2.1. Single Reaction
Perhaps the simplest autocatalytic reaction ([
6], see Introduction) expressed generally in terms of atom-conserving components (that is not in just general symbols of substances) is written as
(Because the discussed thermodynamic approach is a mathematical theory, which views stoichiometric equations as equations, the “
” symbol is used instead of the double arrow common in kinetics). The autocatalysis is seen in the fact that two reactant
molecules are produced, while one of them is consumed in the forward direction of (R1). Traditionally, the rates in the forward and reversed directions would be written in this way:
The thermodynamic methodology starts with writing down the compositional matrix and finding its rank (for details, see [
9], pp. 150–151, [
12]). Numbering the “atoms” as
and components as
, the matrix has the dimension
and is
Its rank
. The total number of the components in this reacting mixture is
thus, the number of the independent reactions is
[
9] (p. 153), [
12]. The stoichiometric matrix
of the only one independent reaction is of the dimension
and should fulfill the condition
[
13]. The entry
is the stoichiometric coefficient of
in the reaction
p. If the matrix
is written generally as
the condition results in:
The reaction R1 calls for two stoichiometric coefficients of that are forbidden by (3), and this reaction cannot be considered and selected as an independent reaction in the discussed reacting mixture. Further, from (3), it is followed that , i.e., the components and should be stated at the opposite sides of the stoichiometric equation (of independent reaction). Nevertheless, the thermodynamic methodology still enables to formulate rate equation with the autocatalytic step.
Suitable and allowable selection of the independent reaction is
Its equilibrium constant is (ideal system, unit standard concentration)
The rate of this reaction (
) as a function of temperature and concentrations, as stated in the Introduction, is first approximated by a polynomial of the third degree in concentrations, which is sufficient to obtain meaningful results (the first or second degree does not lead to the occurrence of an autocatalytic term in the resulting rate equation). The simplification procedure gives the following thermodynamic polynomial (for details, see
Supplementary Material):
(The subscripts of the polynomial coefficients reflect the powers of the concentrations in the corresponding term, e.g.,
; see also ref. [
14,
15]).
As suggested earlier [
10,
11], the individual terms in the thermodynamic polynomial are interpreted in the view of the traditional mass-action kinetics as representing steps in the reaction scheme hidden in the polynomial. The scheme corresponding to (5) is:
The selected independent reaction (R2) is recovered as (R3a), the supposed autocatalytic step (R1) as (R3c). Although this autocatalytic step is excluded by the starting linear algebra of stoichiometry, cf. discussion below (3), the methodology of the thermodynamic polynomial allows its presence in the rate equation. Regardless of the occurrence of the single independent reaction, its reaction rate (5) contains up to four additional reaction steps, forming the 5-step scheme (R3). Among them, there are additional autocatalytic steps like (R3d) or (R3e). The full scheme (R3) should be viewed as a mathematical result, i.e., a mathematically allowable set of steps in kinetic equations. Some or even many of these steps could not be those that real-chemistry should state or detect, i.e., which steps are reliable and which remain as “pure mathematics”. This is illustrated after Equation (6); for more detailed discussion, see ref. [
14].
From the linear algebra of the stoichiometry of the studied reaction mixture (for details, see [
9], p. 154, [
12]), it is followed that
where
is the component rate. When only (R1) really occurs, the mass-action rate equation is obtained
. If both (R1) and (R2) occur, then
.
The thermodynamic methodology presented in this paper includes also proper transformations of rate equation into the function of chemical potentials or affinities. We have illustrated both ways in the following simplified thermodynamic polynomial:
The transformation to the function of chemical potentials is straightforward and is based on the relationship between the equilibrium constant and standard chemical potentials:
The traditional model of chemical potential as a function of concentration (in ideal systems):
where unit standard concentration will be used. The result is
The transformation to the function of affinities requires the knowledge of the relationships between chemical potentials and affinities. Note that in the proper mathematical derivation, not only the traditional chemical affinity but also another affinity, called the constitutional affinity, appears [
13]. There is only one chemical affinity (of the only one independent reaction) in our system defined as (cf. [
9], p. 181):
There are two constitutional affinities (
) defined by (cf. [
9], p. 182)
where
(the contravariant metric tensor) is obtained as an inversion of the covariant metric tensor
[
9] (pp. 295–296), [
12]. The basis vectors are defined (see [
9], p. 152)
and, in our case, have the components:
. Thus,
Now it is not difficult to see that
Combining (11) and (14), we obtain
Introducing (15) into (10), we obtain the reaction rate as a proper function of affinities:
Note that in equilibrium, where , the rate is zero, as expected.
Entropy inequality, or the second law of thermodynamics, puts a restriction on reaction rates expressed as functions of affinities [
9] (p. 211). In our case, this restrictive condition is
Evidently, (17) is fulfilled when
. Elaborating on this condition enables to derive additional restrictions on rate coefficients. Carrying out the partial derivative, we come to the following restriction:
This condition can be further modified substituting from (14), (8), and (4):
Inequality (19) should be valid for any equilibrium (any equilibrium concentrations at a given temperature). Using previously published theorem [
15], we conclude that
and
. The non-negativity of rate coefficients is consistent with (6) and (7) and the traditional kinetic view on positive rate constants (rate coefficients). This kinetic tradition is followed here, naturally, as a condition to fulfill the entropy inequality (the second law of thermodynamics).
2.2. Lotka’s Scheme
This scheme, mentioned in the Introduction, comprises three steps. Houston states [
3], p. 70: “Although Lotka mechanism does not (…) correspond to any observed chemical system, its simple mechanism illustrates the basic principles in the more complex oscillatory system.” Here, it serves a very similar purpose—to illustrate the performance of the thermodynamic approach in the case of problematic reaction schemes. In terms of atom-conserving components, Lotka’s scheme could be written as
These steps retain the principal features of Lotka’s steps. Step (R4.1) represents an autocatalytic step for
whereas (R4.2) represents an autocatalytic step for
(together with the consumption of
); (R4.3) is the consumption of
and is identical with the third Lotka’s step. This reacting mixture is composed of four atoms,
, forming seven (
) components,
,
; the compositional matrix is
Its rank (
) is equal to four. Consequently, there are
independent reactions in this mixture. They can be selected as follows:
The corresponding stoichiometric matrix
fulfills the condition
, as can easily be checked. As in the previous example, no component can be on both sides of the stoichiometric equation of independent reactions at the same time.
To have autocatalytic steps in the resulting rate equation (thermodynamic polynomial), a third-degree-approximating polynomial should again be used in this case. In this example, the general rate function is a vectorial function [
9] (pp. 153-154), whose components are the rates of the individual independent reactions:
. Both the initial and final (simplified) polynomials are lengthy, and more details on their derivation and full forms are given in
Supplementary Material. Here, we reproduce the reduced version of the final thermodynamic polynomial (the rate equation), which retains only the terms corresponding to the independent reactions, or the steps of Lotka’s scheme in the form of classical mass action kinetics (superscripts at concentrations or equilibrium constants mean powers):
Note that the vectors contain the rate coefficients (constants) corresponding to () individual independent reactions (indicated in superscripts); for example, ; refers to their equilibrium constants. The presence of the autocatalytic terms in (22) is evident, though autocatalytic steps are not among the independent reactions.
The transformation of the rate Equation (22) to the function of chemical potentials is:
In this example, there are three chemical affinities of the independent reactions (
) and four constitutional affinities (
). Their links to the chemical potentials are shown in
Supplementary Material, together with the full expression for the rate as a function of affinities. Here, the entropic inequality condition, cf. also (17), is a quadratic form in chemical affinities [
9] (p. 211):
Thus, this quadratic form is positive semidefinite. The well-known (Sylvester) theorem of linear algebra states that in this case, all major sub-determinants of the matrix belonging to this quadratic form are non-negative. The matrix can be found in
Supplementary Material; here, we use only the first sub-determinant and the condition corresponding to it:
From it, we have finally:
Using the same theorem as above, published in [
15], it follows that
and
. Stating that the numbering of (independent) reactions makes no difference, we can derive, after proper renumbering, additional results:
,
,
.
Now we return to the rate Equation (22) and write explicitly the equations for the rates of the independent reactions, following from it but retaining only those terms that correspond to the tradition of mass action kinetics (i.e., selecting the remaining
equal to zero):
The component rates are given, according to the stoichiometric matrix (21), by relations:
,
,
,
,
,
,
. Selecting
we obtain, from (27), traditional rate equations corresponding exactly to Lotka’s scheme (R4):
where
,
, and
. Rate Equations (27) and (28) are all consistent with the restrictions on their rate constants derived above on the basis of the entropic inequality. Thus, traditional phenomenological mass-action kinetics is again shown to be consistent with the non-equilibrium thermodynamics theory of reaction kinetics.