The Marcus Equation

The barrier to thermodynamically unfavorable deprotonation of carbon acids (AGfl, Fig. 1.1) in water is equal to the sum of the thermodynamic barrier to proton transfer (AG°) and the barrier to downhill protonation of the carbanion in the reverse direction (AGr Eq. (1.2)). The observation of significant activation barriers AGr for strongly thermodynamically favorable protonation or resonance stabilized carbanions shows that there is some intrinsic difficulty to proton transfer. The Marcus equation defines this difficulty with greater rigor as the intrinsic barrier A, which is the activation barrier for a related but often hypothetical thermoneutral proton transfer reaction (Fig. 1.2B) [46]. 

The Marcus equation was first formulated to model the dependence of rate constants for electron transfer on the reaction driving force [47-49]. Marcus assumed in his treatment that the energy of the transition state for electron transfer can be calculated from the position of the intersection of parabolas that describe the reactant and product states (Fig. 1.2A). This equation may be generalized to proton transfer (Fig. 1.2A) [46, 50, 51], carbocation-nucleophile addition [52], bimolecular nucleophilic substitution [53, 54] and other reactions [55-57] by assuming that their reaction coordinate profiles may also be constructed from the intersection of 

Many laboratories, including our own, have used the Marcus equation empirically as a relatively simple and convenient framework for describing the differences in the intrinsic difficulty for related reactions, after correction for differences in the reaction thermodynamic driving force. This has led to the determination of the Marcus intrinsic barriers for a variety of proton transfer reactions by experiment and through calculations [58-65], This compilation of intrinsic reaction barriers represents an attempt to compress an essential feature of these kinetic barriers to a single experimental parameter. An examination of the substituent effects on these intrinsic barriers has provided useful insight into the transition state for organic reactions [66], 

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If the intrinsic barrier AGq could be independently estimated, the Marcus equation (5-69) provides a route to the calculation of rate constants. An additivity property has frequently been invoked for this purpose.For the cross-reaction... 

There are several equations other than the Marcus equation that describe rate-equilibrium relationships. Murdoch writes all of these equations in the general form... 

For a system describable with a single progress variable, we derived the Marcus equation, Eq. (5-76). 

Let X be the normalized progress variable in a system subject to the Marcus equation (Eq. 5-69), so jc = -t- AG°/8AGo, where has the significance of a in Eq. (5-67). Then deduce this equation, which describes the energy change, relative to the reactant, over the reaction coordinate ... 

In Eq. (7-21), AGo is the intrinsic barrier, the free energy of activation of the (hypothetical) member of the reaction series having AG" = 0. It is evident that the Marcus equation predicts a nonlinear free energy relationship, although if a limited... 

This discussion of sources of curvature in Br insted-type plots should suggest caution in the interpretation of observed curvature. There is a related matter, concerning particularly item 5 in this list, namely, the effect of a change in transition state structure. Br nsted-type plots are sometimes linear over quite remarkable ranges, of the order 10 pK units, and this linearity has evoked interest because it seems to be incompatible with Marcus theory, which we reviewed in Section 5.3. The Marcus equation (Eq. 5-69) for the plot of log k against log K of the same reaction series requires curvature, the slope of the plot being the coefficient a. given by Eq. (5-67). A Brjinsted plot, however, is not a Marcus plot, because it correlates rates and equilibria of different reactions. The slope p of a Br nsted plot is defined p = d log kobs/d pK, which we can expand as... 

The latter is, except for a couple of terms related to solvent reorganization, the Marcus equation. The central idea is that the activation energy can be decomposed into a component characteristic of the reaction type, the intrinsic activation energy, and a correction due to the reaction energy being different from zero. Similar reactions should have similar intrinsic activation energies, and the Marcus equation obeys both the BEP... 

Actually the assumptions can be made even more general. The energy as a function of the reaction coordinate can always be decomposed into an intrinsic term, which is symmetric with respect to jc = 1 /2, and a thermodynamic contribution, which is antisymmetric. Denoting these two energy functions h2 and /zi, it can be shown that the Marcus equation can be derived from the square condition, /z2 = h. The intrinsic and thermodynamic parts do not have to be parabolas and linear functions, as in Figure 15.28 they can be any type of function. As long as the intrinsic part is the square of the thermodynamic part, the Marcus equation is recovered. The idea can be taken one step further. The /i2 function can always be expanded in a power series of even powers of hi, i.e. /z2 = C2h + C4/z. The exact values of the c-coefficients only influence the... 

Table 15.2 Comparing experimental activation barriers with those calculated by the Marcus equation...

Again this averaging procedure can only be expected to work when the reactions are sufficiently similar . This is difficult to quantify a priori. The Marcus equation is therefore more a conceptual tool for explaining trends, than for deriving quantitative results. 

Equation (3.34) without the H X ) and the H 2la terms is identical to the Marcus equation for methyl transfer reactions (Ref. 13). This equation predicts, at the range ( AG0 < a), a linear relationship between AAG0 and AAg by... 

The Marcus equation says that the overall AG for a one-step reaction is ... 

One type of process that can successfully be treated by the Marcus equation is the Sn2 mechanism (p. 390)... 

When R is CH3 the process is called methyl transfer. For such reactions, the work terms and are assumed to be very small compared to AG° and can be neglected, so that the Marcus equation simplifies to... 

The Brpnsted equations relate a rate eonstant k to an equilibrium constant K. In Chapter 6, we saw that the Marcus equation also relates a rate term (in that case AG ) to an equilibrium term AG°. When the Marcus treatment is applied to proton... 

The Brpnsted law is therefore a special case of the Marcus equation. 

Equation (3.44) (in the Arrhenius form) is usually called the Marcus equation [74,75]. A special feature of the Marcus equation is that it predicts the parabolic dependence of the activation energy AEa on the free energy change AG/, that is, AEa is related to the free energy change AG/ in a parabolic form. 

In the previous section we have shown that the Marcus equation can be derived from Eq. (3.40). In this section, other forms of rate constants used in literatures will be derived. Notice that at T = 0, Eq. (3.40) reduces to... 

Bunting and Kanter have developed a modified form of the Marcus equation to treat the changes in intrinsic barrier A observed for deprotonation of /J-keto esters and amides.81 It would be useful to consider similar modifications of the Marcus equation to model the variable intrinsic barriers observed for carboca-tion-nucleophile addition reactions. 

The first attempt to describe the dynamics of dissociative electron transfer started with the derivation from existing thermochemical data of the standard potential for the dissociative electron transfer reaction, rx r.+x-,12 14 with application of the Butler-Volmer law for electrochemical reactions12 and of the Marcus quadratic equation for a series of homogeneous reactions.1314 Application of the Marcus-Hush model to dissociative electron transfers had little basis in electron transfer theory (the same is true for applications to proton transfer or SN2 reactions). Thus, there was no real justification for the application of the Marcus equation and the contribution of bond breaking to the intrinsic barrier was not established. 

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