CLASSICAL (LINEAR) IRREVERSIBLE THERMODYNAMICS 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 (r_i), the ‘‘flux’’, may be written, close to equilibrium, in linear form

r_i¼X^R

j¼1

a_ijA_j; i ¼ 1; 2; . . . ; R ð5:1Þ

where R is the total number of independent reactions and a_ijare the phenom-enological coefficients. The law of mass-action is used in the form

r_i¼k_iY

and m goes through all reactants, n through all products and k through both these kinds of constituents;
’s are the stoichiometric coefficients and c’s are the concentrations.

Using the definition of affinity in terms of chemical potential and the classical relationship between chemical potential and concentration, the following equation is obtained:

r_i¼o_i½1  l_iK_iðc^oÞ
ⁱ expðA_iyRT Þ ð5:4Þ where K_i is the equilibrium constant, c^o the standard concentration, and

_i¼P

k
_ki. It can be shown that the multiplicative factor at the exponential is equal to one. Close to equilibrium (jA_iyRT j 5 1; index ‘‘eq’’), the exponential may be expanded in a series retaining only the first member. The linear phenomenological relation is finally obtained:

r_i¼ ðo^eq_i yRT ÞA_i ð5:5Þ

(note that o_i, once more, was not expanded in contrast to the affinity), which is a special case of Eq. (5.1). Haase is very careful to identify kinetic and thermo-dynamic equilibrium constants and also to ignore standard concentrations when substituting concentrations for activities, cf. Eq. (5.4).

In the flow-through systems, where the spatial distribution of variables must also be taken into account, the following expression for the local entropy production (u) is derived:

X_ij ð1y2Þðq
_iyqz_jþq
_jyqz_iÞ; i; j ¼ 1; 2; 3 ð5:9Þ and J_Qis the density of heat flow, J
_;kare the densities of diffusion flows, o_r is the rate and A_rthe affinity of the r-th reaction, P_ij¼P_jiare the frictional forces, K_k the external molar force acting on the k-th component, m_k its chemical potential and
_iis the component of the barycentric rate vector v, z’s are spatial coordinates. The gradient in Eq. (5.8) is taken at constant temperature.

The so-called dissipative function C is interpreted with the aid of the concepts of fluxes and forces, viz. ‘‘phenomenological relationships’’ among them are sought. To this end, forces are considered to be independent, and fluxes dependent, variables. Their mutual functional connections are formulated with a rather intuitive use of linear isotropic function representation, here called Curie’s principle. First, the tension term is excluded from the considerations, giving nothing new to chemical kinetics, as the relevant phenomenological relation is again postulated to be:

o_r¼X

s

a_rsA_s ð5:10Þ

where index s also refers to all reactions and a’s are the proportionality (phenomenological) coefficients.

Second, the tension term is taken into account but together with only the chemical rate term from the dissipative function (5.6) and not also with the other two, transport terms:

C ¼X

r

o_rA_rþX³

i¼1

X³

j¼1

P_ijX_ij 0 ð5:11Þ

This is vaguely substantiated by Curie’s principle and should probably be understood as the fact that the vectorial linear isotropic function depends only on vectors whereas the tensorial function may depend also on scalars.

Consequently, the reaction rate is written (better speaking, represented) as:

o_r¼X

s

a_rsA_sL_rdiv v ð5:12Þ

(L_r is the proportionality, phenomenological, coefficient) because div v is considered to be the trace of tensor X given by Eq. (5.9). Eq. (5.12) is claimed to be a generalization of Eq. (5.10). It can be interpreted as a warning that the chemical reaction rate may be affected by viscous processes. The functional

dependence, Eq. (5.12), of the former on the latter is more a matter of interpretation than of exact proof.

Very recently, Cukrowski and Kolbus [97] published another paper utilising flux and force. They found a new, ‘superior’ force, which enables one to use linear flux-force over a wider range (i.e. farther from equilibrium) than usually expected. The new force is defined as the difference between the reactive absolute activities of reactants and products. The absolute activity (of compo-nent i, l_i) was introduced into thermodynamics by Fowler and Guggenheim [98]

as the exponential of chemical potential:

l_i¼expðm_iyRT Þ ð5:13Þ

The reactive absolute activity is defined by Cukrowski and Kolbus using the

‘‘reactive chemical potential’’ m^re_i , which they define as the difference between the chemical potential and its equilibrium (‘‘eq’’) value:

m^re_i ¼m_im^eq_i ð5:14Þ

The reactive absolute activity of component i is then defined as follows:

l^re_i ¼expðm^re_i yRT Þ ð5:15Þ

and the reactive absolute activity of reactants (suffix R) or products (suffix P) as products:

l^re_R ¼ Y

reactants

l^re_i ; l^re_P ¼ Y

products

l^re_j ð5:16Þ

The new force is then X ¼ l^re_Rl^re_P. Cukrowski and Kolbus [97] then present several examples of model reactions, which were analyzed using the new ‘force’.

Their approach is in fact another exercise in combining the traditional mass-action law and the traditional expression for chemical potential as a function of concentration similar to those reviewed in Parts 2.1, 3.1, or 3.2. What is important and new in this approach is the stress on equilibrium and its use as a referential state. It will be seen later that the equilibrium also has similar importance in rational thermodynamic theories. A similar approach, but with no such stress on equilibrium, was presented by Parmon [99,100].

5.2 Tackling mass-action non-linearity and Onsager reciprocity

Anderson and Boyd [101] extended the flux-force approach to the nonlinear area and focused on the so-called Onsager’s reciprocity relations, another popular theme of CIT. They suppose that the reaction rate (flux) J depends on the appropriate force X in the following way:

J ¼ L X þ MX² ð5:17Þ

where L and M are again proportionality, phenomenological, coefficients.

Further they take over the general rate equation:

r ¼ kbY^m

i¼1

ða_iÞ^aik⁰bY^m

i¼1

ða_iÞ^a0i ð5:18Þ

where a’s are the activities and a’s are the reaction orders and not stoichiometric coefficients. Combination of this equation and the condition of a vanishing rate in equilibrium, together with the definition of affinity (A) by Eq. (2.5)₁and with common expression for the relationship between chemical potential and activity, gives the final rate equation:

r ¼ kbY^m

i¼1

ða₁Þ^ai

" #

6½1  expðnAyRT Þ ð5:19Þ

where

n ¼ ða_i⁰a_iÞy
_i ð5:20Þ

Factors in Eq. (5.19) are expanded in Taylor series and the form of Eq. (5.17) is thus obtained as follows:

r ¼ kbY^m

i¼1

ða^eq_i Þ^ai

" #

6½nðAyRT Þ þ nðd₁ny2ÞðAyRT Þ²þ    ð5:21Þ

where

d₁¼q Y^m

i¼1

ða_iya^eq_i Þ^ai

" #

yqðAyRT Þ_A¼0 ð5:22Þ

and index ‘‘eq’’ refers to equilibrium.

The authors then assume that Onsager’s reciprocity relations hold only in the linear range. However: ‘‘Even in the linear regime...the coefficients (in Eq.

(5.21)) are functions of the equilibrium state of the system. This implies that phenomenological coefficients, measured in one reaction mixture, cannot be applied directly to another.’’ The authors therefore conclude that the language of nonlinear thermodynamics is not suitable for chemical kinetics.

The authors agree that there is no ‘‘thermodynamic’’ substitute for the established, mostly empirical, rate equations and Eq. (5.21) serves only as a connection between this tradition and the CIT approach. It could be that the phenomenological coefficients measured and general values obtained, perhaps Eq. (5.17) might substitute the traditional rate equation, with the phenomen-ological coefficients playing the role of rate constants.

Bataille et al. [102] also dealt with Onsager’s relations. They state that linear flux-force relations are not adequate for reaction kinetics and try to discover an extension of Onsager’s reciprocal relations into the non-linear domain. They start with the generalized rate equation for reaction a (J_a) in a rather unusual form but close to Eq. (5.18):

J_a¼k^_aY^m

i¼1

ðf_iÞ
^i k^þ_aY^m

i¼1

ðf_iÞ
^þi ð5:23Þ

where f_idenote fugacities and
^_i ,
^þ_iare the (positive) stoichiometric coefficients for the backward and forward directions, respectively. Three things are then to be checked:

(a) the J ’s can be expressed in terms of the A’s (affinities) and the thermostatic state variables,

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

(c) approximation of the J ’s by linear functions of the A’s over a sufficiently small neighborhood of the equilibrium values A_a gives

J_a&X^R

b¼1

L_abA_b

with satisfaction of the Onsager reciprocity relations L_ab¼L_ba.

Whereas the last two conditions are quite simply satisfied, the first one requires more elaborate treatment. Using affinity definition (2.5)₁and introdu-cing forward and backward affinities such that A_a¼A^þ_a A^_a, Eq. (5.23) is modified:

J_a¼k_aexpðA^_ayRT Þ½1  expðA_ayRT Þ ð5:24Þ From Eq. (5.24) it is clear that the first condition transforms to the question whether the backward affinity can be expressed in terms of the A’s and the thermostatic state variables. This leads to the standard task of linear algebra, viz.

finding (general) solutions of the system of equations X^m

i¼1

_aim_i¼ A_a; a ¼ 1; . . . ; R ð5:25Þ

As there are more unknowns (m_i) than given (A_a) quantities, the number of solutions is infinite. Anyway, the ‘‘unknowns’’ may be expressed from the system with the aid of affinities or other unknowns, which are themselves functions of state variables. Consequently, condition (a) is confirmed. After introducing the general solution into the rate equation (5.24), it is immediately found that Onsager’s relations

ðqJ_ayqA_bÞ_{p;T ;b}

l¼ ðqJ_byqA_aÞ_{p;T ;b}

l ð5:26Þ

(p is pressure and b_l denotes set of parameters in the general solution of the system of algebraic equations) are not generally valid in the non-linear domain.

The authors point out that Edelen’s generalized dissipation potential [103] is still applicable in this domain and its symmetry relations hold as well.

5.3 Hungarian contribution I – Lengyel

Several papers have appeared from the Hungarian school based on Gyarmati’s

‘‘integral principle of thermodynamics.’’ Gyarmati’s approach is, in principle, a certain reformulation of the irreversible thermodynamic approach into the terms of variational principles. Its application to chemically reacting systems in general is described in Sa´ndor’s papers [104,105].

The first contribution dealing with kinetics in more detail is probably the paper by Lengyel and Gyarmati [106,107]. It is interesting to cite the authors’

motivation: ‘‘This consistency (between kinetics and thermodynamics) is both theoretically and practically important also from the aspect of reaction kinetics.

If we can show this consistency, then the whole phenomenological theory of chemical reactions will become a special, but organic, branch of non-equilibrium thermodynamics in the same way as the theory of chemical equilibria has become a special chapter of thermostatics as a result of Gibbs’ work. From the practical point of view the description in non-equilibrium thermodynamics not only offers an alternative description of chemical reactions but can complete the Guldberg – Waage theory. We think that reaction kinetics describe only the concentrations as a function of time but the reaction heats involved in the reaction, i.e. energetics, are not included in the description. In non-equilibrium thermodynamics this inclusion is quite natural; moreover, if the equivalence of both theories can be assumed, then stationary states, the stability and evolution of open kinetic systems, may become objects of exact studies...To illustrate...let us assume that we could show the consistency of nonlinear thermodynamics and the nonlinear theory of chemical kinetics. In this case, instead of the Guldberg – Waage form of the kinetic equations, the consistent differential equations of the nonlinear thermodynamic theory have to be solved.’’

Although the authors present basic thermodynamic equations, including Gyarmati’s principle, in the introduction, the procedure adopted is standard, close to that given in the papers described above. The authors write the Guldberg – Waage law, in this case with molar fractions. Further, the expression of chemical potential in term of molar fraction in ideal systems and the traditional definition of affinity are used and combined with the Guldberg – Waage law to arrive at the general equation

J_r¼J^e_rj_rðA₁;. . . ; A_RÞ ð5:27Þ

for the rate (J_r) of reaction r (which are R in total), called the non-linear phenomenological equation. The novelty in this general equation is the particular representation in

a) the components’ deviations from equilibrium

Dn_i¼n_in^e_i; i ¼ 1; . . . ; m ð5:28Þ

(n’s are mole numbers and ‘‘e’’ refers to equilibrium) which can be introduced into the Guldberg – Waage law through rewriting it into the form:

J_r¼k⁰_rY

where the prime denotes the forward, and double prime the backward, reaction direction and
’s are the stoichiometric coefficients, n the total mole number and J^e_r¼k⁰_rY

i⁰

ðn^e_i0ynÞ
^{i 0 ;r} ¼k⁰⁰_rY

i⁰⁰

ðn^e_i00ynÞ
^{i00 ;r} ð5:31Þ

b) the reaction’s deviations from equilibrium

Dn_i¼X^R

r¼1

_i;rDx_r ð5:32Þ

(x_r is extent of reaction r and D has the same meaning as in Eq. (5.28)) which, after introducing into Eq. (5.30), give:

J_r¼J^e_r Y

c) and introducing the ‘‘absolute affinities’’

L_r¼ Y be considered to be a system of algebraic equations which can be solved for the deviations of the extent of reaction from equilibrium:

Dx_r¼f_rðL₁;. . . ; L_RÞ or Dx_r¼j_rðA₁;. . . ; A_RÞ ð5:35Þ Substituting (5.35) into (5.33), the general Eq. (5.27) is obtained.

Using the absolute affinities, ‘‘general non-linear constitutive equations between reaction rates and affinities’’ are derived in the form

J_r¼J^e_rðL_r1Þu⁰⁰_rðA₁;. . . ; A_RÞ ð5:36Þ where

u⁰⁰_r ¼Y

i⁰⁰

1 þX^R

r¼1

_i00;rDx_ryn^e_i00

!
_{i00 ;r}

ð5:37Þ

and is a function of all affinities as indicated in Eq. (5.36). Onsager’s relations in the linear approximation close to equilibrium are then proved for both stoichiometrically independent and dependent reaction systems.

The authors then present several examples in which they also test the

‘‘Rysselberghe generalized reciprocity relations’’. These were postulated by Rysselberghe [108,109] to be valid for the non-linear equations

J_k¼L_kkA_kþL_klA_lþL_kkkA²_kþL_kklA_kA_lþL_kllA²_l ð5:38Þ in the form

qJ_kyqA_l¼qJ_lyqA_k ð5:39Þ

originally set forth by Pe´ne´loux [110,111]. As this attempt was unsuccessful, the authors conclude that Rysselberghe’s relations are inconsistent with classical chemical kinetics. The authors speculate that the cause lies in the improper choice of thermodynamic forces and some new parameter should be sought instead of affinities. They also stress that, for example, Gyarmati never identified the thermodynamic forces in chemical kinetics with affinities. As he also never gave any specification of these general forces to kinetics, finding the right forces remains an unresolved task.

Summarizing, classical kinetic or thermodynamic quantities and relations are combined and subjected to the interpretation within the flux-force frame-work. As this works in the linear domain only, new, superior ‘forces’ should be found, without asking whether the flux-force approach is correct, necessary or of any practical use, at all. The proclaimed practical aim is not demonstrated even in the linear domain where the flux-force interpretation is satisfactory. The practical value of Eq. (5.34) is questionable especially with regard to the fact that each extent of reaction contains only those moles which have reacted just in that reaction.

The ‘right’ forces are claimed to be found in subsequent papers by Lengyel [112,113], the first one being, in fact, a shortened version of the second. Moreover, it is stated that the mass-action law was deduced from Gyarmati’s governing principle of dissipative processes. This principle reformu-lates the results of CIT in terms of variational principle. Locally, it asserts that the density (o) of the so-called Onsager – Machlup function

o ¼s  c  j ð5:40Þ keeps its extremum value at any point in the system. In other words, its variation (do) is always and everywhere vanishing:

do ¼ ds  dc  dj ¼ 0 ð5:41Þ

In the equations, s is the density of the local entropy production rate, c and j are the so-called dissipation functions (in the forms of densities) or potentials.

Function c is said to depend on all (independent) forces whereas j depends on all fluxes. Consequently, the former is sometimes called the force potential, the latter the flux potential.

The dissipation functions are selected by Lengyel to be

c ¼ 2C ¼ 2X^S

In these equations S is the total number of reactions and Q the number of independent reactions. Parameter l_t comes from nothing more than the mass-action law written in the form

J_t¼ k

(c’s are concentrations) and transformed to the form J_t¼ J The prime or left-to-right arrow, and double prime or right-to-left arrow, represent forward and reverse reaction directions respectively, N’s are the orders (not stoichiometric coefficients).

Coefficient g_t
results from the relations between the dependent and independent (marked by an asterisk) reactions:

J*
¼X^S

t¼1

g_t
J_t;
¼ 1; . . . ; Q ð5:46Þ

Quantities X’s are related to affinities:

A_t⁰¼TX_t⁰; A⁰⁰_t ¼ TX⁰⁰_t; t ¼ 1; . . . ; S ð5:47Þ (all stoichiometric coefficients are considered positive).

Introducing variations of the dissipation functions (5.42) and (5.43), together with the variation of s (which will be discussed later) into Eq. (5.41), the following equation is obtained:

X^Q

As all the varied variables are mutually independent, it follows that the expressions in brackets vanish. Combining the two equal-to-zero equations for J*_u, relation (5.49) follows:

J*_u ¼X^S which is said to be nothing more than the Guldberg – Waage mass-action law.

However, the whole deduction suffers from several deficiencies.

The forms of dissipation functions (5.42) and (5.43) are not proved but stated. They are given in such a way to obtain immediately (5.49) after introducing them into (5.41). This is no deduction but a tautology. Of course, Eqs (5.42) and (5.43) do have physical motivation. It stems from the well-known Eq. (5.44), which was transformed to Eq. (5.45) using the classical relation for the chemical potential in ideal systems. It should be pointed out also that the postulate of expressing the rate as a difference between the forward and backward rate was introduced. Only stoichiometrically independent reactions are considered. They are selected from the whole reaction set by means of relations (5.42) and their affinities are used to express forward and backward rates of independent reactions, e.g. (cf. also Eq. (5.45)):

J

Now, the inspiration for the flux potential is clear.

The origin of the force potential is more unclear. From the inversion of expressions for both forward and backward rates in Eq. (5.45), the author finds relations: which, as he states, satisfy the reciprocal relations. Relations of Eq. (5.51) appear in flux potential (5.43). In this case no attempt is made to use only independent reactions or their affinities and no explanation is given as to why the minus sign in the second expression in (5.51) is not retained in Eq. (5.43).

The tautology is even deeper. The author starts from the mass-action law, either in the form (5.44) or (5.45), to discover it again after several lines of manipulating with it. To recover the desired result, multiplication by 2 is necessary not only in (5.42) and (5.43) but also in the entropy production density. This is achieved in a particularly intriguing manner. The author states that the local entropy density (s) is the function of some set of ‘‘independent extensive state’’ variables

x₁;. . . ; x_i;. . . ; x_f ð5:52Þ

The partial time derivative of this function is given by

qsyqt ¼X^f

i¼1

G_iqx_iyqt ð5:53Þ

and may be used in the general entropy balance equation of CIT:

qsyqt þdivJ_s¼s_s ð5:54Þ

in which J_sis the entropy flux density and s_sits source density. Similar balance equations are supposed to be valid also for the independent variables (5.52).

Combining all balances, the following expression for the entropy production rate is found:

where J_i are current densities and s_isource densities of variables (5.52) in their balances like (5.54). The author claims that the density of entropy production can be written in the form

s_s¼X^f

where m’s are chemical potentials and
’s (positive) stoichiometric coefficients.

Note that the postulate x_i:c_i was introduced. The part of the entropy production rate due to the chemical reactions (the second term in (5.56)) can be expressed using only the independent reactions:

X^S

In the variation condition (5.48), both versions from Eq. (5.60) are summed forming 2s! So, in fact, 2s is used in the Onsager – Machlup function (5.40) instead of s.

Further, there is an obscurity with the independent variables. Initially, it is stated that entropy density is a function of variables (5.52), which are also subjected to the balance equations like (5.54). Later, they are identified with the volume concentrations, see above. However, it is then declared that there are two complete sets of independent variables, viz.

X₁⁰;. . . X_u⁰;. . . ; X_Q⁰; X⁰⁰₁;. . . X⁰⁰_u;. . . ; X⁰⁰_Q ð5:61Þ and

J

?

1;. . . ; J^?_t;. . . ; J^?_S; ^/J₁;. . . ; J^/_t;. . . ; J^/_S ð5:62Þ

which can be used alternatively to express the part of entropy production caused by the chemical reactions, see (5.60). No explanation for this transformation is given. It should be probably understood as a sudden return to the flux-force area. These new sets are used in the variation condition, however, not alternatively, but all-at-once with no explanation, again.

The author also claims that whereas overall reaction rates of all elementary reactions may be dependent, this is not true for the backward and

The author also claims that whereas overall reaction rates of all elementary reactions may be dependent, this is not true for the backward and

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