannot be reformulated into the differential form (5.72). Shiner therefore states that for stationary states far from equilibrium only algebraic symmetry (5.71), in contrast to the differential symmetry (5.72) found close to equilibrium, is valid. Differential symmetry is thus less stringent. The ﬁnal form of resistances (not reproduced here) contains rather complicated combinations of stoichiometric coefﬁcients and (‘‘Guldberg – Waage’’) expressions for forward and backward rates, resulting, in fact, from the material balance. The proof includes one essential point. It is supposed that chemical potentials of some species are controlled from some ‘‘external source’’ and are held at constant values equal to the values of the source (index ‘‘ex’’): mk ¼ mex k for some k

ð5:73Þ www.scilet.com

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This artiﬁcial-looking presumption should probably represent the continuous inand outﬂow in the stationary system. It is not proved nor is its reality discussed. It may refer to the equality of activities (or concentrations) inside the stationary system and in incoming and outcoming ﬂows, which would mean the nonreacting system! Anyway, the result should be considered to be valid only for those stationary systems, in which Eq. (5.73) is valid. Shiner’s analysis is another example of using the well-known kinetic equation in a certain thermodynamic formalism. The usefulness of such derived symmetry relations for practical kinetics is unclear. 5.5 Bro¨nsted re-discovered? An example of an approach based on very unclear justiﬁcation is Grigin’s paper [116]. His aim is to derive a Bro¨nsted-type relationship between rate and equilibrium constant from the generalized thermodynamic force for chemical reactions. The starting point is again the classical rate equation, and the reaction rate is considered to be also the thermodynamic ﬂow. Therefore, another, now thermodynamic, equation should be found for it. It is stated that this equation will be in the form of product I ¼ Rf of the two functions. The ﬁrst one (R) depends on the concentrations, the second one ( f ) on the difference between the chemical potentials of reactants and products (Dm). The author very vaguely and inadequately explains (not proves) why this is possible even when the chemical potential is a function of concentrations. It does not even prevent him from introducing the dependence of the difference between chemical potentials on the concentrations into the supposed form of the second function: f ðDmÞ ¼ expðb1 DmyT Þ expðb2 DmyT Þ

ð5:74Þ

(b1 and b2 are proportionality coefﬁcients); this equation is another premise. The resulting power-law equation immediately leads to the form of the ﬁrst function (for the model reaction A þ B ¼ C): a

R ¼ lðcA cB Þa1 cC2

ð5:75Þ

(l and a’s are again proportionality coefﬁcients). For some unexplained reason, the powers of cA and cB are not independent but equal. Comparing the powers of the corresponding concentrations (or their products) in both terms (separately!) of the initial Guldberg – Waage and resulting ‘‘thermodynamic’’ rate www.scilet.com

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equations, it is revealed that the powers are expressible through some common parameter a. Finally the following equation is obtained: ð5:76Þ IðDmÞ ¼ l cA cB expbð1 aÞDm yT c cC expðaDm yT Þ (this is the correct form of the misprinted Eq. (12) in ref. [116]) where ‘‘ ’’ indicates the standard state). This enables the formulation of the ‘‘connection between the kinetic and thermodynamic reaction parameters’’: ?

p ¼ l expbð1 aÞDm yT c;

/

p ¼ l expðaDm yT Þ

ð5:77Þ ?

/

So, now the chemical potential appears directly in the rate constant p or p (despite the author’s nomenclature naming it as the probability instead of the

rate constant)! Knowing the still intriguing parameters l, a it would be possible to calculate the values of rate constants!! A Bro¨nsted-type relationship then easily follows: /

p ¼ lKa

ð5:78Þ

where K is the equilibrium constant. It is evident from this short report that the whole deduction is depreciated by several ad hoc, non-justiﬁed presumptions or steps. 5.6 Hungarian contribution II – Ola´h Ola´h has developed ‘‘thermokinetics’’ which is claimed to be a general kinetic theory of physico-chemical phenomena [117 – 122]. It is based on some general features of equations of motion. Thermokinetics is not, in fact, a new or deductive theory but another (re)interpretation of well-known concepts and approaches. We will brieﬂy review its main points without evaluation and concentrate on its application in chemical kinetics. Ola´h calls classical (equilibrium) thermodynamics thermostatics and by (non-equilibrium) thermodynamics, he understands classical irreversible thermodynamics [117,121]. While the former works with extensive state properties or their densities and suitable potentials, the latter introduces ﬂuxes and forces. Thermokinetics adds so-called partial ﬂuxes in contrast to the net ﬂuxes of CIT. Partial ﬂuxes mean that every process is treated separately in the forward and reverse directions and the ﬂuxes are considered as the most valuable contributions of thermostatics. Probably the most important equations are the so-called ‘‘constitutive relations of thermokinetics’’, i.e. relations between partial ﬂuxes www.scilet.com

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and (classical) potentials. The functional relationships are not clearly stated but in any case are supposed to be invertible, thus partial ﬂuxes may be functions of potentials or vice versa. Because of the principle of microreversibility, opposite partial ﬂuxes of the same process should be equal at equilibrium. From this statement it is ‘‘proved’’ (no discussion of this proof is given here) that both ﬂuxes have the same canonical functional form [117]. As noted, thermokinetics was inspired by physics where in the case of conservative ﬁelds the force (X) may be given by appropriate potential (F) [121]: X ¼ m grad F

ð5:79Þ

where m is the proportionality factor or the ‘‘charge’’. The force acts only on the movement of objects carrying the charge. The differential of the potential is related to the force by: m dF ¼ X dr

ð5:80Þ

where r is the positional vector. In the case of gravitational ﬁeld and potential, it follows from Newton’s law that m dF ¼ mv dv ¼ mv2 d ln v:m2 d ln where v2 ¼ ð21 ; 22 ; 23 Þ. If the potential acting on the object is the resultant of two potentials (e.g. gravitational and electrostatic), Ola´h supposes that it is given by m dF ¼ m2 d ln ¼ Q1 dF1 þ Q2 dF2

ð5:81Þ

In thermodynamics, the following potentials are considered: 1yT ; PyT ; mi yT (P is the pressure, m’s are the chemical potentials). Charges (Qi ) are not derived or deduced from the theory but rather selected in such a way that some wellknown relation describing a particular process is obtained. It is stated (neither proved nor derived) that charges relate motion with forces by relations analogous to Eq. (5.81): X R d ln jz ¼ Qzk dFk

ð5:82Þ

k

where R is the universal gas constant and jz is the partial ﬂux of the z-th process. Ola´h states [117] that analysis of known rate equations (here lies the source of the interpretative nature of his approach) shows that partial ﬂuxes can be written as products of factors depending on various potentials. For instance, mass ﬂuxes www.scilet.com

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( j) depend on chemical and thermal potentials: j ¼ j0 jm ðmyT Þjq ð1yT Þ

ð5:83Þ

(parentheses just show the independent variables) where the thermal factor jq is given by jq ¼ T n expðE* yRT Þ

ð5:84Þ

which is equivalent to the thermal charge according to Eq. (5.78) in the form: Qq ¼ E* þ nRT

ð5:85Þ

(n is the exponent from (5.84), and not the number of moles). Equation (5.82) is used in ref. [118] to derive reaction rate equations. It is stated without proof (only by comparison with the mass-action law) that, in the case of chemical reaction, the partial ﬂuxes are the forward and backward reaction rates with stoichiometric coefﬁcients being the appropriate charges [120]. However, in ref. [118] it is further stated that, due to the stoichiometric constraints, the charge changes to one and the potential to quantity Fr (instead of mi yT ), which is given below. An additional partial ﬂux is the ﬂux of the transported energy with charge given by Eq. (5.85) where E* is now the activation energy (subscript ‘‘act’’). Introducing the charges and potentials into Eq. (5.82), we obtain: R d ln jr ¼ dFr þ ðEact þ nRT Þdð1yT Þ

ð5:86Þ

and after integration: jr ¼ j0r T n expðEact yRT Þ expðFr yRÞ

ð5:87Þ

Finally, after subtracting expressions (5.87) for the forward and backward P reactions and substitution from the statement Fr ¼ k;r mk yT , the following k equation is obtained: " ! !# X ðþÞ X ðÞ 0 n Jr ¼ jr T expðEact yRT Þ exp k;r mk yRT exp k;r mk yRT ð5:88Þ k

k

ðÞ where ðþÞ k;r ; k;r are the stoichiometric coefﬁcients of the k-th component in the r-

th reaction in the forward ‘‘( þ )’’ and backward ‘‘( )’’ directions, respectively. Another version of potentials is given a few paragraphs later. It originates from the dependence of chemical potential on concentration www.scilet.com

Thermodynamics and mass-action chemical kinetics

mk ¼ mk þ mEk þ RT ln ck

61

ð5:89Þ

(‘‘E’’ means excess) and the Gibbs – Duhem equation d ðmk þ mEk ÞyT ¼ Uk dð1yT Þ þ V k dðPyT Þ

ð5:90Þ

and becomes at constant PyT (or constant V in gases): ðþÞ

dFðþÞ ¼ Ur dð1yT Þ R d ln r

Y ðþÞ ckk;r

ð5:91Þ

k

ðÞ

dFðÞ ¼ Ur dð1yT Þ R d ln r

Y ðÞ ckk;r

ð5:92Þ

k

By inserting (5.91) and (5.92) into Eq. (5.86), the mass-action equation can be derived, e.g.: Y ðþÞ Y ðþÞ jðþÞ ¼ jr0 T n exp ðEact U ðþÞ ckk;r ¼ kðþÞ ckk;r r r ÞyRT r k

ð5:83Þ

k

Once more, the potentials (5.91), (5.92) are neither derived nor proved but selected to obtain the desired results. Ola´h also claims [118] that potentials and partial ﬂuxes can be derived from the entropy and entropy dissipation function, respectively. The potentials are given by the partial derivatives of entropy (S) with respect to all densities X dS ¼ Fi dci ; Fi ¼ qSyqci ð5:94Þ i

i.e. by the Gibbs equation. The entropy dissipation function is introduced in ref. [119] as X ji Fi ð5:95Þ DS ¼ i

dDS ¼

X i

ji dFi þ

X Fi dji :dDFS dDjS

ð5:96Þ

i

Because the entropy dissipation function is deﬁned using partial ﬂuxes, the derivation of them from this equation is tautological. We can only ﬁnd the identity www.scilet.com

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Miloslav Pekarˇ

ji ¼ qDFS yqFi

ð5:97Þ

and also Fi ¼ qDjS yqji

ð5:98Þ

To conclude, this approach is another combination of the mass-action law, reaction isotherm and chemical potential outlined in Part 1. No fundamentally new deductions or proofs were achieved, only old facts were given new interpretations.

6. EXTENDED IRREVERSIBLE THERMODYNAMICS Extended irreversible thermodynamics (EIT) continues its role in the ﬂux-forces arena. Its principal contribution lies in the extension of the (classical) set of independent variables (speciﬁc internal energy, speciﬁc volume, mass fractions) by some of the so-called ﬂuxes (e.g. heat ﬂux or stress tensor). Thus, applications of EIT to chemical kinetics is usually nothing more than the introduction of some chemical ﬂux among independent variables. Perhaps the ﬁrst contribution of EIT to the analysis of thermodynamicskinetics relationships was the paper by Garcı´ a-Colı´ n and de la Selva [123]. They suppose that there exists some function (Z), in fact the non-equilibrium entropy, of the following variables: Z ¼ Zðe; ; ci ; J; Jd ; pÞ

ð6:1Þ

where e is the speciﬁc internal energy, is the speciﬁc volume, ci the mass fraction of component i (i ¼ 1, 2 in ref. [123] for simplicity), J is the chemical ﬂux and Jd the diffusive ﬂux of one of the species, and p is the trace of the viscous tensor. Partial derivatives occurring in the total differential of this function are either expressed by relations resembling relations of classical reversible thermodynamics or modelled by relations suitable for further developments: qZyqe ¼ Y1 ;

qZyq ¼ PY1 ;

qZyqJ ¼ ar Y1 ;

qZyqci ¼ Mi Y1

qZyqJd ¼ ad Y1 ;

qZyqp ¼ ap Y1

ð6:2Þ ð6:3Þ

where Y represents the non-equilibrium temperature, P is the non-equilibrium pressure, Mi the molar mass, a’s are the proportionality coefﬁcients. www.scilet.com

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The chemical ﬂux is not explicitly stated but from the symbol used it is clear that the ﬂux is actually the reaction rate. Unfortunately, this is deﬁned using the time derivatives of the mass fractions, which is in the modelled system with diffusion either improper or, at least, not easily applicable to experiment. The partial derivatives are then approximated by their expansion in some of the independent variables, e.g.: Mi Y1 ¼ mi T 1 þ bqðMi Y1 ÞyqJcJ þ bqðMi Y1 ÞyqJd cJd þ bqðMi Y1 Þyqpcpþ þ ð1y2!Þ q2 ðMi Y1 ÞyqJ2 J2 þ ð6:4Þ where T is the (equilibrium) temperature and mi the (classical) chemical potential. The expansions are nothing more than following transformation of functions: Z ¼ Zðe; ; ci ; J; Jd ; pÞ?qZyqx ¼ f ðJ; Jd ; pÞ þ Cx where x represents any variable from the set fe; ; ci ; J; Jd ; pg and Cx the relevant classical term. This transformation is substantiated by stating that for the classical case, i.e. for the disappearance of extending, ﬂux variables, classical expressions like qZyqe ¼ T 1 should be obtained. Why this equation cannot be arrived at by disappearing corresponding partial derivatives in full functional representation is not explained. Moreover, functions f ðJ; Jd ; pÞ look like a McLaurin series expansion and Cx is the equilibrium expression for the appropriate partial derivative of entropy. Thus, the approximation of partial derivatives is an expansion around equilibrium. It is therefore not clear where the partial derivatives in this expansion should be evaluated, as at equilibrium they should vanish, i.e. be equal to zero. Consequently, Eqs (6.2) – (6.4) should be considered only as a speciﬁc model and the whole analysis is valid only for systems complying with this model. Which real systems or materials correspond to the model is not discussed in the original paper. Another particular model in this work is the expression for the entropy ﬂux, which is constructed just as the sum of the diffusion ﬂux, the only one vectorial independent variable, multiplied successively by some of the scalar independent variables and a term which should again probably resemble some classical term: JZ ¼ Y1 ðM1 M2 ÞJd þ b01 JJd þ b02 pJd þ www.scilet.com

ð6:5Þ

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Miloslav Pekarˇ

(b’s are proportionality coefﬁcients). This should be viewed only as a speciﬁc example of representation of the linear isotropic vectorial function. The tacit construction of models continues: the next model is the representation of the entropy source (s) as a nonlinear isotropic scalar function: s ¼ Jd Xd þ JXr þ pX

ð6:6Þ

where the ‘‘generalized forces’’ are deﬁned as Xd ¼ m20 Jd þ m21 JJd þ

ð6:7aÞ

Xr ¼ mr1 J þ mr2 J2d þ mr3 J2 þ mr4 J3 þ mr5 p

ð6:7bÞ

X ¼ m1 p þ m2 J þ

ð6:7cÞ

Then, a preliminary result is derived – an equation for evolution of the chemical ﬂux, i.e. an equation with the material derivative of J (J_ ). This equation is solved, or approximated to successively higher orders in J, by some strange procedure referring to the stationary state. The ﬁnal result, Eq. (6.8) below – ‘‘general phenomenological relation between the rate of the reaction and the chemical afﬁnity’’ – is a mere summation of several of these approximations and is not proved for consistency with the initial expression for J_ . This general phenomenological relation expresses reaction rate as a function of powers of afﬁnity: J¼

ðrAymr1T Þðmr3 r2A2 Þyðm3r1 T 2 Þð1y2Þ 1 ðqar1 yqc1 Þþ2 ðqar1 yqc2 Þ ðr2A2 Þyðm3r1 T 3 Þþ þ ð2m2r3 ymr1 Þ mr4 ðr3 A3 Þyðm3r1 T 3 Þ ð6:8Þ here, r is the density, A the afﬁnity, i is the product of the stoichiometric coefﬁcient and molar mass of the component i. The afﬁnity is introduced through the classical deﬁnition (2.5)1, which is also used in CIT, supposing the same concentration dependence. There is no extended approach. More peculiar is the way that led to the power law of Eq. (6.8). This was not due to the speciﬁc claims of EIT but just due to the models introduced and used in an unusual manner: 1.

powers of J , which are the causes of later powers of A in the ‘‘general phenomenological relation’’, are introduced due to the model (6.7b),

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2.

afﬁnity is originally introduced, in fact, just as a ﬁrst – classical, equilibrium (!) – member of a series approximation (6.4),

3.

non-equilibrium corrections, introduced into this approximation are ignored in the above-mentioned stationary state analysis,

4.

powers of J , which should be the other non-equilibrium corrections in model (6.7b), are systematically expressed in this analysis in powers of afﬁnity despite its status given under 2 above. There are other unclear points. During the constructing of the models, it

is several times stated, but never proved, that the models should reduce to the standard form of CIT. Rate (chemical ﬂux) is ﬁnally expressed as a function of afﬁnity, which itself is a function of chemical potential, which itself is a function of concentration. Although the chemical ﬂux, or reaction rate, is included among the independent variables, the dependence of the rate on other variable(s), usually afﬁnity, is sought. The next work of the same authors gives only moderate progress. The motivation for the EIT approach is stated in ref. [124]: ‘‘For many years linear irreversible thermodynamics has been the only theory available to account for the empirical kinetic mass-action law (KMAL) as a ﬂux-force relation between the chemical rate J and the afﬁnity A namely, J & ½expðAyRT Þ 1. In spite of the fact that such a relation is a nonlinear one, it has been shown that at least for the reaction B þ C , D þ E, and using a kinetic theory model, the entropy source JAyT is consistent with it. Therefore the rate J is interpreted as a thermodynamic ﬂux of the same footing as the heat ﬂux, the diffusion ﬂux, and the stress tensor. Thus, KMAL has been viewed as a constitutive relation, analogous to Fourier’s heat equation, Fick’s equation for diffusion, and the Newton – Navier equation for the transmission of momentum. On the other hand, the coupling between the chemical rate with its generating forces, namely A, and the divergence of the hydrodynamic velocity satisfy Onsager’s reciprocity theorem in the linear approximation only.’’ A paper entitled simply ‘‘Consistency of the Kinetic Mass Action Law with Thermodynamics’’ [125] starts with the function Z ¼ Zðe; ; ci ; q; J; Ji Þ

ð6:9Þ

thus, instead of the stress tensor, the heat ﬂux q is considered; e is the internal energy density, the other symbols have the same meaning as in Eq. (6.1), but the diffusion ﬂuxes (Ji ) are considered for each component i separately. Again, some www.scilet.com

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speciﬁc models of representations of functions of various tensorial orders are introduced. As usual in EIT, the total derivative of the function is written and the partial derivatives with respect to the ‘‘ﬂux variables’’, viz. q; J; Ji are approximated by expansion around the local equilibrium where these variables should, of course vanish. No true Taylor expansion is used, as the ‘‘classical variables’’ (e; ; ci ) are not included, and do not expand anywhere. Rather, a combination of the isotropic function representation and a Taylor series is used, at least for qZyqJ: qZyqJ ¼ T 1 o0 J

ð6:10Þ

(o0 is the proportionality coefﬁcient). This is the ﬁrst model used. The EIT postulate of the entropy balance equation, viz.: r dZydt þ divJZ ¼ sZ

ð6:11Þ

is a necessary intermediate step calling for at least two expressions – for entropy ﬂux (JZ ) and source (sZ ). The entropy ﬂux is represented as an isotropic vectorial function, which is immediately speciﬁed by the following equation X X JZ ¼ ð1yT Þq þ b0 q ðmi yTMi ÞJi þ bi0 Ji ð6:12Þ i

i

(b’s are the proportionality coefﬁcients, which are, in turn, functions of all scalar invariants) again claiming, not proving, that this form is reducible to the normal entropy ﬂux of CIT where ‘‘ﬂux variables’’ can be ignored. This is the second model which is then combined with mass, energy, and entropy balances and time derivative of Z, coming from the ﬁrst model (6.10). An expression for entropy production then results as follows: sZ ¼ "

# X q grad T 1 þ grad b0 þ ða0 yT ÞðdqydtÞ þ ðg0i yT ÞðdJi ydtÞ þ i

"

# X X Ji grad ðmi yTMi Þ þ grad bi0 þ ða0i yT ÞðdqydtÞ þ ðgij yT ÞðdJj ydtÞ þ þ i

j

X X ð1yT ÞJi Fi þ b0 div q þ bi0 div Ji þ AJyT þ o0 JT 1 ðdJydtÞ ð6:13Þ þ i

i

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where a’s and g’s are the coefﬁcients from equations analogous to Eq. (6.10) but for the partial derivative with respect to q and Ji , respectively; mi is the (classical) chemical potential, Fi the external force on i per unit mass and A is the (classical) afﬁnity. It is claimed that entropy production is a scalar function of deﬁning scalar variables. Therefore, it can be represented as an isotropic function. The representation is again rather speciﬁc: X sZ ¼ P 0 þ X q q þ Xi Ji ð6:14aÞ i

X q ¼ x0 q þ

X

xi Ji

ð6:14bÞ

lij Jj

ð6:14cÞ

i

X i ¼ li q þ

X j

and can be considered as the third model. Symbol P 0 represents a function of all scalar invariants. The two expressions for entropy production, (6.13) and (6.14a), should be consistent. But the consistency is not straightforward and explicit. Therefore, further models have to be invoked. In contrast to the authors’ contention, chemical ﬂux does appear as a multiplicative factor in one of the two equations for entropy production, cf. Eq. (6.13). This appearance is not sufﬁcient and must be supported by extracting the chemical ﬂux from (only some!) scalar coefﬁcients in functional representations. Several additional models can therefore be constructed: b0 ¼ b 00 J;

bi0 ¼ b 0i0 J;

P 0 ¼ JPðJ; . . .Þ

ð6:15Þ

where P in the last equation is again a function of all scalar invariants including J. At last, the desired relation – an equation for the time derivative of chemical ﬂux or the general mass-action law – is obtained: X ðo0 yT ÞðdJy dtÞ ¼ AyT b 00 div q b 0i0 div Ji þ P ð6:16Þ i

Afﬁnity was introduced again due to the reminder of the classical term in the representation of entropy ﬂux. The evolution equation enables, after introducing further models or simpliﬁcations, discussion in the terms ‘‘chemical ﬂux is forced www.scilet.com

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by the afﬁnity’’. It should be stressed that an explicit dependence on afﬁnity is found only for the time derivative of the chemical ﬂux J (which is, perhaps, the reaction rate); the direct relationship between chemical ﬂux (reaction rate) and afﬁnity is obtainable only for the stationary state. To summarize – there are too many models with vague relations to real systems or materials, and too general ﬁnal equations, not containing the main quantity measured by kineticists, namely concentration. Of course, in Eq. (6.16) concentration is hidden in afﬁnity, however, this is not an equation for the reaction rate itself, but for its time derivative. Lebon et al. [126] write that they adopt a position intermediate between classical theory and EIT. Their work was competently criticized by Garcı´ a-Colı´ n [127] to say nothing about its limitation to homogeneous (non-diffusing) mixtures. Instead of the reaction rate, the authors use the degree of advancement (x) deﬁned as x_ ¼ c_ i yi

ð6:17Þ

(ci is the mass fraction of i-th component and i its stoichiometric coefﬁcient). This means that only closed systems are considered. As an independent variable, however, the following difference is used: x ¼ x xe

ð6:18Þ

where xe denotes the equilibrium value. It is assumed that the time evolution of the new variable is given by x_ ¼ jðT ; p; xÞ

ð6:19Þ

(T is temperature, p pressure). It is further postulated that function j is expressed as follows: jðT ; p; xÞ ¼ xcðT ; p; xÞ

ð6:20Þ

During further development, no special irreversible thermodynamical approach is used. Only combinations and manipulations with the postulate, integrated form (6.17), and relations well-known from reversible thermodynamics, are used, viz. the deﬁnition of the relation of chemical potential to the component activity, the deﬁnition of afﬁnity (2.5)1, and the expressing of equilibrium constants by standard chemical potentials. www.scilet.com

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The ﬁnal result is the following equation x_ ¼ oðT ; p; xÞ½1 expðAyRT Þ

ð6:21Þ

containing another function (o) which includes also the function c. Eq. (6.21) is again an expression for the time derivative of a certain reaction rate quantity (degree of advancement), which is claimed to be the standard law of massaction. No explicit rate equation or function was derived. Function c remains undetermined and Garcı´ a-Colı´ n [127] showed how it can be related to phenomenological coefﬁcients of CIT. In summary: no irreversible thermodynamics is utilised and the only new feature is postulate (6.20), which is used several times and coupled with the wellknown relations of classical thermodynamics. This work, as well as other irreversible thermodynamics approaches (e.g. [125]), were brieﬂy criticized by Ross and Garcı´ a-Colı´ n [128]. However, no new ideas were presented, just some reservoirs for reactants and products are introduced with no clear distinction between the reaction Gibbs free energy and the Gibbs free energy of the whole system. The critique of EIT approaches is based on the initial task of EIT – to describe fast processes by introducing new, extending, variables: ‘‘For most reactions, especially in liquids, reaction times are long compared to other relaxation times (vibrations, etc.)... For such cases the condition of local equilibrium holds well; the thermodynamic variables including the progress variable are on the same time scale and there is no need for an extended thermodynamics. That need may arise when the reaction time is more comparable to other relaxation times and the predicted rate coefﬁcients become time-dependent, which expresses the effect of the relaxation of the fast(er) variables of, say, vibrational relaxation, compared to the slow(er) chemical rate.’’ Thus, there is usually no need for incorporating some ‘‘chemical ﬂux’’ among the independent variables. An interesting note is given in the conclusion: ‘‘Furthermore, the identiﬁcation of a generating function (Z above) with an entropy has not yet been justiﬁed.’’ The last contribution from EIT was due to Fort et al. [129] who try to ﬁnd new developments from the same starting point. First, they would like to ascertain whether EIT methodology gives entropy as a sum of its equilibrium value and some non-equilibrium correction also for chemically reacting systems. Second, they support this ﬁnding with deductions from the kinetic theory of www.scilet.com

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gases. As statistical theories are beyond the scope of this review, we will focus mainly on the EIT part. Speciﬁc entropy (s) is, again, considered to be a function of ‘‘classical’’ variables and one extending, which is, of course, the reaction rate J: s ¼ sðu; ; ci ; JÞ; i ¼ 1; 2; . . . ; n

ð6:22Þ

(u is the total speciﬁc internal energy, the speciﬁc volume and ci the mass fraction) and its total differential is constructed; partial derivatives with respect to the classical variables are expressed with the aid of the generalized temperature (Y), pressure (P), and chemical potentials (Zi ) as dictated by the EIT standard procedure. Then, the ﬁrst new postulate or, more appropriately, model is introduced: ðqsyqJÞu;;ci ¼ ðayT ÞJ

ð6:23Þ

where a is some coefﬁcient depending only on the classical variables. This model is substantiated by the traditional claim that the generalized entropy ðsÞ must reduce to the classical one at equilibrium where the rate is zero, which is not proved. Consequently, the generalized local Gibbs equation can be formulated: X ds ¼ ð1yYÞ du þ ðPyYÞ d ðZi yYÞ dci ðayT ÞJ dJ ð6:24Þ i

However, the evolution equation for speciﬁc entropy is restricted to depend on the reaction rate only, which is explained by considering only non-equilibrium processes in an incompressible ﬂuid in the absence of heat and diffusion effects, speciﬁcally: r dsy dt ¼ J½AyT ðaryT Þ dJy dt

ð6:25Þ

In the (second) postulate (6.25), the classical equilibrium deﬁnition of afﬁnity (A) through the classical chemical potentials (not Zi ’s!), cf. Eq. (2.5)1, was applied together with the mass fraction balance, namely dci y dt ¼ ði yrÞJ, where i is the stoichiometric coefﬁcient and r the density. The evolution equation (6.25) is compared with the general law of entropy balance of EIT, see Eq. (7.3) below, and the entropy source (more precisely, the rate of entropy production per unit volume) is then expressed as the right hand side of Eq. (6.25). It should be noted that in contrast to the other EIT approaches, which considered the reaction rate as a part of the entropy ﬂux, here the rate is included in the entropy source! www.scilet.com

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As the second law of thermodynamics calls for a positive value for entropy production, the simplest way to assure this is to have it in only the second power of the reaction rate. Therefore, a third model is proposed: AyT ðarÞyT dJy dt ¼ bJ

ð6:26Þ

This model is particularly convenient for EIT as it is analogous to the Maxwell – Cattaneo equations, which were successfully explained within the EIT approach. With this model, the generalized Gibbs equation (6.24) may be written as X ds ¼ ð1yT Þ du þ ðpyT Þ d ðmi yT Þ dci ðtyrlÞJ dJ ð6:27Þ i

where l ¼ 1yb and t ¼ arlyT , with no explanation as to why the generalized variables were substituted by their classical (equilibrium?) analogues, i.e. temperature (T ), pressure (p) and chemical potential (mi ). Its integrated form is simply expressed as sðu; ; ci ; JÞ ¼ sðu; ; ci Þ ðty2rlÞJ2

ð6:28Þ

And this is all for chemical kinetics. Entropy was, ﬁnally, expressed as its equilibrium value and non-equilibrium correction, which is second order in the reaction rate. The kinetic theory part of this work derives a similar expression for entropy and even the usual proportionality of the rate to the afﬁnity. Interestingly, in the conclusion the authors write: ‘‘...the reaction rate is not a ﬂux in the usual sense because it does not appear as a true ﬂux in the balance equations of mass fractions; instead, it appears as a source term there.’’ Further. ‘‘In spite of this, we have shown how chemical reactions can be included in the much broader framework of EIT.’’ This means that ﬂuxes, forces, sources are interpretations according to some particular motivation and not the results of rigorous deﬁnitions or proofs.

7. COMMON PROBLEMS IN CIT AND EIT APPROACHES Balance equations, well-known from other branches of physics or chemistry, are used in both approaches, of course. They have the general form [130]: ð

qðrbÞ dV ¼ V ðtÞ qt

ð SðtÞ

Jb ? n dS þ

ð

sb dV

V ðtÞ

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ð7:1Þ

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Miloslav Pekarˇ

Here Jb , which may be a vector or tensor, represents the quantity ﬂowing per unit area and unit time (t) through the boundary S of some volume V , and sb is the rate of production or destruction per unit volume of the quantity, which has the speciﬁc value b, r is the mass density and n is the positive unit normal vector to the boundary. Both the volume and its boundary are in general changing in time. The local form of the balance under pertinent conditions is as follows: qðrbÞyqt ¼ div Jb þ sb

ð7:2Þ

It is further supposed that the same balance is valid also for the speciﬁc entropy: qðrsÞyqt ¼ div ðJs þ rs vÞ þ ss

ð7:3Þ

where v is the barycentric velocity. Entropy ﬂux Js is deﬁned by: ð e Js n dS d Sydt ¼

ð7:4Þ

SðtÞ

where superscript ‘‘e’’ refers to the entropy (S) exchanged with the surroundings (balances (7.2) and (7.3) are rather inconsistent). That every approach considering reaction rate as a part of entropy ﬂux is dubious. The reaction rate is certainly not limited to something which only ﬂows through the boundary. The same problem is with the ‘‘chemical ﬂux’’, widely used by LI and EI thermodynamicists, in fact the reaction rate. Considering the reaction (rate) to be some ﬂux is unacceptable. As this is usually no more than a linguistic denomination, it may present no serious problem in the results of particular deductions. Jou et al. [131] state: ‘‘. . . decomposition into thermodynamic ﬂuxes and forces is arbitrary to a certain extent. . .one could permute the deﬁnitions of ﬂuxes and forces.’’ Why use this arbitrary concept, at all? The ﬂux-force concept is a somewhat conﬁning trap of CIT or EIT. Jou et al. [131] write when introducing ﬂux-force conception into balances, particularly into the entropy balance: ‘‘. . . the cause is provided by the driving thermodynamic force, which elicits the effect manifested through the conjugated ﬂux.’’, whereas several pages further on [132]: ‘‘In order to obtain evolution equations for the ﬂuxes compatible with the positiveness of the entropy source, we express the forces as functions of the ﬂuxes.’’ The logic of the cause – effect is suddenly abandoned. Whereas in many other cases it is really only an arbitrariness of improper nomenclature, despite of which realistic equations can be obtained, in the chemically reacting systems this effort of forcing the wellwww.scilet.com

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known equations into the ﬂux-force framework is much less productive. Instead of searching for relationships between ﬂuxes and forces a priori selected to be the reaction rate and afﬁnity, respectively, a description of the actual problems should be sought – what are the correct independent variables in chemically reacting systems. Denbigh in the early 1950’s even argued [133] that the rate of a chemical reaction is not proportional to a thermodynamic force in general, and is primarily determined by the concentrations and not by the difference of chemical potentials. Chemical ﬂux, which is in fact the reaction rate, is systematically introduced among the independent variables. Because concentrations (in the general meaning, not just molar concentrations) are also and traditionally independent variables, it follows that the reaction rate is not dependent on the concentration of the reacting species! This is in strong contradiction with experience and theory of chemical kinetics. A particular example is found in ref. [129]. Requiring Eq. (6.24) to be an exact differential, the following condition, among others, should be fulﬁlled: qðZi yYÞyqJ ¼ q½ðayT ÞJyqci

ð7:5Þ

This condition is written in ref. [129] as qðZi yYÞyqJ ¼ JqðayT Þyqci

ð7:6Þ

i.e. the reaction rate is independent of concentrations! Fortunately, Eq. (7.6) is of no further use in the development given in ref. [129]. The CIT and EIT approaches also seeks evolution equations for perhaps every variable, i.e. also for the chemical ﬂux or reaction rate. Thus, equations for the (material) time derivative of this ﬂux, i.e. of reaction rate, are being derived, sometimes including also powers of the ﬂux. This is unusual in common chemical kinetics where (rate) equations for the time evolution of concentrations are the desired object of study. Further, substantiation of the various postulates for non-equilibrium or extending variables often calls for the necessity of their reduction to classical equations (containing classical variables only) at equilibrium. It seems then that classical variables, concentrations among them, have no effect in the nonequilibrium regime. This is certainly not true for the non-equilibrium reaction rate. www.scilet.com

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The Guldberg – Waage law is still empirical. Instead of implementing it into the theoretical ﬂux-force framework, it might be desirable to derive it, or something analogous to it, from the theoretical thermodynamic framework. To this point, EIT is of much more beneﬁt than CIT. As Bowen has clearly stated in the beginning of his development of thermodynamic mixture theory [134]: ‘‘In a general theory of mixtures . . . it should be possible to prescribe, as independent ﬁelds, . . . certain parameters characterizing the state of any chemical reactions taking place.’’

8. RATIONAL OR CONTINUUM THERMODYNAMICS APPROACHES TO CHEMICAL KINETICS 8.1 Introduction Under this heading, all papers, more or less using balance equations, entropy inequality, and constitutive equations in the continuum approach, are collected. Rational thermodynamics provides an elaborated rigorous framework for the description of various continuum systems. Because of its axiomatic basis and strong mathematical language, it is rather difﬁcult to give here a brief yet exhaustive review. Therefore we will focus only on the most important works in regard to chemical kinetics. A very general overview of basic balance laws (equations), which lie at the heart of all rational thermodynamics treatment for a reacting continuum, was given by Kelly in the 1960’s [135]. The reaction rate, of course, enters into the mass balance but usually in a very general way. Most rational thermodynamics contributions thus have a primarily theoretical value. The integral form of material balance for component a reads [135]: ð ð d ra d ¼ rba d ð8:1Þ dt Va Va where ra is the mass density of constituent a, r the mass density of the reacting P continuum, r ¼ a ra , and rba is the volume supply of mass of species a due to chemical reaction. The differential form of the material balance (8.1) is obtained under certain continuity suppositions as: qra yqt þ div ðra va Þ ¼ rba

ð8:2Þ

where va is the velocity of component a. www.scilet.com

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The reaction rate is ‘‘hidden’’ in the mass supply rba and perhaps virtually no publication gives its explicit functional form or, using the terminology of rational thermodynamics, constitutive equation. 8.2 Bowen lays the foundation stone Bowen [136] treats the model of a non-isothermal reacting mixture with elastic deformations and viscous stresses but without diffusion. Instead of reaction rate or mass supply, he works with the extent of reaction deﬁned using his algebraicstoichiometric formalism [34], which is not identical with the deﬁnition known from (irreversible) classical thermodynamics but closely related to it. It is deﬁned as a vector (j), i.e. for each component of the reacting mixture as the difference between its actual and referential (initial) molar fraction. It is thus neither deﬁned for, nor related to, a particular reaction. It summarizes the effects of all reactions, which may inﬂuence the concentration of a particular component. However, it can be related to reaction rates through the algebraic-stoichiometric formalism [34]; this is not used in ref. [136]. Moreover, Bowen introduces extent of reaction among the independent variables, so this seems to be analogous to the CIT and EIT practices criticized in Part 6. However, concentrations are not explicitly included in the set of independent variables (which can be seen on the right hand side of Eq. (8.3)), the extent of reaction is not a reaction-rate quantity in this case and, in fact, one of the constitutive equations, viz. j_ ¼ wðu; g; F; F_ ; jÞ

ð8:3Þ

resembles traditional rate equations setting the reaction rate (concentration time variation) to be some function of concentrations themselves. In Eq. (8.3), w stands for some function, u is the temperature, g its gradient, F is the gradient of the deformation and the dot means (material) time derivative. Unfortunately, no explicit, practically convenient expression of function w is derived. Bowen arrives, however, at interesting conclusions regarding the relationships between afﬁnity and reaction rate (at equilibrium). Afﬁnity (A) is deﬁned by him as follows: A ¼ Aðu; g; F; F_ ; jÞ ¼ ðqcyqjÞðu; g; F; F_ ; jÞ

ð8:4Þ

and the state of equilibrium (denoted ‘‘ þ ’’) by: g ¼ 0;

F_ ¼ 0;

wðuþ ; 0; Fþ ; 0; jþ Þ ¼ 0 www.scilet.com

ð8:5Þ

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Miloslav Pekarˇ

Function c means free energy as a function of the same variables as the extent of reaction in Eq. (8.3), which is expressed by the last parentheses in Eq. (8.4). Bowen shows that, at equilibrium, the afﬁnity is restricted but it is not necessarily zero. It must vanish if qwyqj is at equilibrium non-zero (regular); this special equilibrium state is called by Bowen ‘classical equilibrium’ and, without additional assumptions, classical equilibrium and Bowen’s equilibrium states are generally different. Further, Bowen tests what the vanishing of afﬁnity implies about the reaction vector and proves that in this state this vector need not be zero. It must vanish (only) if qAyqj is regular in the state deﬁned by (8.5)1,2. It is thus seen that more complicated material systems are richer in equilibrium states and therefore LIT (EIT) approaches are restricted in their generality. The next Bowen paper [137] treats a reacting mixture of elastic materials with diffusion. The extent of reaction is abandoned, as it is not in general deﬁnable in systems with diffusion [34,35]. Further, concentrations (precisely, densities or weight concentrations) are among the independent variables and the same form of constitutive equation as for other dependent variables is formulated for the mass supply (closely related to the reaction rate): ba ¼ f ðy; g; Fb ; Gb ; vb ; gb ; db Þ

ð8:6Þ

where ba ¼ jdet Fa j _ ca

ð8:7Þ

The repeated symbols have the same meaning as above, Fb is the deformation gradient and Gb the second deformation gradient for the constituent b, vb its velocity, gb ¼ rb jdet Fb j, db is the certain gradient of density of component b (rb ) and _ ca is the mass supply for constituent a. Subscripts a and b apply to all constituents of the mixture which are n in total. Thus, in this case, the mixture is not considered and balanced as a whole, as a single-continuum, but as a mixture of continua of individual components. Thus reaction rate is, in fact, one of the proposed dependent variables because it is determined by _ ca , which in traditional chemical kinetics is called the production rate of component a. No further simpliﬁcation of its functional form is obtained. Equations (8.6) and (8.3) are typical examples of (only) results of most rational thermodynamics works related to kinetics. Even for a reacting www.scilet.com

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mixture of two ideal gases with different temperatures, the mass production rate remains as a general function of densities, temperatures and their gradients, gas velocities, and deformation tensors [138]. The consequences of Bowen’s model (8.6) to afﬁnity-rate relationships are similar to the preceding case. Now, the afﬁnity of reaction p is deﬁned as: Ap ¼

n X ma jdet Fa jPpa qCyqga ¼ Ap ðy; Fb ; gb ; db Þ

ð8:8Þ

a¼1

where ma is the molecular weight of constituent a, C is the (inert part of) free energy density, Ppa is the stoichiometric coefﬁcient related to the mass supply through [34] _ ca yma

¼

R X

Ppa jp

ð8:9Þ

p¼1

where R is the number of independent reactions and jp their rates. Although now the afﬁnity is deﬁned as the afﬁnity of a reaction, it is still in principle identical to the deﬁnition in the previous paper and has not the same meaning as its usual, classical counterpart. Using ca ¼ ra yr

ð8:10Þ

and ga ¼ ra jdet Fa j

ð8:11Þ

deﬁnition (8.8) can be transformed to deﬁnition (8.4) for non-diffusing mixtures where concentration changes determined solely by the reaction and extent of reaction can be used: qCyqga ¼ ðqCyqca Þðqca yqga Þ ¼ ðqCyqca Þð1yrjdet Fa jÞ qCyqxa :ðqCyqca Þma

ð8:12Þ

Bowen shows that for the state of weak equilibrium (identical to equilibrium from ref. [136]) deﬁned as þ jp ðyþ ; 0; Fþ b ; 0; 0; gb ; 0Þ ¼ 0;

p ¼ 1; . . . ; R

ð8:13Þ

T afﬁnity need not be zero in general but it must vanish if qjþ p yqga , a ¼ 1; . . . ; n, is regular. This state, where both (8.13) and www.scilet.com

78 þ Ap ðyþ ; Fþ b ; gb ; 0Þ ¼ 0;

Miloslav Pekarˇ

p ¼ 1; . . . ; R

ð8:14Þ

are valid, is called strong equilibrium [35]. Conversely, in an equilibrium state where (8.14) is fulﬁlled, the reaction T þ rate need not vanish but must be zero (only) if qAp yqga is regular. In other T T þ words, if both q jp yqga and qAp yqga are regular at (yþ ; 0; Fþ b ; 0; 0; gb ; 0), þ þ þ þ þ p then jp ðy ; 0; Fþ b ; 0; 0; gb ; 0Þ ¼ 0 if and only if A ðy ; Fb ; gb ; 0Þ ¼ 0 for p ¼ 1; . . . ; R. Interestingly, Bowen asserts that there is no natural relationship between chemical potential and chemical afﬁnities. This contradiction with traditional irreversible thermodynamics originates probably in his deﬁnition of the chemical potential (tensor) Ka : Ka ¼ ca I TTa yra

ð8:15Þ

where ca is the constituent’s partial free energy density and Ta is the partial stress on the a-th constituent. Besides its inherent (third order) tensorial character (in contrast to scalar afﬁnity), this quantity is deﬁned using free energy directly and not only its (concentration) derivative as traditionally. Deﬁnition (8.15) derives from the classical relation between the Gibbs (G) and Helmholtz (F) energies: G ¼ H T S ¼ U þ PV T S ¼ F þ PV

ð8:16Þ

together with the classical equivalence between chemical potential and partial molar Gibbs energy. Deﬁnition (8.15) can be then understood as a generalization of (8.16) to materials, which can support stress. As noted by Nunziato and Walsh [139], this deﬁnition views the chemical potential as reﬂecting any change in the free energy due to the deformation of a constituent, and not due just to the changes in its mass. 8.3 Gurtin re-examines the classical theory Gurtin and Vargas [140] tried to reinvigorate the classical theory of reacting ﬂuid mixtures using an appropriate constitutive model. The mixture is treated as a single body: balance laws for momentum, energy, and entropy are postulated for the mixture as a whole. Their effort was successful using the following independent variables in constitutive equations: speciﬁc volume (), temperature (y), concentrations (in the above mentioned sense; collected in vector c), and www.scilet.com

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gradients of all these quantities. Among others, they proved that in this mixture, stress reduces to a pressure, that free energy, pressure, and entropy depend only on speciﬁc volume, temperature, and concentrations, and are related by the classical relations (e.g. Maxwell or Gibbs relations), and that there exists a chemical potential given as the derivative of the free energy with respect to the corresponding concentration. Regarding our interest in chemical kinetics, the reaction rate, or, more precisely, the mass supply ma (of any constituent a) is considered to be one of the dependent variables possessing thus the standard constitutive equation: ma ¼ _ ma ð; y; c; grad ; grad y; grad cÞ: _ ma ðLÞ;

a ¼ 1; . . . ; n

ð8:17Þ

Unfortunately, the rational thermodynamic procedure again gives no further simpliﬁcation of this general equation. Thus, even with this simple mixture, otherwise obeying classical thermodynamic relations, the reaction rate is not in general a function only of temperature and concentrations as supposed by traditional mass-action kinetics, but also a function of, for example, their gradients. Gurtin and Vargas’ results have consequences also for afﬁnity-based approaches. Expressing the mass supply through the rates of R independent reactions (Jr ): ma ¼

R X ar Jr

ð8:18Þ

r¼1

(ar divided by the molecular mass of constituent a is proportional to the corresponding stoichiometric coefﬁcient) and deﬁning afﬁnity using the chemical potential ma : X ar ma ð8:19Þ Ar ¼ a

the product of mass supply and chemical potential occurring in the ﬁnal entropic inequality can be written as ma ma ¼

R X Jr Ar

ð8:20Þ

r¼1

and the entropic inequality reads: X Jr A r 0 ha grad ma þ ð1yyÞq grad y þ www.scilet.com

ð8:21Þ

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Miloslav Pekarˇ

Here, ha is the relative mass ﬂux of constituent a (diffusive ﬂux) and q is the heat ﬂux. Thus, only, for example, in the non-diffusing mixture with no temperature gradients, the (‘‘de Donder’s’’) relation R X Jr Ar 0

ð8:22Þ

r¼1

is valid (and for independent reactions only). Gurtin and Vargas formulate results for the equilibrium state and its neighbourhood with unusual mathematical care. The equilibrium state is deﬁned as a homogeneous state Lþ , i.e. a state with grad ¼ grad T ¼ grad ca ¼ 0, with the property that ma ðLþ Þ ¼ 0;

_

a ¼ 1; . . . ; n

ð8:23Þ

It is proved that þ 2 ma mþ a ¼ OðjL L j Þ;

ma ma ¼ OðjL Lþ j2 Þ

ð8:24Þ

thus, within the terms of OðjL Lþ j2 Þ, the mass supply and the chemical potential (called the reduced chemical potential by Gurtin and Vargas) are ‘‘orthogonal’’. Equations (8.24) may be viewed as a certain generalization of the equilibrium (equality) version of (3.12). Strong equilibrium is deﬁned by þ 3 ma ðLÞmþ a ¼ OðjL L j Þ

_

ð8:25Þ

as jL Lþ j ?0 in addition to (8.23). These notions of strong equilibrium and equilibrium are somewhat weaker than those given above, yet their consequences regarding the simultaneous vanishing of reaction rate and afﬁnity are similar. Gurtin and Vargas prove that for a strong equilibrium, mass supply depends only on the chemical potential (as jL Lþ j ?0): ma ¼

n X þ 2 tab ðmb mþ b Þ þ OðjL L j Þ

ð8:26Þ

b¼1

or, if the scalar coefﬁcients tab fulﬁl tab ¼ tba , ma ¼

n X tab mb þ OðjL Lþ j2 Þ

ð8:27Þ

b¼1

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þ 2 because tab mþ a ¼ 0. Thus, within the terms of OðjL L j Þ, the reaction rate could be expressed directly through the chemical potentials, or, if the ideal

relation between the chemical potential and concentration is inserted, through the concentrations. Equations (8.26) or (8.27) show that the mass supply of any component is in general determined by the chemical potential of all the other constituents. Further, if a linear relationship between reaction rate and afﬁnity around strong equilibrium is assumed: Jr ¼

R X þ 2 L rs ðAs Aþ s Þ þ OðjL L j Þ

ð8:28Þ

s¼1

then R X þ 2 Jr Aþ r ¼ OðjL L j Þ

ð8:29Þ

r¼1

Equation (8.29) is the ﬁnal result restricting simultaneously the values of reaction rate and (equilibrium) afﬁnity. It should be stressed that this general approach gives no restrictions on the equilibrium value of the afﬁnity. Although the equilibrium criteria are deﬁned with minimum (mathematical) restrictions, they are probably too weak to be used and tested in practice. On the other hand, statements on relationships between reaction rate and afﬁnity reviewed in Section 2 may have very limited validity even if they are deduced for the vicinity of equilibrium only. ‘‘Kinetic’’ results obtained by Gurtin and Vargas are very close to those presented by Gurtin [141] for a mixture of inviscid (zero or near-zero viscosity) ﬂuids treated more generally, i.e. considering each constituent as a single body (although the chemical potential should be deﬁned in a slightly different way). 8.4 Treatments of more complex systems Nunziato and Walsh [139] apply continuum theory to (ideal) multiphase mixtures. In fact, they do not explicitly consider individual phases and multiphasicity is included (among others) by considering the volume fraction of each constituent as an independent kinematical quantity. To this end, an additional balance equation of forces is added. The volume fraction ja is related to the usual continuum mass density ra (called by Nunziato and Walsh partial density) of constituent a through the actual or local density (called by Nunziato and www.scilet.com

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Walsh simply density) ra by the relation: ja ¼ ðma yVtot Þyðma yVa Þ:ra yra

ð8:30Þ

where Vtot is the volume of the total continuum and Va the volume occupied by the constituent a. To imply phase separation, the thermal and mechanical response of the a-th constituent depends only upon the properties of that component. However, the remaining responses, including mass supply, are functions of interactions with all components. Also in this model, the constitutive equation for the mass supply c# a of constituent a does not simplify during further development and remains in the very general form: _ b ; ub ; yb ; Fb ; grad yb ; cb Þ:Fa ðLÞ; c# a ¼ Fa ðjb ; grad jb ; j

a; b ¼ 1; . . . ; n ð8:31Þ

Here the dot means the material time derivative, ub is the diffusion velocity of component b, yb its temperature and Fb its deformation gradient, cb stands for the component mass concentration. The development of entropic inequality naturally leads to the following deﬁnition of chemical potential: ma ¼ qðca ca Þyqca

ð8:32Þ

where ca is the (Helmholtz) free energy of constituent a. Nunziato and Walsh distinguish between ‘‘unsaturated’’ and ‘‘saturated’’ mixtures. In general, the sum of volume fractions is restricted by: 05:

X a 1

ð8:33Þ

a

and saturated mixtures are those where ¼ 1. Nunziato and Walsh deﬁne thermochemical equilibrium (denoted again by ‘‘ þ ’’) in an unsaturated mixture by þ þ ðc# a Þ :Fa ðL Þ ¼ 0;

a ¼ 1; . . . ; n

ð8:34Þ

and þ þ þ þ Lþ ¼ ðjþ b ; grad jb ; 0; 0; yb ; Fb ; 0; cb Þ;

b ¼ 1; . . . ; n

Among other consequences, the following relation is found: www.scilet.com

ð8:35Þ

Thermodynamics and mass-action chemical kinetics n X

þ mþ b ðqFb yqca Þ ¼ 0;

a ¼ 1; . . . ; n

83

ð8:36Þ

b¼1

which means that all the chemical potentials need not be equal at equilibrium. This led the authors to designate the equilibrium deﬁned by Eqs (8.34) and (8.35) as ‘weak’ equilibrium in contrast to the ‘strong’ equilibrium, which was deﬁned by adding the following conditions to Eqs (8.34) and (8.35): þ mþ a ¼ mb ; a 6¼ b;

a; b ¼ 1; . . . ; n

ð8:37Þ

It should be pointed out that the chemical potentials used by Nunziato and Walsh differ from those familiar in classical reversible thermodynamics, as these authors do not explicitly consider individual phases with their own potentials. Moreover, the classical condition of phase equilibrium does not follow (8.37) but states the equality of chemical potentials in all phases for every component. It is not clear how the strong equilibrium condition (8.37) is related to the classical phase equilibrium condition. As a consequence of strong equilibrium in both saturated and unsaturated mixtures, this inequality is also found: ðqFa yqca Þþ 0;

a ¼ 1; . . . ; n

ð8:38Þ

Referring to (8.34) this means that the production rate (rate of formation) of a component has, at strong equilibrium, its maximum value with respect to the concentration of this specie, providing the strong inequality is valid in Eq. (8.38). If the afﬁnity of a reaction is deﬁned as X Ar ¼ Ma Pra ma ð8:39Þ a

where Ma is the molecular weight of constituent a and Pra its stoichiometric coefﬁcient in the (independent) reaction r, then the strong equilibrium conditions guarantee its disappearance. Drumheller and Bedford presented an extension of their variational approach [142] to reacting immiscible mixtures [143]. Their methodology was criticized by Passman, Nunziato and Walsh [144]; the critical remarks as well as Drumheller’s responses [145] are not repeated here but some others related to the scope of this review are added. To incorporate immiscibility, Drumheller and Bedford also introduce local density ra and volume fraction ja , as Nunziato and Walsh (see above), and suppose that the constitutive equation for the free energy www.scilet.com

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of a constituent depends only on the variables pertinent to that constituent alone. The remaining dependent variables are assumed to be functions of variables of all constituents. Among independent variables, the reaction rates of constituents (more precisely, the rates of variation in time constituents, i.e. their material derivatives) are also included. Thus, the authors do not formulate a constitutive equation for the reaction rate or the reaction rate equation. Because (local) densities are also included among the independent variables, the same questions and problems arise regarding the dependence of reaction rate on concentrations, which were discussed in Section 6. Moreover, also the time dependence of the local densities (material derivatives) appear in the set of independent variables. However, the reaction rates, densities and their changes with time are not mutually independent but related through the mass balance equation. This paradox can perhaps be resolved by noting that volume fraction, which also appears in the mass balance after substituting densities by partial densities, is not included among the independent variables. But then it cannot vary independently (of component density) as stated by Drumheller and Bedford (for compressible constituents). A much more acceptable and lucid version of the application of continuum theory to immiscible mixtures was elaborated subsequently by Drumheller [145]. The developments are much more relevant to the principles of immiscible mixture theory than to chemical kinetics themselves, and are not described here. Drumheller considers for simplicity a mixture of a (porous) solid and gas. The constitutive equations are again controlled by the immiscibility postulate (alternatively, the principle of phase separation): the constitutive equations of any constituent depend only upon the variables of this constituent, while the mixture variables (production terms) depend upon all variables. Mass production rates are included among the dependent variables. As Drumheller takes into account only one direct reaction between the gas (G) and solid (S), i.e. _ cG

¼ _ cS , only one constitutive rate equation is necessary, viz.:

_ cS

E E _ E ; G ; d ; r ; T ; G ; v ; v Þ:f ðL Þ ¼ fM ðFS ; AES ; TS ; F_ S ; A S S S G G S G M M G

ð8:40Þ

E

where _ c is the mass production rate, F the elastic part of true deformation gradient, AE the elastic part of distention gradient, T the temperature and G its gradient, d is the damage (represents damage to the solid matrix) and v is the material velocity; symbols S and G refer to the solid and gas constituents, www.scilet.com

Thermodynamics and mass-action chemical kinetics

85

respectively and symbol M stresses the whole mixture. This very general relation is again not simpliﬁed by subsequent development. Using a special constitutive postulate for the energy change of the solid component, and concentrating on only sufﬁcient conditions to fulﬁl the entropy inequality, one of these conditions may be written: _ cS ðGS yTS GG yTG Þ 0

ð8:41Þ

Here, G represents the (equilibrium) Gibbs energy. Condition (8.41) can be realized by the following constitutive postulate for the reaction rate: _ cS

¼ ðGS yTS GG yTG ÞFðLM Þ

ð8:42Þ

where FðLM Þ is an arbitrary positive scalar function of LM . Experience with detonations of granular explosives suggests that in this case the function might be proportional to the pressure. Even under these special conditions, rate equation (8.42) remains too general to be directly applied in practical kinetics. Thomas [146] considers an ‘‘open mass’’ continuum. His approach was motivated by the requirement to describe the evolution of complex material systems composed of constituents in varying physical states. The body of the mixture is taken to correspond to the body of one of the constituents (symbolized by r), preferably solid. This reference constituent deﬁnes the mixture. One of the consequences of this model is the non-constant mass of the whole mixture. While this can be acceptable and understandable, it seems that problems in balances arise. The total mass balance of the mixture is written in integral or local forms as: dr dt

ð

ð r dV ¼

Vr

3 X

dr rydt þ r Vr

! qir yqxi dV ¼

! 3 nþ1 X X qJi yqxi þ r ca _ ca dV

ð Vr

i¼1

i¼1

a¼1

ð8:43Þ or nþ1 X ca _ ca

qryqt þ div ðrvr Þ þ div J ¼ r

ð8:44Þ

a¼1

respectively. Symbol dr stresses the material derivative following the motion of the reference constituent body with volume Vr , r is the mixture density, vector vr the velocity of the reference constituent, vector J is the total diffusion mass ﬂux, ca is the mass fraction of constituent a and _ ca its mass supply rate per unit mass. www.scilet.com

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Miloslav Pekarˇ

It is assumed that _ cnþ1 : _ cr ¼ 0, so the total number of constituents is equal to n þ 1. The right-hand-sides of Eqs (8.43) and (8.44) are zero in the usual ‘‘closed mass’’ systems. The mass supply rate S is considered to be

S¼

nþ1 X ca _ ca

ð8:45Þ

a¼1

and is not in general equal to zero. The non-vanishing ﬁrst term on the right hand side of Eq. (8.43) is perhaps easily acceptable. However, it is not clear how the total mass supply (8.45) accords with simple mass conservation in chemical reactions, which is valid, of course, if we take into account the indestructibility of atoms [34]. This difﬁculty is illustrated in Thomas’s chapter 5, part V [146] where for the closed systems he arrives at div J ¼ rS

ð8:46Þ

and simply states that this cannot be physically enforced in reacting and diffusing mixtures and (only?) non-reacting (S ¼ 0) and diluted (i.e. J & 0) mixtures can be modelled as closed. Clearly, there is some confusion in mass exchange within a real mixture and in the reference-constituent-model for the mixture. It is not clear how in reality the non-reference constituents enter the volume occupied by the reference one (Vr ), react within it but not with the reference constituent, while keeping a non-zero mass supply (8.45). There are other ﬂaws. The independent variables are not clearly stated. At the beginning, the mixture density, species concentrations, diffusional mass ﬂuxes of all constituents, and velocity of the reference constituent are stated to be selected independent variables. Later, in discussion of inequality of entropy, a new and larger set of independent variables is suddenly deﬁned. And yet another set is used on the page before when expressing the derivative of free energy. Some relations, which are in continuum theories proved on the basis of particular constitutive equations and the general validity of entropic inequality, are here postulated and then used in discussion of the consequences of entropy inequality. In this paper the methodologies of rational and classical irreversible thermodynamics are inconsistently combined. Ignoring all the controversial points, let us look at the implications for reaction kinetics and rates. The following relationships for the mass supplies (i.e. equivalents of reaction rates) result in: www.scilet.com

Thermodynamics and mass-action chemical kinetics _

rca _ ca ðX; vÞ ¼ qFðX; vÞyqAa Ua ðX; vÞ;

87

a ¼ 1; . . . ; n

ð8:47Þ

where parentheses only show independent variables. Here F is the so-called generalized dissipation potential, which is determined by the entropy production sS or by the vectors of thermodynamic ﬂuxes JT and forces X as follows: ð1

ð1 sS ðlX; vÞdlyl ¼

FðX; vÞ ¼ 0

X JT ðlX; vÞdl

ð8:48Þ

0

v is the second part of the set of independent variables; details on independent variables can be found in the original source. Quantities Aa are given by the following equation: Aa ¼ ma yT

n X _ cb mb yT v2 y2T þ CyT ;

a ¼ 1; . . . ; n

ð8:49Þ

b¼1 _

_

where ma is the reduced chemical potential deﬁned by ma ¼ qCyqca , C being the _

free energy. Quantity Ua represents the components of the so-called nondissipative thermodynamic ﬂux (for details see again the original source). As the author states in the conclusion, to apply such a complex equation, models for the generalized dissipation potential and non-dissipative ﬂux must be constructed, as well as for the free energy. Thus, within one material model (of open mass continuum), a couple of other (material) models (of classical irreversible thermodynamics) are necessary. The practical value of equations like (8.47) plus (8.48) and (8.49) in chemical kinetics remains remote. Jabbour and Bhattacharya [147] made some compromise between rate equations that are too general and the direct use of simple mass-action kinetics. They proposed a continuum theory for the chemical vapour deposition of thin solid ﬁlms. The rate of reaction in the gas ﬂow is expressed by the mass-action difference of the forward and reverse rates with mass fractions in place of concentrations and with rate constants dependent on temperature and speciﬁc volume. Similarly, the rate of surface reactions is expressed with densities (per unit area) instead of concentrations and rate constants dependent on temperature, surface orientation (described by the unit normal vector) and on chemical composition (!) of the ﬁlm surface. Either rates are introduced a priori, as constitutive postulates. The authors test their consistency with thermodynamics. From the entropic inequality, it follows as sufﬁcient that all rate constants are non-negative. www.scilet.com

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8.5 Mu¨ller’s results Mu¨ller developed a linear approach [148] and went a step further beyond his and Shih-Liu’s preceding work [149]. Mu¨ller considers a mixture of inviscid ﬂuids and discusses only its linear case, i.e. constitutive equations are linear representations of isotropic functions. Mu¨ller operates with ‘‘reaction rate densities’’ Lp (p is the index of independent reactions) which are related to the mass supplies ta (called by him production densities of mass) of components a by: ta ¼

R X

gpa ma Lp

ð8:50Þ

p¼1

where gpa is the stoichiometric coefﬁcient of constituent a in reaction p and ma is the molecular mass of constituent a. The original constitutive equations of a mixture of inviscid ﬂuids [148,149] Lp ¼ Lp ðra ; grad ra ; va ; T ; grad T Þ

ð8:51Þ

are simpliﬁed in the linear approximation to [148,149] Lp ¼ Lp ðra ; T Þ

ð8:52Þ

In fact, Eq. (8.52) a priori postulates what had been proved long before by Samohy´l for his somewhat more general mixture model. Using the deﬁnition of equilibrium as a process of uniform temperature, with the same speed of all constituents and no chemical reactions, it is shown that entropy production vanishes and assumes its minimum at equilibrium (denoted by ‘‘ þ ’’). From this, the equilibrium mass-action relation follows [148,149]: n X

gpa Ma mþ a ¼0

ð8:53Þ

a¼1

where ma is the ‘‘intrinsic’’ chemical potential of constituent a deﬁned in the course of applying entropic inequality to simplify constitutive equations (for details see refs [148,149]). Equation (8.53) provides some constraints on the equilibrium densities of the constituents, leaving only n R of them independent. Due to constitutive equations (8.52), the rate densities should vanish after insertion of equilibrium densities (and temperature) into them. Capitalizing upon (8.53), the constitutive equations are then written as [148] www.scilet.com

Thermodynamics and mass-action chemical kinetics

Lp ¼

R X

L pq ðr1 ; . . . ; rnR ; T Þ

q¼1

n X

gqa Ma ma

89

ð8:54Þ

a¼1

Mu¨ller [148] also presents a simple example of its application – a binary mixture of atomic and molecular oxygen between two inﬁnite parallel plates with several assumptions (e.g. a stationary and one-dimensional process at constant total pressure). Only one reaction, viz. 12 O2 ¼ O is considered and only the linear vicinity of equilibrium is treated. Thus, just one mass balance of components is sufﬁcient, e.g. of atomic oxygen numbered as 1: 1 rþ 1 q yqx1 ¼ m1 L

ð8:55Þ

Here ‘‘ þ ’’ means the equilibrium value, x1 is the coordinate in that one dimension in which the process occurs and 1 is the shortened notation of the (one-dimensional) velocity of constituent 1. In fact, example balances are written in a rather unusual way. The values of some quantities are taken at equilibrium whereas others are not considered to be ﬁxed and probably allowed to evolve not far from equilibrium. The former are called coefﬁcients, see, e.g. density in Eq. (8.55) and appear in balances as multipliers at partial derivatives of the latter. This is what should be understood under the ‘‘linearization about an equilibrium’’ in this example. Thus, when temperature appears in balances as a multiplicative factor, it takes the equilibrium value, whereas when it appears in the partial derivative, it is a general (ﬁeld) variable. The only exception is chemical potential which appears also in the product due to Eq. (8.54) with a non-equilibrium value, see Eq. (8.56) below, because otherwise the whole example would be nulliﬁed (zero reaction rate). Instead of linearization close to equilibrium, the whole procedure should be simply called a model. Introducing constitutive equation (8.54) into Eq. (8.55) we obtain: 2 1 þ rþ 1 q yqx1 L ðm1 Þ ðm1 m2 Þ ¼ 0

ð8:56Þ

(molecular oxygen takes number 2). This balance equation is solved together with the others giving the result pﬃﬃﬃﬃﬃﬃﬃ m1 m2 ¼ b sinh ABx1 ð8:57Þ which could be, in principle, used to compute the reaction rate from Eq. (8.54). Symbols b, A and B stand for rather complicated expressions which can be found in the original reference [148]. However, to compute a reaction rate from www.scilet.com

90

Miloslav Pekarˇ

Eq. (8.54), the (equilibrium) value of coefﬁcient L þ is also needed. As stated by Mu¨ller [148], the only way to get it is by calculation from statistical mechanics. This simple example clearly illustrates the common problems of many theoretical treatments – too many approximations and simpliﬁcations lead to too complicated expressions, which are difﬁcult to use in practice. Mu¨ller’s later treatment goes deeper and develops so-called rational extended thermodynamics including relativistic effects. The principle of relativity was added to the entropy principle (split into the entropy inequality and into stability, i.e. convexity conditions) as one of the universal principles to be satisﬁed by constitutive or ﬁeld equations. Relativity demands form invariance of the equations under Galilean or Lorentzian transformations, depending on whether the theory is non-relativistic or relativistic. Rational extended thermodynamics lies between extended irreversible and rational thermodynamics. It combines adding ﬂux variables, etc. with constitutive equations restricted by entropic inequality and the kinetic theory (of gases). The main substance of its contribution to the kinetics of chemically reacting systems is given in the paper by Kremer and Mu¨ller [150] which is reproduced in a shortened version in the book by Mu¨ller and Ruggeri [151]. A simpler, non-relativistic derivation is given in the paper by Kremer and Mu¨ller [152]. In fact, the results are given only for a non-diffusing binary mixture (of ideal gases). The authors claim that in a reacting mixture, the rest mass is not conserved and there is a mass-defect M given by M¼

X ga ma

ð8:58Þ

a

where ga is the stoichiometric coefﬁcient and ma is the molecular mass of constituent a. The mass-defect determines the heat of reaction which is Mc2 , where c should be the speed of light. The reacting mixture is exempliﬁed by a binary, non-diffusing mixture described by 15 ﬁelds: mixture fugacity (a), chemical afﬁnity (A), temperature (T ), mixture velocity vector (U), stress deviator tensor (t), dynamic pressure (p; non-equilibrium part of the pressure) and heat ﬂux vector (q). Mixture fugacity is deﬁned using equilibrium (!; symbol þ ‘‘ þ ’’) values of the components’ fugacities as a ¼ g1 m1 aþ 1 ¼ g2 m2 a2 and P afﬁnity by A ¼ ga ma aa . The relevant balance equations and constitutive a equations for, among others, reaction rate density (L) are written as www.scilet.com

Thermodynamics and mass-action chemical kinetics

div

X a

ma na U ¼

X

ga m a L

91

ð8:59Þ

a

(na represents the component’s number density) and _

L ¼ Lða; A; T ; U; t; p; qÞ

ð8:60Þ

respectively, where the reaction rate density is deﬁned as in Eq. (8.50) (only one reaction is considered). The complete derivation of further results is not given in the original papers because of the lengthy cumbersome algebra. It is therefore not easy to trace the origin of the varied and complex equations found in them. Nevertheless, some remarks can be made. The principle of relativity requires that, among others, the reaction rate density is an isotropic function of its variables and attention is then restricted to linear functions, i.e. the reaction rate density is a linear function of only the chemical afﬁnity and dynamic pressure. The latter is in fact a ‘‘linearization of the linear isotropic function.’’ After transformation of variables, and the requirement of the general validity of entropic inequality, the latter is simpliﬁed to X l ga m a L l I 0

ð8:61Þ

a

The lambdas (l and elements of the tensor l) belong to the set of Lagrange multipliers introduced as a result of incorporating balance equations into the entropic inequality. Tensor I (of the 3rd order) is called the ﬂux production and it is the divergence of the ﬂux tensor, which is of the 4th order. The next step is to ﬁnd the physical meaning of the formal Lagrange multipliers, i.e. determining them in terms of physical quantities. This can be done (simply) only for nearequilibrium processes. Because at equilibrium, the productions L and I must vanish, it is concluded from (8.61) that l and l also vanish. It is not clear how this conclusion is arrived at, because the reaction rate disappears at equilibrium by deﬁnition, and not due to the vanishing of some Lagrange multipliers. The relevant equations or terms are then linearized near equilibrium. For the reaction rate density, the following equation is ﬁnally derived þ L ¼ ð1yDÞðMc2 ykT Þ ðnþ ð8:62Þ 1 þ n2 Þ div U ð1ykT Þ div q where D stands for www.scilet.com

92

D¼

Miloslav Pekarˇ

2 2

2 Mc Mc þ3ðg1 þ g2 Þ kT kT ð3y4Þg1 g2

2 m2 ðnþ Þ2 þ m22 ðnþ m21 þ m22 2Þ 2 1 1 þ 5 6 þ m1 m2 m 1 m 2 nþ 1 n2

! ð8:63Þ

k being the Boltzmann constant. An alternative expression can be derived from the linear constitutive representation of the reaction rate density mentioned above and approximating the dynamic pressure p by the Maxwellian iteration of the kinetic theory of gases. The result is as follows: þ A p L ¼ ð2lp y3DBp1 ÞðMc2 ykT Þ2 ðnþ 1 þ n2 Þ div U þ ðlA lp B1 yB1 ÞA

ð8:64Þ

where lp and lA are coefﬁcients of the linear representation of the reaction rate density and Bp1 and BA 1 are coefﬁcients from similar linear representation of the ﬂux production tensor I. The resulting Equations (8.62) and (8.64) are really complicated considering that they are only linear, near-equilibrium approximations for nondiffusing, two-component ideal gas mixture with only one reaction, i.e. with some isomerizations at most. Their practical value is perhaps low not only due to this complexity but also due to the following consequences. Equation (8.62) states that, with no expansion or heating, there is no chemical reaction. This must sound rather strange to the ordinary chemical kineticist. Should it mean that in the constant volume adiabatic autoclave there is no possible chemical reaction? More serious is the absence of the very essence of chemical kinetics – the dependence of the reaction rate on the concentrations of reacting species. The only parameters involved are the molar masses and equilibrium particle densities, which are constants under given conditions. There is no change of rate with change of concentrations. Certainly, this is a very serious drawback to the near-equilibrium approximation. Equation (8.64) is simply an attempt to recover a linear phenomenological relation of LIT between reaction rate and afﬁnity with still undetermined coefﬁcients. 8.6 Samohy´l’s achievements Perhaps the most signiﬁcant and successful step towards relating mass-action kinetics and thermodynamics was made by Samohy´l [153 – 155]. Using the standard rational thermodynamics procedure, Samohy´l proved [154 – 156] www.scilet.com

Thermodynamics and mass-action chemical kinetics

93

that, in a mixture of ﬂuids with ‘‘linear transport properties’’, reaction rates (collected in a vector J) are functions of only temperature and the densities of the mixture components (collected in a vector r): J ¼ JðT ; rÞ

ð8:65Þ

The elements of the vector J are the rates of individual reactions, J ¼ ðJ1 ; J2 ; . . . ; Jr Þ. Alternatively, this function can be expressed with (molar) concentrations (c) as independent variables instead of densities: J ¼ JðT ; cÞ

ð8:66Þ

Function (8.66) is approximated by a polynomial of degree M [153,157,158]: J¼

Z X

knb

b¼1

n Y

caba ;

a¼1

n X

nba M

ð8:67Þ

a¼1

This approximation can be viewed as an application of Weierstrass’ theorem on the polynomial approximation of continuous functions. Vector knb ¼ ðk1nb ; k2nb ; . . . ; krnb Þ of polynomial coefﬁcients is to be interpreted as a vector of rate constants and vector nb ¼ ðb1 ; b2 ; . . . ; bn Þ as a vector of reaction orders with respect to components 1; 2; . . . ; n. The number of terms in the polynomial is equal to Z¼

M X ðn þ k 1Þ! k¼0

ð8:68Þ

n!ðn 1Þ!

The fundamental point in further development in this methodology is, in fact, applying the condition of consistency of the kinetic and thermodynamic descriptions of equilibrium. From the kinetic viewpoint J ¼ 0 ðequilibriumÞ

ð8:69Þ

From the thermodynamic viewpoint, equilibrium concentrations must accord with the value of equilibrium constant Kp of the p-th reaction: Kp ¼

n Y pa cPa ;

p ¼ 1; 2; . . . ; n h ðequilibrium concentrationsÞ

a¼1

www.scilet.com

ð8:70Þ

94

Miloslav Pekarˇ

Here, it should be underlined that only independent reactions are considered, which number n h, where h is the rank of a certain matrix describing the atomic composition of the mixture components (see below). Ppa stands for the elements of the stoichiometric matrix, i.e. for the stoichiometric coefﬁcient of component a in reaction p. The consistency condition sets restrictions on the polynomial (8.67). The correct, consistent form of the polynomial is found by the standard rational thermodynamics procedure. First, the independent reactions are selected. Second, their equilibrium constants are used to express some concentrations as functions of the remaining ones. These expressions are substituted into the polynomial (8.67). The modiﬁed polynomial should be zero for arbitrary equilibrium values of the remaining concentrations. The arbitrariness demands further modiﬁcation of the polynomial – some coefﬁcients (rate constants) vanish, some others are related through the equilibrium constants. As the values of the coefﬁcients and equilibrium constants depend only on temperature, these results are valid also out of equilibrium and can be put into the initial (i.e. non-vanishing) polynomial (8.67). The ﬁnal rate equation results, having the same general form for all (independent) reactions. Details on the procedure, and examples of rate equations, can be found elsewhere [153,159]. For the purpose of this review it should be sufﬁcient to note the principal features of this method. These features result when interpreting the polynomial rate equations within the framework of traditional mass-action kinetics; that is, interpreting the polynomial terms as corresponding to individual steps of the reaction mechanism and (the difference of) their forward and reverse rates. Thus, polynomial coefﬁcients are considered to be rate constants. The principal features of Samohy´l’s method are as follows: Rate equations of the mass-action type are directly derived from thermodynamic considerations. The rate of any step may be affected also by species, which do not directly take

part in this step or, in other words, by the rate of (some) other steps; similarly, the effect of an inert additive on the reaction rate can be included naturally. The reaction mechanism, relevant for the description of kinetics, ‘‘automatically’’ follows from the rate equation, contrary to conventional procedure.

The complexity of the mechanism can be changed by the number of components, which consequently determines the number of independent reactions, or by the degree of the approximating polynomial. www.scilet.com

Thermodynamics and mass-action chemical kinetics

95

The resulting rate equations contain only integral-number reaction orders with respect to any component. The procedure starts with independent reactions only, however, the resulting rate equations, in the sense of above interpretation, may also contain other

reactions relevant for the description of the reaction kinetics. The resulting ‘‘kinetic’’ mechanism may be thus different from, and more complex, than the initial, let us say ‘‘thermodynamic’’ one. The procedure does not preclude (‘‘thermodynamically’’) dependent reactions to be kinetically important – reaction steps, which are not among the independent ones may occur in the rate equation. Rational thermodynamics naturally differentiates thermodynamic from kinetic independence. Kinetic equilibrium criteria are fulﬁlled with a more general equation than usual Guldberg – Waage, considering only true thermodynamic equilibrium constants, with no need for any ‘‘kinetic’’ equilibrium constant, given as the ratio of rate constants in the forward and reverse reaction directions. The method can be used to design a suitable mechanism for the reaction under study which, fully and consistently with (irreversible) thermodynamics,

describes the reaction kinetics. Samohy´l’s methodology can be viewed as a purely phenomenological theory of the mass-action law of chemical kinetics giving generally rate equations of type (2.18) or (4.1). It can be also considered as a modern and more general version of van’t Hoff’s equation (4.5). Let us consider just one example – the oxidation of carbon monoxide, CO þ 12 O2 ¼ CO2 , in the presence of some catalyst S. There are supposed to be six components (CO, O2, CO2, S, OS, COS) and, consequently, three independent reactions are possible. There were selected as 1.

O2 þ 2 S ¼ 2 OS

2.

CO þ S ¼ COS

3.

OS þ COS ¼ 2 S þ CO2

The approximation by the second degree polynomial leads to the following the rate equation:

1 1 J ¼ k100100 ðcCO cS K1 2 cCOS Þ þ k100010 ðcCO cOS K2 K3 cCO2 cS Þ 1 þ k010001 ðcO2 cCOS K1 1 K3 cCO2 cOS Þ

www.scilet.com

ð8:71Þ

96

Miloslav Pekarˇ

Thus, the following three reactions ‘‘appear’’ in this approximation necessary for the description of the kinetics of catalytic oxidation of CO: CO þ S ¼ COS CO þ OS ¼ CO2 þ S O2 þ COS ¼ CO2 þ OS As has been stated already, this method was derived for a mixture of ﬂuids with linear transport properties. This model should be plausible for most homogeneous chemically reacting systems but is in general inapplicable to heterogeneous mixtures or for complex (e.g. viscoelastic) ﬂuids. However, conventional phenomenological kinetics of such systems are inspired by equations used in homogeneous cases, anyway, and uses them directly. In other words, the massaction rate equations are frequently used in any system regardless of their phase or rheological state. More general studies indicate that, in more complex systems, the reaction rate can be a function of various deformation (kinematics) variables not usual in chemical kinetics (see also above). To stay within the scope of Samohy´l’s work, let us refer to his treatment of general reacting mixtures [160]. In it, reaction rates remain functions of deformation gradients, second deformation gradients, densities, density gradients, diffusion velocities, and velocity gradients of all components and of temperature and its gradient. Restricting discussion solely to mixtures of ﬂuids brings just moderate simpliﬁcation [161] – reaction rates are functions of densities, density gradients, diffusion velocities, and velocity gradients of all components and of temperature and its gradient. Samohy´l’s results also shed more light on the afﬁnity-reaction rate relation. For the reader’s convenience, it is now necessary to review the basic ﬁndings of linear algebra in chemical stoichiometry [34]. The most essential point is the material balance of the mass of the reacting mixture. If ra indicates the mass source of component a, representing the mass of component a produced or consumed by chemical reactions in unit time and volume, then the usual (molar) reaction rate of component a (Ja ) is deﬁned by: Ja ¼ ra yMa

ð8:72Þ www.scilet.com

Thermodynamics and mass-action chemical kinetics

97

where Ma is the molar mass (weight). The balance of total mass can be written as [35,162,163]: n X

ra ¼ 0

ð8:73Þ

a¼1

Utilising the last two equations, the material balance can be rewritten as: n X

Ja Ma ¼ 0

ð8:74Þ

a¼1

As Eq. (8.74) resembles the relation for the scalar product of two vectors, let us deﬁne an abstract n-dimensional vector space of mixture components U with base ea and reciprocal base ea . From the individual molar masses and reaction rates respectively, vectors M and J are constructed as follows: M¼

n X

M a ; ea ;

J¼

a¼1

n X

Ja ea

ð8:75Þ

a¼1

The material balance (8.74) can be rewritten as M?J ¼ 0. It can be proved [34] that space U can be decomposed into two complementary orthogonal subspaces V and W; the vector of molar weights lies in subspace W and that of reaction rates in subspace V: U ¼ V+W;

V ? W;

M P W;

JPV

ð8:76Þ

Subspace V is called the reaction subspace. The molar weights of components are determined by their composition and atomic weights: Ma ¼

z X

Msa Tsa

ð8:77Þ

s¼1

where z is the number of atoms, Msa is the atomic weight of atom s and Tsa indicates the number of atoms s in component a. Atoms are assumed to be indestructible and are thus conserved in chemical reactions: n X

Tsa Ja ¼ 0;

s ¼ 1; 2; . . . ; z

ð8:78Þ

a¼1

However, not all relations in (8.78) may be independent. Their maximum number is given by the rank of matrix k Tsa k which will be denoted by h. Keeping only linearly independent relations, Eq. (8.78) is transformed into: www.scilet.com

98

Miloslav Pekarˇ

n X Ssa Ja ¼ 0;

s ¼ 1; 2; . . . ; h

ð8:79Þ

a¼1

where the h6n matrix k Ssa k with rank h was constructed by elimination of linearly dependent relations from (8.78), i.e. rows (columns) from matrix k Tsa k. It can be proved [34] that vectors fs ¼

n X

Ssa ea ;

s ¼ 1; 2; . . . ; h

ð8:80Þ

a¼1

form a basis in the subspace W. Let us also select the basis for the subspace V and denote the (reciprocal) base vectors as dp , p ¼ 1; 2; . . . ; n h. As these vectors also belong to the (original) space U, they can be expressed using its basis: dp ¼

n X

Ppa ea ;

p ¼ 1; 2; . . . ; n h

ð8:81Þ

a¼1

Matrix k Ppa k of dimension ðn hÞ6n is a matrix of the stoichiometric coefﬁcients of components a in the independent reaction p. Thus there are n h independent reactions and only they are needed for the kinetic and thermodynamic description. As vectors f s and dp lie in orthogonal (and complementary) subspaces, their scalar product is zero: f s ? dp ¼

n X Ssa Ppa ¼ 0

or

ð8:82aÞ

a¼1

k Ppa k 6 k Ssa kT ¼k 0 k

ð8:82bÞ

Equations (8.82) relate the stoichiometric matrix with matrix k Ssa k describing the atomic composition of the constituents by linearly independent relations. Samohy´l deﬁnes chemical potential as follows: ga ¼ qðrf Þyqra

ð8:83Þ

where r is the mixture density, ra represents the density of component a, and f states for the speciﬁc free energy ( f ) as a function of relevant independent variables (component densities being among them), i.e. f ¼ f (independent variables). The molar chemical potential is then given by ma ¼ ga Ma

ð8:84Þ www.scilet.com

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As chemical potentials are inseparably associated with the components, they should and easily can be introduced into the component space U by deﬁning the vector of molar chemical potential in this space: m¼

n X

ma ea

ð8:85Þ

a¼1

Because the component space is decomposable to subspaces V and W, so is the vector m: m ¼ A þ B;

m P U;

A P V;

BPW

ð8:86Þ

Equation (8.86) expresses the molar chemical potential vector as a function of A and B. Projection A into the reaction subspace can be expressed using its base vectors: A¼

nh X

Ap dp

ð8:87Þ

p¼1

The relation of coordinates Ap to molar chemical potentials can be found using linear algebra and the above deﬁnitions: A p ¼ m ? dp ¼

n X

ma Ppa ;

p ¼ 1; 2; . . . ; n h

ð8:88Þ

a¼1

The ﬁnal expression in Eq. (8.88) is the classical deﬁnition of afﬁnity of the p-th reaction with a reverse sign. Therefore, vector A is called the (chemical) afﬁnity vector and Ap the (chemical) afﬁnity of reaction p. Similar equations can be written for the projection B, lying in the subspace W: B¼

h h X X Bs f s ¼ Bs f s s¼1

Bs ¼ m?f s ¼

ð8:89Þ

s¼1 n X

ma Ssa

ð8:90Þ

a¼1

Now we are ready to ﬁnd the relation between reaction rate and afﬁnity. Samohy´l has proved Eq. (8.78) using rational thermodynamics and a model of ﬂuids with linear transport properties. Supposing the invertibility of another www.scilet.com

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proved function, viz. ga ¼ ga ðT ; rÞ, i.e. assuming the regularity of the matrix k qga yqra k, which was proved, for example, for stable mixtures [164,165], and then using Eqs (8.84), (8.85), densities can be substituted by chemical potentials and function (8.65) transformed to J ¼ JðT ; mÞ

ð8:91Þ

Introducing decomposition (8.86), the following function is obtained: J ¼ JðT ; A; BÞ

ð8:92Þ

Equation (8.92) states that, in general, there is no direct, unambiguous functional dependence of reaction rate on afﬁnity, as noted also by Bataille et al. [102] (cf. Part 5), even in the case of a simple model of linear ﬂuids. Besides afﬁnities, vector B also appears in such a function. The reason as to why the chemical potentials in Eq. (8.91) cannot be directly substituted only by afﬁnities lies in the deﬁnition of afﬁnity itself, Eq. (8.88). The latter equation does not, in general, provide an unambiguous expression for molar chemical potential ma as a function of afﬁnities; see the example below – the ﬁrst equality in (8.98a). This conclusion was arrived mathematically rigorously and not by making ad hoc and a priori statements on the functional dependence. The following example will aid better understanding. In the reaction mixture of NO2 and N2O4 there is only one independent reaction possible [154,155]. Let us select it as N2 O4 ¼ 2 NO2 and index it by 1. Using Samohy´l’s method of polynomial approximation with a second degree polynomial, the following rate equation is obtained [154,155]: J1 ¼ k01 c2 k01 K1 c21

ð8:93Þ

(i.e. virtually the usual mass-action expression) where NO2 is numbered as 1 and N2O4 as 2 and K is the equilibrium constant of the selected independent reaction. Unfortunately, an explicit particular expression for the equality in (8.86) is not known and no useful explicit relation (8.92) can be found from (8.91). Instead we are to use some model equation relating chemical potential and concentrations. The ﬁrst choice we have available is the common relation for ideal mixtures: ma ¼ ma ðT Þ þ RT ln ca

ð8:94Þ www.scilet.com

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where the standard chemical potential is a function of temperature only and concentrations are, in reality, divided by the standard value 1 mol dm 3. In fact, this model has already been used in Eq. (8.93) where the equilibrium constant was deﬁned generally by: RT ln Kp ¼

n X

ma Ppa

ð8:95Þ

a¼1

and transformed by (8.94) into: Kp ¼

n Y Ppa ðcþ aÞ

ð8:96Þ

a¼1

where symbol ‘‘ þ ’’ again indicates the equilibrium value. Using Eqs (8.94) and (8.95), chemical potential can be introduced into the rate equation (8.93) giving: J1 ¼ k01 expðm2 yRT Þ expðm2 yRT Þ expð2m1 yRT Þ ð8:97Þ which is a function of the form (8.91). Equations (8.88), (8.90), (8.89), and (8.93) give: A1 ¼ 2m1 m2 ;

B1 ¼ ðm1 þ 2m2 Þy5

ð8:98aÞ

m1 ¼ ð2y5ÞA1 þ B1 ;

m2 ¼ ð1y5ÞA1 þ 2B1

ð8:98bÞ

Combining (8.98b) and (8.97), the ﬁnal rate equation follows: J1 ¼ k01 expðm2 yRT Þ expð2B1 yRT Þ expð4A1 y5RT Þ expðA1 yRT Þ 1

ð8:99Þ

This is the rate equation of type (8.92), i.e. a reaction rate expressed as a function of afﬁnity and not of afﬁnity and concentration as in the case of equations like (2.4) or (2.6). Note that at reaction equilibrium, deﬁned as A1 ¼ 0, the reaction rate really vanishes, and note that rational afﬁnities are deﬁned with the opposite sign compared to the classical deﬁnition. The ﬁrst equality in (8.98a) clearly illustrates that there is no inversion function mi ¼ mi ðA1 Þ. Expression (8.99) is more complex than the usual classical analogue (2.4) or (2.6) and also contains, as emphasised above, quantity B. Let us compare the rational derivation with that based on the classical reaction isotherm. Using Eqs (8.88), (8.94), and (8.95), we obtain for the present example: A1 ¼ 2m1 m2 ¼ 2m1 m2 þ RT lnðc21 yc2 Þ ¼ RT ln K þ RT lnðc21 yc2 Þ ¼ RT lnðc21 yKc2 Þ www.scilet.com

ð8:100Þ

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Because there is no presumption as to the reverse rate constant and the relation between the ratio of rate constants and equilibrium constants in rational theory, there is no equation like (2.4) or (2.6). Of course, we can easily derive, combining Eqs (8.93) and (8.100), the rate equation J1 ¼ k01 c2 ½1 expðA1 yRT Þ but this is not an equation of type (8.92). The equilibrium implications of the rational rate equation (8.93) are as follows: 1 þ 2 þ 1 þ 2 0 ¼ k01 cþ 2 k01 K ðc1 Þ ) 0 ¼ c2 K ðc1 Þ

ð8:101Þ

and only the relation for equilibrium constant (8.96) is thus recovered. The ﬂaws of the (classical) reaction isotherm-afﬁnity-based procedures in chemical kinetics emphasised in Part 1 are conﬁrmed once more. Classical afﬁnity approaches are limited only within the model of the reverse rate constant and the identity between the thermodynamic equilibrium constant and the ratio of the forward and reverse rate constants. Rational thermodynamics offers a different, more general alternative - there is no need for any assumption about reverse rate constants, instead equilibrium constants are used in rate equations directly, and consistency between thermodynamic and kinetics is achieved straightforwardly simply through this thermodynamic quantity with no additional presumptions on a ‘‘kinetic equilibrium constant’’. Eq. (8.99) is a fully equivalent version of Eq. (8.93), of course, within the model used for chemical potential (8.94) and equilibrium constant (8.95). However, it is clear that the latter equation is much more useful for practical kinetics and evaluation of kinetic data. The algebraic essence of the conservation of mass and atoms conceals another interesting result which is ignored in equilibrium and other nonequilibrium theories. Because the reaction rate vector J is located in the reaction subspace V, whereas vector B is in its orthogonal counterpart W, the scalar product m?J is identically equal to A?J. Thus, vector B is ‘‘invisible’’ in the product m?J and also therefore in the modiﬁed form of the entropy inequality: A?J ¼

nh X

Ap Jp 0

ð8:102Þ

p¼1

(again, note that rational afﬁnities are deﬁned with the opposite sign in comparison to the traditional deﬁnition) which is often the starting point for deductions on afﬁnity-rate relationships. Flux-forces approaches of linear or extended irreversible thermodynamics usually postulate independent variables, and the idea of the afﬁnity as the variable (force) determining the reaction rate www.scilet.com

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(ﬂux) comes from entropy inequality of the type of Eq. (8.102). Consequently, vector B is overlooked as the other determining quantity in LIT (EIT). What is the meaning of quantity B? Eq.(8.90) indicates that it is also closely related to the molar chemical potential. It is located in the same subspace as the molar weights of the components where, on the other hand, reaction rates are not located. Thus, whereas afﬁnities relate the chemical potentials of individual components to the reactions (and their stoichiometry), in which the components take part, vector B relates the chemical potentials to the composition of the components. Matrix k Ssa k appearing in Eq. (8.90) followed from the atomic composition of the individual components, see text following Eq. (8.92). This matrix represents the atomic composition in linearly independent relations, i.e. using independent (pseudo)atomic substances. For instance, in the example discussed of the mixture with nitrogen oxides, the composition matrix k Tsa k reads (s ¼ 1 for nitrogen and s ¼ 2 for oxygen): k Tsa k¼

1 2 2 4

ð8:103Þ

and its rank is 1 and only one (pseudo)atomic substance exists and is necessary for the description of the mixture. Thus, the matrix k Ssa k can be selected as k Ssa k¼ ½1 2

ð8:104Þ

and represents the composition of nitrogen oxides in terms of the pseudoatomic substance NO2. Then B1 ¼ m1 þ 2m2 (and also B1 ¼ 5 B1 , cf. Eq. (8.98a)); recall that m1 is the chemical potential of NO2 and m2 of N2O4. Another selection of the matrix k Ssa k, viz. k Ssa k¼ ½2 4 gives the composition in terms of the pseudoatomic substance N1=2O and has perhaps only theoretical signiﬁcance. Whereas the afﬁnity combines chemical potentials according to the stoichiometric coefﬁcients of reactions in which individual components (may) take part, quantity B combines them according to the numbers of atomic or pseudoatomic substances which are contained in the individual components. The number of afﬁnities is equal to the number of (independent) reactions, cf. Eq. (8.101), the number of components of vector B, is equal to the number of (pseudo)atomic substances, cf. Eq. (8.102). Afﬁnity can be therefore viewed as the afﬁnity or potential of a particular specie in a particular reaction, whereas quantity B represents the afﬁnity or potential of (pseudo)atomic substances www.scilet.com

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themselves, which are in the number necessary and sufﬁcient for the thermodynamic description. Let us add two more examples in which there are no pseudo- but only real atomic substances, i.e. in which h ¼ z. The ﬁrst one is the simple combination D þ E ¼ DE, where n ¼ 3, h ¼ z ¼ 2; with numbering of atoms as 1 ¼ D, 2 ¼ E, and constituents as 1 ¼ D, 2 ¼ E, 3 ¼ DE matrices read:

1 0 1 k Tsa k¼ : k Ssa k ð8:105Þ 0 1 1 The components of B are then given by: B1 ¼ m1 þ m3 :mD þ mDE ;

B2 ¼ m2 þ m3 :mE þ mDE

ð8:106Þ

and we can consider B1 :BD ; B2 :BE . The second example refers to the abovementioned mixture involved in carbon monoxide oxidation. Here, n ¼ 6, h ¼ z ¼ 3, the atoms are numbered: 1 ¼ C, 2 ¼ O, 3 ¼ S, components: 1 ¼ CO, 2 ¼ O2, 3 ¼ CO2, 2 1 0 k Tsa k¼ 4 1 2 0 0

4 ¼ S, 5 ¼ OS, 6 ¼ COS, and the matrices are selected as: 3 1 0 0 1 2 0 1 1 5: k Ssa k ð8:107Þ 0 1 1 1

The components of B are then determined by: B1 ¼ m1 þ m3 þ m6 :mCO þ mCO2 þ mCOS B2 ¼ m1 þ 2m2 þ 2m3 þ m5 þ m6 :mCO þ 2mO2 þ 2mCO2 þ mOS þ mCOS B3 ¼ m4 þ m5 þ m6 :mS þ mOS þ mCOS

ð8:108Þ

and, in fact, B1 :BC , B2 :BO , B3 :BS . Equations (8.106) or (8.108) combine the constituents’ chemical potentials with respect to the individual atomic substances according to the number of particular atomic substances in all constituents. Vector B refers to the atomic constitution of the components in a similar or, so to speak, reciprocal manner as does vector M. The latter expresses the molecular weight as the respective sum of atomic weights, the former is the respective sum of chemical potentials of all constituents which contain the relevant atomic substance. Within this framework, what counts are not individual chemical potentials but their combinations, partly according to the individual reactions and partly according to the content of the (pseudo)atomic substances in the individual constituents. This is the essence of the decomposition in (8.99). Eq. (8.102) states that only the ﬁrst www.scilet.com

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part is signiﬁcant and acting in non-equilibrium effects described by the entropy inequality. Coordinates Bs of quantity B can be perhaps called the constitutive afﬁnity and B itself the constitutive afﬁnity vector; consequently, the molar chemical potential vector is decomposed into the chemical and constitutive afﬁnity vectors, see Eq. (8.99). To conclude, nearly all classical afﬁnity approaches to rate equations are valid only within the model of identity between thermodynamic and kinetic equilibrium constants, which is (at least one) necessary condition for deducing equations like (2.4), (2.6) or (2.7). This need not mean that such a model is incorrect, inappropriate, and useless. However, Eq. (8.92) and also other rational thermodynamic results given above, cast serious doubt upon nonequilibrium attempts to ﬁnd the reaction rate as a function of (only chemical) afﬁnity. In his typically cautious procedure, Haase [96] was very careful and avoided this model, see Eqs (5.2), (5.4), still persisting in afﬁnity as the quantity determining the reaction rate, cf. Eq. (5.1).

9. CHEMICAL POTENTIAL MODEL This short section concerns almost all reviewed thermodynamic treatments. It refers to expressing the dependence of chemical potential (m) on concentration or, generally, on activity (a) for the i-th component: mi ¼ m þ RT ln ai , cf. Eq. (2.5). This expression is commonly introduced as a ‘‘well-known’’, ‘‘common’’, ‘‘usual’’ or even as a deﬁnition of chemical potential and it looks as though its origin has been forgotten. Let us recall it. Chemical potential is, in equilibrium thermodynamics, deﬁned as, among others, the partial derivative of the Gibbs energy (G) at constant temperature (T ) and pressure (P): m ¼ ðqGyqnÞT ;P

ð9:1Þ

where n is the molar quantity. Considering volume work only and reversible, equilibrium processes only, the classical ﬁrst and second laws of thermodynamics can be combined giving, for example, the following equation for the differential of Gibbs energy: dG ¼ S dT þ V dP

ð9:2Þ

(S is the entropy, V is the volume) from which it follows that Gibbs energy is a function of temperature and pressure and also: www.scilet.com

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Miloslav Pekarˇ

V ¼ ðqGyqPÞT

ð9:3Þ

From Eq. (9.3) we have: Vm ¼ ðqmyqPÞT

ð9:4Þ

Integration of Eq. (9.4) from P to P with the ideal gas state equation results in: mðT ; PÞ ¼ mðT ; P Þ þ RT lnðPyP Þ

ð9:5Þ

This procedure can be easily extended to mixtures, which introduces no essential change to our discussion. Equation (9.5) is the basis for all expressions for the dependence of chemical potential on concentrations, fugacities or activities, and serves to deﬁne ideal gas, ideal or real mixtures. It is extended to the liquid state by means of vapour-liquid phase equilibrium. Thus, even the expression mi ¼ m þ RT ln ai should be viewed as a model, and not as a deﬁnition of chemical potential. It can, at most, be considered as a deﬁnition of some material system. This model comes from, and depends on, the ideas of equilibrium thermodynamics. Its validity in non-equilibrium theories and systems should be carefully tested. All theories using it have their validity restricted to the range of validity of this model. Of course, the validity of this model in equilibrium systems has been conﬁrmed exhaustively. Even the very deﬁnition (9.1) should be re-checked for non-equilibrium states. It is applicable only when it is proved that the Gibbs energy is a function of temperature, pressure and composition only. The chemical potential model is also included in Samohy´l’s methodology reviewed in the preceding section. Samohy´l deﬁned chemical potential by Eq. (8.83) where function f is not a function of pressure. Samohy´l proved that, in the simple model of linear ﬂuids, the free (and Gibbs) energy is a function of temperature and component densities only. Introducing classical (thermodynamic) pressure, even into this simple material model, is not straightforward [166,167]. The main cause is rooted in that the true (measured) pressure is generally not equal to the thermodynamic pressure P except at equilibrium. Independently of this model, Samohy´l has proved that reaction rate can be expressed as a function of chemical potentials (and temperature) only, at least in a stable mixture of ﬂuids with linear transport properties. As mentioned in the preceding section, matrix k qga yqra k (a is the constituent index) is regular in this www.scilet.com

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mixture when it is stable and enables one to substitute (molar) chemical potentials, cf. Eq. (8.84), in place of densities into the functional dependence of reaction rate (8.65). It should be stressed that the dependence of reaction rate on chemical potential was thus proved using the reversibility of the relevant function and not by stating this relationship a priori or by stating that free (Gibbs) energy is the driving force for reaction and its rate. Within rational thermodynamics there is no proof that reaction rate is a function of free energy (and temperature, at least) even in the simple case of linear ﬂuids, i.e. no proof exists of the invertibility of function f ðT ; rÞ or f a ðT ; rÞ. Probably, these functions are not invertible, because partial free energy ( fa ) is not ‘‘mixture invariant’’ [166 – 168] in contrast to chemical potential. 10. CONCLUSIONS Chemical kinetics seems to survive as the touchstone of all various thermodynamic theories. Thermodynamic approaches to the mass-action law hitherto achieved are still not very satisfactory. Speciﬁcally, the traditional mass-action law – an expression for the reaction rate as the difference between ‘‘forward’’ and ‘‘reverse’’ rates – is not discovered, recovered, proved. Very often, this law is only introduced in some way into the framework of a particular thermodynamic theory. The notable exception (perhaps the only one) is Samohy´l’s work on rational thermodynamics of ﬂuids with linear transport properties, although it is still more a matter of mathematical approximation and qualitative interpretation of approximating polynomials. Many (irreversible) deductions rely on afﬁnity as a device to relate reaction rate and thermodynamics. However, we have clearly shown that afﬁnity is not well suited for this purpose, particularly as a variable which should determine the value of a rate of reaction. Its real practical value consists, eventually, in determining how far the actual concentrations are from their equilibrium values. Most ‘‘thermodynamic’’ kinetic equations have only theoretical value. They are too complicated to be useful for practical kinetics and contain parameters whose values are difﬁcult to ﬁnd. Interestingly,

all

non-equilibrium

thermodynamic

approaches

to

chemical kinetics demonstrate the power of equilibrium. Even in Samohy´l’s approach, just equilibrium constants and the universality of equilibrium enabled arrival at ﬁnal, mass-action-type rate equations. www.scilet.com

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Samohy´l’s rational thermodynamic procedure also implies a possible change in current paradigms in the reaction mechanism-reaction rate arena. Traditionally, based on (kinetic) experiments, a reaction mechanism is proposed and from it reaction rates are formulated. Samohy´l’s methodology leads to the opposite approach. Knowing the species involved in the reacting mixture, (polynomial) rate equations are derived and, by interpreting them within the common mass-action law framework, an appropriate mechanism is found. Indeed, usually we do not measure mechanisms and even not just the reaction rates but we do measure concentrations of the determined reacting species. Samohy´l’s way thus looks more natural. Modern thermodynamic treatments show that chemistry is not just some simple generalization of mechanics or dynamics. It seems even that there is still not full agreement which variables are pertinent for a full and correct description of chemically reacting systems or, in the narrower sense, reaction rates occurring in these systems generally, i.e. not only in isotropic or homogeneous mixtures. Resolving this task calls not only for rigorous theoretical considerations but also for detailed experimental data on reaction kinetics in spatially inhomogeneous systems where mechanical (hydrodynamic, deformation) variables may also inﬂuence rates of chemical reactions. Future work is necessary to clarify also the role of ‘‘constitutive afﬁnity’’ (vector B from Part 8.) in chemical potential and reaction rate, the applicability of the traditional (equilibrium) expression relating chemical potential and activity in non-equilibrium states as well as using activity and activity coefﬁcients determined at equilibrium in non-equilibrium states. To conclude, from all the approaches discussed, only rational thermodynamics was able to come really close to the traditions of chemical kinetics and to derive the classical kinetic mass-action law. The other theories simply only implement it into their frameworks. ACKNOWLEDGEMENT I thank all authors of every work reviewed here, despite the existence of some inevitable criticism. Their ideas enabled me to better understand the role and signiﬁcance of thermodynamics in chemical kinetics. Particularly I am indebted to Ivan Samohy´l for many valuable discussions, and not only on his rational thermodynamics method. This work was partially supported by the Ministry of Education of the Czech Republic, project. No. MSM 0021630501. www.scilet.com

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