Concept of Activation Energy in Unimolecular Reactions

Abstract

An exact expression is derived, valid for all pressure regions, for the activation energy of a unimolecular reaction. Because of the reaction, the distribution of reactant molecules among vibrational levels deviates from a Boltzmann distribution; this deviation is explicitly taken into account. In the high-pressure limit, the expression reduces to a well-known form: Eact=〈EK〉 / 〈K〉−〈E〉, due originally to Tolman. In the low-pressure limit our expression contains terms which do not appear in Tolman's formalism. These are a term EM which depends on the deviation from a Boltzmann distribution, and a term EQ which arises from the thermal average of energy-dependent vibrational cross sections. It is shown that the low-pressure expression for Eact does indeed yield an energy fairly close to the energy at which conversion from reactant to product occurs. An approximate analytic expression for Eact is obtained for the case that the cross sections for vibrational energy transfer can be approximated by a truncated harmonic oscillator model. A numerical study of the thermal decomposition of N2O, using SSH theory to evaluate collision rates, shows that both EM and EQ make important contributions to the final value of Eact. It is also shown that the steady-state approximation can lead to serious error in calculating low-pressure rate constants and activation energies at high temperatures.