Theory of relaxation spectra; the kinetics of coupled chemical reactions

Abstract

The analysis of multi-step reaction kinetics—an important problem in the study of biological systems—is inherently complex and requires the techniques of matrix algebra. The recent development of chemical relaxation methods allows the experimental investigation of such systems, even when some or all of the individual steps are extremely rapid. A general treatment of the theory of chemical relaxation in multi-step systems of any order and any degree of complexity is presented in this paper. The theory is developed in terms of the groups of chemical constituents that participate in the elementary reactions. A general derivation is given for the form of [ K L ], the matrix which determines the kinetic behavior of the system near the stationary state. A proof is given that the eigenvalues of [ K L ], the negative reciprocals of the relaxation times, are all either negative or zero if detailed balance (microscopic reversibility) holds at the stationary state. The number of zero eigenvalues is shown to be determined by constraints related to conservation of mass for the system and by linear dependences among the groups of reactants. Diagonalization of [ K L ] expresses the behavior of the system in terms of normal coordinates, each with its own relaxation time. Apparent oscillations, due to the superposition of relaxation curves for the normal coordinates, are discussed.