Molecular Imaging. Markus Rudin

for ΔF ^∗

There is an infinite number of pairs of intermediate states X^∗ and X which could satisfy the free energy restriction (2.24). All of the pairs have the same charge distribution but the r-dependent P_u(r) is different for different pairs. The problem is now of determining the P_u(r) of that pair of “groups of atomic configurations” that “contribute to the macroscopic or ‘thermodynamic’ properties of the transition state TS” [19]. Such a pair, subject to the free energy restriction (2.24), is characterized by the minimum free energy F^∗, because we are looking for that thermodynamic state which has the maximal probability of formation from the reactants.⁽⁷⁾ In terms of atomic configurations, one can say that many of them can satisfy the energy restriction, but only a group of atomic configurations contribute to the macroscopic or “thermodynamic” properties of the TS, namely those which minimize its free energy of formation from the reactants [19]. I give here the final results for P_u and F^∗. They allow the calculation of ΔF^∗, which is one of the major results of the Marcus theory.

      The work required to charge up two conducting spheres of charges and and of radii a₁ and a₂ in the dielectric medium is given by⁽⁸⁾:

      Similarly for

      Marcus obtained for the P_u(r) and the F^∗:

      Using the preceding results Marcus obtained the famous formula for ΔF^∗:

      where we have used the conservation of charge relation and R is the mean activation distance in the activated complex.

      In order to properly appreciate the earlier astonishing final result, the reader should consider that it was obtained by three very clever steps. In the first one, M. discovered the beautiful general formulas (2.13) and (2.14) (appearing in the demonstration earlier in the form of Eq. (2.22)) for the electrostatic free energies F^∗ and F. He then devised the minimization procedure to get the formulas for the free energies of the activated complexes. Finally, starting from an expression for F^∗ depending on E^∗, P_u, and E_c he ended up—as if by magic—with a formula of great simplicity and elegance and of immediate applicability depending on the charges the geometry of the system, (a₁, a₂, R), the dielectric properties of the medium (D_op and D_s), and the thermodynamic quantities ΔF⁰ and T ΔS_e which appear through the Lagrangian multiplier m obtained from the formula:

      Let us use it to calculate the value of m for the electron exchange reaction:

      We see that for this case each term on the RHS of Eq. (2.28) is zero. On the LHS, the terms in parenthesis are different from zero, Δe = 1 and so Eq. (2.28) can only be satisfied if 2m + 1 = 0, that is, if

      To understand what this means, let us consider again formula (2.26) for the P_u which minimizes F^∗.

      We see that if

      that is, the value of P_u which minimizes F^∗ is the one that would be in equilibrium with a hypothetical electric field equal to the arithmetic average of the fields E^∗ and E. And is consistent with the intuitive idea that the P_u of the TS would be in equilibrium with fictitious charges on the ions which would be averages between initial and final charges!

       ^∗ Proof of Eqs. (2.26), (2.27), and (2.28)

At the minimum of F^∗, we shall have δF^∗ = 0 (M. uses here the Calculus of Variations, elements of which can be found in books of Advanced Calculus or of Mathematical Methods of Physics and Chemistry, for example in Refs. [20–24] and, more extensively, in Ref. [25]).

      M. expresses the variation δF^∗ as function of the variations δP_u of the function P_u.

      The charges are fixed, so is fixed and Computing δF^∗ from Eq. (2.13) one obtains:

      We have seen that so that:

      δF^∗ depends on the variations of P_u and E^∗. But we know that P_u