Molecular Imaging. Markus Rudin

Eq. (2.27), in the first term, the one of the Coulombic interaction between ions, and in the second, related to the barrier due to difference in solvation of the ions. For an electron exchange between positive ions, we see that a decrease of R increases the contribute to ΔF^∗ in the first term and decreases it in the second. If the second term in Eq. (2.27) is greater than the first, so that the major barrier to reaction lies in the difference of solvation about reactants, the reaction will occur most readily for the smallest possible R. As a matter of fact, until now, when the ions come close to each other, we have only considered the overlap of their electronic orbitals. But another kind of overlap should be considered: that of their solvation atmospheres, because, according to M., when the solvation atmospheres overlap they become more similar to each other so as to lower the solvation barrier to reaction and to favor the ET process.

      In summary, while in theory the most appropriate value of R is to be found maximizing the overall rate constant with respect to R, limitations in the quantitative knowledge of the electronic jump process suggested to Marcus in 1956 to consider as the most suitable value for R its minimum value, that is, the sum of the radii of the separated reactants:

      2.10.The Radius a

      The radii a₁ and a₂ of the reacting ions enter the expression for rate constant k₁ through ΔF^∗, Eq. (2.27). M. considers in its first paper the radii for simple cations and anions and for complex cations and anions.

      In the case of simple cations, the radius is taken as the sum of the crystallographic radius plus the diameter of a solvent molecule, water in particular, since the cations have a fairly tightly bound hydration layer. The recipe follows the previously assumed model of a reactant considered as a sphere formed by the ion inside a spherical shell made up by an innermost layer of dielectrically saturated solvent molecules, outside of which the solvent is assumed to be dielectrically unsaturated.

      In the case of simple anions, this recipe needs to be modified because the solvent in the first layer appears in this case not to be completely saturated. After this warning, M. did not give a recipe for this case in Ref. [13]. The topic of effective polarizing radius will be introduced in the “Effective Radii” and “Ionic Radii” sections of Chapter 3.

      In the case of complex ions such as or the degree of dielectric saturation in the first solvation layer is much less than for monoatomic ions because the orienting effect of the ions’ electric field varies roughly as the inverse square of the distance from the center of the ion. In this case then the radius of the ion is to be considered as only that of the naked ion.

       NOTES

      1.M. will give in his 1965 paper an expression for k_bi given by

      M: “The 1965 expression is derived from statistical mechanics, the other, is an approximate macroscopic expression. In the 1965 expression there are all sorts of microscopic details that are not present in the 1956 paper.”

      2.M’s correction of the statement on p. 970 of [1], “We shall treat an ion plus its rigid, saturated dielectric region as a conducting sphere of radius a.”:

      M: “I think it is unfortunate that I called it a conducting sphere. . . in the later papers of course I didn’t have the system as a conducting sphere.”

      NOTE: M. believes that he used that model in the first two papers because he was under the influence of Born’s work on the charging of ions, where they were considered as conducting spheres.

      3.M: “Both α_e and α_u depend on temperature. α_e is really related to an optical dielectric constant, so it depends very little on temperature, but it depends a little. α_u depends upon a difference of dielectric constants, one of which is the static dielectric constant and so it definitely depends on temperature.”

      4.M: “The polarization in an equilibrium system is always proportional to the electric field, so that means that as soon as you specify the polarization in an equilibrium system, you specify the electric field everyplace. In a non-equilibrium system that is not true.”

      5.Q: The macroscopic system is formed by the couple of the reactants plus all the other solvent molecules and ions surrounding them, so in theory the potential is exercised by all of the 10²³ molecules and ions which make up the macroscopic system. But in practice how many are the solvent molecules and ions interacting with the couple of electron exchanging reactants?

      M: “I’d say, maybe a few thousands, because Coulomb forces are long range, and of course it depends on how dilute the solution is. But it is true that usually you have solvent there which dampens the Coulomb forces, so the distance of the solvent molecules and ions affecting the reacting pair doesn’t go beyond a few Debye lengths, and you can call mesoscopic the system that would cover most of the relevant part of the system. The main advantage of considering it would be for numerical calculations, you wouldn’t have to use 10²³ coordinates.”

      6.Q: In the M16 paper, you say nothing about the nature of the reaction coordinate but looking at the nonequilibrium free energy formula one sees that at constant E_c the functional F depends on the function P_u, that is, apparently you tacitly consider the P_u polarization as the ET reaction coordinate.

      M: “In the 1956 paper the reaction coordinate is really P_u(m). m is really the reaction coordinate. You see, P_u varies all over the place, every point has a different value of P_u, but those P_u’s are connected in a way in a form of reaction coordinate with that m, the Lagrangian multiplier, so really if one would ask what really was the reaction coordinate in the 1956 paper, a single coordinate, you could say that m was the coordinate. m = 0 when the function P_u is everywhere appropriate to the reactants, if it’s m = −1 then the function everyplace is appropriate to the products, but in the TS has a value given by that (2m + 1)λ formula. . . so m is really the reaction coordinate in the first paper, although I didn’t say it explicitly, and it describes how P_u changes along the reaction coordinate everyplace. . . m can serve as a reaction coordinate, in the 1956 paper, but taking on for the actual system the value given by −(2m + 1)λ = ÄG⁰.”

      Q: But what exactly is a reaction coordinate?

      M: “Every coordinate that leads from reactants to products, and of course that means that it could be some terrible reaction coordinate and the whole idea is to try to find the best within some framework of reaction coordinates. Nothing unique. I mean, there are different reaction coordinates, then better ones, then better ones. . . and getting the best one is a big question and one may not be able to do it. So one does the best one can. One should speak