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    REVIEW

    Thermodynamics and foundations of mass-action

    kinetics

    Miloslav Pekar ˇ*

    Institute of Physical and Applied Chemistry, Faculty of Chemistry, Brno University of 

    Technology, Purkyňova 118, 612 00 Brno, Czech Republic.

    E-mail: [email protected]

    ContentsABSTRACT

    1. INTRODUCTION   5

    2. CLASSICAL BACKGROUND   6

    2.1. Reaction isotherm   6

    2.2. Thermodynamic consistency of rate equations   9

    3. AFFINITY AND CHEMICAL KINETICS   13

    3.1. De Donder as originator   13

    3.2. Successors to De Donder   15

    3.3. Garfinkle’s original approach   233.4. Critical slowing; linearity testing   26

    3.5. Summary   30

    4. ACTIVITIES IN CHEMICAL KINETICS   31

    5. CLASSICAL (LINEAR) IRREVERSIBLE THERMODYNAMICS   40

    5.1. Fundamentals   40

    5.2. Tackling mass-action non-linearity and Onsager reciprocity   44

    5.3. Hungarian contribution I – Lengyel    46

    5.4. Onsager far from equilibrium   555.5. Bro ¨ nsted re-discovered?    57

    5.6. Hungarian contribution II – Olá h   58

    6. EXTENDED IRREVERSIBLE THERMODYNAMICS   62

    7. COMMON PROBLEMS IN CIT AND EIT APPROACHES   71

    8. RATIONAL OR CONTINUUM THERMODYNAMICS

    APPROACHES TO CHEMICAL KINETICS   74

    8.1. Introduction   74

    8.2. Bowen lays the foundation stone   758.3. Gurtin re-examines the classical theory   76

    Progress in Reaction Kinetics and Mechanism. Vol. 30, pp. 3–113. 2005

    1468-6783# 2005 Science Reviews

    3

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    8.4. Treatments of more complex systems   81

    8.5. Mu ¨ ller’s results   88

    8.6. Samohý l’s achievements   92

    9. CHEMICAL POTENTIAL MODEL   105

    10. CONCLUSIONS   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 thermody-

    namics. Special attention is devoted to affinity, the applications of activities in

    chemical kinetics and the importance of chemical potential. The review showsthat 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 possiblereaction mechanism consistent with mass conservation and thermodynamics. It

    further shows that affinity alone cannot determine the reaction rate and should

    be supplemented by a quantity provisionally called constitutive affinity. Future

    research should focus on reaction rates in non-isotropic or non-homogeneous

    mixtures, the applicability of traditional (equilibrium) expressions relating

    chemical potential to activity in non-equilibrium states, and on using activities

    and activity coefficients determined under equilibrium in non-equilibrium states.

    Prog React Kinet Mech 30:3-113 (c) 2004 Science Reviews

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    KEYWORDS:   activated complex, activity, affinity, chemical potential,continuum thermodynamics, equilibrium constant, extended irreversiblethermodynamics, Guldberg – Waage law, ionic strength, irreversible thermo-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 moregeneral 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 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 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 ¼Xni¼1

    iAi   ð2:1Þ

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

    for reactants) it is written as follows:

    DGr

     ¼DGr

     þ RT    lnY

    n

    i¼1a

    i

    i  :DGr

     þ RT    ln Qr

      ð2:2

    Þwhere   Qr   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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    DGr ¼ RT    ln K þ RT    lnYni¼1

    ci

    i  ¼  RT    ln K þ RT    lnYni¼1

    ð k?

    k/

    y k?

    k/

    Þcii  ¼

     RT    ln K

     þ RT    ln

    ½ðk?

    yk/

    Þðr/

    y r?

    Þ ð2:3

    Þ( k?

    ; k/

    are the rate constants in respective directions). Identifying the thermo-

    dynamic with the kinetic ( k?

    yk/

    ) equilibrium constant, the final equation results:

    DGr ¼ RT    lnð r?

    y r/Þ ð2:4Þ

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

    A ¼ Xni¼1

    imi ¼ Xni¼1

    ðimi þ  iRT    ln aiÞ ¼ DGr  RT    ln 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   r?½1  expðAyRT  Þ ð2:6Þ

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

    expðAyRT  Þ ¼ 1  ðAyRT  Þy1 þ ðAyRT  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?¼ k?  f  f ðciÞ;   r/¼k/  f bðciÞ ð2:8Þ

    where   ci,   i ¼ 1; . . . ; n, represent the concentrations of reacting species, to beconsistent 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 f  f ðciÞ;   r/¼ f bðciÞ 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  f y f b ¼ 1;   r

    ?y r/41 when DGr50   ð2:10Þ

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

     f  f y f b ¼ ðk?yk/ÞY

    i

    cii" #z

    ;   where  k?yk/ ¼ Kz 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 ðci; T  Þ. Two equilibrium conditions must besatisfied:

    Qr ¼ KðT  Þ and  f  ¼ 1 ðequilibriumÞ ð2:12Þ

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

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

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

    ðQr; T  

    Þ ¼F

    ðQryK

    Þ and F

    ð1

    Þ ¼ 1

      ð2:14

    ÞThe condition given by Manes   et al ., see Eq. (2.11), is then considered as a

    special case:

    FðQryKÞ ¼ ðQryKÞz 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  ; u jÞ ¼ FðQryK; u jÞ and Fð1; u jÞ ¼ 1   ð2:16Þ

    where  u j  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 whichis divisible by the function (QryK  1) in the neighbourhood of  QryK ¼ 1: f   1 ¼ ðQryK  1ÞCðci; T  ; u jÞ ð2:17Þ

    It should be added that in his proof the invertibility of the function ðQryKÞðci; T  Þis 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 ¼k?

    jYmi¼1

    aaii    k

    /

    jYmi¼1

    aa 0ii   ð2:18Þ

    where  j   is some function of activities,  ai, of reacting species, and  ai   and  a0i   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/

    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 ionicstrength,   etc. leaving the rate constants dependent only on temperature. At

    equilibrium, the following relation is valid:

    k?

    yk/

    ¼Yni¼1

    aða 0aiÞi;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?

    yk/

    ¼ f ðKÞ ð2:20Þ

    As the equilibrium activities of all species except one may be selected arbitrarily,

    it is shown that function  f   inevitably has the form   f ðKÞ ¼ Kz wherez ¼ ða 0i  aiÞyi;   i ¼ 1; . . . ; n   ð2:21Þ

    Condition (2.11)2   was derived also by Van Rysselberghe [28] after introducing

    affinity defined using chemical potential, Eq. (2.5)1   and its dependence on

    activity,   cf . Eq. (2.5)2, into the general mass-action law, Eq. (2.18). However,

    this law should be now formulated with stoichiometric coefficients as exponents

    at activities, moreover, it was also supposed that only one reaction step is

    kinetically significant and the overall affinity is a  g-multiple of the affinity of this

    step. Under these conditions,   z ¼ 1yg. In fact, this is another example of application of the reaction isotherm in the mass-action law.

    Boudart [29] joined equations (2.4) written for elementary steps of a

    reaction with Temkin’s theory of stationary reaction rates. The following

    equation for the ratio of overall reaction rates in both directions is thus

    obtained:

    r?

    y r/ ¼ expðAysRT  Þ ð2:22Þ

    where s  is the average stoichiometric number and  A  the affinity. Using again the

    reaction isotherm-based argument, another relation between the rate and

    equilibrium constants is obtained:

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    k?

    yk/

    ¼ K1ys ð2:23Þ

    All the consistency tests seek, from the mass-action law type rate equation,

    relations between the equilibrium constant and ratio of rate constants. A general

    ‘‘consistency’’ criterion, which does not refer to any particular rate equation, has

    been presented by Corio [30]. Function  u  is defined

    u ¼ KYnri¼1

    cii;reactant Yn

    i¼nrþ1c

    i

    i;product   ð2:24Þ

    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 otherhand, the kinetic condition may be written as  r ¼ 0. These two conditions can beinterpreted 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 pointonly, 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

    Þy

    ðquyqc1

    Þ ¼ ðqryqc2

    Þy

    ðquyqc2

    Þ ¼ ¼ ðqryqcn

    Þy

    ðquyqcn

    Þ ð2:25

    ÞUsing Eq. (2.24) these equations become:

    ðciyiÞðqryqciÞ ðciþ1yiþ1Þðqryqciþ1Þ ¼ 0   ð2:26Þ

    or, alternatively

    ciðqryqciÞ ¼ li   ð2:27Þ

    where  l   is a negative constant.

    Equations (2.26) or (2.27) represent the consistency condition to be

    fulfilled by any rate equation (expression for   r) to be consistent with 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 ¼ yXni¼1

    ðimiÞ ð2:28Þ

    where   y   is some positive function of concentrations and   mi   are the chemicalpotentials. 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 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 U*

    ðp; T  ; xÞ, the following relation for the differential of heat(Q) was derived:

    dQ ¼ hT  xdp þ CpxdT     rpT  dx   ð3:1Þ

    where

    hT  x ¼ ðqU*

    yqpÞT  ;x þ pðqV  *

    yqpÞT  ;xCpx ¼ ðqU*yqT  Þp;x þ pðqV  *yqT  Þp;x

    rpT   ¼ ðqU*

    yqxÞp;T   þ pðqV  *

    yqxÞp;T  

    ð3:2Þ

    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 afunction of the same variables. Thus

    dQ 0 ¼ h 0T  x dp þ C 0px dT     r 0pT    dx   ð3:3Þ

    where

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    h 0T  x ¼ T  ðqS*

    yqpÞT  ;x  hT  xC 0px ¼ T  ðqS

    *

    yqT  Þp;x  Cpxr 0pT   ¼ T  ðqS

    *

    yqxÞp;T   þ rpT  

    ð3:4Þ

    From Eq. (3.3) de Donder derived

    dQ 0y dx ¼ h 0T  x dpy dx þ C 0px dT  y dx  r 0pT     ð3:5Þ

    Next he introduced the key hypothesis which is neither well substantiated nor

    supported: the derivative dQ 0ydx 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 independentvariables 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 0C 0px ¼ 0

    ð3:6Þ

    and defined affinity as

    A

     ¼ dQ 0ydx:

    r 0xy;

      ð3:7

    Þwhere   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 thefirst 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; niÞ, using theextent of reaction as de Donder suggested, we obtain:

    dG ¼X

    i

    mi dni ¼X

    i

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

    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.

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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 Gibbsenergy and reaction rate at equilibrium simultaneously, they arrived at an

    equation valid sufficiently close to equilibrium:

    r ¼ xða j; bkÞ DG   ð3:12Þ

    where the proportionality factor represents:

    xða j; bkÞ ¼ ½qryqðDGÞa2 ;a3 ;...;am ¼ ½qryqðDGÞa1 ;a3 ;...;am ¼  . . . ¼ ½qryqðDGÞa1 ;a2 ;...;am1ð3:13Þ

    and depends on full sets of  a j  and bk. In the derivation, the implicit assumption

    on the invertibility of the reaction Gibbs energy function is hidden. Theirthermodynamic 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-idealsystems 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 Cl2 ¼ 2  NOCl   ð3:14aÞ

    2 NOCl þ I2 ¼ 2 NO þ 2 ICl   ð3:14bÞ

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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 DG6¼r   , i.e. theactivation Gibbs energy, which might be questionable.

    Boudart shows in several papers more precisely the potential practical

    value of affinity-containing equations in chemical kinetics. He distinguishes [40]

    between the de Donder inequality:

    Ar 0   ð3:15Þ

    and de Donder equation:

    lnð r?y r/Þ ¼ AyRT     ð3:16Þ

    Because Eq. (3.15) is valid for the overall reaction process, it may explain why

    some reaction steps may occur against the ‘‘thermodynamic direction’’ [41]. For

    instance, two reactions may occur simultaneously even when

    A1r150   ð3:17Þ

    providing that

    A1r1 þ A2r240   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 Ai, it follows that:

    Ai40;   Airi40   ð3:20Þ

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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

    R1 ¼ 2 I1R2 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?

    1aR1ð1  z1Þ ð3:22Þ

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

    z1 ¼ a2I1yðK1aR1Þ ð3:23Þ

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

    reversibility (z) is given by

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    z ¼ z1z2z3 ¼ aPyðKaR1 aR2Þ ð3:24Þ

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

    r ¼   k?1aR1 ½1  aPyðKaR1 aR2 Þ ð3:25Þ

    This result can be derived by the usual procedure without reversibility or de

    Donder relations. Actually, in this example, the rate is given by:

    r ¼ r1 ¼   k?

    1aR1  k/

    1a2I1

    ð3:26Þ

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

    K2 ¼ aI2yðaI1 aR2Þ;   K3 ¼ aPyðaI1 aI2 Þ ð3:27Þit can be easily derived:

    a2I1 ¼ aPyðaR2 K2K3Þ 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)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 concentra-

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    tions (more precisely, surface coverages and partial pressures) instead of 

    activities.

    Timmermann [49] asserts that he obtained the general formula relating

    reaction rate and affinity, and a general and rigorous statement of the

    thermodynamic restrictions on reaction rate is thus given. His proof is based

    only on the argument that the rate of increase of the extent of reaction has a

    unique value independent of the particular language used to describe the

    reaction and the affinity. However, the key point of his proof is unclear.

    Timmermann defines the gross reaction rate (r) as the rate of increase of the

    extent of reaction (x):

    r ¼ dxydt ¼ dniyði dtÞ ð3:29Þ

    where   ni   is the amount of substance   i    in the whole system and   i   its

    stoichiometric coefficient. Timmermann further states that the gross rate is

    generally not determined in a kinetic experiment. Instead, an intensive quantity

    is measured, which is the gross rate normalized to some extensive reference

    quantity. Two from several of Timmermann’s examples are reproduced here.

    The most common reference quantity is the volume of the system (V ) and the

    intensive reaction rate is then expressed as:

    rc ¼ ryV     ð3:30Þ

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

    rm ¼ ryðn0M0Þ ð3:31Þ

    where  n0   is the mole number of the solvent and  M0   its molar mass. Clearly,

    rcV   ¼ rmn0M0   ð3:32Þ

    Timmermann then combines the classical mass-action rate equation

    rc ¼   r?

    c    r/

    c, where   r?

    c ¼   k?

    c

    Pic

    ii   and   r

    /

    c ¼   k/

    c

    P jc

     j

     j   (i  runs through reactants,

     j  through products), with the classical definition of affinity  A ¼ Pk

    kmk  (k runs

    through both reactants and products). Chemical potential (mk) is expressed also

    traditionally, mk ¼ mok þ RT    lnðgkckycoÞ where ‘‘o’’ denotes the standard state andgk  is the activity coefficient on the molarity scale. Timmermann finally arrives at

    the following expression:

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    rc ¼   r?

    c   1 k/

    cKgðcoÞ

    k?

    cQk

    gk

    k

    expðAyRT  Þ

    0

    BB@

    1

    CCAð3:33Þ

    where  Kg  is the thermodynamic equilibrium constant on the molarity scale and

     ¼P k. He states that   r   cannot depend on the particular language used todescribe 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/

    cKg

    ðco

    Þ

    k?cQ

    k

    gk

     1  ð

    3:34Þ

    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:

    rm ¼   r?

    m   1 k/

    mKjðmoÞ

    k?

    m

    Qk

    jk

    k

    expðAyRT  Þ

    0BB@

    1CCA 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

    Qk

    gk

    k

    ¼ k/

    mKjðmoÞ

    k?

    m

    Qk

    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 thereaction isotherm written in terms of affinity (A) instead of the Gibbs energy

    (and with concentrations approximating to activities) [50]:

    _AA ¼ RT  X

    i

    ð2i yciÞðdciyi dtÞ ð3:37Þ

    (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

    the affinity decay rate ( _AA) is obtained:

    r ¼ ð _AAyRT  ÞyX

    i

    2i yci   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):

    _AA ¼ Arð1yt  1ytKÞ ð3:39Þwhere  Ar   and tK  are 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:   ln

    ½ðtytK

    Þ  exp

    ð1

      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 equa-

    tion 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 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 k1cAdcBydt ¼ k1cA  k2cBdcCydt ¼ k2cB

    ð3:41Þ

    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:

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    dcAydt ¼ k1c0A expðk1tÞdcBydt ¼ k1c0A expðk1tÞ  k1k2c0A½expðk1tÞ  expðk2tÞyðk2  k1ÞdcCydt

     ¼ k1k2c

    0A

    ½exp

    ðk1t

    Þ  exp

    ðk2t

    Þy

    ðk2

      k1

    Þ

    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   k24 k1   can thelast 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  tK, i.e. the

    most probable time of attaining equilibrium. As any chemical reaction proceeds

    from some initial concentrations and time to equilibrium concentrations and

    time, it may be expected that such ‘‘scaling to equilibrium’’ will work.

    3.4  Critical slowing; linearity testing

    Affinity- and reaction isotherm-based approaches have found some popularity

    in the interpretation of the slowing down of chemical reactions near some critical

    point, see   e.g. refs [59 – 62]. Actually, the ‘‘linear’’ relationship (2.7) is used

    [59,60] for qualitative interpretations, not for quantitative evaluations. Recently,Kim and Baird [62] reported even a speeding up near the critical point. Several

    approximations are used, the nature of which is clearly seen from an inspiring

    older work by Meixner [63]. Meixner claims that the close-to-equilibrium

    reaction rate is expressed as  dxydt  and given by:

    dxydt ¼ eðT  ; ; xÞAðT  ; ; xÞ ð3:43Þ

    where x is the extent of reaction, e  is the proportionality coefficient dependent on

    temperature (T  ), specific volume () and extent of reaction, and  A  is the affinity

    determined by the same set of variables. First, Meixner states that the close-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 eðqAyqxÞT  ;½x  xeðT  ; 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  RT    ln   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

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    Table 1 Decay of logarithm and its argument

    r!

    y r

      lnð r!y rÞ1.0000000000E þ 10 23.031.0000000000E þ 09 20.721.0000000000E þ 08 18.421.0000000000E þ 07 16.121.0000000000E þ 06 13.821.0000000000E þ 05 11.511.0000000000E þ 04 9.2101.0000000000E þ 03 6.9081.0000000000E þ 02 4.6051.0000000000E þ 01 2.3031.1000000000E þ 00 9.531E-021.0100000000E þ 00 9.950E-031.0010000000E þ 00 9.995E-041.0000001000E þ 00 1.000E-071.0000000001E þ 00 1.000E-101.0000000000E 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 RT    ln K  RT    ln Q ¼ RT    ln KyQ ¼ RT    lnð r? y   r/Þ ¼ RT    ln½ðrþ   r/Þy   r/ ¼¼ RT    lnðry   r/ þ1Þ:RT    lnðx þ 1Þ ¼ RT  ðx  x2y2 þ x3y3  x4y4 þ Þ

    ð3:45Þ

    The expansion in Eq. (3.45) is valid only for 15x 1. From Eq. (3.45) it isbetter 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 theoverall 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 valuesof its argument (x40) in the following way:

    ln x ¼ 2ðy þ y3y3 þ y5y5 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 RT    ln   r? y   r/¼ RT  2 ð r? y   r/ 1Þyð r? y   r/ þ1 þ h i ¼ 2RTryðr þ 2   r/Þ þ

    ð3:47Þ

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

    x   0.01 0.1 1 2

    expðxÞ   0.99005 0.90484 0.36788 0.135341  x   0.99000 0.90000 0.00000   1.000001  x þ x2y2   0.99005 0.90500 0.50000 1.000001  x þ x2y2  x3y6   0.99005 0.90483 0.33333   0.33333

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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   r/¼ const., which is easily imagined to be fulfilled inpractice, 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 (partialpressures, 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 zis 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 asa transformation of the function   r ¼ f ð r?;  r/Þ   to function  r ¼ gð r?; AÞ  using thereaction 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 kcH2O2 cHþ cBr , they found a concentration-independent rateconstant 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 concentra-

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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-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:

    K ¼Y

    products

    ci

    i

    Yreactants

    c j j   ð4:3Þ

    from which he formulates the equilibrium condition:

    K

    Yreactantsc j j   ¼ Yproducts

    ci

    i   ð4:4Þ

    and on its basis he claims that the reaction rate should be proportional to the

    appropriate difference:

    r ¼ k KY

    reactants

    c j j  

    Yproducts

    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   ð4:6Þshould be replaced by the equation

    r ¼ kcAcBð f A f By f A ?BÞ ð4:7Þ

    where   f i   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 ¼k?

    lYm

    i¼1a

    i

    i    k/

    l Yn

    i¼mþ1a

    i

    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/¼ expðAyRT  Þ ð4:9Þ

    and tries to show its validity for elementary reactions in non-ideal systems. To

    achieve this goal he uses traditional expressions for the dependence of chemical

    potential on concentration and the mass-action law in the usual, concentration

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    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 thermo-

    dynamics 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 argu-

    ments 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.

    r ¼k?

    l? Ym

    i¼1a

    i

    i    k/

    l/ Yn

    i¼mþ1a

    i

    i   ð4:10Þ

    is superfluous because   l?

    ¼l/

    . Unfortunately, his proof lacks its claimed general

    validity as has been shown by Samohýl (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.

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    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

    1½1 þ   2½2?½6¼?3½3 þ  4½4 ð4:11Þ

    In transition state theory, the reactants are considered to be in equilibrium with

    the transition state ([6¼]). The true thermodynamic equilibrium constant is thengiven by

    K?¼ a6¼yða11  a22 Þ ð4:12Þ

    The reaction rate is proportional to the   concentration   of transition state,

    r?¼ ? c6¼. Expressing activity as the product of activity coefficient (g) and relativeconcentration, 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

    ?

    ca21  yg6¼:   k?

    a1

    1  a2

    2  yg6¼   ð4:13Þ

    By the principle of microscopic reversibility, the reaction must proceed in the

    reverse direction via the same transition state [86]. Therefore the products are

    also in equilibrium with the same transition state:

    K/

    ¼ a6¼yða33  a44 Þ ð4:14Þ

    and by analogy:

    r/¼ /K/

    ca33  a4

    4  yg6¼:   k/

    a33  a4

    4  yg6¼   ð4:15Þ

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    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:

    a3

    3  a4

    4  yða11  a22 Þ ¼K?

    y K/

    ¼ ðequilibriumÞ constant   ð4:16Þ

    The entire analysis could thus be valid only for equilibrium where the overall

    rate is zero! Introducing Eq. (4.16) into the generalized rate equation (4.8), we

    obtain:

    r ¼ a

    11  a

    22 ð k?

      k/

    K

    ?

    y K

    /

    Þyg=:ka

    11  a

    22  yg=   ð4:17Þ

    This generally gives non-zero equilibrium rate unless   k?

    ¼k/

    K?

    y K/

    , which leads to

    ?¼ /. Otherwise, Eq. (4.17) would give the very strange result that the overallrate 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

    ?

    ca11  a2

    2  yg=? ;   c

    =/ ¼K

    /

    ca33  a4

    4  yg=/   ð4:18Þ

    The final result is:

    r ¼k?

    a1

    1  a2

    2  yg=?   k

    /

    a3

    3  a4

    4  yg=/   ð4:19Þ

    which is, in fact, Hall’s general mass-action law (4.10). The same result can be

    obtained considering different activity coefficients,   i.e. different activities of a

    common transition state in the forward and reverse directions. Both different

    transition states and different activities sound rather strange and illustrate the

    problems which are encountered when applying transition state theory to

    reactions occurring simultaneously in both directions out of equilibrium.

    Considering different transition states in different directions of the same reaction

    may violate microscopic reversibility. It might be therefore supplemented by the

    hypothesis that the transition states are different in non-equilibrium states only,

    and become identical when equilibrium is attained.

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    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, demon-

    strated 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 parti-

    tioning. 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 experi-

    ments 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

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    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 ¼XR j¼1

    aijA j;   i ¼ 1; 2; . . . ; R   ð5:1Þ

    where   R   is the total number of independent reactions and  aij   are the phenom-

    enological coefficients. The law of mass-action is used in the form

    40 Miloslav Pekar ˇ

    www.scilet.com

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    ri ¼ kiY

    m

    cmim   k 0iY

    n