Dynamical treatment of unimolecular decomposition reactions. The RRKM formula

Abstract

To explore the range of validity of the RRKM theory of unimolecular reactions, we present a completely dynamical derivation of the RRKM expression for the decomposition rate of isolated molecules. This derivation avoids the usual equilibrium statistical assumptions and expresses the conditions for validity of the RRKM theory in terms of fundamental, static, molecular properties. To carry out this derivation we apply a treatment of energy transfer and decomposition which combines the Wigner–Eisenbud R-matrix approach to scattering with a technique we previously developed for studying the internal dynamics of nonreactive molecules. We obtain a molecular dissociation rate which agrees with the predictions of microcanonical transition state theory by introducing conditions which ensure statistical equilibration of all states describing the molecular fragments in close proximity of each other. We verify that, under the conditions of our derivation, the distribution of product states is statistical, i.e., in agreement with the implications of simple phase space theory, when the RRKM unimolecular rate formula is valid. Our treatment relies on a number of assumptions which may be regarded as sufficient (although not necessary) conditions for RRKM behavior. These include assumptions concerning the relative magnitudes of various interaction matrix elements (and thus the relative rates of intramolecular relaxation, passage along the reaction coordinate, and decomposition), the properties of intramolecular potential energy surfaces, and the energy dependence of certain quantities. One assumption belonging to the last category may not be entirely valid in all relevant cases but can probably be relaxed considerably. Other assumptions seem plausible but ultimately remain somewhat uncertain due to our ignorance about detailed features of potential energy surfaces for polyatomic molecules.