Bimolecular Master Equations for a Single and Multiple Potential Wells with Analytic Solutions

The analytic solutions, that is, populations, are derived for the K-adiabatic and K-active bimolecular master equations, separately, for a single and multiple potential wells and reaction channels, where K is the component of the total angular momentum J along the axis of least moment of inertia of the recombination products at a given energy E. The analytic approach provides the functional dependence of the population of molecules on its K-active or K-adiabatic dissociation, association rate constants and the intermolecular energy transfer, where the approach may complement the usual numerical approaches for reactions of interest. Our previous work, Part I, considered the solutions for a single potential well, whereby an assumption utilized there is presently obviated in the derivation of the exact solutions and farther discussed. At the high-pressure limit, the K-adiabatic and K-active bimolecular master equations may each reduce, respectively, to the K-adiabatic and K-active bimolecular Rice-Ramsperger-Kassel-Marcus theory (high-pressure limit expressions) for bimolecular recombination rate constant, for a single potential well, and augmented by isomerization terms when multiple potential wells are present. In the low-pressure limit, the expression for population above the dissociation limit, associated with a single potential well, becomes equivalent to the usual presumed detailed balance between the association and dissociation rate constants, where the multiple well case is also considered. When the collision frequency of energy transfer, Z_LJ, between the chemical intermediate and bath gas is sufficiently less than the dissociation rate constant k_d( E' J' K') for postcollision ( E' J' K), then the solution for population, g( EJK)₊, above the critical energy further simplifies such that depending on Z_LJ, the dissociation and association rate constant k_r( EJK), as g( EJK)₊ = k_r( EJK)A·BC/[ Z_LJ+ k_d( EJK)], where A and BC are the reactants, for example, relevant for O₃ formation from O + O₂ + Ar up to ∼100 bar; otherwise, additional contributions from postcollision are present and especially relevant at high pressures. In the aforementioned regime Z_LJ < k_d( E' J' K) the physical connection of recombination rate constants, k_rec based on either utilizing population from the master equation approach or a collision based bimolecular RRKM theory is traced and elucidated analytically that the rate constants are equal. Recombination rate constants, k_rec, based on the population, are also given and considered for an adiabatic or active K. For example, for O₃ formation in Ar bath gas, the K-adiabatic-based k_rec for O₃ yields 4.0 × 10^-34 cm⁶ molecule^-2 s^-1 at T = 300 K and 1 bar, in agreement with the experimental value, where the contribution from the population of metastable ozone is discussed and the adiabaticity of K highlighted. A facile numerical implementation of the formalism for g( EJK) and k_rec for O₃ is noted. The application of the expressions to ozone recombination as a function of pressure and temperature is given elsewhere, beyond the selection considered here, for studying the physical features, including the contributions from the K and intermolecular energy transfer to the k_rec, and the puzzles reported from experiments for this reaction.