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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
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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
k¼
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
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ri ¼ kiY
m
cmim k 0iY
n