Molecular Imaging. Markus Rudin

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       CHAPTER 3

       Extensions, Electron Transfer at Electrodes, Applications

After the fundamental results of the wonder year 1955 (published ten months later in 1956), M. developed his theory and applied it to a variety of chemical and electrochemical systems. This chapter is devoted to the description of results mainly reported in Refs. [1–7], which span the years 1956–1964. I have also freely resorted to the reviews in Refs. [8–13], altogether a real “Guide of the Perplexed” [14]. Among the early applications are comparisons of calculated and experimental data, examples of how to use the theory to investigate reaction mechanisms and of how to systematize and correlate experimental data. These last characteristics have been typical of chemical theories like, say, the theory of the periodic system of elements or that of electrode potentials.

       3.1.Theoretical Equations

      In Chapter 1, the Landau–Zener–Stueckelberg–Majorana Equation was introduced, giving the probability P of a nonadiabatic transition in the narrow avoided crossing region.

      If a transition from the R curve on the left of the col in Fig. 1.4a to the R on the right occurs, the ET does not occur because the system remains on the reagents R set of configurations. The probability that the system remains on the lower potential energy surface (PES) passing, through the col of the surface, to the products P set of nuclear configurations is γ = 1 − P. Marcus denotes with κ the nuclear velocity weighted average of γ [11, p. 181]. The formula for k_rate is written by M. as:

      The same equation can obviously be written as

      where ΔF^* refers now to a mole of ET reactions.

      The M. theory in its early formulation is adiabatic, that is, κ is supposed equal to 1. Note that even if κ were 0.01 the value of would only increase by kT ln 100, that is, 2.5kcal mol⁻¹ at 0^∘C. The symbol ρ is “the ratio of the root mean square fluctuations in separation distance in the activated complex to the root mean square fluctuations of a coordinate for motion away from the intersection surface... and should have a value of the order of unity.” [11, p. 178]

      M. introduces in Ref. [1] the symbol λ:

      in terms of which

      which shows very neatly the two components of the activation free energy:

(i) the reorganization free energy m²λ from an equilibrium state at distance of approach R and (ii) the Coulombic interaction at R, with the usual meaning of the symbols. R is the average distance between the centers of the reactants in the activated complex [6]. Note that in the following the symbol R is also used for the ideal gas constant.

      The energy of activation of a bimolecular reaction is defined in chemical kinetics as:

      ΔS^* is expressed in thermodynamics in terms of ΔF^* and T as:

      Introducing Eq. (3.1; with ) into the defining equation of E_a and neglecting the small dependence of Z on T one gets (using the symbol k_bi for k_rate and R instead of k):

The terms ΔS^* and ΔF^* are not the same as the usual free energy and entropy of activation ΔF^‡ and ΔS^‡ as in, for example, Ref. [15,