“Here, a central ion just outside the double layer is denoted, in its initial electronic state, by A and in its final state by B. As noted earlier, step (4.20) involves a suitable reorganization of the solvent and ionic atmosphere and (if necessary) a suitable penetration of the electrical double layer. Step (4.21) is the actual ET itself, and step (4.22) involves a reversion of configuration of solvent and atmosphere to one in equilibrium with the new charge on the central ion. Step (4.22) also involves a motion away from the electrode. As in the redox theory, the reverse of Eq. (4.22) occurs but it needs not be considered in the computation of kf . Steady-state considerations for cX* and cX” (where the c’s are concentrations) “lead to the relation [4, p. 969]:
As discussed later, when the probability of ET in the lifetime of the intermediate state X* (10−13 sec) is large, kf is about half of k1. k1 depends on the free energy of formation of X* from the initial state in reaction (4.20). We proceed first to the calculation of this free energy change and later to a discussion of the evaluation of the rate constants.”
The treatment given here by Marcus in 1957 is today considered by him only as a formalism that helped him to get started, just a step in an evolution, “a thing that may be interesting from the point of view of how thoughts developed but not for presentation of a way of thinking. That was history, I’m not thinking anymore in those terms.” (M). I shall then not discuss the rate constants of the elementary steps and I refer the reader to Chapter 6 for the modern treatment.
4.12.Free Energy of Formation of State X*
“The electrostatic contribution, ΔF*, to the free energy of formation of state X* from state A will be different from zero because of
(1)The work which may be required to transport the central ion from a point just outside the double layer to some particular point in it (if necessary)
(2)The work required to reorient the solvent molecules and the ionic atmosphere to a nonequilibrium configuration, for this position of the central ion
Let
where
E*(r) = electric field in state X*. It equals −∇φ*.
φ*(r) = inner potential in state X* (cf. Eq. (4.1))
αe = E-type polarizability = (Dop − 1)/4π
αu = U-type polarizability = (Ds − Dop)/4π
ci (r) = concentration of ions of type i in states X* and X
ni in the solution divided by the latter’s volume V
Pu(r) = U-type polarization in states X* and X
χ = potential drop at electrode–solution interface due to an oriented solvent dipolar layer”
Note that ci(r) and Pu(r) in state X* are the same as those in state X because of the constant atomic configuration restriction.
Equation (4.24) and subsequent equations treat χ as being independent of the average electrode charge density.” However, M. shows that “the final equations” (4.31)–(4.33) in the following “are unchanged even ifχ were a function of this quantity” [2, pp. 186–187].
4.13.Constraint Imposed by the Constant Energy — Constant Atomic Configuration Restriction
“The free energy of formation of X* from A consists of the electrostatic contribution ΔF* and of a term, described later, associated with the localization of the center of gravity of the central ion in a narrow region near the electrode. (3) The free energy of formation of X from B contains an electrostatic term