Systems of first-order chemical reactions

Abstract

We consider the classical model in chemical kinetics of a system of n species in which each species is converted to every other species by a first-order reaction. Solutions to the initial-value problem are given in matrix form and the properties of the n × n matrix K representing the system are analysed. For arbitrary (i.e. non-negative) values of the first-order rate constants, zero is an eigenvalue, and the other eigenvalues are complex with negative real parts. Thus, in this case the system generally oscillates to equilibrium. However, if the principle of microscopic reversibility is applied, and if each species is converted directly to every other species, then the system cannot oscillate but must converge “exponentially” to equilibrium. We discuss when K is diagonalizable, and we calculate a bound for the eigenvalues of K. Special forms of K, corresponding to special systems of reactions, are also examined; these include reactions in the configuration of a “chain”, a “cycle”, a “node” and reactions comprising combinations of these. We find again that if the principle of microscopic reversibility is rigorously applied then oscillations cannot take place, but that if this principle is not applied then oscillations may take place. The system of rate equations considered can be used to model various chemical, physical and biological phenomena.