Other analyses of reaction rates, h f

There have been several other methods for estimating the Noyes h(f) expression. Those of Northrup and Hynes [103] and Kapral and coworkers [37, 285, 286] are discussed briefly below. Northrup and Hynes used the diffusion equation to describe the probability that a reactant was extant when the separation between reactants was r at a time t. To represent reaction, a sink term was included so that an equation of the form of eqn. (185) is appropriate 

Rather than solve this equation by standard techniques and develop the connection between rate coefficient and density, p, as originally done by Smoluchowski, Northrup and Hynes used projection operator techniques to obtain the probability that a reactant pair survives at a time t after formation, P t) = /drp(r, t) as usual. They found that the survival probability satisfies an equation (which is derived in Appendix D) 

Northrup and Hynes [103] have developed these ideas further and provided a direct comparison with the molecular pair analysis of Noyes (see Chap. 6, Sect. 2.3). They showed that the various parameters introduced by Noyes, and used also by other authors, can be defined by functions familiar in the diffusion equation analysis. Consequently, there seems no specific reason to advocate using the molecular pair approach in future. Those interested in this subject are urged to consult this excellent and readable article by Northrup and Hynes [103]. 

In the simplest case, where propagation of reactants is diffusion-like, the Collins and Kimball rate coefficient, eqn. (25), is obtained [285]. 

The great advantage of this approach is the ease with which other effects and complications can be included and at a level of description appropriate rather than as heuristic additions to the diffusion equation. For instance, the corrections to the motion of the reactants to account for departures from purely diffusive motion can easily be incorporated. The rate kernel appropriate for pure diffusion and improved propagation are 

---