thermodynamics and mass action law.pdf

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REVIEW

Thermodynamics and foundations of mass-action

kinetics

Miloslav Pekarˇ*

Institute of Physical and Applied Chemistry, Faculty of Chemistry, Brno University of Technology, Purkynˇova 118, 612 00 Brno, Czech Republic.

E-mail: [email protected]

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 thermody-namics. 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 thermody-namic 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 thermody-namics. Most published ‘‘thermodynamic’’ kinetic equations are too compli-cated 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.

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KEYWORDS: activated complex, activity, afﬁnity, chemical potential, continuum thermodynamics, equilibrium constant, extended irreversible thermodynamics, Guldberg – Waage law, ionic strength, irreversible dynamics, kinetic law, mass-action, Onsager reciprocity, rational thermo-dynamics, 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 ‘‘(reac-tion) 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 (indepen-dent) 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 phenomenolo-gical kinetics which is the primary motivation of this review. Only phenomen-ological 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 thermo-dynamic 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.

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Chemical kinetics and thermodynamics are usually considered as two independent disciplines describing reacting systems. Thermodynamics is said to state the conditions for the running and equilibrium of chemical reactions, while giving no information on how fast this all happens. The latter is the domain of kinetics. This review should further demonstrate that the relationships between thermodynamics and kinetics are much closer and that even from solely thermodynamic theories, some inferences on reaction rates can be obtained.

Boyd [21] notes that, in contrast to thermodynamics, the kinetic descrip-tion of a reacdescrip-tion 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 thermody-namics 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

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each of the reacting species? Or is it some other concentration (e.g. molar fraction) or thermodynamic (chemical potential, activity) variable? These questions are not (sufﬁciently) answered by (classical) thermodynamic theory. Kinetic experience tells us that just the molar concentration is a very important variable, and that the rate can be expressed as the difference of two terms containing small powers of the molar concentrations.

Denbigh further states that thermodynamics places only two require-ments 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

0 ¼X

n

i¼1

iAi ð2:1Þ

(iis the stoichiometric coefﬁcient, which is positive for products and negative

for reactants) it is written as follows:

DG_r¼DG_rþRTlnY n i¼1 ai i:DG rþRT ln Qr ð2:2Þ

where Q_r is called the reaction quotient and DG

r 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

concentra-tions. If the forward and backward reaction rates (r with respective arrow) are expressed according to the Guldberg – Waage law with orders equal to stoichio-metric coefﬁcients, the reaction isotherm can be modiﬁed as follows:

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DG_r¼ RT ln K þ RT lnY n i¼1 ci i ¼ RT ln K þ RT ln Yn i¼1 ðk ? k / y k ? k / Þci i ¼ RT ln K þ RT ln½ð k ? y k/Þð/ry r?Þ ð2:3Þ ( k ? ; k /

are the rate constants in respective directions). Identifying the thermo-dynamic with the kinetic ( k

?

y k/) equilibrium constant, the ﬁnal equation results:

DG_r¼RT lnð r?y r/Þ ð2:4Þ

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

A ¼ X n i¼1 _im_i¼ X n i¼1 ð_im i þiRT ln aiÞ ¼ DGrRTln Qr ð2:5Þ

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. dimen-sional 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 ?r½1 expðAyRT Þ ð2:6Þ

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

expðAyRT Þ ¼ 1 ðAyRT Þy1 þ ðAyRT Þ2y2 ) r % r?AyRT ð2:7Þ A linear relationship between reaction rate and afﬁnity is thus obtained.

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

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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 ? ¼k ? ffðciÞ; r / ¼k / fbðciÞ ð2:8Þ

where ci, i ¼ 1; . . . ; n, represent the concentrations of reacting species, to be

consistent with the thermodynamic equilibrium condition (and constant), the ratio of forward ( k

?

) and reverse ( k

/

) rate constants must be equal to the equilibrium constant.

Manes et al. [24] correct the conclusions of Gadsby et al. The rates for opposing reactions are formulated as

r

?

¼f_fðc_iÞ; /r¼f_bðc_iÞ ð2:9Þ

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

at equilibrium : r?y r/:f_fyfb¼1; r ?

y r/41 when DG_r50 ð2:10Þ

In order to fulﬁl these conditions it is sufﬁcient to assume, for example, that

ffyfb¼ ðk ? y k / Þ Y i ci i " #z ; where k ? y k / ¼Kz ð2:11Þ

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

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

Q_r¼KðT Þand f ¼ 1 ðequilibriumÞ ð2:12Þ

A sufﬁcient condition for this is that f be expressible as a function of Qrsuch that

f ðci; T Þ ¼ FðQr; T Þand FðK; T Þ ¼ 1 ð2:13Þ

A necessary and sufﬁcient condition for Eqs (2.13) to hold could be that FðQ; T Þ be expressible as a function of Q_ryKðT Þ such that

FðQ_r; T Þ ¼FðQ_ryKÞ and Fð1Þ ¼ 1 ð2:14Þ

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

FðQ_ryKÞ ¼ ðQ_ryKÞz ð2:15Þ

In a subsequent paper [26], Hollingsworth states that the conditions (2.14) are not necessary although sufﬁcient. He presents other sufﬁcient conditions:

f ðci; T ; ujÞ ¼FðQryK; ujÞand Fð1; ujÞ ¼1 ð2:16Þ

where uj stands for a set of non-thermodynamic variables. Hollingsworth then

shows that the necessary condition when f has continuous derivatives of all orders at QryK ¼ 1 is: it must be possible to express ( f 1) as a function which

is divisible by the function (Q_ryK 1) in the neighbourhood of QryK ¼ 1:

f 1 ¼ ðQ_ryK 1ÞCðc_i; T ; u_jÞ ð2:17Þ

It should be added that in his proof the invertibility of the function ðQ_ryKÞðci; 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: r ¼ k ? jY m i¼1 aai i k / jY m i¼1 aa 0 i i ð2:18Þ

where j is some function of activities, a_i, of reacting species, and a_i and a0

i 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

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(see below). Boyd reproduces it [21] in more general form with k

?

j and k

/

j0_,

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 j0_{. 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 ? y k / ¼Y n i¼1 aða0aiÞ i;eq ð2:19Þ

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

?

y k/¼f ðKÞ ð2:20Þ

As the equilibrium activities of all species except one may be selected arbitrarily, it is shown that function f inevitably has the form f ðKÞ ¼ Kz_where

z ¼ ða0_ia_iÞy_i; i ¼ 1; . . . ; n ð2:21Þ

Condition (2.11)2was 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/¼expðAysRT Þ ð2:22Þ

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

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k

?

y k

/

¼K1ys ð2:23Þ

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

u ¼ KY nr i¼1 ci_i;reactant Y n i¼n_rþ1 ci i;product ð2:24Þ

where n_rsymbolizes the number of reactants and c_irepresent 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:

ðqryqc₁Þyðquyqc₁Þ ¼ ðqryqc₂Þyðquyqc₂Þ ¼ ¼ ðqryqcnÞyðquyqcnÞ ð2:25Þ

Using Eq. (2.24) these equations become:

ðc_iyiÞðqryqciÞ ðciþ1yiþ1Þðqryqciþ1Þ ¼0 ð2:26Þ

or, alternatively

c_iðqryqc_iÞ ¼l_i ð2:27Þ

where l is a negative constant.

Equations (2.26) or (2.27) represent the consistency condition to be fulfilled by any rate equation (expression for r) to be consistent with thermo-dynamics 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

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r ¼yX

n

i¼1

ð_im_iÞ ð2:28Þ

where y is some positive function of concentrations and m_i 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 non-rigorous 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 thermo-dynamics in the formdU 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 U

*

ðp; T ;xÞ, the following relation for the differential of heat (Q) was derived: dQ ¼ h_T_xdp þ C_pxdT r_pTdx ð3:1Þ where h_T_x ¼ ðqU * yqpÞT ;xþpðqV * yqpÞT ;x Cpx ¼ ðqU * yqT Þp;xþpðqV * yqT Þp;x rpT ¼ ðqU * yqxÞp;T þpðqV * yqxÞp;T ð3:2Þ

De Donder also supposed that the second law of thermodynamics could be written (according to Clausius) as TdS dQ:dQ0_{0 and that entropy was a}

function of the same variables. Thus

dQ0¼h0_T_xdp þ C0_pxdT r0pTdx ð3:3Þ

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h0_T_x ¼T ðqS * yqpÞ_{T ;x}h_T_x C0_px ¼T ðqS * yqT Þ_p;xCpx r0 pT ¼T ðqS * yqxÞ_p;T þr_pT ð3:4Þ

From Eq. (3.3) de Donder derived dQ0_{y dx ¼ h}0

Txdpy dx þ C0pxdT y dx r0pT ð3:5Þ

Next he introduced the key hypothesis which is neither well substantiated nor supported: the derivativedQ0_{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

h0Tx¼0

C0px¼0

ð3:6Þ

and deﬁned afﬁnity as A ¼dQ0ydx: r0

xy; ð3:7Þ

where xy stands for the two (constant) independent variables other than the extent of reaction.

The reason why de Donder’s affinity often ‘‘works’’ lies probably in that it is applied under conditions where some quantities are constant, as indicated by Eq. (3.7) so the conditions (3.6) are superfluous. Further, affinity can be related to the chemical potential which is also defined by several alternative relations under conditions of constant various pairs of independent variables while not changing its value. For example, the affinity of a reaction is simply given by the first relation in (2.5). Expressing the total differential of the Gibbs energy as a function of temperature, pressure and composition, G ¼ G

*

ðT ; p; n_iÞ, using the extent of reaction as de Donder suggested, we obtain:

dG ¼X

i

m_idn_i¼X

i

_im_idx ¼ A dx ðconstant T and pÞ ð3:8Þ

As at constant temperature and pressure, heat is identical with the change of enthalpy (H),dQ0_{¼ dG under these conditions and Eq. (3.7) is derived with no}

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

As regards chemical kinetics, de Donder deduced from the second law and (3.8)

0 dQ0_{ydt ¼ AðdxydtÞ:Ar} _ð3:9Þ

where r is the reaction rate.

3.2 Successors to De Donder

Most applications of afﬁnity in chemical kinetics are, in fact, deductions based on the reaction isotherm outlined in Section 1.

The ﬁrst work from this area is probably the paper by Prigogine et al. [36], which also refers to de Donder’s work. They started from the assumption that both reaction rate (r) and afﬁnity (A) depend on the same variables (xi) and

that the function for afﬁnity is invertible in at least one variable. Substituting this variable in the function for reaction rate, the following relationship results:

r ¼ f ðx1; x2;. . . ; xl; AÞ ð3:10Þ

(variable xlþ1 was substituted). At equilibrium, both reaction rate and afﬁnity

vanish. Expanding the function in (3.10), a close-to-equilibrium linear relation-ship is thus obtained:

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(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 (a_j; j ¼ 1; 2; . . . ; m) and that the reaction rate depends on the same variables and some added, ‘‘non-thermodynamic’’ 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 xða_j; b_kÞ DG ð3:12Þ

where the proportionality factor represents:

xðaj; bkÞ ¼ ½qryqðDGÞa2;a3;...;am ¼ ½qryqðDGÞa1;a3;...;am ¼. . . ¼ ½qryqðDGÞa1;a2;...;am1 ð3:13Þ and depends on full sets of ajand 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-thermo-dynamic 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 concentra-tions of components in a reacting mixture of NO, Cl2, NOCl, I2, and ICl. Two

reactions were considered, viz.:

2 NO þ Cl₂¼2 NOCl ð3:14aÞ

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Of course, affinities were not measured but calculated from the reaction isotherm and concentration profiles. Rates of reactions (3.14a) and (3.14b) were taken as time derivatives of the chlorine and iodine concentrations. It was found that, for some portions of a run of the reacting system, the linear relationship is valid. In general, however, it was not verified as well as the so-called 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 DGrð:AÞ with DG

6¼

r, i.e. the

activation Gibbs energy, which might be questionable.

Boudart shows in several papers more precisely the potential practical value of afﬁnity-containing equations in chemical kinetics. He distinguishes [40] between the de Donder inequality:

Ar 0 ð3:15Þ

and de Donder equation:

lnð r?y r/Þ ¼AyRT ð3:16Þ

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

A₁r₁50 ð3:17Þ

providing that

A₁r₁þA₂r₂40 ð3:18Þ

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

r ¼ r?_i/r_i40 ð3:19Þ

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

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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 non-negative 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:

z_i¼expðA_iyRT Þ ¼ Y j aij j ! yKi ð3:21Þ

where A_i is the step afﬁnity, a_j 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)2is used in the analysis and to calculate the

reversibility. The rate equation is not derived but step rates are stated as mass-action 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

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rate constants are bounded by the total equilibrium constant. However, if it is realized that rate constants of each step are related by the kinetic equilibrium constant of the step, it immediately follows that only three kinetic parameters are necessary (and selectable independently).

Reversibilities for each step are calculated from experimental data. Steps with close-to-one reversibility are (quasi-)equilibrated. If there is a step with reversibility far from a zero value, then this step is considered to be rate determining, and the overall reaction reversibility is equated to its reversibility whereas the other reversibilities are identiﬁed with unity. The overall rate is set equal to the rate-determining step rate. The whole procedure closely resembles the classical Langmuir – Hinshelwood – Hougen – Watson approach. This is felt also by the author as he states that his approach is advantageous because it provides the means to derive the overall reaction rate from the more general case where multiple steps are not in quasi-equilibrium. In fact, this means only that equilibrium constants of equilibrated steps, together with the overall equilibrium constant given as appropriate product of steps equilibrium constants, are used to eliminate intermediate activities.

Let us illustrate this approach by the simple example of the three-step mechanism

R₁¼2 I1

R₂þI₁¼I₂ I₁þI₂¼P

of the overall reaction R₁þR₂¼P

The rate of the ﬁrst step can be expressed as [44]:

r₁¼ k

?

1aR₁ð1 z1Þ ð3:22Þ

where z₁ is given as follows from Eq. (3.21): z₁¼a2_I

1yðK1aR1Þ ð3:23Þ

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

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z ¼ z₁z₂z₃¼aPyðKaR₁aR₂Þ ð3:24Þ

and z₂, z₃ are in this case equal to unity, it follows that r ¼ k

?

1aR₁½1 aPyðKaR₁aR₂Þ ð3:25Þ

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

r ¼ r₁¼ k ? 1aR1k / 1a 2 I1 ð3:26Þ

From equilibrium constants of (quasi-)equilibrated steps 2 and 3: K₂¼a_I

2yðaI1aR2Þ; K3¼aPyðaI1aI2Þ ð3:27Þ

it can be easily derived:

a2I₁¼aPyðaR₂K2K3Þ ð3:28Þ

Introducing Eq. (3.28) into Eq. (3.26) and using the kinetic deﬁnition of equilibrium constant K₁ and the relation K ¼ K₁K₂K₃, 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 equili-brium constants of these steps to eliminate some (intermediates) concentra-tions. 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)2but 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

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concentra-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 ¼dxydt ¼ dn_iyðidtÞ ð3:29Þ

where n_i 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:

rc¼ryV ð3:30Þ

When the molality (m) reference basis is selected, we have:

r_m¼ryðn₀M₀Þ ð3:31Þ

where n₀ is the mole number of the solvent and M₀its molar mass. Clearly,

r_cV ¼ r_mn₀M₀ ð3:32Þ

Timmermann then combines the classical mass-action rate equation r_c¼?r_c/r_c, where r?_c¼ k ? c P ic i i and r / c¼k / c P jc j

j (i runs through reactants,

jthrough products), with the classical deﬁnition of afﬁnity A ¼ P

k

_km_k (k runs through both reactants and products). Chemical potential (m_k) is expressed also traditionally, m_k¼mo

kþRT lnðgkckycoÞwhere ‘‘o’’ denotes the standard state and

gkis the activity coefﬁcient on the molarity scale. Timmermann ﬁnally arrives at

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rc¼ r ? c 1 k / cKgðcoÞ k ? c Q k gk k expðAyRT Þ 0 B B @ 1 C C A ð3:33Þ

where Kgis the thermodynamic equilibrium constant on the molarity scale and

¼P_k. He states that r cannot depend on the particular language used to describe the intensive reaction rate (i.e. on the referential quantity), conse-quently, the factor in Eq. (3.33) must be the same for every kinetic description, that is unity: k / cKgðcoÞ k ? c Q k gk k ¼1 ð3:34Þ

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:

rm¼ r ? m 1 k / mKjðmoÞ k ? m Q k jk k expðAyRT Þ 0 B B @ 1 C C A ð3:35Þ

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 / cKgðcoÞ k ? c Q k gk k ¼ k / mKjðmoÞ k ? m Q k jk k ð3:36Þ

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.

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3.3 Garﬁnkle’s original approach

Yet another approach to affinity in relation to reaction kinetics was presented by Garfinkle. Actually, he takes the time derivative (symbolized by a dot) of the reaction isotherm written in terms of affinity (A) instead of the Gibbs energy (and with concentrations approximating to activities) [50]:

_ A

A ¼ RTX

i

ð2_iyciÞðdciyidtÞ ð3:37Þ

(_i is the stoichiometric coefficient and c_i 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 the affinity decay rate ( _AA) is obtained:

r ¼ ð _AAyRT ÞyX

i

2iyci ð3:38Þ

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):

_ A

A ¼ Arð1yt 1ytKÞ ð3:39Þ

where Arand tKare parameters to be determined. The latter is called the

most-probable 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,

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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: ln½ðtyt_KÞ expð1 tyt_KÞ, 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 equa-tion that represents the overall chemical reacequa-tion under observaequa-tion 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),I₂þSt ?IStI with a velocity deﬁned as

r ¼d½Stydt ¼ d½I₂ydt ¼ d½IStIydt ð3:40Þ

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

dcAydt ¼ k1cA

dc_Bydt ¼ k₁c_Ak₂c_B dc_Cydt ¼ k2cB

ð3:41Þ

where k₁is the rate constant of the stepA ?B and k₂of the stepB ?C. It is clear that the time derivatives are not in general equal, which is even more evident after inserting the analytical solutions:

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dcAydt ¼ k1c 0

Aexpðk1tÞ

dc_Bydt ¼ k₁c0_Aexpðk₁tÞ k₁k₂c0_A½expðk₁tÞ expðk₂tÞyðk₂k₁Þ dc_Cydt ¼ k1k2c0A½expðk1tÞ expðk2tÞyðk2k1Þ

ð3:42Þ

where 0 in the superscript denotes the initial concentration. So there is, in general, no simple single rate expression for the overall stoichiometric transfor-mation A ?C and no identity dcAydt ¼ dcCydt. Only when k24k1 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 indepen-dent 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

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embodying the reaction quotient and the equilibrium constant, which, in turn, contains equilibrium concentrations, the latter through the parameter t_K, 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 asdxydt and given by:

dxydt ¼ eðT ; ; xÞAðT ; ; xÞ ð3:43Þ

where x is the extent of reaction, e is the proportionality coefficient dependent on temperature (T ), specific volume () and extent of reaction, and A is the affinity determined by the same set of variables. First, Meixner states that the close-to-equilibrium 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 eðqAyqxÞ_{T ;}½x x_eðT ; Þ ð3:44Þ

Why the dependence on the extent of reaction is suppressed only in the first function from (3.43), and why only the second one, affinity, is expanded, is neither explained nor discussed. Coefficient e in (3.44) is thus effectively a constant, which is stated, e.g. by Procaccia and Gitterman [60], as a fact at the outset.

Kim and Baird [62] present a more correct derivation and expand, in fact, both functions in (3.43). In the end, however, they retain only the terms of first order and arrive at Eq. (3.44) once more. From their procedure, the motivation for Meixner’s inconsequent treatment of functions can be clarified a little. From Eq. (2.7) it is clear that coefficient e is the forward reaction rate [62], which is

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non-zero at equilibrium in contrast to the affinity. Consequently, the first term in the forward rate (or coefficient e) expansion is non-zero whereas that in the affinity expansion vanishes.

What does an approximation like (3.44) using the equilibrium forward rate as a constant not-far-from-equilibrium mean in reality? From the more general Eqs (2.4) or (3.16), it is seen that within this approximation, the affinity at a given temperature is given by const RTln r/. All affinity and, conse-quently, 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 approxima-tion 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 x0ðT ; Þ where e: r

?

is not constant and the subscript ‘‘0’’ denotes some point within this interval?

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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

Table 1 Decay of logarithm and its argument r ! y r lnð r!y rÞ 1.0000000000E þ 10 23.03 1.0000000000E þ 09 20.72 1.0000000000E þ 08 18.42 1.0000000000E þ 07 16.12 1.0000000000E þ 06 13.82 1.0000000000E þ 05 11.51 1.0000000000E þ 04 9.210 1.0000000000E þ 03 6.908 1.0000000000E þ 02 4.605 1.0000000000E þ 01 2.303 1.1000000000E þ 00 9.531E-02 1.0100000000E þ 00 9.950E-03 1.0010000000E þ 00 9.995E-04 1.0000001000E þ 00 1.000E-07 1.0000000001E þ 00 1.000E-10 1.0000000000E þ 00 0.000

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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 RTln K RT ln Q ¼ RT ln KyQ ¼ RT lnð r?y r/Þ ¼RT ln½ðrþ r/Þy r/ ¼ ¼RTlnðry r/þ1Þ:RT lnðx þ 1Þ ¼ RT ðx x2y2 þ x3_{y3 x}4_{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 y3y3 þ y5_{y5 þ Þ;} _{where y ¼ ðx 1Þyðx þ 1Þ} _ð3:46Þ

In our case x: r?y r/. From Eq. (3.46) then follows:

A ¼ RTln r?y r/¼RT 2 ð r?y r/1Þyð r?y r/þ1 þ

h i

¼2RTryðr þ 2 r/Þ þ ð3:47Þ Table 2 Comparison of exponential and the ﬁrst three terms of its series expansion

x 0.01 0.1 1 2

expðxÞ 0.99005 0.90484 0.36788 0.13534

1 x 0.99000 0.90000 0.00000 1.00000

1 x þ x2_y2 _0.99005 _0.90500 _0.50000 _1.00000

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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/r¼const., 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/Þto function r ¼ gð r?; AÞ using the reaction isotherm. Table 1 clearly illustrates that afﬁnity by itself is a proble-matic measure or determining quantity for reaction rate because it does not vary too much when the rate undergoes steep changes and vice versa. Afﬁnity or reaction Gibbs or free energy alone does not determine the reaction rate, or kinetic ‘‘driving force’’. Water synthesis from molecular oxygen and hydrogen is a notoriously well-known example – its (standard) reaction Gibbs energy amounts to several hundreds kJ but its reaction rate is negligible unless some external catalytic action is introduced. It follows from the reaction isotherm that any reaction mixture containing only reactants possesses in zero time an

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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 kcH₂O₂cHþc_Br, 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

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concentra-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 kaa_wab_hac_s ð4:1Þ

where aw is the activity of water, ah the activity of hydrogen ion, and as the

activity of sucrose (in Scatchard’s terminology, k is only a proportionality factor), Scatchard asks in what units should it be expressed, having to be measured by the number of molecules which react in unit time. The answer is not straightforward. First, the concept of a semi-ideal solution is introduced, which principal characteristic is that the activity of each component is proportional to the molar fraction of that component which actually exists in the solution. In fact, this means that thermodynamic environment is independent of concentra-tion. 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 kaa_wab_hðC0_s xÞyCor

dxydt ¼ kaa_wab_hðC0_s xÞ ð4:2Þ

(C0

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Although both Scatchard’s suppositions are rather operational and apparently formal, they are much better than simple replacement of (dimen-sional) concentrations with (non-dimen(dimen-sional) activities. The total concentration Chas 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: K ¼ Y products ci i Y reactants cj j ð4:3Þ

from which he formulates the equilibrium condition:

K Y reactants cj j ¼ Y products ci i ð4:4Þ

and on its basis he claims that the reaction rate should be proportional to the appropriate difference: r ¼ k K Y reactants cj j Y products ci i ! ð4:5Þ

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

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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 concentration-based 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 kcAcB ð4:6Þ

should be replaced by the equation

r ¼ kcAcBðfAfByfA ? BÞ ð4:7Þ

where fi represents the activity coefﬁcient of, and A ? B denotes, the critical

complex. Why should the rate be just inversely proportional to the activity coefficient of the activated complex is explained by Bro¨nsted only by rather unclear physical reasoning, with no unambiguous proof being given. The inverse proportionality should make explicit, according to Bro¨nsted, that only those few reactant molecules possessing a sufficiently high activity to build up very unstable, i.e. a very ‘active’ activated complex. Thus, Bro¨nsted tried to formulate mathematically the decelerating effect of the necessity of existence of an activated complex with high ‘activity’. The two meanings of ‘activity’ are thus confused – that of high ‘reactivity’, which is rather vague, and that of the precisely-defined thermodynamic quantity.

The vagueness of Bro¨nsted’s reasoning prompted another Scandinavian, Bjerrum, who presented the whole matter more precisely two or three years later [78,79]. In fact, he made the same hypothesis as did formerly Arrhenius, and later Eyring and collaborators, in absolute reaction rate theory. Bjerrum supposed that Bro¨nsted’s activated complex is in equilibrium with the reactants, and that the reaction rate is directly proportional to its concentration. Expressing the activated complex concentration in terms of the thermodynamic equilibrium constant containing the products of concentration and activity coefﬁcient then resulted in a rate equation like Eq. (4.7). Bjerrum supported his argument with some ideas from kinetic-statistical theory.

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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 mass-action 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 ¼ k ? lY m i¼1 ai i k / l Y n i¼mþ1 ai i ð4:8Þ

(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/¼expðAyRT Þ ð4:9Þ

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

thermodynamics and mass action law.pdf

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Thermodynamics and foundations of mass-action

kinetics

Miloslav Pekarˇ*

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