Kinetics of a Sequence of First-Order Reactions

Abstract

Solutions are obtained for the finite set of coupled rate equations ∂Ci / ∂t = αi,i−1Ci−1 + αi,iCi + αi,i+1Ci+1 (i = 0, ···, N), where αi,j are given in general as αi,i−1 = A, αi,i+1 = B, αi,i = − (A + B), except that α0,0 = − α1,0 = − a, αNN = − αN−1,N = − b, α0,−1 = αN,N+1 = 0. Asymptotic expressions are given for the approach to equilibrium as a function of the various rate parameters and the chain length N. For large N, we find that if A < B, the eigenvalue spectrum approaches a continuum, and the approach to equilibrium is described by a simple relaxation time λ1 ≃ (A1/2 − B1/2)2. However, if A(1 − a / A)2 > B, the system exhibits a peculiar eigenvalue spectrum, and the relaxation is characterized by two distinct and well-separated relaxation times, λ1 and λ2 = − a{1 − B[A(1 − a / A}2]−1).