The Rates of Complex Processes

In the previous section, we have referred to the way in which the overall rate of the reaction depends on the chemical potential of the transition state when rates for identical processes are compared under identical conditions on different substrates. It was also implied that an equilibrium could be assumed between processes taking place before the rate-determining step in simple cases. This concept will be examined in more detail in this section for the case of more complex processes in which parallel reaction pathways can and do occur. One of the most characteristic (and one of the simplest and most studied) of such processes is that of hydrogen evolution, where three possible steps involving adsorbed species are generally considered to occur. These are known respectively as the discharge or Volmer process, the electrochemical desorption or Heyrovsky process, and the hydrogen atom combination (Tafel) reaction, as follows (written in the cathodic direction)  

The first two are electron transfer steps, the third chemical. All of the above can, in principle, contribute to the rate of the overall process. In general, however, a combination of any two will predominate. Since the rate constant of (49) is independent of potential, whereas the rate constants of the other two are potential dependent, the pair of reactions predominating may change as the potential is varied on any given surface. The rates of the individual processes will be given by the expressions 

In the steady state, we may assume, as usual, that dS/dt = 0. This gives us, in the general case. 

The real solution to this quadratic gives a general expression for 6, but it is not very useful unless certain simplifying assumptions are made. The simplest of these assumptions is to regard the process as taking place in a potential region where the combination reaction is unimportant, i.e., to ignore and Ph2 3 compared with the other rate terms. We then obtain  

Using Eq. (54), and counting the cathodic direction positive, the overall rate is then equal to 

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