Diffusion controlled reactions collision frequencies

For most termination reactions, the rate constant is determined by the collision frequency of radicals (diffusion controlled reactions), and the reduced mass does not become independent of radical size as in Eq. (P6.1.3). Hence the assumption that kt is independent of size is usually less valid. Nevertheless this assumption must be made in order to obtain tractable results. 

X 10 M" s . This value is equal to that calculated theoretically for a diffusion controlled reaction occurring when collisions are at high frequencies. This is not the case for the cytochrome hi core - cytochrome c complex since the affinity of the latter complex is found equal to 2.5 pM while the dissociation constant of the flavocytochrome hi - cytochrome c complex is equal to 0.1 pM. 

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)]. 

Because of the limitations imposed by activity coefficients and specific interactions, a precise quantitative check of experimental data against the collision formula presented here is not possible. However, the frequency factors of bimolecular reactions which are diffusion-controlled (i.e., those which occur on nearly every collision) such as free radical recombinations,... 

Marcus12 and others13 extended this model to include reactions in which electron transfer occurred during collisions between the donor and acceptor species, that is, between the short-lived Dn—Am complexes. In this context, electron transfer within transient precursor complexes ([Dn — A" in Scheme 1.1) resulted in the formation of short-lived successor complexes ([D(, + — A(m 1)] in Scheme 1.1). The Debye-Smoluchowski description of the diffusion-controlled collision frequency between D" and A " was retained. This has important implications for application of the Marcus model, particularly where—as is common in inorganic electron transfer reactions—charged donors or acceptors are involved. In these cases, use of the Marcus model to evaluate such reactions is only defensible if the collision rates between the reactants vary with ionic strength as required by the Debye-Smoluchowski model. The requirements of that model, and how electrolyte theory can be used to verify whether a reaction is a defensible candidate for evaluation using the Marcus model, are presented at the end of this section. 

If the activation threshold Ajyw vanishes or is very low, just about every encotmter will lead to a reaction. Hence, it is not the height of this potential threshold but the frequency of collisions that then determines the conversion rate. In this case, the concentration of the transition complex can remain far below its equilibrium value because continued supply is stalled, while decomposition continues taking place. Reactions of this kind are said to be dijfusion-controlled (or diffusion-limited) because their collision frequency is dependent upon the diffusion rate (diffusion velocity) of the partners involved. Bimolecular reactions in water and similarly viscous liquids are of this type if the activatimi threshold sinks tmder the third or fourth rung of our potential ladder, meaning that Ajyw <20 kG (see Fig. 18.2). Because diffusion in solid substances proceeds incomparably slowly, almost all the bimolecular reactions in such an environment are diffusion-controlled. 

To estimate the rate constant for a reaction that is controlled strictly by the frequency of collisions of particles, we must ask how many times per second one of a number n of particles will be hit by another of the particles as a result of Brownian movement. The problem was analyzed in 1917 by Smoluchowski,30/31 who considered the rate at which a particle B diffuses toward a second particle A and disappears when the two codide. Using Fick s law of diffusion, he concluded that the number of encounters per milliliter per second was given by Eq. 9-26. 

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