The Smoluchowski rate coefficient

The current of B reactants diffusing towards the A reactant is calculated by substituting eqn. (16) for the density distribution into the expression for the particle current, eqn. (7). As noted above, this is dependent upon the separation between A and B. Very close to A, all the B diffusing toward A will react with A. The number of B molecules diffusing towards A per second at the encounter separation is also the rate of reaction of B reactants with any one A reactant. 

This is the usual form of rate expression for a reaction A + B Products 

To convert this to units of dm3 mol-1 s 1, eqn. (19) should be multiplied by 103iV where N is Avogadro s number. In the long time limit, the steady-state rate coefficient is found to be 

With typical values for R and D as above, the Smoluchowski rate coefficient (19) is shown in Fig. 3 for a range of times. The time dependence of the rate coefficient is due to the transient concentration of B in excess of the steady-state concentration. As the density distribution of eqn. (16) relaxes to the steady-state distribution (17), so the rate coefficient decreases, because at longer times, B has to diffuse further to A on average. The magnitude of the rate coefficient ( 1010 dm3 mol-1 s-1) is large. In some reactions, the mutual diffusion coefficient of reactants may be nearer 5 x 1CT9 m2 s 1, and the rate coefficient is 3 x 1010 dm3 mol-1 s-1. Under such circumstances, diffusion-limited reactions proceed very rapidly. It is likely that the rates of most chemical reactions are slower than the diffusion-limited rate. Only the most rapid molecular chemical reactions are faster than diffusion-limited rates. Some typical reactions are discussed in Sect. 2 and will be reconsidered in Sect. 5 and later in the volume. 

The concentrations of the majority species B needed to observe the time-dependent component of the rate coefficient of eqn. (19) can be estimated. Integrating eqn. (18) to find the time dependence of the minority species A (for instance the fluorophor) gives 

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Fig. 3. The Smoluchowski rate coefficient, eqn. (18), for a diffusion-limited reaction with the mutual diffusion coefficient Z) = 10-9m2s 1 and the encounter distance R = 0.5 nm.

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. 

In Chaps. 3 and 4, estimates of encounter distances and mutual diffusion coefficients from similar experiments to those of Buxton et al. [18] are discussed. The complications to the analysis of diffusion-controlled rate processes in solution when the reactants interact strongly with one another or the reaction can occur over distances much larger than typical encounter distances do not lead to markedly different time-dependent rate coefficient expressions to the Smoluchowski form. Indeed, replacing R in eqn. (29) by an effective encounter distance, Reff, allows the compactness of the Smoluchowski rate coefficient to be extended to other situations. Means of estimating Reff are discussed in Chaps. 3, 4, 5 (Sect. 4.3), 8 (Sect. 2.6) and 9 (Sects. 4 and 6). 

Fig. 17, The correction factor to the Smoluchowski rate coefficient for reaction between an isotropic reactant, B, and an axially symmetric reactant, A, which has a cap of reactivity of the spherical surface, subtending a semi-angle of 77/9, 77/2, 577/6 at the sphere s centre. The reaction radius is R and the radius of B is rB. Translational and rotational diffusion coefficients are given by eqn. (118), for reactions with G = 1, i.e,...

There is sufficient and convincing experimental evidence available already to support the need to consider activation and diffusion processes simultaneously. For instance, in Chap. 2, Sects. 5.2 and 5.4, mention was made of other instances where the reaction rate had been measured and found to be slower than anticipated from the Smoluchowski rate coefficient [eqn. (19)]. Using the Collins and Kimball expression enabled the workers to obtain reasonable estimates of the rate coefficients of encounter pair reactions. There is still some degree of uncertainty that the slower than expected reaction rate might not be attributable to partial... 

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. 

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