Impulsive Collision Model for the Dissociation of Diatomic Molecules

We investigate the dissociation rate of a dilute solution of diatomic molecules in an inert gas. The diatomic molecule is assumed to dissociate when its vibrational energy exceeds the dissociation energy E. A classical impulsive collision model is used for the interaction between the diatomic molecules and the solvent gas which is treated as a temperature bath at temperature T. The diatomic molecule is also simplified by treating it as one dimensional, thus neglecting the rotational degrees of freedom, and assuming that its translational degree of freedom is always in equilibrium when it is not neglected entirely. Still further simplification is achieved by considering only cases where the effect of the solvent gas may be represented by a transition rate between the vibrational energy states of the diatomic molecule, and the distribution function for vibrational energies ε≦E is approximately given by the equilibrium distribution F₀(ε). We find generally (i.e., when the assumptions stated in the last sentence hold, but independent of our model) that when a diatomic molecule with vibrational energy E is very likely to lose (rather than gain) energy in a collision with a gas atom, the dissociation rate k(E) is given by the expression k(E)=β^-1[τ(E)]^-1F₀(E), where τ(E) is the mean time between collisions for a molecule with vibrational energy E and β^-1 is Boltzmann's constant times T. For the model considered in this paper, this will be the case when E≫(γβ)^-1 where γ is the ratio of the mass of a gas atom to that of a diatomic molecule. The expression for k(E) then assumes the simple form, k(E)=αAcβ^-1e^-βE[a(E)/Z(E)], where A is the cross-sectional area for a collision, c is the concentration of gas atoms, Z(E) is the vibrational partition function for the bound states, a(E) is the distance between the minimum and maximum value of the vibrational coordinate when the molecule is on the threshold of dissociation, and α is a constant of order unity. We also treat the case when γ≪1, which leads to a Fokker-Planck type equation for the distribution function from which k(E) is found for βE≫1. A quasi-quantum-mechanical calculation for k(E) is also presented and leads to the same results as the classical calculation.