Analytic solution of relaxation in a system with exponential transition probabilities. II. The initial transients

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Abstract
Using an exponential transition probability model normalized in the (0,∞) energy domain, we have obtained an analytical solution for the time‐dependent population density below threshold, c (x,t), in the form of the eigenfunction expansion where x is internal energy, t is time, A0 and Aj are constants that depend on initial conditions, S and Rj are solutions of a determinant of a matrix of coefficients and k0−1 and τj are the relaxation times, the lowest of which (subscript 0) represents the reciprocal of the steady‐state rate constant. From c (x,t) we then obtain all other time‐dependent properties such as the non‐steady‐state rate constant and average energy, as well as incubation times and dead times for both number density and vibrational energy. Calculations relative to shock tube decomposition of N2O, CO2, and O2 in inert gas are compared with experiment, with generally good results. For the triatomics, average energy transferred per collision, as calculated from the experimental relaxation time, compares well with that calculated from the Schwartz–Slawsky–Herzfeld theory. The calculated diatomic rate constants (but not the relaxation and incubation times) are too low. Calculations relative to shock tube decomposition of cyclopropane are compared with numerical calculations of Malins and Tardy using a stepladder model. It is concluded that non‐steady‐state effects are unlikely in the cyclopropane shock tube work, and that diatomic rate constants are sensitive to rotational energy transfer.