Molecular Imaging. Markus Rudin

      The cross relation is:

       and are the work terms for the isotopic exchange reactions, denote the equilibrium constant and the work terms of Eq. (3.80).

      When like in the case of charge interchange among aquo ions, the formula simplifies to:

      For a somewhat more accurate comparison, k₁₂ may be estimated from k₁₁ and k₂₂ using the complete Eqs. (3.1, 3.17, 3.19) and noting that λ₁₂ = (λ₁₁ + λ₂₂)/2. When the work terms are negligible, we have:

      where ln f = (ln K₁₂)²/4 ln(k₁₁k₂₂/Z²) [6, p. 856].

The cross relation was studied very extensively by N. Sutin and coworkers [42, 43].

In the words of Henry Taube: “The Marcus correlation is a powerful one, leading as it does to a calculation in most instances of a specific rate to an order of magnitude or so” [44]. It is “a touchstone of normal behavior” for outer sphere activated complexes [45].

       3.15.Correlation between Isotopic Electron Exchange Rate and Corresponding Electrochemical Rate Constants

      “For an isotopic exchange reaction between ions differing only in valence state, ΔF⁰ = 0, w^r = w^p, and hence m = −0.5 in Eq. (3.19). In the ‘exchange current’ of the corresponding electrochemical system η_a = 0 by definition, and m = −0.5, if the work term w^r − w^p is small. The λ₁’s and λ₂’s in Eq. (3.78) are all equal” [6, p. 854]. Considering Eqs. (3.78) and (3.83) “It then follows that:

      (= or < according as the reactant can or cannot penetrate the solvent layer adjacent to the electrode). From a physical viewpoint, the factor of two enters in the exchange system because two ions and their solvation shells are undergoing rearrangement in forming the activated complex, while in the electrochemical system there is but one such a particle.”

      *Demonstration by Marcus: “In solution and if a₁ = a₂

      When r = 2a

      For the electrode

      Where R = distance from the ion to its charge image. So

      (equal 2a when the ion is in contact with the surface of the electrode). For the case that R = 2a we have

      which is 1/2 of the value in Eq. (1).”

      Marcus’ comment: “Let’s pick the simplest mode, namely that you have that dielectric continuum, there are no subtleties about saturation, and things like that. Now, if the ion can get up and touch the electrode, then the equality sign would prevail, just looking at the equations. If the ion can’t get that close, then that means that it doesn’t interact much with its image charge, and its image charge would cancel some of the field, and that’s all part of this inequality here, so the image charge is cancelling less if the ion can’t come up close, because of some absorbed layer, and then in that case, because of the image effect, for R bigger than 2a the λel is bigger.”

      “It thus follows that:

      when w^r and w^p are small in both the ex and el experiments. From Eq. (3.1), we then expect that:

      where k_ex and k_el are in units of liter mole sec⁻¹ and cm sec⁻¹, respectively. Another factor tending to favor the ‘>’ sign is the existence, if any, of inactive sites due, say, to any strongly absorbed foreign particles. More recently it has been concluded theoretically that under conditions neither the earlier deduction of this relation nor that of the 0.5 slopes of the ΔF^* plots should be affected if one or both of the reactants form relatively weak complexes with other ions. The ΔF^*’s are then corresponding to the pseudo-rate constants, ‘constants’ which depend on the concentrations of these other ions.” [6, pp. 854–855].

       3.16.Chemical and Electrochemical Transfer Coefficients

      When the work terms can be made small by using high electrolyte concentration, or when they are essentially constant, one may draw from Eq. (3.25) the following variations in the plots of ΔF^* versus ΔF⁰ and of ΔF^* versus −nFη_a:

      1.In the oxidation—reduction reaction of a given reagent with a series of related compounds such that the reactions’ ΔF⁰ is essentially the only parameter varied, a plot of ΔF^* versus ΔF⁰ and hence of log k versus log K should be linear with a slope of 0.5 for ΔF⁰’s satisfying Eq. (3.26).

      2.In the electrochemical case, the corresponding plot of ΔF^* versus −nFη_a (or of −RT/nF ln k versus electrode potential), the electrochemical transfer coefficient, should also be linear with a slope of 0.5.

      By analogy, M. calls the slope of the ΔF^* versus ΔF⁰ plot in case (1) the “chemical transfer coefficient” of the reaction [6].