Molecular Imaging. Markus Rudin

      with e_i given by Eq. (3.55). “If denotes the free energy change for unit concentration of reactants at a separation distance R, it is related to the same quantity at infinite separation ΔG^o′ by an equation similar to Eq. (3.47a),” namely:

      Reactants and products have the same distribution of configurations (the same set of values of q^‡) on the intersection hypersurface of Fig. 3, and the same potential energy, U^r = U^p, averaged over a distribution of such configurations. Being the distribution of configurations and momenta, the same for reactants and products on the intersection hypersurface, the entropy is also the same and consequently the free energy of the reactants is there the same as that for the products.

      We can write:

      where ΔG^r is given by Eq. (3.61) and ΔG^p has the same expression with e_i replaced by

      In order to find and that is, the fictitious charges of the activated complex that would be in electrostatic equilibrium with the correct and real nonequilibrium P_u, M. minimizes ΔG^r in Eq. (3.61) subject to the constraint imposed by Eq. (3.63):

      Following the recipe of the calculus of variation, one multiplies the second equation by a Lagrange multiplier m and adds the equation so multiplied to the first, introducing then expressions such as Eq. (3.64) for δG^r and δG^p into Eq. (3.65). Setting finally the coefficients of and equal to zero, one finds:

      Introducing this result into Eqs. (3.54) and (3.56), one obtains:

      and

      where

      and the charge transferred. Solving for m from Eq. (3.68), we have:

      The free energy barrier to reaction ΔG^* consists of two terms: the work term w^r to bring the reactants together and ΔG^r. Introducing m in Eq. (3.67), one obtains:

      Turning now to the electrode case, the electrostatic potential in step 1 is given by Eq. (3.50) with e₂ replaced by the image charge of reactant 1 inside the electrode, that is, e₂ → −e₁. “The image charge ensures that the potential given by Eq. (3.50) is constant on the surface of the electrode where r₁ = r₂.” The derivation parallels the one above, where R denotes now the distance from ion 1 to its image, namely twice the distance to the electrode. W_I and W_II are computed considering that in the electrode case, only ion 1 needs to be charged, the image being only a fictitious ion (cf. Ref. [27], p. 124). Expressions similar to Eqs. (3.67)–(3.71) are ultimately found, but now we have:

      and

      that is, one obtains for the electrode case:

      where F is the Faraday.

This type of simplified derivation of the earlier results was given earlier in Ref. [10] and made more readily available in a review which is also an excellent review of the theoretical literature up to its time [41]. In order to consider simultaneously these fluctuations in solvent polarization and those in reactants’ vibrations, one could do so: add to the right side of Eq. (3.61) the term and add to the corresponding expression for ΔG^p. One proceed then as before using Eqs. (3.64), (3.65) but now Eq. (3.64) contains an extra term (because

      and δΔG^p contains an