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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.






2.1. Reaction isotherm 2.2. Thermodynamic consistency of rate equations

6 9


4. 5.



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





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










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

74 75 76


Miloslav Pekarˇ

9. 10.

8.4. Treatments of more complex systems


8.5. Mu¨ller’s results


8.6. Samohy´l’s achievements



105 107


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

Thermodynamics and mass-action chemical kinetics


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.


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

Thermodynamics and mass-action chemical kinetics


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



(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



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:


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


? /

( 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


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 


n X

ði mi þ i RT ln ai Þ ¼ DGr  RT ln Qr



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


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


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

Thermodynamics and mass-action chemical kinetics


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


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


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


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


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


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


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


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


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


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


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



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

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




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


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


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


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:


Miloslav Pekarˇ

? /

k yk ¼ K1ys


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 


n Y

i ci;product


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


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


or, alternatively ci ðqryqci Þ ¼ li


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

Thermodynamics and mass-action chemical kinetics


n X ði mi Þ




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


where *






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


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




Miloslav Pekarˇ *

h 0T x ¼ T ðqS yqpÞT ;x  hT x *

C 0px ¼ T ðqS yqT Þp;x  Cpx



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


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


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


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



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.

Thermodynamics and mass-action chemical kinetics


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


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


(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



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


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


2 NOCl þ I2 ¼ 2 NO þ 2 ICl


Thermodynamics and mass-action chemical kinetics


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


and de Donder equation: ? /

lnð r y r Þ ¼ AyRT


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


providing that A1 r1 þ A2 r2 40


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


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

Ai ri 40



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



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

Thermodynamics and mass-action chemical kinetics


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


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


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


Miloslav Pekarˇ

z ¼ z1 z2 z3 ¼ aP yðKaR1 aR2 Þ


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

r ¼ k 1 aR1 ½1  aP yðKaR1 aR2 Þ


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


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

K3 ¼ aP yðaI1 aI2 Þ


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


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

Thermodynamics and mass-action chemical kinetics


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


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


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


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


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,


? rc


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:


Miloslav Pekarˇ





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



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



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




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



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



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.

Thermodynamics and mass-action chemical kinetics


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



(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



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


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,


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


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


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:

Thermodynamics and mass-action chemical kinetics


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


 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


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


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


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

Thermodynamics and mass-action chemical kinetics


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?


Miloslav Pekarˇ Table 1

Decay of logarithm and its argument




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

Thermodynamics and mass-action chemical kinetics


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





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



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


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

Thermodynamics and mass-action chemical kinetics


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


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


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


(C0s is initial sucrose concentration).

Thermodynamics and mass-action chemical kinetics


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


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



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


 cj j K reactants



 ci i products


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.


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


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


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.

Thermodynamics and mass-action chemical kinetics


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



(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


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


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


n Y

ai i


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.

Thermodynamics and mass-action chemical kinetics


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


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


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


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


and by analogy: /



r ¼  K c a33 a44 yg6¼ : k a33 a44 yg6¼



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


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 /





The final result is: ?


r ¼ k a11 a22 yg ?  k a33 a44 yg / =



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.

Thermodynamics and mass-action chemical kinetics


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


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



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

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





where o i ¼ ki


cmmi ; li ¼ ki 0 yki



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


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


(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


i¼1 j¼1

where XQ ¼ ð1yT ÞgradT


Xk ¼ Kk  ðgradmk ÞT



Miloslav Pekarˇ

Xij  ð1y2Þðqi yqzj þ qj yqzi Þ;

i; j ¼ 1; 2; 3


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



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


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



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

Thermodynamics and mass-action chemical kinetics


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


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


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


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


lre j



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].


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


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




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



where n ¼ ðai 0  ai Þyi


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   



where "

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

and index ‘‘eq’’ refers to equilibrium.


Thermodynamics and mass-action chemical kinetics


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



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,


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


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


ab Ab


with satisfaction of the Onsager reciprocity relations L



ba .


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


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



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


(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.

Thermodynamics and mass-action chemical kinetics


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


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


(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:


Miloslav Pekarˇ

Jr ¼ k 0r


ðnei 0 ynÞi 0 ;r



ðni 0 ynei 0 Þi 0 ;r  k00r


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



giving " Jr ¼


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

# ð5:20Þ


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



the reaction’s deviations from equilibrium

Dni ¼

R X i;r Dxr



(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






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




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


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



Thermodynamics and mass-action chemical kinetics




R X 1þ i00 ;r Dxr ynei00



!i00 ;r ð5:37Þ


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


in the form qJk yqAl ¼ qJl yqAk


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


Miloslav Pekarˇ



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


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




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



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



(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


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



Thermodynamics and mass-action chemical kinetics


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

At 0 ¼ TXt 0 ;

t ¼ 1; . . . ; S


(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





" Q X

gtu lt exp


J* þ 2


!# 0

gt X yR

¼1 S X


gtu lt exp 




dXu 0 !#

gt X00 yR



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



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 


!# gt X00 yR


u ¼ 1; . . . ; Q


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)):


Miloslav Pekarˇ


J u* ¼



gtu J t ¼



gtu lt exp



! gt X 0 yR


u ¼ 1; . . . ; Q



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


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


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

f X

Gi qxi yqt



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


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



Thermodynamics and mass-action chemical kinetics


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





ðXt 0 þ X00t Þð J t  J t Þ



because of Gi ¼ mi yT A 0t ¼



 0i;t mi ;

A00t ¼



si ¼

00i;t mi




ð00i;t   0i;t ÞJt



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



Ju* ðXt 0 þ J00t Þ



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


and ?



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




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



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


lki Ji ;




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




lki ð J i  J i Þ;




This equation can be modified as ?


J k  Jk ¼



lki J i 




lki J i ;




Stating ?

Jk ¼

R X i¼1



lki J i ; J k ¼



lki J i ;




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.

Thermodynamics and mass-action chemical kinetics


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


ðai Þi;r  kbr



ðai Þi;r



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


Miloslav Pekarˇ

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


(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


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



the symmetry means Rij ¼ Rji


or qXi yqJj ¼ qXj yqJi


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


Thermodynamics and mass-action chemical kinetics


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


(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


(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


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


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

Thermodynamics and mass-action chemical kinetics


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


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


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


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



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


Miloslav Pekarˇ

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


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


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


(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


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


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


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

Thermodynamics and mass-action chemical kinetics

mk ¼ mk þ mEk þ RT ln ck



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


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

dFðþÞ ¼ Ur dð1yT Þ  R d ln r

Y ðþÞ ckk;r




dFðÞ ¼ Ur dð1yT Þ  R d ln r

Y ðÞ ckk;r



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



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



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


Miloslav Pekarˇ

ji ¼ qDFS yqFi


and also Fi ¼ qDjS yqji


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


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.

Thermodynamics and mass-action chemical kinetics


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



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


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


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


X ¼ m1 p þ m2 J þ   


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),

Thermodynamics and mass-action chemical kinetics



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


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


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


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


Miloslav Pekarˇ

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


(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


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


(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


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


Thermodynamics and mass-action chemical kinetics


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


xi Ji


lij Jj



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


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


Miloslav Pekarˇ

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


(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


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


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


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.

Thermodynamics and mass-action chemical kinetics


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


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


Miloslav Pekarˇ

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


(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


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


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!

Thermodynamics and mass-action chemical kinetics


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


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


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


sb dV

V ðtÞ



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


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


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



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

Thermodynamics and mass-action chemical kinetics


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


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


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.


Miloslav Pekarˇ

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


where va is the velocity of component a.

Thermodynamics and mass-action chemical kinetics


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


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


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

F_ ¼ 0;

wðuþ ; 0; Fþ ; 0; jþ Þ ¼ 0



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


where ba ¼ jdet Fa j _ ca


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

Thermodynamics and mass-action chemical kinetics


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



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



Ppa jp



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


and ga ¼ ra jdet Fa j


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


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


 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

78 þ Ap ðyþ ; Fþ b ; gb ; 0Þ ¼ 0;

Miloslav Pekarˇ

p ¼ 1; . . . ; R


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


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


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

Thermodynamics and mass-action chemical kinetics


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


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



(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



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



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



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


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

ma ma ¼ OðjL  Lþ j2 Þ


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



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



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

n X tab mb þ OðjL  Lþ j2 Þ



Thermodynamics and mass-action chemical kinetics


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



then R X þ 2 Jr Aþ r ¼ OðjL  L j Þ



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


Miloslav Pekarˇ

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


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


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



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


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:


Thermodynamics and mass-action chemical kinetics n X

þ mþ b ðqFb yqca Þ ¼ 0;

a ¼ 1; . . . ; n




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


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


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


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



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,

Thermodynamics and mass-action chemical kinetics


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


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



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 r dV ¼


3 X

dr rydt þ r Vr

! qir yqxi dV ¼

! 3 nþ1 X X  qJi yqxi þ r ca _ ca dV

ð Vr




ð8:43Þ or nþ1 X ca _ ca

qryqt þ div ðrvr Þ þ div J ¼ r



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.


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

nþ1 X ca _ ca



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


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:

Thermodynamics and mass-action chemical kinetics _

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


a ¼ 1; . . . ; n


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



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


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.


Miloslav Pekarˇ

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


gpa ma Lp



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


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


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



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]

Thermodynamics and mass-action chemical kinetics

Lp ¼


L pq ðr1 ; . . . ; rnR ; T Þ


n X

gqa Ma ma




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


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


(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


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



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

Thermodynamics and mass-action chemical kinetics


X a

ma na U ¼


ga m a L




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

L ¼ Lða; A; T ; U; t; p; qÞ


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



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


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


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]

Thermodynamics and mass-action chemical kinetics


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


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


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




n Y

caba ;


n X

nba  M



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


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


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




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.

Thermodynamics and mass-action chemical kinetics


 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


CO þ S ¼ COS


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



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


Thermodynamics and mass-action chemical kinetics


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

ra ¼ 0



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

Ja Ma ¼ 0



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 ;


n X

Ja ea



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;



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



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



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:


Miloslav Pekarˇ

n X Ssa Ja ¼ 0;

s ¼ 1; 2; . . . ; h



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



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



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




k Ppa k 6 k Ssa kT ¼k 0 k


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


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


Thermodynamics and mass-action chemical kinetics


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



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;



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



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



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


s¼1 n X

ma Ssa



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


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


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


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


(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


Thermodynamics and mass-action chemical kinetics


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



and transformed by (8.94) into: Kp ¼

n Y Ppa ðcþ aÞ



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


m1 ¼ ð2y5ÞA1 þ B1 ;

m2 ¼ ð1y5ÞA1 þ 2B1


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


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



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


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



(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

Thermodynamics and mass-action chemical kinetics


(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


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


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


Miloslav Pekarˇ

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


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


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

Thermodynamics and mass-action chemical kinetics


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


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


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


Miloslav Pekarˇ

V ¼ ðqGyqPÞT


From Eq. (9.3) we have: Vm ¼ ðqmyqPÞT


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


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

Thermodynamics and mass-action chemical kinetics


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,






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.


Miloslav Pekarˇ

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.

Thermodynamics and mass-action chemical kinetics


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