Approximations to the rate constant for dissociation of diatomic molecules

Abstract

A number of results which pertain to the rate of collisionally induced gas phase dissociation of diatomic molecules were derived. In the standard model for this process collisions bring about random transitions between internal states of the diatom and ultimately lead to dissociation. The rate constant kd for such a model is simply related to λ1, the smallest eigenvalue of a matrix κ which is expressed in terms of rate constants for detailed transitions between states of the diatom. A standard approximation for λ1 is the solution of the linearized characteristic equation for κ. It was shown that the kd so obtained is very nearly the same as the kd obtained from the steady state approximation. If the states of the diatom are assumed to form a continuum, then κ becomes the kernel of an integral operator. Relations between discrete and continuum models were established. It was shown that equivalent diffusion equations which have been derived for the continuum case have analogs in the discrete case. These discrete analogs pertain to stepladder models, models for which transitions are allowed only between adjacent states. It was further shown how discrete (quantum) models and continuum (classical) models are linked by quasicontinuum models which, while still discrete, are formulated in terms of elements of the continuum rather than in terms of individual quantum states. Some of the foregoing results were illustrated by a treatment of the separable exponential model. Parameters α and β for this model have a straightforward physical interpretation. The rate constant and the state distribution for the model are obtainable in closed form via the steady state approximation. Their dependence on α and β is readily interpretable physically. The state distribution for the model was shown to be potentially useful in variational calculations of kd for other models. In such calculations, α and β would be chosen so as to maximize the variational lower bound to λ1 for the model in question.

Studies of a model of hydrogen recombination

“…The monotonic increase of p with decreasing 0 for the separable exponential model cannot be explained in this way (4). The present study goes beyond ref.…”

“…on rate constants of a change in level density has been examined for the Morse oscillator model (8) and for the so-called separable exponential model (4). In both studies the preexponential factor of kd was found to increase with increasing level density, but it is not evident that the cause of the increase is the same in both cases.…”

“…4 where xi is the fraction of molecules in level i, and xi,, is the fraction of molecules which level i would contain if a Boltzmann distribution were maintained. For the Morse oscillator model xi,, is given by…”

“…tion has been proposed (4) wherein the distribu-length (4)(5)(6). Less attention has been paid to the tion of level populations is assumed to have a accuracies of the second (3,4) and the third (4) plausible functional form which is characterized of the just mentioned approximations. by two parameters.…”

Some properties of the mean first-passage time for thermal decomposition

“…At the same time, our knowledge of the transition probabilities that determine the rate of decomposition of the reactant is often not good enough to justify anything more than an approximate solution. Two common approximations to the non-equilibrium rate constant, k0t , are the equilibrium and steady-state rates of activation, ko eq and ko s~, respectively [4][5][6][7]. As will be argued below, the steady-state rate constant is a lower bound to k0*, while the equilibrium rate constant is known to bean upper bound, although not a very good one.…”