^{*}

Edited by: Ramesh L. Gardas, Indian Institute of Technology Madras, India

Reviewed by: Evgeni B. Starikov, Independent Researcher, Germany; Miguel Rubi, Universitat de Barcelona, Spain

*Correspondence: Miloslav Pekař

This article was submitted to Physical Chemistry and Chemical Physics, a section of the journal Frontiers in Chemistry

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Recently, a method based on non-equilibrium continuum thermodynamics which derives thermodynamically consistent reaction rate models together with thermodynamic constraints on their parameters was analyzed using a triangular reaction scheme. The scheme was kinetically of the first order. Here, the analysis is further developed for several first and second order schemes to gain a deeper insight into the thermodynamic consistency of rate equations and relationships between chemical thermodynamic and kinetics. It is shown that the thermodynamic constraints on the so-called proper rate coefficient are usually simple sign restrictions consistent with the supposed reaction directions. Constraints on the so-called coupling rate coefficients are more complex and weaker. This means more freedom in kinetic coupling between reaction steps in a scheme, i.e., in the kinetic effects of other reactions on the rate of some reaction in a reacting system. When compared with traditional mass-action rate equations, the method allows a reduction in the number of traditional rate constants to be evaluated from data, i.e., a reduction in the dimensionality of the parameter estimation problem. This is due to identifying relationships between mass-action rate constants (relationships which also include thermodynamic equilibrium constants) which have so far been unknown.

Investigating impacts of thermodynamics on kinetics of chemical reactions is an area of unflagging interest and continuous research. Due to the inherent non-equilibrium nature of ongoing chemical reactions, especially the non-equilibrium thermodynamics brings significant progress to the aim to put thermodynamics and kinetics in a common framework (for example, Pagonabarraga et al.,

Recently, a paper was published describing the detailed (non-equilibrium) thermodynamic analysis of chemically reacting mixtures with consequences for kinetic models (Pekař,

The abovementioned approach (Pekař,

The method of this (and the preceding) paper is of an

The main steps of the method are briefly as follows. First, the number of independent reactions is determined on the basis of the list of components of a reacting mixture and their atomic compositions (Bowen, _{1}, _{2},…, _{n}) =

Here, _{1}, _{2},…, _{R}), _{i} is the molar concentration of component _{β1},ν_{β2},…, ν_{βn}) contain polynomial powers and are also used as subscripts to index various vectors of polynomial coefficients (

The equilibrium condition is applied and the polynomial is modified to a simplified final form, called the thermodynamic polynomial, which contains also thermodynamic equilibrium constants; no reversed rate constants are used. Although the second law was already applied in the application of results of non-equilibrium thermodynamics mentioned above, there is still a condition resulting from this law which is not used and is ignored in other works. This condition refers to reaction rates expressed as functions of (chemical) affinities (^{p}) and reads (Pekař and Samohýl,

Equation (2) is a negative semidefinite quadratic form in affinities and the well-known conditions for a quadratic form to be negative semidefinite can be applied. These conditions give restrictions on rate coefficients (rate constants) in the thermodynamic polynomial; in other words, they put additional conditions on the thermodynamic consistency of rate equations. However, the transformation of the above function of concentrations,

The method described in previous works (Pekař and Samohýl,

The first reacting system is a simple mixture of two isomers, A and B. Chemically, the simplest reaction here is assumed to be in the form:

which, as written, invokes first order kinetics. Its second order version can be expressed in traditional kinetics as

In this mixture, only one independent reaction is possible; the vector

where the vectors

The second analyzed system is composed of two single components, A and B, and their compound AB. Here, the combination reaction

occurs naturally and is of the second order (forward direction). Also here, only one independent reaction exists—the most natural selection is just the reaction (R2). The first-degree thermodynamic polynomial vanishes in this reacting mixture; consequently, only the second-degree (or possibly higher) polynomial is reasonable. It has the following very simple form

which resembles the traditional mass-action expression;

The third system is just the mixture of three isomers (A, B, and C) analyzed in the previous work, but, here, modeled with the second degree thermodynamic polynomial:

Here, two independent reactions are possible and are selected (Pekař, _{1} and _{2}, resp. The vectors _{1}, _{2}). The first two terms on the right hand side of (5) represent the first-degree thermodynamic polynomial which was analyzed in the previous work (Pekař,

Traditionally, “triangular” reactions, (R3), are assumed to occur in this mixture, one of them being dependent.

The last (fourth) system is similar and is that suggested by the reviewer of the previous work. Here, also, a compound of two isomers is included and the mixture thus contains the following components: A, B, AB, BA where (only) AB and BA are isomers. Analogically to (R3), a triangular scheme (R4) is suggested here in traditional kinetics.

The number of independent reactions is still two. The second-degree thermodynamic polynomial is:

The vectors _{1} and _{2}, resp.

Now, the results of the thermodynamic analysis of these four reacting systems are discussed. Crucial for this analysis is the application of condition (2).

The first-degree thermodynamic polynomial contains only one rate coefficient (also called here “rate constant”), namely _{10} (the bold symbol is not retained due to the one-dimensionality of _{10} from (3), similarly for other one-dimensional quantities). The transformed thermodynamic polynomial corresponding to the function

where ° denotes the standard state and

and from this it follows that _{10} > 0 (_{10} = 0 makes no sense in a reacting mixture), which is consistent with A being the reactant and with its component rate ^{A} being equal to –^{A} represents reactant A's formation rate. Thus, the traditional mass-action kinetics expressed in the form of the first-degree version of (3), including the sign of the rate constant, is fully consistent with non-equilibrium thermodynamics—particularly, with entropic inequality (the second law). It should be noted that the presented method does not inherently include restrictions on the non-negativity of concentrations; these should be added as additional constraints.

In the case of the second-degree polynomial, the transformation to the function

which should be fulfilled for an arbitrary equilibrium concentration. This enables the following theorem^{1}

where

Note that the theorem can be easily modified to

The theorem thus gives:

The first condition (12) is the same as that found above for the first-degree polynomial. Thus, both polynomials give consistent conditions for the first order rate constant (_{10}). Referring to terminology introduced in a previous work (Pekař, _{10} is a proper coefficient, while _{20} and _{02} are examples of coupling coefficients. However, in this case, the coupling coefficients are of a somewhat different type than in Pekař (

Let us further suppose that

Then, condition (13) is fulfilled (also) for

This, with inequality symbols only in (12) and (14), gives the traditional expression for ^{A} (i.e., ^{A} = –

As stated above, the first degree thermodynamic polynomial vanishes here. The second degree polynomial in terms of affinities is as follows (for derivation see

Condition (2) is detailed in _{110} ≥ 0, which is fully consistent with the fact that, for example, ^{A} = −^{A} represents reactant A's formation rate and

Because the zero rate constant is impossible for A really reacting with B, the positiveness of the rate constant, traditionally supposed in mass-action kinetics, is shown here to be a condition for consistency with thermodynamics (the second law).

In this example, the thermodynamic polynomial contains only (one) proper coefficient, cf. (4).

Our previous work (Pekař,

The relationship between the traditional rates and the rates based on our thermodynamic approach are (for details see

which is more complex than without the second order (degree), cf. Pekař (

To be able to find restrictions on the traditional rate constants, the thermodynamic polynomial (5) should retain only those second degree terms which correspond to the reactions between identical isomers. In other words, we select _{110} = _{101} = _{011} =

together with the additional restrictions:

(note that, for example,

This is similar to the first order scheme (R3), where two first order rate constants were also not independent (Pekař, _{ij} = 0, then from _{i}_{2} in (17) it is very easy to derive an additional, “detailed balance” condition for the second order rate constants:

which, exactly as in the case of the first order (Pekař,

The transformation of (5) into the function of affinities is very complex in this example, and can be found in _{110} = _{101} = _{011} =

With this modification, condition (2) leads to restrictions on two (proper) first order rate constants which are fully consistent with those obtained previously with the first-order polynomial (Pekař,

Further, the following explicit restrictions for three second order constants can be derived from (2):

All three are coupling constants but the latter two couple an independent reaction with its double. There are no similar simple restrictions on the remaining three rate constants which are of a coupling type (coupling with rates of reactions which do not belong to the selected independent reactions). They should conform to a more complex condition given in

Note that the first term on the right hand side of (6) represents the (only) first order contribution and corresponds to the reaction AB = BA (one of the two selected independent reactions). If only the first degree thermodynamic polynomial is considered, condition (2) gives the following restrictions (see

This means that _{1} = 0, which is consistent with the fact that the first selected independent reaction (A+B = AB) is of the second order. Thus, our analysis independently underlines the fact that the whole triangular scheme (R4) cannot be modeled solely by a first order kinetic model. The second (inequality) condition, which in real reactions is ^{AB} = –_{2}.

All terms in (6) formally form the following reaction scheme (Pekař,

1. AB = BA,

2. A + B = AB,

3. 2 AB = 2 BA,

4. 2 AB = AB + BA,

5. A + AB = A + BA,

6. B + AB = B + BA.

Only the first two reactions directly reflect what chemists would expect for the triangular scheme (R4) (the third reaction in this triangle is not independent). Therefore, similarly as in the preceding triangular example, we will restrict our discussion to the simplified polynomial. This also enables comparison with traditional mass-action kinetics. Thus, we put

The vector of reaction rates expressed as a function of affinities is then:

Condition (2) then gives the following simple and explicit restrictions (see

Both constants in (30) are of the proper type. The conditions in (30) are consistent with the fact that ^{AB} = _{1} − _{2}. The first condition in (30) also corresponds to the second condition in (28) found for the first order model. The restrictions on the remaining two (coupling) rate constants are not as simple, again. They should satisfy the following condition (see

Note, that due to (30), the right hand side of (31) is positive, as required.

Also in this case we can derive additional constraints on the rate constants of the traditional mass-action kinetic model of scheme (R4)—see

(the numbering of the rate constants corresponds to the numbers shown in scheme (R4)). Thus, the dimensionality of parameter estimation problems is reduced by two.

It can also be easily verified that the traditional mass-action rate equations of scheme (R4) also lead to a ”detailed balance” constraint on their rate constants, very similar to that of the “first order” triangular scheme (R3). This is another dimensionality reduction. The number of independent rate constants is therefore three, the same as in the case of the first order scheme (R3) analyzed in our previous work (Pekař,

Second-order kinetic models—or, more specifically, second-degree thermodynamic polynomials—further demonstrated the power of the presented thermodynamic analysis and also revealed where simple, specific results (restrictions) cannot be expected. Second-degree polynomial models led to more complex results arising from thermodynamic condition (2), but in all cases, regardless of the degree (order) and overlooking some of the simplifications made above, simple sign restrictions were found for proper rate constants (coefficients). In contrast, restrictions on coupling constants were usually weaker, giving no strict restrictions on their sign, thus allowing them more “freedom.” This means more freedom in kinetic coupling between different reactions forming a reaction scheme, i.e., in the kinetic effects of other reactions on the rate of some reaction in a reacting system; at the same time, zero coupling (zero coupling rate constants) is not excluded by these final thermodynamic restrictions.

The comparison of thermodynamic polynomials with traditional mass-action rate models revealed constraints on the rate constants of the latter which could not be revealed using the traditional approach. The thermodynamic methodology presented in this work thus allows a reduction in the number of kinetic parameters to be evaluated from data, i.e., a reduction in the dimensionality of the parameter estimation problem.

In this analysis, the universality and consistency of the presented method was thus further underlined. It not only allows rate equations to be derived on the basis of results of non-equilibrium thermodynamics but also enables the derivation of constraints on their rate coefficients which are necessary for full consistency with entropy inequality (the second law of thermodynamics). To this end, the proper transformation of rate equations to functions of chemical and constitutive affinities should be performed.

The author confirms being the sole contributor of this work and approved it for publication.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at:

^{1}The theorem and its proof are provided by Vít Samohýl.