Extended irreversible thermodynamics (EIT) continues its role in the flux-forces arena. Its principal contribution lies in the extension of the (classical) set of independent variables (specific internal energy, specific volume, mass fractions) by some of the so-called fluxes (e.g. heat flux or stress tensor). Thus, applications of EIT to chemical kinetics is usually nothing more than the introduction of some chemical flux among independent variables.
Perhaps the first contribution of EIT to the analysis of thermodynamics-kinetics relationships was the paper by Garcı´a-Colı´n and de la Selva [123]. They suppose that there exists some function (Z), in fact the non-equilibrium entropy, of the following variables:
Z ¼ Zðe; ; ci; J; Jd;pÞ ð6:1Þ
where e is the specific internal energy, is the specific volume, ci the mass fraction of component i (i ¼ 1, 2 in ref. [123] for simplicity), J is the chemical flux and Jd the diffusive flux of one of the species, and p is the trace of the viscous tensor. Partial derivatives occurring in the total differential of this function are either expressed by relations resembling relations of classical reversible thermodynamics or modelled by relations suitable for further devel-opments:
qZyqe ¼ Y1; qZyq ¼ PY1; qZyqci¼ MiY1 ð6:2Þ
qZyqJ ¼ arY1; qZyqJd¼adY1; qZyqp ¼ apY1 ð6:3Þ where Y represents the non-equilibrium temperature, P is the non-equilibrium pressure, Mi the molar mass, a’s are the proportionality coefficients.
The chemical flux is not explicitly stated but from the symbol used it is clear that the flux is actually the reaction rate. Unfortunately, this is defined using the time derivatives of the mass fractions, which is in the modelled system with diffusion either improper or, at least, not easily applicable to experiment.
The partial derivatives are then approximated by their expansion in some of the independent variables, e.g.:
MiY1¼miT1þ bqðMiY1ÞyqJcJ þ bqðMiY1ÞyqJdcJdþ bqðMiY1Þyqpcpþ þ ð1y2!Þ q 2ðMiY1ÞyqJ2
J2þ ð6:4Þ
where T is the (equilibrium) temperature and mi the (classical) chemical potential. The expansions are nothing more than following transformation of functions:
Z ¼ Zðe; ; ci; J; Jd;pÞ?qZyqx ¼ f ðJ; Jd;pÞ þ Cx
where x represents any variable from the set fe; ; ci; J; Jd;pg and Cxthe relevant classical term. This transformation is substantiated by stating that for the classical case, i.e. for the disappearance of extending, flux variables, classical expressions like qZyqe ¼ T1should be obtained. Why this equation cannot be arrived at by disappearing corresponding partial derivatives in full functional representation is not explained. Moreover, functions f ðJ; Jd;pÞ look like a McLaurin series expansion and Cx is the equilibrium expression for the appropriate partial derivative of entropy. Thus, the approximation of partial derivatives is an expansion around equilibrium. It is therefore not clear where the partial derivatives in this expansion should be evaluated, as at equilibrium they should vanish, i.e. be equal to zero. Consequently, Eqs (6.2) – (6.4) should be considered only as a specific model and the whole analysis is valid only for systems complying with this model. Which real systems or materials correspond to the model is not discussed in the original paper.
Another particular model in this work is the expression for the entropy flux, which is constructed just as the sum of the diffusion flux, the only one vectorial independent variable, multiplied successively by some of the scalar independent variables and a term which should again probably resemble some classical term:
JZ¼ Y1ðM1M2ÞJdþb01JJdþb02pJdþ ð6:5Þ
(b’s are proportionality coefficients). This should be viewed only as a specific example of representation of the linear isotropic vectorial function.
The tacit construction of models continues: the next model is the representation of the entropy source (s) as a nonlinear isotropic scalar function:
s ¼ JdXdþJXrþpX ð6:6Þ
where the ‘‘generalized forces’’ are defined as
Xd¼m20Jdþm21JJdþ ð6:7aÞ
Xr¼mr1J þmr2J2dþmr3J2þmr4J3þmr5p ð6:7bÞ
X¼m1p þ m2J þ ð6:7cÞ
Then, a preliminary result is derived – an equation for evolution of the chemical flux, i.e. an equation with the material derivative of J ( _JJ). This equation is solved, or approximated to successively higher orders in J, by some strange procedure referring to the stationary state. The final result, Eq. (6.8) below –
‘‘general phenomenological relation between the rate of the reaction and the chemical affinity’’ – is a mere summation of several of these approximations and is not proved for consistency with the initial expression for _JJ. This general phenomenological relation expresses reaction rate as a function of powers of affinity:
J ¼
ðrAymr1T Þðmr3r2A2Þyðm3r1T2Þð1y2Þ 1ðqar1yqc1Þþ2ðqar1yqc2Þ
ðr2A2Þyðm3r1T3Þþ þ ð2m 2r3ymr1Þ mr4
ðr3A3Þyðm3r1T3Þ ð6:8Þ
here, r is the density, A the affinity, i is the product of the stoichiometric coefficient and molar mass of the component i.
The affinity is introduced through the classical definition (2.5)1, which is also used in CIT, supposing the same concentration dependence. There is no extended approach. More peculiar is the way that led to the power law of Eq.
(6.8). This was not due to the specific claims of EIT but just due to the models introduced and used in an unusual manner:
1. powers of J , which are the causes of later powers of A in the ‘‘general phenomenological relation’’, are introduced due to the model (6.7b),
2. affinity is originally introduced, in fact, just as a first – classical, equilibrium (!) – member of a series approximation (6.4),
3. non-equilibrium corrections, introduced into this approximation are ignored in the above-mentioned stationary state analysis,
4. powers of J , which should be the other non-equilibrium corrections in model (6.7b), are systematically expressed in this analysis in powers of affinity despite its status given under 2 above.
There are other unclear points. During the constructing of the models, it is several times stated, but never proved, that the models should reduce to the standard form of CIT. Rate (chemical flux) is finally expressed as a function of affinity, which itself is a function of chemical potential, which itself is a function of concentration. Although the chemical flux, or reaction rate, is included among the independent variables, the dependence of the rate on other variable(s), usually affinity, is sought.
The next work of the same authors gives only moderate progress. The motivation for the EIT approach is stated in ref. [124]: ‘‘For many years linear irreversible thermodynamics has been the only theory available to account for the empirical kinetic mass-action law (KMAL) as a flux-force relation between the chemical rate J and the affinity A namely, J & ½expðAyRT Þ1. In spite of the fact that such a relation is a nonlinear one, it has been shown that at least for the reactionB þ C , D þ E, and using a kinetic theory model, the entropy source JAyT is consistent with it. Therefore the rate J is interpreted as a thermo-dynamic flux of the same footing as the heat flux, the diffusion flux, and the stress tensor. Thus, KMAL has been viewed as a constitutive relation, analogous to Fourier’s heat equation, Fick’s equation for diffusion, and the Newton – Navier equation for the transmission of momentum. On the other hand, the coupling between the chemical rate with its generating forces, namely A, and the divergence of the hydrodynamic velocity satisfy Onsager’s reciprocity theorem in the linear approximation only.’’
A paper entitled simply ‘‘Consistency of the Kinetic Mass Action Law with Thermodynamics’’ [125] starts with the function
Z ¼ Zðe; ; ci; q; J; JiÞ ð6:9Þ
thus, instead of the stress tensor, the heat flux q is considered; e is the internal energy density, the other symbols have the same meaning as in Eq. (6.1), but the diffusion fluxes (Ji) are considered for each component i separately. Again, some
specific models of representations of functions of various tensorial orders are introduced.
As usual in EIT, the total derivative of the function is written and the partial derivatives with respect to the ‘‘flux variables’’, viz. q; J; Ji are approximated by expansion around the local equilibrium where these variables should, of course vanish. No true Taylor expansion is used, as the ‘‘classical variables’’ (e; ; ci) are not included, and do not expand anywhere. Rather, a combination of the isotropic function representation and a Taylor series is used, at least for qZyqJ:
qZyqJ ¼ T1o0J ð6:10Þ
(o0 is the proportionality coefficient). This is the first model used. The EIT postulate of the entropy balance equation, viz.:
rdZydt þ divJZ ¼sZ ð6:11Þ
is a necessary intermediate step calling for at least two expressions – for entropy flux (JZ) and source (sZ).
The entropy flux is represented as an isotropic vectorial function, which is immediately specified by the following equation
JZ¼ ð1yT Þq þ b0q X
i
ðmiyTMiÞJiþX
i
bi0Ji ð6:12Þ
(b’s are the proportionality coefficients, which are, in turn, functions of all scalar invariants) again claiming, not proving, that this form is reducible to the normal entropy flux of CIT where ‘‘flux variables’’ can be ignored. This is the second model which is then combined with mass, energy, and entropy balances and time derivative of Z, coming from the first model (6.10). An expression for entropy production then results as follows:
sZ¼
where a’s and g’s are the coefficients from equations analogous to Eq. (6.10) but for the partial derivative with respect to q and Ji, respectively; miis the (classical) chemical potential, Fithe external force on i per unit mass and A is the (classical) affinity.
It is claimed that entropy production is a scalar function of defining scalar variables. Therefore, it can be represented as an isotropic function. The representation is again rather specific:
sZ¼P0þXqq þX
i
XiJi ð6:14aÞ
Xq¼x0q þX
i
xiJi ð6:14bÞ
Xi¼liq þX
j
lijJj ð6:14cÞ
and can be considered as the third model. Symbol P0represents a function of all scalar invariants.
The two expressions for entropy production, (6.13) and (6.14a), should be consistent. But the consistency is not straightforward and explicit. Therefore, further models have to be invoked. In contrast to the authors’ contention, chemical flux does appear as a multiplicative factor in one of the two equations for entropy production, cf. Eq. (6.13). This appearance is not sufficient and must be supported by extracting the chemical flux from (only some!) scalar coefficients in functional representations. Several additional models can therefore be constructed:
b0¼b00J; bi0¼b0i0J; P0¼JPðJ;. . .Þ ð6:15Þ where P in the last equation is again a function of all scalar invariants including J.
At last, the desired relation – an equation for the time derivative of chemical flux or the general mass-action law – is obtained:
ðo0yT ÞðdJy dtÞ ¼ AyT b00div q X
i
b0i0div JiþP ð6:16Þ
Affinity was introduced again due to the reminder of the classical term in the representation of entropy flux. The evolution equation enables, after introducing further models or simplifications, discussion in the terms ‘‘chemical flux is forced
by the affinity’’. It should be stressed that an explicit dependence on affinity is found only for the time derivative of the chemical flux J (which is, perhaps, the reaction rate); the direct relationship between chemical flux (reaction rate) and affinity is obtainable only for the stationary state.
To summarize – there are too many models with vague relations to real systems or materials, and too general final equations, not containing the main quantity measured by kineticists, namely concentration. Of course, in Eq. (6.16) concentration is hidden in affinity, however, this is not an equation for the reaction rate itself, but for its time derivative.
Lebon et al. [126] write that they adopt a position intermediate between classical theory and EIT. Their work was competently criticized by Garcı´a-Colı´n [127] to say nothing about its limitation to homogeneous (non-diffusing) mixtures.
Instead of the reaction rate, the authors use the degree of advancement (x) defined as
_xx ¼ _cciyi ð6:17Þ
(ci is the mass fraction of i-th component and i its stoichiometric coefficient).
This means that only closed systems are considered. As an independent variable, however, the following difference is used:
x ¼ x xe ð6:18Þ
where xedenotes the equilibrium value. It is assumed that the time evolution of the new variable is given by
x_
x ¼ jðT ; p; xÞ ð6:19Þ
(T is temperature, p pressure). It is further postulated that function j is expressed as follows:
jðT ; p; xÞ ¼ xcðT ; p; xÞ ð6:20Þ
During further development, no special irreversible thermodynamical approach is used. Only combinations and manipulations with the postulate, integrated form (6.17), and relations well-known from reversible thermodynamics, are used, viz. the definition of the relation of chemical potential to the component activity, the definition of affinity (2.5)1, and the expressing of equilibrium constants by standard chemical potentials.
The final result is the following equation x_
x ¼ oðT ; p; xÞ 1 expðAyRT Þ½ ð6:21Þ
containing another function (o) which includes also the function c. Eq. (6.21) is again an expression for the time derivative of a certain reaction rate quantity (degree of advancement), which is claimed to be the standard law of mass-action. No explicit rate equation or function was derived. Function c remains undetermined and Garcı´a-Colı´n [127] showed how it can be related to phenom-enological coefficients of CIT.
In summary: no irreversible thermodynamics is utilised and the only new feature is postulate (6.20), which is used several times and coupled with the well-known relations of classical thermodynamics.
This work, as well as other irreversible thermodynamics approaches (e.g. [125]), were briefly criticized by Ross and Garcı´a-Colı´n [128]. However, no new ideas were presented, just some reservoirs for reactants and products are introduced with no clear distinction between the reaction Gibbs free energy and the Gibbs free energy of the whole system. The critique of EIT approaches is based on the initial task of EIT – to describe fast processes by introducing new, extending, variables: ‘‘For most reactions, especially in liquids, reaction times are long compared to other relaxation times (vibra-tions, etc.)... For such cases the condition of local equilibrium holds well; the thermodynamic variables including the progress variable are on the same time scale and there is no need for an extended thermodynamics. That need may arise when the reaction time is more comparable to other relaxation times and the predicted rate coefficients become time-dependent, which expresses the effect of the relaxation of the fast(er) variables of, say, vibrational relaxation, compared to the slow(er) chemical rate.’’ Thus, there is usually no need for incorporating some ‘‘chemical flux’’ among the independent variables. An interesting note is given in the conclusion: ‘‘Furthermore, the identification of a generating function (Z above) with an entropy has not yet been justified.’’
The last contribution from EIT was due to Fort et al. [129] who try to find new developments from the same starting point. First, they would like to ascertain whether EIT methodology gives entropy as a sum of its equilibrium value and some non-equilibrium correction also for chemically reacting systems.
Second, they support this finding with deductions from the kinetic theory of
gases. As statistical theories are beyond the scope of this review, we will focus mainly on the EIT part.
Specific entropy (s) is, again, considered to be a function of ‘‘classical’’
variables and one extending, which is, of course, the reaction rate J:
s ¼ sðu; ; ci; JÞ; i ¼ 1; 2; . . . ; n ð6:22Þ
(u is the total specific internal energy, the specific volume and ci the mass fraction) and its total differential is constructed; partial derivatives with respect to the classical variables are expressed with the aid of the generalized tempera-ture (Y), pressure (P), and chemical potentials (Zi) as dictated by the EIT standard procedure. Then, the first new postulate or, more appropriately, model is introduced:
ðqsyqJÞu;;c
i¼ ðayT ÞJ ð6:23Þ
where a is some coefficient depending only on the classical variables. This model is substantiated by the traditional claim that the generalized entropy ðsÞ must reduce to the classical one at equilibrium where the rate is zero, which is not proved. Consequently, the generalized local Gibbs equation can be formulated:
ds ¼ ð1yYÞ du þ ðPyYÞ d X
i
ðZiyYÞ dci ðayT ÞJdJ ð6:24Þ
However, the evolution equation for specific entropy is restricted to depend on the reaction rate only, which is explained by considering only non-equilibrium processes in an incompressible fluid in the absence of heat and diffusion effects, specifically:
rdsy dt ¼ J AyT ðaryT Þ dJy dt½ ð6:25Þ
In the (second) postulate (6.25), the classical equilibrium definition of affinity (A) through the classical chemical potentials (not Zi’s!), cf. Eq. (2.5)1, was applied together with the mass fraction balance, namely dciy dt ¼ ðiyrÞJ, where i is the stoichiometric coefficient and r the density.
The evolution equation (6.25) is compared with the general law of entropy balance of EIT, see Eq. (7.3) below, and the entropy source (more precisely, the rate of entropy production per unit volume) is then expressed as the right hand side of Eq. (6.25). It should be noted that in contrast to the other EIT approaches, which considered the reaction rate as a part of the entropy flux, here the rate is included in the entropy source!
As the second law of thermodynamics calls for a positive value for entropy production, the simplest way to assure this is to have it in only the second power of the reaction rate. Therefore, a third model is proposed:
AyT ðarÞyT dJy dt ¼ bJ ð6:26Þ
This model is particularly convenient for EIT as it is analogous to the Maxwell – Cattaneo equations, which were successfully explained within the EIT approach.
With this model, the generalized Gibbs equation (6.24) may be written as ds ¼ ð1yT Þ du þ ðpyT Þ d X
i
ðmiyT Þ dci ðtyrlÞJdJ ð6:27Þ
where l ¼ 1yb and t ¼ arlyT , with no explanation as to why the generalized variables were substituted by their classical (equilibrium?) analogues, i.e.
temperature (T ), pressure (p) and chemical potential (mi). Its integrated form is simply expressed as
sðu; ; ci; JÞ ¼ sðu; ; ciÞ ðty2rlÞJ2 ð6:28Þ And this is all for chemical kinetics. Entropy was, finally, expressed as its equilibrium value and non-equilibrium correction, which is second order in the reaction rate. The kinetic theory part of this work derives a similar expression for entropy and even the usual proportionality of the rate to the affinity.
Interestingly, in the conclusion the authors write: ‘‘...the reaction rate is not a flux in the usual sense because it does not appear as a true flux in the balance equations of mass fractions; instead, it appears as a source term there.’’
Further. ‘‘In spite of this, we have shown how chemical reactions can be included in the much broader framework of EIT.’’ This means that fluxes, forces, sources are interpretations according to some particular motivation and not the results of rigorous definitions or proofs.