Critical analysis of the thermodynamics of reaction kinetics

Our objective is to show the weakness of the recent thermodynamics of chemical reactions. We show that such a thermodynamic theory of chemical reactions, which could be similar to the generalized Onsager’s theory in thermodynamics, is not reality at the moment.


INTRODUCTION
The thermodynamics of the reaction kinetics is presently limited on the theory of ideal gases or their mixtures. Consequently it describes the reality to the same degree as the ideal gas describes it. However, complex biological phenomena can be described [i], [ii] and some approaches give realistic results despite this general simplification [iii].
The law of mass action, which is the basis of the chemical reaction kinetics, is not generally valid. It is true that the constitutive equation of reaction speed is approximated well by the product of the power of concentrations, but the the stoichiometric ratio as the power of the concentration does not surely fit to reality. Additionally the law of mass action is a non-linear constitutive equation which does not fit the generalized Onsager's symmetry relations. This is because Onsager's theory is based on constitutive equations of the exchange of current fluxes and their interactions with thermodynamic forces [iv]. However the chemical reactions are based on sources and products of balance equations. Because the resulting values satisfy the mass-conservation rules, and therefore include both negative and positive terms, this excludes the existence of the dissipative potential.

THE BASIS OF REACTION KINETICS
We study a system having multiple chemical components which could interact in stoichiometric chemical reactions. Let the number of the reactions be R. The molar number of chemical reactions derived from two sources: the environmental inand ex-fluxes and internal sources, counting the chemical components that are used up and those that are produced. Consequently the molar number has a balance equation: The equations of the stoichiometric reactions are: ,..., ,..., ,..., ,...,  1 1   (5) where j  is the j-th reaction coordinate and dt d j  is its reaction speed.
The mass-conservation principle is of course valid in the chemical reactions too. In our case, this means that the consequent mass balance of all the reactions must not have a source, so: where i M is the molar-mass of the i-th components.

HOMOGENEOUS SYSTEMS
Chemistry works with concentrations, so we use these too temporarily.
When the system has volume V, the molar concentration of the i-th components is This equation must be completed by constitutive equations to determine the time-dependence of the reactions.

THE CONSTITUTIVE EQUATIONS
The constitutive equation for the concentration flux By understanding these principles in more depth, the reaction speed will be connected to a measurable value. Based on (4) we find a connection between the concentration speed (7)) and the reaction speed.
According to (4), the i-th component of the change of speed of the molar-concentration in the j-th reaction is: (14) from which the constitutive equations are: For example, let us study a homogeneous system with four components This system has four concentration balances, four concentration productions, but only one reaction coordinate and one characteristic reaction speed.
This reaction speed could be expressed with any of the four concentration productions according to (14).
Let us study the concentration production of chemical component 1 A . In equilibrium we could derive two kinds of speeds.
One reaction speed is in the direction of the upper arrow (forward) while the other one is in the direction of the lower arrow (backward). The mass conservation is valid, so the consequent reaction speed is: J u l y 24, 2 0 1 5 Hence the consequent reaction speed is the sum of the forward and backward reactions.
It is zero in equilibrium, so: where K is the equilibrium constant.
Examples of the reaction kinetics of homogeneous systems.

Single-component system without reservoir reaction
In this case, the concentration balance (7) has no production term, only flux, so b) When the system has a material transfer contact with a reservoir according to (10), then from (20) we get: Its solution is: Note that the boundary condition has a definite influence on the result.

Production of component A from component X by decay
In this case the system has a reaction: The reaction speed according to (13) is In equilibrium: Then we get: is a predetermined constant value, the equilibrium value is: Introducing this, we again obtain (32) with other parameters as before: Its solution has the form shown in (33).

Population dynamics
The reaction kinetics of autocatalytic systems shown above can be connected to the Volterra-Lotka-Glaser populationdynamics theory [v]. Its basis is that, in the case of a population of N individuals, the kinetic equation with a growth rate r. This analogy can be made only when the feeding has infinite capacity. When the capacity is finite, the growth rate decreases by the growth of the number of the given population. In this case, , where 0 N is the upper limit of the population size. In this case, the population dynamics could be described by which has the identical form to (30).
Consequently the population dynamics can be modelled well by Equation (28) of the autocatalytic processes. The challenge however is that Equation (30) was derived while considering at least two independent constitutional conditions, and it is not yet clear how these were fixed here.

A Gray-Scott-model
We can show the asymmetry in consequence of the mass conservation. The model investigates the following processes in a homogeneous system [vi]: Here the first equation describes the situation when the catalyzer B produces component B from A. The second equation is the degradation of the catalyzer. The balance of P is undetermined so it has no dynamical description. The reaction speed of the decay of the component A is (13), and so the concentration production of A is according to (14). The reaction speed of the component B catalyzer according to (13) . Hence the balances of concentration are: J u l y 24, 2 0 1 5 We found that one of the productions is opposite to the other one. a) When the system is in material exchange contact with the reservoir described by (10), then from (20) we get: When the parameters of the material transfer are identical, we get the equations of the Gray-Scott model: The thermodynamics have not been taken into consideration until this point. Now we give the further development of the thermodynamic reaction kinetics.

THERMODYNAMIC BASIS OF REACTION KINETICS
The first law of thermodynamics is formulated: From the equations (41) and (42), the first law of thermodynamics is derived: The Gibbs relation for multicomponent systems is the unification of the first and second principles of thermodynamics. Its formulation with the molar numbers is: Introducing the following denotations (affinities), we get the form of the first law of thermodynamics formulated by molecular numbers for chemical reactions: is the affinity of the j-th chemical reaction. The value  is the so-called reaction coordinate.
The affinity and the form of the irreversible heat in the next formula (49) were precisely formulated by De Donder [vii]: In consequence of (49), that the consequent which, in the case of a close system, is: J u l y 24, 2 0 1 5 The free enthalpy G decreases during the chemical reaction and reaches a minimum in equilibrium. The necessary condition of the equilibrium is: Dividing (38) by the volume V of the system and using the balance of molar concentration (7), we get:

CONTINUOUS SYSTEMS -EQUATION OF REACTION DIFFUSION
Morphogenesis is the embryonal development of the structure of an organism. Its principal result is the explanation of the complex structures starting from a single cell. The embryonal development has two additional mechanisms: the movement of the cell and the specialization of its function. These mechanisms are influenced by various chemical components produced in the cell and liberated afterwards. These chemical products are the morphogens. The morphogens diffuse freely from one cell to another through their membranes, and their intracellular concentration affects the development of the cell. The first mathematical model of the morphogenesis was made by Turing [xiii]. His model was made with two morphogens, a and b. Transport of these is defined by their concentration gradient. However the diffusion is not the only mechanism of the variation of morphogen concentration in the cell. The morphogens can chemically react with each other, depending on their intracellular concentrations. The time-variation of morphogens can be described by coupled equations of reaction-diffusion. Study of the system of equations showed that when the diffusion constants differ between the morphogens then the initial perturbation of the morphogen concentration shows a stable periodic pattern.
In the thermodynamic formulation Let us next calculate the mass balance of the individual components of a static, continuous, multi-phase system. The mass of components can be changed for two reasons: A mass-current flux exists on the surface of the system. This is only conductive in the static system, which is connected to diffusion.
There are chemical reactions inside the system, which could be the source or sink of the mass of various components.
In a system which has volume V and a mathematically closed surface we consider the mass Applying the Gauss theorem: This could be written for any volume V, so the differential mass balance is:

NON-EQUILIBRIUM THERMODYNAMICS OF MULTICOMPONENT SYSTEMS
For the actual form of the entropy balance, we start from the Gibbs relation: from which the entropy current density is: and the entropy production is: In the case of an isothermal system, two term remains for investigation, of which the term which describes the chemical reactions was discussed above. The term which describes the diffusion is

REACTION-DIFFUSION EQUATION
For simplicity, let us limit the calculation to two components. We now have two mass current densities and two forces: The system is isothermal, so the chemical potentials depend only on the concentrations. Because the chemical reactions were described in the ideal-gas approximation, we apply that here too. Then the chemical potentials of the various components depend only on their own concentrations, while cross-effects are neglected, and using the form , 76) J u l y 24, 2 0 1 5

Homogenous equilibrium solution
When (76) exists then the solutions satisfy the equations: The  is a bifurcation parameter that drastically changes the character of the solution at the point accompanied by a critical value c k connected to F: The Jacobi matrix of the right side of the equation system is: