Another situation in which the solvent can have a profound influence on the rate constant occurs when the energy of solvation differs substantially for reactants and products. This situation occurs frequently in electron transfer reactions, since as the electron moves from the donor site to the acceptor site the structure of the solvent must adjust to accommodate the new charge distribution. Since all oxidation- reduction reactions in solution involve the transfer of an electron, the determination of the rate constant for electron transfer reactions is an extremely important chem- ical problem. This problem has been considered in detail by Marcus," and we develop here a simplified derivation of his result^.^ It can be shown that the more complete derivation gives the same answer in the limit when the distance between donor and acceptor sites never becomes too small.
The overall reaction that we would like to consider can be symbolized by the following scheme:
D + A + (DA) + (D+A-) + D+ + A-,
where the species in parentheses represent having the donor and acceptor at a dis- tance short enough so that the electron can be transferred, and (DA) and (DfA-) represent this "contact pair" before and after transfer. In many oxidation-reduction reactions, the overall rate constant is limited by the rate for the electron transfer, so we will concentrate on this step of the process. In some cases, D and A are differ- ent sites on the same molecule, so that, again, the overall rate constant is deter- mined by the rate of the electron transfer.
Consider an electron located on the donor molecule, which itself is surrounded by a number of solvent molecules. The energy of the electron will depend on the nuclear positions of all the atoms in the donor and solvent molecules, so that there will in general be 3N - 6 nuclear coordinates, where N is the total number of atoms. Under the assumption that the electron moves much more rapidly than the nuclei, the energy of the system can be adequately approximated by calculating the electronic energy for a each possible nuclear configuration. We now imagine how this energy varies along a particular coordinate, one of the 3N - 6 coordinates. The coordinate we will choose to examine is the reaction coordinate, the one whose nuclear displacements would lead along a minimum energy path from the nuclear configuration of (DA) to that of (D+A+).
bR. Marcus, J. Chem. Phys. 24, 966 (1956); ibid., 24,979 (1956); ibid. 26, 867 (1957); ibid. 26, 872 (1957); Disc. Farad. Soc. 29,21(1960); J. Phys. Chem. 63,853 (1963); J. Chem. Phys. 38,1858 (1963); ibid., 39, 1734 (1963); ibid., 43,679 (1965).
CI am grateful to Prof. A. C. Albrecht for providing this derivation and to M. Stimson for bringing it to my attention.
1 Donor \
Reaction coordinate
II Figure 5.4
Energy dependence as a function of reaction coordinate for electron on donor or acceptor.
When the electron is on the donor, the energy will be a minimum at a particu- lar location along the reaction coordinate; let us arbitrarily label this as position zero along that coordinate, as shown in Figure 5.4. The parabolic curve labeled
"donor" shows how the energy of the system might vary with displacement when the electron is on the donor. If the electron were on the acceptor, the energy of the system would be different; its minimum will in general be at a different location, say x", along the reaction coordinate, and the energy of the minimum will differ from that of the donor by AGO, the free energy of the reaction. Note that, as drawn in the figure, AGO is negative (the products are more stable than the reactants), so that the positive energy difference between the minima of the two parabolas is -AGO. The parabolic curve labeled "acceptor" shows how the energy of the system might vary with displacement when the electron is on the acceptor. If we assume, as did Marcus, that the coupling between the donor and acceptor electronic energy states is weak, then the energy of the transition state will be given by the point of intersection between the two curves. The key to determining the rate constant for the reaction is to find the value of AGt in the figure. From equation 3.23, we know that the rate for the process is given simply by kET(T) = (kTlh)exp(-AG'lkT).
We now suppose that the curves describing how the energy changes with posi- tion along the reaction coordinate can be approximated by parabolas, both for the donor and for the acceptor; ịẹ, we will assume that E = x2 for both parabolậ^ It can be shown that this approximation is equivalent to the full theory developed by Marcus in the limit when the donor and acceptor sites are not too close together. Let us label by x' the reaction coordinate position where the donor and acceptor parabola intersect. From the point of view of the donor curve, the value of AG' is simply AGf = x t 2 . Let us also define Em = xU2 as the value, relative to its minimum, of the
dActually, we need only assume that E a x2; e.g., E = Cx2 = fix)^. The arguments given in the text are then appropriate provided that we then transform variables so that the reaction coordinate, now already plotted in arbitrary units, is plotted in units of fi times the current arbitrary unit.
Section 5.3 Reactions of Charged Species in Solution acceptor parabola at the location of the minimum energy for the donor. This energy, called the reorganization energy, is the energy required to reorganize the nuclei of the acceptor and its surrounding solvent into the configuration of the donor and its surrounding solvent in the absence of back transfer of the electron. We now calcu- late the energy of the intersection point for the two parabola above the minimum energy for the acceptor. As measured from the bottom of the acceptor parabola, this energy is -AGO + AGf = (x" - x ' ) ~ = xw2 - 2xtx" + xt2. Substituting AG+ for x t 2 and Em for xtt2, we obtain
Finally, noting again that Em = xV2, we obtain the final result for G+:
The rate constant for the electron transfer reaction is thus
The form of equation 5.38 makes an interesting prediction about the rate con- stant for the reaction. If Em is large and 4G0 is positive or just slightly negative, then (Em + AGO) will be positive and the rate constant will be relatively small. Fig- ure 5.5 shows the positions of the parabolic curves for the same value of Em (the same displacement between the two parabolas) and for values of AGO ranging from positive in panel (A) to increasingly negative values in panels (B)-(D). In panel (A) AGO > 0, while in panel (B) AGO < 0; both panels have Em + AGO > 0. Thus, the rate should be small for panel (A) and a little larger for panel (B). As 4G0 becomes increasingly negative, eventually (Em + AGO) will become zero, and the rate con- stant will become a maximum; this situation is shown in panel (C) of the figure. If 4G0 becomes even more negative, then E + AGO will be negative and its square will again increase. Equation 5.38 predicts that the rate constant will then actually decrease with increasing free energy change, -AGO. The reason why is shown in panel (D) of the figure, where it can be seen that the activation energy for the elec- tron transfer reaction has now increased. The region of free energy change over which the rate constant decreases as the reaction releases more free energy is called the "Marcus inverted region."
That a rate constant should decrease with increasing exothermicity was quite counterintuitive at the time Marcus put forth his model, and the existence of the Marcus inverted region was in doubt for nearly thirty years until G . Closs, J. R.
Miller, and their coworkers reported the confirming set of experiment^.^ The results of their measurements are shown in Figure 5.6, which plots the logarithm of the
eJ. R. Miller, L. T. Calcaterra, G. L. Closs, J. Am. Chem. Soc. 106,3047 (1984); G. L. Closs, L. T. Cal- caterra, N. J. Green, K. W. Penfield, and J. R. Miller, J. Phys. Chem. 90,3673 (1986); G. L. Closs and J. R.
Miller, Science 240,440 (1988).
H Figure 5.5
Intersecting donor and acceptor parabolas for increasingly negative values of AGO going from panels (A) to (D). Note that activation energy for the reaction is a minimum in panel (C), but that it increases in going from (C) to (D).
1) Figure 5.6
Intramolecular electron transfer rate constants as a function of free energy change. The transfer occurs from biphenyl anions to the eight acceptors attached at A in the structure shown.
From G. L. Closs, L. T. Calcaterra, N. J. Green, K. W. Penfield, and J. R. Miller, J Phys. Chem. 90, 3673 (1986). Reprinted with permission from The Journal of Physical Chemistry. Copyright 1986 American Chem- ical Society.
Section 5.4 Experimental Techniques energy transfer rate constant versus -AGO for a series of molecules consisting of a donor group separated from various acceptor groups by a rigid spacer molecule. By varying the composition of the acceptor and measuring the rate of electron transfer following pulse radiolysis to produce the radical anion, the authors were able to see how the electron transfer rate varied with increasing -AGO. As shown in the fig- ure, the rate did indeed decrease with increasing -AGO in accordance with the Marcus theory (solid line). Similar experiments were performed by other group^.^
While the Marcus theory is now the accepted standard for electron transfer reac- tions,g it should be noted that this important area of chemistry is still one of active research.
5.4 EXPERIMENTAL TECHNIQUES
It should come as no surprise that some reactions in solution are very rapid. Reac- tions involving oppositely charged ions, reactions in which the solvent participates as a reactant, or reactions involving the motion of light particles such as protons or electrons might reasonably be expected to proceed quickly. For such rapid processes, which often occur more rapidly than the reactants can be mixed, special methods are necessary to determine reaction rate constants.