Smoluchowski encounter-rate equation

As in the case of hydrogen exchange discussed in chapter 2 (Section l.F), it is possible to apply the Smoluchowski equation [Eq. (2.1)] to calculate the encounter rate. For nitration of heteroaromatic substrates, Ridd took a fixed value of 6 for the ratio of the ions (63JCS4204), thus simplifying the expression to Eq. (3.17). Since values of ii at various temperatures are well known, the calculation of A(enc.) at various acidities... 

Smoluchowski, who worked on the rate of coagulation of colloidal particles, was a pioneer in the development of the theory of diffusion-controlled reactions. His theory is based on the assumption that the probability of reaction is equal to 1 when A and B are at the distance of closest approach (Rc) ( absorbing boundary condition ), which corresponds to an infinite value of the intrinsic rate constant kR. The rate constant k for the dissociation of the encounter pair can thus be ignored. As a result of this boundary condition, the concentration of B is equal to zero on the surface of a sphere of radius Rc, and consequently, there is a concentration gradient of B. The rate constant for reaction k (t) can be obtained from the flux of B, in the concentration gradient, through the surface of contact with A. This flux depends on the radial distribution function of B, p(r, t), which is a solution of Fick s equation... 

In the limit as ftact the rate of reaction of encounter pairs is very fast. The Collins and Kimball [4] expression, eqn. (25), reduces to the Smoluchowski rate coefficient, eqn. (19). Naqvi et al. [38a] have pointed out that this is not strictly correct within the limits of the classical picture of a random walk with finite jump size and times. They note the first jump of the random walk occurs at a finite rate, so that both diffusion and crossing of the encounter surface leads to finite rate of reaction. Consequently, they imply that the ratio kactj TxRD cannot be much larger than 10 (when the mean jump distance is comparable with the root mean square jump distance and both are approximately 0.05 nm). Practically, this means that the Reii of eqn. (27) is within 10% of R, which will be experimentally undetectable. A more severe criticism notes that the diffusion equation is not valid for times when only several jumps have occurred, as Naqvi et al. [38b] have acknowledged (typically several picoseconds in mobile solvents). This is discussed in Sect. 6.8, Chap. 8 Sect 2.1 and Chaps. 11 and 12. Their comments, though interesting, are hardly pertinent, because chemical reactions cannot occur at infinite rates (see Chap. 8 Sect. 2.4). The limit kact °°is usually taken for operational convenience. 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

It is interesting to note that eqn. (190) is reminiscent of the steady-state Collins and Kimball rate coefficient [4] [eqn. (27)] with kact replaced by kacig R) and 4ttRD by eqn. (189). Equation (190) for the rate coefficient is significantly less than the Smoluchowski rate coefficient on two counts hydrodynamics repulsion and rate of encounter pair reaction. Had experimental studies shown that a measured rate coefficient was within a factor of two of the Smoluchowski rate coefficient, it would be tempting to invoke partial diffusion control of the reaction rate. The reduction of rate due to hydrodynamic repulsion should be included first and then the effect of moderately slow reaction rates between encounter pairs. 

The partition ratio kBl(k d + kB) defines the efficiency of product formation from the encounter complex (see also Section 3.7.4). For the limiting case k dobserved rate constant of reaction approaches the rate constant of diffusion, kx kd. In 1917, von Smoluchowski derived Equation 2.27 from Fick s first law of diffusion for the ideal case of large spherical solutes. 

For a particle radius of 200 A, using Smoluchowski s equation (with k( jff = 2 x 10-5 cm-1), Meisel calculated a rate constant of k = 3.0 x 1011 M 1s 1 for the encounter of 1-hydroxy 2-ethyl radicals with the Au sols. 

There is a close parallel between this development and the microscopic theory of condensed-phase chemical reactions. First, the questions one asks are very nearly the same. In Section III we summarized several configuration space approaches to this problem. These methods assume the validity of a diffusion or Smoluchowski equation, which is based on a continuum description of the solvent. Such theories will surely fail at the close encounter distance required for reaction to take place. In most situations of chemical interest, the solute and solvent molecules are comparable in size and the continuum description no longer applies. Yet we know that these simple approaches are often quite successful, even when applied to the small molecule case. Thus we again have a microscopic relaxation process exhibiting a strong hydrodynamic component. This hydrodynamic component again gives rise to a power law decay in the rate kernel (cf. 

Reactions between species, where the interaction energy is large compared with thermal energies, is markedly different from those reactions where no such interaction occurs. The energetics of reaction of encounter pairs, the timescale for approach of reactants, and the relative importance of other factors are all changed. In principle, these modifications to reaction processes enable more information to be obtained about the whole range of factors complicating any analysis of diffusion-limited reaction rates. However, in practice, the more important factors (such as initial distribution of pair separations, hydrodynamic repulsion, and electric field-dependent mobility) are of themselves unable to explain all the differences between experimental results and theoretical predictions. There is a clear need for further work. Finally, it can be remarked that when interactions between reactants are specifically included in an analysis of these rates of reaction in solution, the chosen theoretical techniques has been almost exclusively the Debye—Smoluchowski equation... 

Application of the diffusion law to rates of encounter and of chemical reaction the Smoluchowski equation... 

If chemical reaction occurs at every encounter, kr> is also the rate constant for reaction. Equation (2.1) is often known as the Smoluchowski equation. 

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