Mean-First-Passage Times and the Collision Theory of Bimolecular Reactions

A general collision theory is outlined for the kinetics of reaction of molecules which are dilutely dispersed in an inert gas, reaction being the result of binary collisions between the reacting molecules and the inert gas molecules. It is assumed that the products of reaction are instantaneously removed from the system. The mean‐first‐passage time for the transition from reactant to product states is expressed in the classical theory in terms of the solution of an integral equation in which the kernel is the transition probability per unit time between two states of the reacting molecule. As shown by Kim, when a rate constant exists (which is not always the case) it is the reciprocal of the mean‐first‐passage time. In a one‐dimensional system, or in a system of two or three dimensions where the interaction between colliding molecules has a finite range, there exists a finite collision number Z which is independent of the internal state of the reacting molecule. In such a case there is a Liouville‐Neumann solution of the integral equation which allows the mean‐first‐passage time to be expressed as a sum of contributions corresponding to failure to react after no collisions, after one collision, after two collisions, etc. The equilibrium hypothesis is analyzed and it is seen what the precise nature of that approximation is. The theory is illustrated with each of two transition kernels, both satisfying the conditions of completeness and detailed balance. The first is a very simple but physically unrealistic one in which the transition probability depends only on the final state but not on the initial state, and here it is found that the equilibrium hypothesis yields the exact rate constant without error. The second illustration uses the previously calculated transition probability for a one‐dimensional string oscillator suffering impulsive collisions with particles of mass identical to its own, and counts the oscillator as ``dissociated'' when its energy exceeds a certain critical energy ε^*. Here, as is obvious, the equilibrium hypothesis is correct when ε^*≫kT, and it is found that in this range the rate constant for dissociation is ${\mathrm{Z(kT/\pi \epsilon }}^{*}{)}^{\frac{1}{2}}\exp {\mathrm{(-\epsilon }}^{*}\mathrm{/kT)}$ . It is shown in an Appendix that if instead of just formally calling the oscillator dissociated when its energy exceeds ε^*, one actually specifies that the string is then ruptured, the basic transition probability is altered, and the equilibrium hypothesis then yields for the dissociation process the even simpler rate constant Z exp(—ε^*/kT).