Rate kernel analysis

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... 

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). 

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... 

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. 

In the analysis of the other term in (8.3), we recall that for small solute densities we only need the solute-density-independent rate kernel. Thus, using number conservation again, we may write... 

The dynamic process that enter into the rate kemal expression [(9.46) or (8.9)] are, of course, those that have been included in the kinetic equation, as discussed briefly in Section VII. We discuss now the specific processes, which are relevant for the rate kernel, in more detail. The kinetic theory expression contains all the collision events that one might anticipate would be important for liquid state reactions. The analysis of the rate kernel in the limit where velocity relaxation effects are neglected bears a strong similarity to the derivation of Stokes law from kinetic theory, and we also explore this relationship. 

Although the analysis in terms of the propagators for independent motion gL is convenient for displaying the content of the kinetic theory expression for the rate kernel, calculations based on (10.4), which contains the propagator for the correlated motion of the AB pair, are probably more convenient to carry out. In kinetic theory, such rate kernel expressions are usually evaluated by projections onto basis functions in velocity space. (We carry out such a calculation in Section X.B). Hence the problem reduces to calculation of matrix elements of (coupled AB motion in a nonreactive system) and subsequent summation of the series. This emphasizes the point that a knowledge of the correlated motion of a pair of molecules for short distance and time scales is crucial for an understanding of the dynamic processes that contribute to the rate kernel. 

We very briefly consider the effects of the terms that were neglected in the course of projecting onto the diffusion mode. Once again, the analysis closely parallels the derivation of Stokes law. To explicitly consider the effects of nonhydrodynamic (nondiffusion mode) states on the rate kernel, we use an analysis similar to that of van Beijeren and Dorfman and introduce a projection operator... 

This reformulation in terms of diffusive propagation and microscopic dynamics in the boundary layer is reminiscent of Noyes s encounter formulation that we briefly described earlier. Now each diffusive encounter is interrupted by sequences of collisions and very short excursions into the fluid. The analysis of nonhydrodynamic effects on the rate kernel can, therefore, be discussed naturally in terms of the encounter formalism. 

There are other ways of analyzing nonhydrodynamic contributions. Projections onto finite sets of velocity states, in combination with kinetic modeling techniques, have proved useful in the analysis of the small molecule velocity autocorrelation function. These techniques can also be used to calculate the rate kernel. ... 

Before closing this section, we should remark that although this analysis of velocity relaxation effects has focused on a simple collision model, we expect that the detailed structure of the rate kernel for short times will depend on the precise form of the chemical interactions in the system under consideration. It is clear, however, that a number of fundamental questions need to be answered before more specific calculations can be undertaken form the kinetic theory point of view. 

Because of the general difficulty encountered in generating reliable potentials energy surfaces and estimating reasonable friction kernels, it still remains an open question whether by analysis of experimental rate constants one can decide whether non-Markovian bath effects or other influences cause a particular solvent or pressure dependence of reaction rate coefficients in condensed phase. From that point of view, a purely... 

Population-balance analysis has been adapted to both coalescence and dispersion of drops in numerous papers by Calabrese, Ramkrishna, and Tavlarides. The analyses with these tools have led to a considerably better understanding of breakage kernels, breakage rates, coalescence efficiency, and collision rates. However, the description and use of these tools goes beyond the scope of this chapter. For a detailed understanding, see Ramkrishna [66]. 

An automated NIRS system capable of scanning individual grains containing late-instar larvae of S. oryzae, R. dominica, or S. cerealella at the rate of 15 kernels/min has been developed in the United Kingdom (Chambers et al., 1998). The system was effective and could detect the infestation irrespective of the type/class of wheat, its protein content (range 11.32 16.2%) and moisture content (range 10.0 13.2%). The minimum detectable size of the insects by NIRS varied between species. As identified by x-ray analysis, the NIRS system has been shown to detect R. dominica as small as 1.1 mm2 with 95% level confidence, whereas for S. oryzae it was 2.0 mm2, and for S. cerealella 2.7 mm2. For a particular insect species, the accuracy of detection increases as insect development proceeds. Accordingly, in S. oryzae the accuracy of detection of first instar larvae was 10%, second instar larvae 24%, third instar larvae 82%, fourth instar larvae 95%, and the accuracy was 100% for pupae and adults. 

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