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Molecular Imaging. Markus Rudin
with ei 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 ΔGo′ 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, Ur = Up, 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 ΔGr is given by Eq. (3.61) and ΔGp has the same expression with ei 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 Pu, M. minimizes ΔGr 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 δGr and δGp 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 wr to bring the reactants together and ΔGr. 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. WI and WII 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 ΔGp. One proceed then as before using Eqs. (3.64), (3.65) but now Eq. (3.64) contains an extra term
(because
and δΔGp contains an