The effect of vibrational-rotational disequilibrium on the rate constant for an atom-transfer reaction

We report the first realistic quantitative calculation of the competition between reaction and vibrational-rotational energy transfer in respectively perturbing and reestablishing the internal-state equilibrium of reactants and products in a typical fast, second-order bimolecular reaction. The reaction considered is Cl + HBr → HCl + Br at 300 K, including V-V,R,T; V-V,T; V-R,T; V-T; R-R,T; and R-T energy-transfer processes. The 110-state nonlinear master equation is solved by a variety of techniques, including numerical integration, iteration to a quasi-steady state, eigenvalue-eigenvector analysis of the symmetric matrix obtained by linearization in the deviations of the state concentrations from their values at chemical equilibrium, and eigenvalue-eigenvector analysis of the nonsymmetric matrix obtained by linearization in the deviations of the state concentrations from their values for typical experimental initial conditions. We show that vibrational-rotational nonequilibnum effects are less than 1% under typical experimental conditions (i.e., initial rate studies in the absence of significant back reaction), but that the conventional picture of reactive time scale > vibrational relaxation time scale > rotational time scale is not valid. The eigenvalue of the rate matrix corresponding to the eigenvector that carries most of the reactive flux is the fourth or fifth smallest nonzero eigenvalue, in contrast to the conventional picture in which it is the smallest nonzero eigenvalue. As a consequence, after the early-time quasi-steady state in which the local equilibrium rate coefficient is observable (within 1%), the reaction rate coefficient decreases and phenomenological analysis of the net rate of change of the concentration of reactants may show up to four additional quasi-steady states involving successively slower reaction rates before chemical equilibrium is finally achieved.