Molecular Imaging. Markus Rudin

say, and a center of gravity term equal to that noted previously. As in the redox theory, it follows from assumption (a) given earlier that no energy change and no configurational entropy change accompany the formation of state X from state X*. The electronic entropy change arising from any possible change in the electronic degeneracy of the central ion and of the electrode is zero or negligible. Accordingly, X* and X have the same free energy.”

      Note that the equality of the free energies of X* and X have been deduced from the equality of their energies and of their entropies.

      “The net free energy change in forming B from A is therefore

      Independently, this free energy change of reaction (4.19) can be written as the sum of the following terms:

      (a)The change in chemical potential of ions A and B and of the electrons in electrode M. This is where n is the number of electrons transferred in reaction (4.21).

      (b)The change in free energy due to the transfer of charge e−e* from a metal of inner potential φ_M to a solution of inner potential φ_S, e* and e denoting the charges of ions A and B, respectively. This term is (e*−e)(φ_M −φ_S) which we shall denote by (e* −e) Δφ”.

      “If electrode M were in electrochemical equilibrium with the actual concentrations of A and B just outside the double layer, then the sum of these two terms would be zero.”

      Denoting the corresponding value of Δφ as Δφ′ we would then have:

      For Δϕ ≠ Δϕ′, there wouldn’t be such an equilibrium and the difference:

      defines the activation overvoltage.

      For the net free energy change ΔF*− ΔF written as sum (a) + (b) we have:

      Introducing Δφ′ in the above equation using Eq. (4.26) we have:

      “It may be remarked that μ_B − μ_A is related to its standard value and Δφ to its standard value Δφ⁰, by Eqs. (4.28) and (4.29), where the f’s denote activity coefficients and the c^S’s denote concentrations just outside the double layer:

      [2, pp. 187–188].

      4.14.Minimization of ΔF * Subject to the Constraint Imposed by Eq. (4.27)

      The Final Theoretical Equations for ΔF *

      We shall not delve in the details of the minimization process of ΔF* which is analogous to the one described in Chapters 2 and 3 for the redox theory. I only remark that Marcus considers now not only the configuration of the solvent molecules in the activated complex, but also the configuration of the ionic atmosphere so that ΔF* is minimized now not only with respect to arbitrary variations δP_u(r) of the orientation–atomic polarization, but also with respect to arbitrary variations δc_i(r) of ionic concentrations in the ionic atmosphere. There is then now a new condition of constraint, that is, besides the condition:

      there is the new condition of fixed number of ions n_i in the solution, expressed as:

      There are correspondingly two Lagrange multipliers: the m, already met in Chapter 2, and a new one, − ln l_i. In terms of the multipliers, the solvent and ionic configuration of the pair of intermediate states X* and X is obtained so that:

      We see that the concentration of ions at distancer from the central ion is obtained by Boltzmann weighting the total number of ions n_i with the expression in the fraction which depends on the inner potential φ* + m(φ* − φ).

      Notice that Marcus uses here the condition of constraint (4.30) instead of the condition δF* − δF = 0 which was used in the redox case but the two conditions are really the same because ΔF* = and

Marcus evaluates then ΔF* introducing what he will later name in Ref. [20] the equivalent equilibrium system (e.e.s). Such a system is introduced here as a hypothetical system having fictitious charges on the central ion and on the electrode which are in equilibrium with the configuration of solvent and ionic atmosphere characterized by the same P_u(r) and c_i(r) as those of states X* and X and which were not in equilibrium with the real charges of X* and X. The electrostatic free energy of this hypothetical system is determined and used to calculate the final theoretical equations for ΔF*. The hypothetical equilibrium system is characterized by having an inner potential φ^† = φ* + m(φ* − φ), ionic surface charge σ^† = σ* + m(σ* − σ) and, likewise, E^† = E* + m(E* − E), α_u(E* +m(E* −E), and e^† = e* +m(e* −e), where e^† is the charge on the central ion in the hypothetical system.

      Note that it is not

      The final theoretical equations for ΔF*