Molecular Imaging. Markus Rudin

of temperature, ΔS^* is obtained from Eqs. (3.4) and (3.6):

      When only one electron is transferred, the term (Δe)²/4 will generally be appreciably less than and we would then have:

The first term of Eq. (3.15) is the entropy change which occurs when two ions approach each other under electrostatic equilibrium conditions. The lowering of ΔS^* for ions of the same sign when they get close to each other, that is, the entropy change which results when two ions are brought together is due to the enhanced local electric field causing an ordering of the solvent molecules, that is, when and are like charges, ΔS^* < 0. This is demonstrated in Section 3.5 because it may well not be immediately evident. In the opinion of J. Randles, later seen to be incorrect, for instance, in a discussion in Ref. [5, p. 246]: “with like charges the solvent molecules do not know which way to turn: I would expect that this would represent an increase in entropy.” A correct interpretation is given there by Marcus’ answer to Randles’ remark.

      The second term in the equation is the entropy change resulting from the reorganization of the solvent molecules which accompanies the formation of the nonequilibrium state from the equilibrium one at distance R. The smallness of the second term relative to the first is due to a cancellation effect, because accompanying the solvent reorganization there is a decrease of entropy of solvation around one ion which approximately cancels the increase around the other: the solvent near the more highly charged reacting ion becomes less oriented while it becomes more oriented near the less charged ion.

       3.5.*Demonstration That the Solvation Entropy of Ions of Like Sign Decreases with Their Distance [5, p. 246]

      The free energy of Coulombic repulsion between two ions of like sign and charge is ΔF = q²/D_s R.

      Moreover, the free energy difference between two states of a system at the same temperature is ΔF = ΔU − T ΔS. If we now show that for our system the ratio we will have demonstrated that ΔS^* < 0 and that ΔS depends on R as ΔF does.

      Recalling that ΔS = −(∂ΔF/∂T) we have:

      and for water at room temperature we have

      Marcus’ comment: “D decreases with increasing T so the RHS of your equation above is positive, for water at room temperature it is 1.1. This makes physical sense, ΔS will get smaller with increasing T (less ordered) and ΔF will get larger with increasing T (smaller D), so the net result for the LHS of eq 1 is positive.”

      3.6.The Work Terms w and wp (w^* and w, wr, and wp)

      The work term is “the free energy change when the reactants are brought together to the separation distance R” [18, p. 690], it is the work required to bring the reactants from infinity to their separation distance R. It was initially taken as e₁e₂/D_s R, which is the Coulombic work to bring the reactants together at the distance R at infinite dilution. M. subsequently considered the possibility of polar (e.g., electrostatic) and nonpolar contributions to the work, and a more realistic situation in which the reactants were in a solution with some electrolyte concentration. For a discussion of the most appropriate value of R, see Section 2.9 of Chapter 2. One should also consider that “When R becomes large κ tends to zero and when R is small the van der Waals’ repulsion makes F^*(R) large” [18, p. 685]. The symbol w was then used for the earlier work term “in the prevailing medium,” and the symbol −w^p was used for the work required to separate the products from R to infinity in that same medium.

      Using the new symbols, Eq. (3.4) is written as:

      and Eq. (3.3) as:

      with m equal to:

      Eq. (3.14) for ΔF^* for isotopic exchange reactions can be written as:

      and Eq. (3.15) becomes:

      I have used here the symbols w and w^p instead than the symbols w* and w used earlier. A certain confusion may arise because in terms of the older symbolism Eq. (3.15) is written as:

      Moreover, w^* is also indicated by Marcus as w^r. One finds then the following couples of symbols for the work terms in Marcus’ papers:

      In the following only the symbols w^r and w^p will be used. M. uses also sometimes the symbol F and sometimes the symbol G for the free energy.

      3.7.Inner and Outer Contributions to λ

      In the first formulation of the theory (1956), the ion was treated as a sphere inside of which no changes in interatomic distances occurred during the reaction. This assumption was later eliminated when the theory was extended to include the effect of changes in bond distances and bond angles in the inner coordination shell of each reactant [6, p. 853]. In the earlier equations, λ will then be in general equal to: