Solutions of the generalized rate law for vibrational relaxation of a pure diatomic gas

Abstract

The generalized rate law for the relaxation of the vibrational energy of a pure diatomic gas, AB, derived earlier, is solved analytically for a variety of initial conditions corresponding to shock tube, laser-excited fluorescence, and chemical activation experiments. The resulting expressions can be used to easily predict whether a given system will relax according to a V–V or a T–V mechanism or both. The initial conditions and the molecular anharmonicity are shown to be as important, if not more important, for this purpose than the ratio of T–V and V–V rate constants. Behind shock waves the energy relaxes exponentially with a T–V time constant. The initial distribution remains Boltzmann. In laser or chemical activation experiments the energy does not relax exponentially, leading to phenomenological time "constants" [Formula: see text] or [Formula: see text] which are not constant in time and prevent direct comparisons with shock tube data. It is only after an incubation period during which the vibrational energy is redistributed via V–V processes that the energy then exchanges with translational energy and decays. Prescriptions are given to extract T–V and V–V rate constants from such data. The initial degree of laser excitation, a, and the time regime probed, t/τ, must be known for this purpose. However, when direct overtone excitation is used, a careful choice of α can lead to extraction of the T–V constant directly. Even though the vibrational energy itself does not relax exponentially, it is shown that the mean energy, [Formula: see text], and the mean squared energy, [Formula: see text], relax in such a way that the quantity [Formula: see text] does decrease exponentially with a time constant very closely related to the V–V rate constant for 2AB(ν = 1) → AB(ν = 2). A short survey of various laser and chemical excitations in the literature is presented and analyzed in terms of initial conditions. In general, the larger the degree of excitation and the higher the quantum numbers of the excited levels, the more V–V character does the energy relaxation have.