Theoretical Rate Constant for Thermal Unimolecular Reactions in a Multilevel System

Abstract

A macroscopic unimolecular rate constant is given here in terms of the detailed microscopic rates of transport and relaxation in a many-quantum-level system. A rate equation for each quantum level is written considering all input and output processes. The resulting set of equations constitutes a master equation. The conventional unimolecular rate constant is then the lowest eigenvalue to the relaxation problem defined by the master equation. This is then a general prescription for the theoretical treatment of unimolecular behavior based only on the assumed mechanism and explicitly including the multilevel behavior of real systems undergoing chemical reaction. General methods for solving this eigenvalue problem are discussed. These methods are illustrated with the master equation for a generalized Lindemann mechanism. The lowest eigenvalue for this mechanism is obtained by an iteration procedure. The zero-order result of the iteration is shown to be equivalent to the rate constant derived from a steady-state approximation. Both the zero-order and exact expressions for the lowest eigenvalue are given in terms of a computationally useful many-shot expansion. The lowest eigenvalue result then constitutes an exact and computationally simple version of the unimolecular rate constant which is an unambiguous consequence of the microscopic parameters inserted into the chosen relaxation mechanism. Since the particular mechanism used here is also the point of departure in the derivation of the well-known RRKM theory of unimolecular reactions, the lowest eigenvalue solution is directly compared with the RRKM rate constant. The additional restrictive conditions for agreement between the two rate constants are enumerated. Some generalized expressions for mean first passage times in unimolecular rate theory are also derived and expressed in terms of many-shot solutions. These latter expressions are computationally useful for extremely rapid reactions.