Diffusion-limited reactions, Smoluchowski equation

When there are two or more reactants diffusing throughout space, the motion of each reactant influences that of all the others due to the solvent being squeezed from between the approaching reactants. The effect of this hydrodynamic repulsion on the rate of a diffusion-limited reaction was discussed in Chap. 8, Sect. 2.5. In this section, this discussion is amplified. First, the nature of the hydrodynamic repulsion is discussed further and then a general diffusion equation for many particles is derived. The two-particle diffusion equation is selected and solved subject to the usual Smoluchowski initial and boundary conditions to obtain the rate coefficient. Finally, this is compared with the rate coefficients in the absence of hydrodynamic repulsion and from experiments. 

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... 

While this is natural in view of the complexity of more detailed approaches to diffusion-limited reaction rates (e.g. kinetic theory), it is nevertheless a mute point as to whether the Debye—Smoluchowski equation represents an adequate description of diffusion and drift of interacting species in solution. 

The upper boundary of the reaction rate is reached when every collision between substrate and enzyme molecules leads to reaction and thus to product. In this case, the Boltzmann factor, exp(-EJRT), is equal to lin the transition-state theory equations and the reaction is diffusion-limited or diffusion-controlled (owing to the difference in mass, the reaction is controlled only by the rate of diffusion of the substrate molecule). The reaction rate under diffusion control is limited by the number of collisions, the frequency Z of which can be calculated according to the Smoluchowski equation [Smoluchowski, 1915 Eq. (2.9)]. 

In other studies, Jonah et al. [116] measured the rate of reaction of the hydrated electron with Cd and Cu cations. They noted a decreasing rate coefficient with increasing ionic strength. In all cases, the rate was slower than that based on the Debye- Smoluchowski equation [68], eqn. (51), but greater than or equal to the corrected rate coefficient using the Bronsted- Bjerram correction [eqn. (58)]. In fact, Jonah et al. found that the rate coefficient for reaction of hydrated electrons with pure Cd(C104 )2 or Cu(C104 )2 follows that predicted by Coyle et al. [94] where no ionic atmosphere has developed around e q. Jonah et al. pointed out that such a situation was improbable (see Sect. 1.6). Furthermore, no hydrodynamic correction was made to the rate coefficient, which would lower the expected value by 20%. Jonah et al. [119] showed that the observed rate for reaction of e q with HsO was about one third of the expected Debye—Smoluchowski diffusion-limited rate (see the Debye... 

There are some limits where the Smoluchowski s equation can be solved by simple methods. First one has to discuss the kernels. Consider a diffusion-controlled reaction where the clusters grow in time. Then the cluster size increases and the typical radius of a cluster can be written as R(i) where df is the fractal dimension of the cluster containing i monomers. With increasing size the diffusion constant decreases according to Stoke s law and one can assume a power law for the diffusion constant of the cluster, i.e. The reaction constants are then of the... 

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. 

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. 

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