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

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 (〈Δ E d 〉, −〈Δ E 〉, and 〈Δ E 2 〉 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; −〈Δ E 〉 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., k 0 ( T )/ k r ( T ) ≠ K eq ( T ), where k 0 ( T ) and k r ( T ) are the dissociation and recombination rate coefficients and K eq ( T ) is the equilibrium constant.