Diffusion or random walk approximations for h f

If this form for h(t) is substituted into eqn. (191) with a constant of proportionality, say a, the rate coefficient is infinite This is a similar problem to that noted for the diffusion equation analysis where the Smoluchowski theory gives an infinite rate coefficient at the initial time 

The best form of h(t) which can be deduced from the random walk or diffusion equation has been discussed by Berg [278], Pagistas and Kapral [37] and Naqvietal. [38]. They noted that the encounter pair (effectively 

Over long times, this displays the limiting form characteristic of the diffusive recombination of radicals. The various forms of h(t) which have been developed are shown in Fig. 40. There are significant differences between these forms and, in particular, the form of h(f) at short times must be ) where n — 1. The partially reflecting form [eqn. (194)] is satisfactory as its limiting short-time dependence is so too is the 

Noyes random flights form of h(t), though its theoretical justification is limited. The purely diffusive form of h(t), eqn. (193), is an unnecessary contrivance. 

It is interesting to substitute the partially reflecting boundary condition form of h(t) of eqn. (194) into the rate expression deduced by Noyes [eqn. (191)]. Integrating (191) from — to t, leads to a rate coefficient [278] 

---