Rate kernel

As an example of the application this work, Kapral [285] and Pagistas and Kapral [37] have considered the reaction rate between iodine atoms (or some other similar species) effectively distributed uniformly in solution. They compared their calculations with those of the diffusion equation analysis and with the molecular pair approach rather than compare rate coefficients, Kapral [285] compared the rate kernels (which are approximately the time derivatives of rate coefficients). Over long times, these kinetic theory and molecular pair rate kernels both reduce to the typical form of the Smoluckowski rate kernel. However, with parameters such as R — 0.43 nm and D = 6 x 10 9m2s 1, the time beyond which the rate kernels of kinetic theory and the Smoluchowski theory are in reasonably close agreement is 20 ps, a time much longer than the velocity... 

Fig. 40. The Noyes h(f) expression (i.e. rate kernels) of eqn. (191) with estimates...

It should be remarked that differences between various rate kernels are more marked than the differences between the respective rate coefficients (because rate kernels are approximate derivatives of the rate coefficients). To observe any difference between experimental results and one or other of the theories would require accurate measurements ( 2%) at times of... 

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... 

Since the complications due to solvent structure have already been discussed, the remainder of this chapter is mainly devoted to a discussion of the complications introduced into the theory of reaction rates when the collision of solvent molecules does not lead to a complete loss of memory of the molecules about their former velocity. Nevertheless, while such effects are undoubtedly important over some time scale, the differences noted by Kapral and co-workers [37, 285, 286] between the rate kernel for reaction estimated from the diffusion and reaction Green s function and their extended analysis were rather small over times of 10 ps or more (see Chap. 8, Sect. 3.3 and Fig. 40). At this stage, it is a moot point whether the correlation of solvent velocity before collision with that after collision has a significant and experimentally measurable effect on the rate of reaction. The time scale of the loss of velocity correlation is typically less than 1 ps, while even rapid recombination of radicals formed in close proximity to each other occurs over times of 10 ps or more (see Chap. 6, Sect. 3.3). 

In developing this rate kernel [eqn. (303)]. Kapral noted that one term could be ignored because it was small. This is a factor which allows for non-equilibrium flow of A and B towards each other. 

This rate kernel may now be incorporated into the Noyes expression for the rate coefficient [see eqn. (191)], where ft[( ) = h(f), providing that the rate kernel varies much more rapidly than does the concentration 8nA change with time. The rate coefficient is... 

As a final point, Kapral has discussed the higher collision events where multiple collisions between A and B occur and a near equilibrium spatial distribution is not maintained. He found that the rate kernel was of the form... 

On Laplace inversion and then inserting the rate kernel into the Noyes expression for the rate coefficient [eqn. (191)], the rate coefficient is seen to be exactly that of the Collins and Kimball [4] analysis [eqn. (25)]. It is a considerable achievement. What is apparent is the relative ease of incorporating the dynamics of the hard sphere motion. The competitive effect comes through naturally and only the detailed static structure of the solvent is more difficult to incorporate. Using the more sophisticated Gaussian approximation to the reactant propagators, eqn. (304), Pagistas and Kapral calculated the rate kernel for the reversible reaction [37]. These have already been shown in Fig. 40 (p. 219) and are discussed in the next section. 

With this extension to the analysis by Kapral, the anslysis leads to a result for the observed rate kernel, which is... 

Pagistas and Kapral have also considered the dependence of the rate kernel (and hence coefficient also) on the density of the B species [37]. They found that the steady state rate kernel is... 

Smoluchowski aggregation rate kernels modified to account for hetero-aggregation in the presence of repulsive barriers become... 

Thus the rate kernel plays a central role in the description of relaxation processes in the chemically reacting system, and a good deal of our attention is devoted to this quantity in the sequel. 

The half-sided Fourier transform of the rate kernel for the absorption of A,... 

The rate coefficient then follows by taking the w=0 limit of the rate kernel... 

The l-space expression for the rate kernel then follows after inversion of the half-sided Fourier transform... 

Note that the rate kernel has a long time tail that decays as t... 

This equation may be formally solved to yield an expression for the rate kernel with the general form of (3.7), (cf. Northrop and Hynes ),... 

The results given above are the correlation function expressions for the rate coefficient, which we wished to obtain. The last line also serves to define the quantity k z the rate kernel. As noted earlier, this quantity is central to discussion of reaction rate theory. Here and in the sections that follow, we attempt to elucidate its structure. 

The rate kernel is again most conveniently obtained by taking the Laplace transform of (6.21),... 

The rate kernel can therefore be written as the sum of a -independent part... 

Solvent effects enter through the potential of mean force and the activation energy they may cancel or nearly cancel in the expression for kj (cf. Northrup and Hynes ). The collision frequency per unit density of B, ab(8 b /Mab) 1 expression for k° for a bimolecular reaction, takes the place of the frequency Wq in (3.23) for an isomerization reaction. This analysis shows that the transition state expression for the rate coefficient appears in this theory as a singular contribution to the rate kernel for the hard-sphere model of the reaction. 

If the reaction can not be modeled by an impulsive collision event, as for atomic recombination or some isomerization reactions, then is zero by time reversal symmetry as discussed earlier. Nevertheless, the structure of the rate kernel has the same qualitative structure as described above. To... 

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