Generalized rate law for vibrational relaxation of a pure diatomic gas

Abstract

The master equation for the vibrational relaxation of a pure gas of diatomic molecules AB is reduced to a simple analytical rate law. Anharmonicity is accounted to first order, and both T–V and near-resonant V–V energy transfer processes are included with the limitation that Δν = ± 1. L and au–Teller type transition probabilities are used to scale the rate constants. The rate law consists of a pair of simultaneous first order non-linear differential equations — one for the mean vibrational energy, [Formula: see text], and one for the mean squared vibrational energy [Formula: see text]; or equivalently a non-linear second order differential equation for [Formula: see text], with respect to time, t, plus an algebraic equation for [Formula: see text] These lead to[Formula: see text]where χe is the anharmonicity factor, N the molecular concentration, νe,. the spectroscopic vibrational frequency; ν′ = νe (1 − χe); ν″ = νe. (1 − 3χe); [Formula: see text]; 1/τ = Nk1.0(1 − e−hν″/KT); k1.0 the rate constant for the process AB(ν = 1) + AB(ν) → AB(ν = 0) + AB(ν); and [Formula: see text] the rate constant for the process 2AB(ν = 1) → AB(ν = 0) + AB(ν = 2). It is shown that the Bethe–Teller law, [Formula: see text], is valid only in the limit of zero anharmonicity or slow V–V processes, or when the initial population is Boltzmann, such as in shock tube experiments. Furthermore, a population distribution which is initially Boltzmann will remain so; whereas a non-Boltzmann distribution rapidly becomes a Boltzmann distribution on a time scale determined by the sum of T–V and V–V rate constants. The present study allows one to gauge the importance of two common assumptions: the validity of the Bethe–Teller law and the existence of a Boltzmann distribution or vibrational temperature during the relaxation.