Statistical Mechanical Theory of Bimolecular Reaction. Derivation of the Arrhenius Law. Characterization of a Catalyst

A statistical mechanical theory is developed for gas-phase bimolecular (binary) reactions: AB ⇌ A + B. Based on the activated-complex hypotheses, and expression for the association rate constant k_a is derived: k_a = k*_{AB* → AB} (QAB*/q_A+B), where k*_{AB* →AB} is the rate constant for the transition from the activated state AB* to the molecular state AB, and qx is the equilibrium occupation probability for state X. The three states (AB, AB*, A + B) are defined by three regions of the energyseparation plane for the relative motion of the reactant pair (A, B). If the interatomic potential has a critical barrier ϵ_c at separation r_c and an attractive well with depth ϵ_b (ϵ_c, ϵ_b > k_BT) computations of the q_AB*/q_{A +B} generate an Arrhenius-Boltzmann factor exp (—ϵ_c/k_BT). The virtual rate constant k*_{AB* →AB} is calculated by assuming that the reactant pair reaching the activated state AB* with the separation r < r_c and the energy E > _c moves on to the molecular state AB only if it loses part of its radial kinetic energy with the aid of third body (catalyst or collision) and is trapped by the potential well. With no catalysts present, this constant is approximately given by k*_{AB* →AB} =πd′²v′ R′⁻³, where v′ is the thermal center-of-mass speed, and d′ and R′ are respectively the collision-sphere radius and the mean distance between AB* and any molecule (A, B, AB or AB*). For binary dissociation, the rate constant k_d is given by k_d = k*_{AB* →A+B} q_AB*/q_AB), which generates exp —(ϵ_c + ϵ_b)/k_BT]. A catalyst for binary reaction is assumed to act as a mediator, facilitating the energy exchange between the radial and rotational modes of motion. Additionally for association only, it also acts as a confinement agent, preventing the pair from flying away from each other. Connections with the collision theory and the activated complex theory are discussed critically.