Marcus expression

Occasionally, the successful application of the Marcus expressions (5.35) and (5.37) to a reaction can support its designation as outer-sphere. The reduction of a series of substituted benzenediazonium salts by Fe(CN)5 and (Me5cp)2pe conforms to the simple Marcus expression and represents supporting evidence for the formulation of these reactions as outer sphere (or non-bonded electron transfer in organic systems)... 

Early attempts at observing electron transfer in metalloproteins utilized redox-active metal complexes as external partners. The reactions were usually second-order and approaches based on the Marcus expression allowed, for example, conjectures as to the character and accessibility of the metal site. xhe agreement of the observed and calculated rate constants for cytochrome c reactions for example is particularly good, even ignoring work terms. The observations of deviation from second-order kinetics ( saturation kinetics) allowed the dissection of the observed rate constant into the components, namely adduct stability and first-order electron transfer rate constant (see however Sec. 1.6.4). Now it was a little easier to comment on the possible site of attack on the proteins, particularly when a number of modifications of the proteins became available. 

The electron self-exchange rate constants evaluated by the Marcus expressions (using cross-reaction data) and those determined experimentally differ in the following cases. Give possible reasons for these differences. 

Fig. 9.25. (e) The hydrogen evolution reaction overthe Tafel relation which is linear over eleven orders of magnitude (experimental points). Curved line Marcus expression with assumption of harmonic oscillators. (Reprinted from J. O M. Bockris and S. U. M. Khan, Quantum Electrochemistry, Plenum 1979, p. 228.)... 

The increase in double bond character is assumed to increase the intrinsic barrier for reaction at the a-carbon atom. As this increase is greatest for the thermodynamically least stable (CF3-substituted) carbocation, changes in thermodynamic driving force and intrinsic barrier oppose each other. The constancy of the values of kn2o thus reflects a change in intrinsic barrier overriding the second and third terms in the Marcus expression of Equation (20). This is a more radical effect than the lesser variation preserving the linearity of the plots for the reaction families in Fig. 3 (p. 77), for which only the third term is overridden. 

In conclusion, it can also be pointed out that in principle a large value of A is itself sufficient to account for an extended linear free energy relationship. However, as Mayr has noted this is only true if the slope of the plot is O.5.238 Moreover, if the Marcus expression offers a quantitative guide to the degree of curvature of a free energy relationship (and it is by no means clear that it does),228 it is evident that the intrinsic barriers to reactions of carbocations with familiar nucleophiles are insufficiently large to account for the lack of curvature. 

Notwithstanding the possibility of variation of an intrinsic barrier within a reaction series, for comparisons between different reactions it is often convenient to assume that an unmodified Marcus expression applies. This approximation is justified partly by the high intrinsic barriers and small amounts of curvature characteristic of most reactions at carbon, including reactions of carbocations. The Marcus relationship then provides a common framework for comparisons between reactions based on the measurement of even a single combination of rate and equilibrium constants. Thus, calculation... 

Marcus expression for the rate constants is consistent (when tr 2) with the description of the solvent isotope effect in terms of fractionation factors. 

Rehm and Weller, 1969). Estimates of kd k Jkmn kd gave a value of 0.25, and with the AG given by (73) log fcobs vs. AG° plots can be obtained. Figure 7 shows two curves AG (0) = 2.4 and 7.2 kcal mol-1] of this type together with two curves calculated from (74) using the Marcus expression for AG instead, but with all variables identical. The similarities and differences commented upon already are clearly visible. 

With these reservations in mind, we can compare observed and calculated rate constants. Where appropriate, the full Marcus expression, which includes the log f term (Equation 9), has been used in the calculation of the electron transfer rates. 

The second property expected for non-equilibrium processes is the lack of dependence (Fig. 2.6, curve 1) or weak dependence (curve 2) of the experimental rate constant of ET in both Marcus regions (inverted and non-inverted), compared to that predicted by the classic Marcus expression (curve 3). 

In his 1961 paper,67 Hush modified his outer sphere energy equation by changing the (1 - l/s0) term to the Marcus expression (1/n2 - 1/e0) because the electron transition time, though longer than that under FC conditions, would still be short compared to solvent molecule motion. This proposition will be discussed later. The q value... 

One should also mention the so-called mean spherical approximation (MSA) treatment of solvent reorganization [25]. McManis and Weaver [125] considered how the solvent radius and dielectric parameters affect the electron transfer within the frame of this theory. The frequency dependence of the effective radius should cause significant deviations from the Marcus expression for the activation energy of... 

In equations used in the analysis there appear the Pekar factor, y, and the second factor of the Marcus expression for the electrode kinetics, which we denote by g ... 

Despite its utility at room temperature, simple Marcus theory cannot explain the DeVault and Chance experiment. All Marcus reactions have a conspicuous temperature dependence except in the region close to where AG = —A. Marcus theory does not predict that a temperature-dependent reaction will shift to a temperature-independent reaction as the temperature is lowered. Hopfield proposed a quantum enhancement of Marcus theory that would permit the behavior seen in the experiment [11]. He introduced a characteristic frequency of vibration hco) that is coupled to electron transfer, in other words, a vibration that distorts the nuclei of the reactant to resemble the product state. This quantum expression includes a hyperbolic cotangent (Coth) term that resembles the Marcus expression at higher temperatures, but becomes essentially temperature independent at lower temperatures. Other quantized expressions, such as a full quantum mechanical simple harmonic oscillator behavior [12] and that of Jortner [13], give analogous temperature behavior. 

The molecular solvent contribution, Aso1 is often described by modelling the solvent as a dielectric continuum which leads to the standard Marcus expression [36] or by a more up-to-date liquid state theory such as integral equation technique [37] or computer simulations [38],... 

Figure 4.9 Variation of In with -AG° predicted by the classical Marcus expression, eq. 4.31. The maximum rate is attained when AG° = -A. Where this maximum is high, the rate remains high over a range of AG° values, and can be substantial even for endergonic reactions (shown by the dashed line).

The classical Marcus expression, eq. 4.31, has its limitations exactly because it is classical. It also has great strengths. It predicts the inverted region. It provides a powerful physical model of ET (and PET), which will enable us to draw inferences about optimal conditions for PET in Section 4.9. It also indicates that varying the medium (and hence A) for a reaction of given -AG° will markedly change k. A reaction with small -AG°... 

The classical Marcus expression, eq. 4.31, assumes the internal modes of R and P to behave classically and is primarily aimed at adiabatic reactions with /c i = 1, offering no method of evaluating /Qi < 1 for nonadiabatie transfer between more distant or weakly coupled donors and acceptors. This requires nonclassical approaehes, to which we now turn. Semiclassical approaches, dealt with in this section, incorporate the quantisation of the internal modes of D and A but retain the classical treatment of the surrounding medium as a structureless dielectric continuum... 

The high-temperature limit semiclassical Marcus expression for the rate of ET in homogeneous solution, eq. 4.47, and the expressions for the rate of ET at an electrode (eqs. 4.63 and 4.64) were derivedby different approaches, which makes it hard to see that there is a link between them. Here we demonstrate this link by deriving eq. 4.47 using the overlapping density-of-states approach used to derive eqs. 4.63 and 4.64. 

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