Moment Equations for Vibrational Relaxation Coupled with Dissociation

Abstract

Sets of equations for the moments of the vibrational energy distribution for a diatomic species in a gas are derived, considering simultaneous dissociation and vibrational relaxation. In first approximation a set of two coupled equations are obtained, one for the number of diatomic molecules and the other for the vibrational energy per molecule. The latter contains two correction terms to the Bethe—Teller equation, due to the coupling. The next higher approximation gives three coupled equations, in which the second moment (dispersion) of the distribution function is taken into account. In these equations the dissociation rate ``constant'' is a function of both the average vibrational energy and the second moment. The recombination process is described by three constants which define the over‐all rate, the average vibrational energy, and the dispersion about the average energy of just recombined molecules. The starting point in the development is a gain—loss equation for occupation numbers of vibrational levels. The assumption of harmonic vibrations is used. In order to close the set of equations in any order of approximation, those terms involving higher moments are determined by application of a general method due to Jaynes which gives the least biased estimate in view of the information contained in the moments considered. Consideration of the steady state in first approximation leads to Onsager reciprocal relations and, in second approximation, to more complicated relations.