Detailed Balance - Wegscheider's Conditions For The Generalized Mass Action Law

Wegscheider's Conditions For The Generalized Mass Action Law

In chemical kinetics, the elementary reactions are represented by the stoichiometric equations

where are the components and are the stoichiometric coefficients. Here, the reverse reactions with positive constants are included in the list separately. We need this separation of direct and reverse reactions to apply later the general formalism to the systems with some irreversible reactions. The system of stoichiometric equations of elementary reactions is the reaction mechanism.

The stoichiometric matrix is, (gain minus loss). The stoichiometric vector is the rth row of with coordinates .

According to the generalized mass action law, the reaction rate for an elementary reaction is

where is the activity of .

The reaction mechanism includes reactions with the reaction rate constants . For each r the following notations are used: ; ; is the reaction rate constant for the reverse reaction if it is in the reaction mechanism and 0 if it is not; is the reaction rate for the reverse reaction if it is in the reaction mechanism and 0 if it is not. For a reversible reaction, is the equilibrium constant.

The principle of detailed balance for the generalized mass action law is: For given values there exists a positive equilibrium with detailed balance, . This means that the system of linear detailed balance equations

is solvable . The following classical result gives the necessary and sufficient conditions for the existence of the positive equilibrium with detailed balance (see, for example, the textbook).

Two conditions are sufficient and necessary for solvability of the system of detailed balance equations:

1. If then (reversibility);
2. For any solution of the system

the Wegscheider's identity holds:

Remark. It is sufficient to use in the Wegscheider conditions a basis of solutions of the system .

In particular, for any cycle in the monomolecular (linear) reactions the product of the reaction rate constants in the clockwise direction is equal to the product of the reaction rate constants in the counterclockwise direction. The same condition is valid for the reversible Markov processes (it is equivalent to the "no net flow" condition).

A simple nonlinear example gives us a linear cycle supplemented by one nonlinear step:

There are two nontrivial independent Wegscheider's identities for this system:

and

They correspond to the following linear relations between the stoichiometric vectors:

and .

The computational aspect of the Wegscheider conditions was studied by D. Colquhoun with co-authors.

The Wegscheider conditions demonstrate that whereas the principle of detailed balance states a local property of equilibrium, it implies the relations between the kinetic constants that are valid for all states far from equilibrium. This is possible because a kinetic law is known and relations between the rates of the elementary processes at equilibrium can be transformed into relations between kinetic constants which are used globally. For the Wegscheider conditions this kinetic law is the law of mass action (or the generalized law of mass action).