Stochastic Theory of Vibrational Relaxation and Dissociation

Abstract

The familiar theories of unimolecular reactions are based on the assumptions of strong collisions, a randomizing redistribution of internal energy, and an a posteriori choice of a single dissociation coordinate. In this paper we replace these starting points by introducing a model with a set of transition probabilities defining a Markov chain whose time evolution describes the individual bond breaking and the inter‐ and intramolecular energy transfer. We provide for the dissociation of all the bonds at all times and explicitly specify the couplings of the internal degrees of freedom. The differential‐difference equations for the over‐all relaxation process are transformed by the standard generating function technique to a single partial differential equation which can be solved exactly by the method of characteristics. This procedure displays the competing effects of the vibrational interactions and the individual dissociation energies and suggests that it is possible to determine a priori that bond which is most likely to break first. We treat in particular two fundamentally different classes of unimolecular reaction: (i) collisional activation and (ii) photochemical excitation at low pressures. In the former case we present a formal justification of the basic assumptions implicit in the familar theories of thermal dissociation. In the latter case, however, where the initial distribution of internal energy is highly nonuniform and the molecular relaxation processes occur on a time scale shorter than the mean time between collisions, we show that the details of the vibrational interactions and dissociation energies must be taken into account in order to properly describe the bond‐breaking kinetics. Finally, the closed‐form expressions obtained in the exact solution of our Markov chain allow for a simple discussion of the “random lifetime assumption.”