Marcus relation

We were also curious to extend the Marcus relation a bit further to see if we could somehow eliminate the need to calculate two very expensive ion-molecule clusters for each crossreaction required to obtain AE. As Professor Brauman pointed out, the energy difference between the transition state and the separated reactants can be used as a measure of the overall efficiency of the reaction for structurally similar reactions. 

The extent to which the radicals react according to Eqs. 6 or 7 depends on the nature of Ri, Ra, and R3. If Ri = Rj = H and R3 = H through NO2, the ratio (6) (7) > 20. The addition reactions observed with these systems are characterized by strongly negative activation entropies, which can be rationalized in terms of immobilization of water molecules by the positive charge at C in the transition state [15]. That the transition state for addition has pronounced electron-transfer character concluded from the fact [15] that the rate constants for addition depend on the reduction potential of the nitrobenzene in a way describable by the Marcus relation for outer-sphere electron transfer. 

Figure 3 Driving force (-AG°et) dependence of intramolecular ET rate constants in ZnP-Cso (CS white circles CR white squares), Fc-ZnP-Ceo (black circles), FC-H2P-C60 (black triangles), ZnP-H2P-Ceo (black squares), and Fc-ZnP-H2P-C6o (white triangles). The lines represent the best fit to the Marcus relation, [Eq. (1)] (see text). (From Ref. 47.)...

The kinetics of the reduction of perruthenate(VII) by [FefCbOe]" and [W(CN)g]" and the oxidation of ruthenate(VI) by [Mo(CN)g] and [Ru(Cb06] have been studied in aqueous alkaline solutions. The cross-reaction data have been treated according to the Marcus relations and yield a self-exchange rate constant of 10 s at 25.0 °C and 1.0 M ionic strength for the... 

Similar arguments apply to the six a-carboxy-substituted ketones that have been studied by Kresge and coworkers (entries acetoacetate to oxocyclobutane-2-carboxylate in Table 1). Kresge already noted that the rate constants kucK observed for the uncatalyzed ketonization of some of these compounds would give unrealistically high calculated values for k e near or above 1011 m-1 s-1 using Equation (18). Indeed, these calculated values of k are about two orders of magnitude above those expected from the Marcus relation except that for 4,4,4-trifluoroacetate. The rate constants k c observed for the formation of these a-carboxy-substituted ketones are, however, close to those expected for the protonation of the neutral enols by water, k = kf. 

The acidity constants of protonated ketones, pA %, are needed to determine the free energy of reaction associated with the rate constants ArG° = 2.3RT(pKe + pK ). Most ketones are very weak bases, pAT < 0, so that the acidity constant K b cannot be determined from the pi I rate profile in the range 1 < PH <13 (see Equation (11) and Fig. 3). The acidity constants of a few simple ketones were determined in highly concentrated acid solutions.19 Also, carbon protonation of the enols of carboxylates listed in Table 1 (entries cyclopentadienyl 1-carboxylate to phenylcyanoacetate) give the neutral carboxylic acids, the carbon acidities of which are known and are listed in the column headed pA . As can be seen from Fig. 10, the observed rate constants k, k for carbon protonation of these enols (8 data points marked by the symbol in Fig. 10) accurately follow the overall relationship that is defined mostly by the data points for k, and k f. We can thus reverse the process by assuming that the Marcus relationship determined above holds for the protonation of enols and use the experimental rate constants to estimate the acidity constants A e of ketones via the fitted Marcus relation, Equation (19). This procedure indicates, for example, that protonated 2,4-cyclohexadienone is less acidic than simple oxygen-protonated ketones, pA = —1.3. 

However, with this E° value it is impossible to fit the data of entry no. 1 of Table 15 to the Marcus relation. Only by using a considerably higher E° for the SO4VSO4- couple, 3.08 V (or, of course, a set of lower values for ArH), can any reasonable fit be obtained. We then also have to postulate a rather small X value, 8.0 kcal mol, for the reaction. Knowing that X for self-exchange reactions of the compound types involved is around 10 kcal mol-1 (see Table 7) and assuming it to be small, 10 kcal mol-1, for SO4VSO4- too we can calculate X to be 9.0 kcal mol-1. 

Entry no. 2 presents another problem in that the electrostatic correction term for AG° in the solvent used, acetic acid, is very large, —15.3 kcal mol-1 (see Table 3 Z,Z2 = —2, rn = 7 A). Again, E° = 2.52 V is far too small to give any reasonable fit to the Marcus relation. With E° = 3.08 V, the result is at least consistent with that of entry no. 1. Entries nos. 3 and 4 have been treated with A values for self-exchange reactions of pyridine and acetate equal to 10 and 20 kcal mol-1 respectively fccalc comes out at 1010 M 1 s irrespective of the choice of E° = 2.52 or 3.08 V. For entry no. 5 X (CH3OH) was assumed to be > 20 kcal mol-1 and the quoted value of kcalc is estimated with E° = 3.08 V. It thus represents a maximum value and the reaction is certainly not feasible as an electron-transfer step. 

The pathways involving Mn04 ((Eq. (2)) and protonation products of [MiCNlg] are thermodynamically the least favorable and make only a small contribution to the total reaction rate. The Marcus relation... 

Fig. 6. Marcus relation Free energies of activation as a function of the driving force (terms (A) + (B) in text) corrected for electrostatic work terms (cyano complexes reaction with hydrazine ( ) methylhydrazine (O) 1,2-dimethylhydrazine ( ). Adapted with permission from Dennis et al. (52). Copyright 1987, American Chemical Society.

The relation shown here suggests that a measure of the intrinsic rate of a reaction, corrected for its driving force, is given by AF = AF — 0.5 AF°. This amounts to taking the average AF for the reaction in the forward and reverse directions. Under conditions where Marcus relation ki2 = ( 11 12 12/) (42) is applicable, AF = (AF ii-f AF 22 — RT In /). Values for these intrinsic activation free energies are given in Table III. This table shows some patterns of relative reactivity as well as some apparent anomalies. For example, the relative intrinsic... 

A recent study of Kiefer and Hynes49 used an EVB formulation, with a continuum treatment of the solvent, in an attempt to derive an LFER for PT reactions. Unfortunately, they assumed that a Marcus relation was never actually derived for PT reactions, apparently overlooking all the above works. Furthermore, their derivation ignored the crucial effect of Hy. Nevertheless, it is encouraging to see again a... 

The over-bar notation for the Bronsted coefficient and its derivative, e.g. a , is introduced to distinguish it from the inverse coupling length used in Section 10.3) This result is closely related to, but more fundamentally based than the well-knoivn and ividely employed (cf Refs. [9c, 12[) Marcus relation [19]... 

If the interpolation functions in Eq. (19.5) take the values = AE J4AE fi- and fi this relationship between the barrier and the overall energy change for reaction becomes the Marcus relation, Eq. (19.6), whose range of applicability is... 

It is well known [6, 14a, 27] that the Marcus relation may be derived from a model of intersecting parabolas for the reactant and product energy curves (Fig. 19.2). A parabola provides an unrealistic description of the energy profile for extension of an A-H bond far from its equilibrium length and towards complete dissociation a Morse curve offers a better description. It is perhaps not too widely appreciated [28, 29] that transformation of a Morse function (Eq. (19.7)) from bond length (r) to bond order (n) coordinates (Eq. (19.8)) yields a parabola (Eq. (19.9)). 

The Marcus relation, Eq. (19.6), is clearly not a linear relationship between the activation energy and the reaction asymmetry but a quadratic one. The first derivative of AEt with respect to is equivalent to the Bronsted coefficient a in Eq. 

The first three terms on the right-hand side correspond to the Marcus relation for the nonadiabatic case where there is no coupling between the diabatic energy states (i.e. E = 0 at all values of the reaction coordinate). The fourth and fifth terms reflect the effect of the adiabatic coupling of the two surfaces on the transition state and reactant state, respectively, and < /2 AGjxn + 4AGin,l. ... 

In his theoretical treatment of outer-sphere electron transfer reactions, Marcus related the free energy of activation, AG1, to the corrected Gibbs free energy of the reaction, AG°, via a quadratic equation (Equation 1.4).2 4 13... 

Weber, K. Creager, S.E. Voltammetry of redox-active groups irreversibly adsorbed onto electrodes. Treatment using the marcus relation between rate and overpotential. Anal. Chem. 1994, 66, 3164. 

---