Distribution on Heterogeneous Surfaces

Let us assume that a single-step reaction, for instance, reaction IV, 1 occurs on a heterogeneous surface. The proportionality that is found for many electrocatalytic reactions between / and A for heterogeneous surfaces of solid metals suggests that the sites on which each of the rate constants and 2 has the same value are uniformly distributed. The smallest apparent area A for which this statement still holds depends upon the structure and composition of the surface in question. In a crude approximation this area may be equal to the average grain size. If there is a finite number of types of sites with rate constants and fc2,v the rate of the reaction in electric units is  

The distribution of the /c s can be expressed as a periodic function of appropriate coordinates, for instance x and y for a plane surface. 

The rate constants satisfy Eq. IV, 3 if AGf is replaced by AGf, . Different values of the rate constant result from a variation of + C and ( M 0s) according to Eq. IV, 5 and IV, 6. It follows from the discussion in section 1 of chapter IV that only reactions with strong interaction between the transition state and the surface have to be considered. If (0M 0s) is not constant in the case of weak interaction it can only be due to a non-uniform current distribution. 

In order to obtain a solution of the problem, the Poisson equation has to be solved with Eq. 1 as the boundary condition at the test electrode and a second, possibly simpler, boundary condition at the counter electrode. If concentration changes are involved, the equations for mass transport have also to be taken into account [7]. In this case the activities in Eq. 1 are replaced according to Eq. 2. 

Here /j designates the individual activity coefficient of the species j. Assuming that /j does not depend upon Cj because of a suitable composition of the electrolyte, the product of activity coefficient and rate constant is frequently combined to a new rate constant with the unit A/M. The computation outlined depends strongly upon the assumption of the distribution function for the rate constants across the surface. 

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