Solution of Some One- and Two-Dimensional Master Equation Models for Thermal Dissociation: The Dissociation of Methane in the Low-Pressure Limit

Abstract

Using three formulations of the master equation (ME), we have investigated theoretically the dissociation of methane in the low-pressure limit. The three forms of the ME are as follows: (1) A one-dimensional model in which E, the total energy, is the independent variable (the E model). (2) The two-dimensional strong-collision-in-J model of Smith and Gilbert (Int. J. Chem. Kinet. 1988, 20, 307−329) in which ε, the energy in the active degrees of freedom, and J, the total angular momentum quantum number, are the independent variables (the ε,J model). (3) A two-dimensional variant of the ε,J model in which E and J are the independent variables (the E,J model). The third form of the ME is the most physically realistic, and for this model we investigate the dependence of values of the energy transfer moments (〈ΔEd〉, −〈ΔE〉, and 〈ΔE2〉1/2) deduced from experiment on assumed forms of the energy transfer function, P(E,E‘), and on temperature. All three moments increase as the temperature rises; −〈ΔE〉 increases from 20−25 cm-1 at 300 K to 110−120 cm-1 at 4000 K. The variation in the energy transfer moments with the form of P(E,E‘) depends on the particular moment and the temperature, but generally the variation is not greater than 25%. For the same input to the models, the E and E,J models give similar values of the rate coefficient at high temperature, implying that the rotational degrees of freedom behave increasingly as if they are active as temperature is increased. For T > 3000 K, the dissociation perturbs the equilibrium energy distribution of the molecule so much that the detailed-balance condition begins to fail; i.e., k0(T)/kr(T) ≠ Keq(T), where k0(T) and kr(T) are the dissociation and recombination rate coefficients and Keq(T) is the equilibrium constant.