Smoluckowski equation

Both negative ions Fe(CN) and 3-5 dinitrobenzoate are much slower in silica compared to water but not as slow as in NaLS where the rates were too slow to measure accurately. The decreased reaction rates are due to repulsion of the anionic quencher by the anionic silica particles on NaLS micelles.It is possible to use the Debye modification of the Smoluckowski equation to explain these data. 

Another most important question in anomalous dielectric relaxation is the physical interpretation of the parameters a and v in the various relaxation formulas and what are the physical conditions that give rise to these parameters. Here we shall give a reasonably convincing derivation of the fractional Smoluckowski equation from the discrete orientation model of dielectric relaxation. In the continuum limit of the orientation sites, such an approach provides a justification for the fractional diffusion equation used in the explanation of the Cole-Cole equation. Moreover, the fundamental solution of that equation for the free rotator will, by appealing to self-similarity, provide some justification for the neglect of spatial derivatives of higher order than the second in the Kramers-Moyal expansion. In order to accomplish this, it is first necessary to explain the concept of the continuous-time random walk (CTRW). 

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... 

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See also in sourсe #XX -- [ ]

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