Molecular Imaging. Markus Rudin

      “where m satisfies the equation

      and where

      In these equations, Δe = e* − e, the charge transferred to the electrode in reaction (4.19); a denotes the effective radius of the central ion, discussed earlier; R/2 is the distance of the ion to the electrode surface in the state X*; w* and w denote the work required to transport the central ion from the body of the solution to a distance R/2 from the electrode surface when the ion is its initial and final state, respectively; the remaining quantities have been defined previously” [2, pp. 190–191].

      I want to stress that ΔF* = ΔF*(R), i.e., ΔF* depends, through λ, on the distance of the central ion from the electrode.

      “The term w* can be evaluated by solving the usual Poisson–Boltzmann equation [3] when the central ion is in the body of the solution, and then introducing these solutions for the electrostatic potential into the appropriate equations for the electrostatic free energy of the system (Eqs. (4.16) and (4.18)). The difference in electrostatic free energy is w*. Similarly, w can be computed from analogous equations for the central ion B.

However, a considerably simpler though less rigorous procedure has generally been assumed for calculating the work required to transport an ion from the body of the solution to some distance R/2 from the electrode. The assumption is generally made that this work equals the charge of the central ion multiplied by the difference in the value of the electrostatic potential at R/2 and in the body of the solution (i.e., outside the double layer), this potential being computed in the absence of the central ion. This particular potential can be inferred from measurements on the electrical double layer by various methods [14, 21]” [2, p. 192].

      4.15.Dependence of χ on Mean Electrode Charge Density

In the earlier theory, “it has been assumed for simplicity that χ, the potential drop due to electrode–solvent and solvent–solvent interactions at the interface, was independent of the mean electrode charge density, say. Recent work suggests, however, that the degree of orientation of the solvent molecules next to the electrode (and hence χ) may vary with this quantity [22, 23].”

      Marcus considers then the possible modification of the theory for the case in which χ is not just a constant, but a function and concludes that “when a few electrons are transferred in some process” as in our case “the average electrode charge density is affected only to a negligible extent. In such cases, χ is a constant for all states in the process.” M. then shows that χ considered either constant or depending on does not appear in the final Eqs. (4.31)–(4.33).

      “Accordingly, we infer that any effect of a change in degree of orientation in the solvent layer next to the electrode is a more indirect one. It might affect the ‘dielectric constant’ in the vicinity of the electrode, particularly the contribution from orientation polarization. But, we observe from Eq. (4.33), unless D_S is close to D_op, changes in the former have very little effect on ΔF*.

      Again, it might affect to some extent the distance of closest approach of the central ion. In some studies of the equilibrium properties of the electrode double layer, Grahame [22, 23] has shown that a self-consistent interpretation of the data can be obtained assuming such an effect. Extremely interesting inferences were drawn about the behavior of the solvent in this region toward ions of different size. It is clear that an analogous study of the electrode kinetics of simple electron transfers at various electrode charge densities should be very interesting. At certain charge densities, it appeared from the equilibrium studies, there is no oriented solvent layer. The interpretation of kinetic data obtained under such conditions would be correspondingly simplified” [2, pp. 204–205].

       4.16.Presence of Fixed, Adsorbed Ions in the Electrical Double Layer

      If fixed adsorbed ions are present in the double layer [3], one should consider their contribution to the electrostatic free energy

      “If q_k denotes the charge of the kth fixed ion and if r_k and denote the distance from the kth ion and from its image to the field point, then the contribution to the potential arising from all fixed ions is, in an equilibrium polarization system,

      the summation being over all fixed ions.

      This can be shown to add the following terms to in Eq. (4.16):

      where a_k is the radius of the kth fixed ion; R_k, r_jk, R_jk, ρ_k, and denote the distance from this ion to its electrical image, to the jth fixed ion, to the latter’s electrical image, to the central ion and to the latter’s electrical image, respectively. The first term in Eq. (4.35) describes the interaction of the mobile ions with the fixed ions.

      In Eqs. (4.17) and (4.18), φ_S now represents the potential in the body of the solution due to all ions, fixed and mobile, of the double layer and so includes Eq. (4.34). Accordingly, qφ_S in Eqs. (4.17) and (4.18) now includes the last term of