Applications of the Marcus Equation

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. 

The applicability of the Marcus equation (Eq. 14) to MCET reactions was examined using a ferrocene-naphthoquinone dyad (Fc—NQ) as discussed below (Scheme 19)(121,122). No ET from the Fc to NQ moiety occurs in Fc—NQ with a... 

Application of the Marcus equation for electron transfer affords the electron exchange rate of the molybdenum radical/anion couple. The value is fcge = 3 X 10 L mol s. The high value argues that very little nuclear reorganization is needed to add an electron to the SOMO of the 17e radical. 

The data summarized in Table 2 indicate that the series of cyanine borates with the n-butyltriphenylborate counterion possess a negative AGei value, i.e., they fulfill the basic Rehm Weller requirements for effective photoinduced electron transfer processes. Extending this to a practical application of the Marcus equation (see... 

The first application of the Marcus equation (3e) to methyl transfers was the study by Albery and Kreevoy (4, 5) based on data in solutions. In this study, the thermodynamic data of Abraham and McLennan (6) were used, and the identity barriers were more or less treated as adjustable parameters. The treatment was qualitatively plausible, but the quantitative identity barriers were not completely convincing. An important study (7) showed the application of the Marcus equation to methyl transfer in the gas phase. 

The reactions of the photochemically prepared [U02] ion with a series of chromium, ruthenium, and cobalt complexes have been studied/ An application of the Marcus equation to the cross-reaction data yields a [U02] self-exchange rate constant in the range of 1-15 s The reactions with the chromium and... 

Reactions of cobalt complexes of macrocyclic tetramine ligands with various couples (including the highly oxidizing MnCl /, hitherto little studied) have been reported. Inner-sphere and outer-sphere reactions involving, for example, [Ru(NH3)6py] +/ + have been characterized. A free-energy correlation for reactions between [Co(N4)(OH2)2] with inner-sphere oxidants shows the expected slope 9AG /3AG 0.5 down to a limit of AG a 7 kcal mol, which is believed to represent diffusion control. Application of the Marcus equations (1) and (3) to reactions of both types leads to an assessment of the factors con-... 

The requirements for application of the Marcus equation to reactions other than electron transfer are expressed in precise and succinct form hy J. Jortner, Faraday Discuss. Chem. Soc. 74 (1982) 306, 307. See also the contributions to that Discussion by R.A. Marcus, p. 306 and J.R. Murdoch, pp. 297 seq. 

KAB 98] Kabatc J., Pietrzak M., Paczkowski J., Cyanine borates revisited. Application of the Marcus equation for the description of the kinetics of photoinitiated free radical polymerization . Macromolecules, vol. 31, pp. 4651-4654, 1998. 

S. Scheiner and X. Duan, Applicability of the Marcus equation to proton transfer in symmetric and unsymmetric systems, J. Mol. Struct. (Theochem), 285 (1993) 27. 

The reversibility of the [Os(bpy)3]3+/2+ couple makes it useful for the determination of the electron self-exchange rates of other couples by application of the Marcus cross-reaction equation. Recently, this has been applied to the oxidation of S032- to S042- (622). The new rate constant for this reaction of 1.63 x 107 M-1 sec-1 is consistent with the... 

The theoretical results obtained for outer-sphere electron transfer based on self-exchange reactions provide the essential background for discussing the interplay between theory and experiment in a variety of electron transfer processes. The next topic considered is outer-sphere electron transfer for net reactions where AG O and application of the Marcus cross reaction equation for correlating experimental data. A consideration of reactions for which AG is highly favorable leads to some peculiar features and the concept of electron transfer in the inverted region and, also, excited state decay. 

Most systematic studies on gas-phase SN2 reactions have been carried out with methyl halides, substrates which are free of complications due to competing elimination. Application of the Marcus rate-equilibrium formalism to the double-minimum potential energy surface led to the development of a model for intrinsic nucleophilicity in S 2 reactions233. The key quantities in this model are the central energy barriers, Eq, to degenerate reactions, like the one of equation 22, which are free of a thermodynamic driving force. 

Arnett and coworkers later examined the reaction of lithium pinacolone enoiate with substituted benzaldehydes in THE at 25 °C. The determination of the heat of reaction indicated that the Hammett p value for the process is 331. Although the aldol reaction was instantaneous in THF at 25 °C, the reaction with o- or p-methylbenzaldehyde could be followed using a rapid injection NMR method in methylcyclohexane solvent at —80 °C. Application of Eberson s criterion based on the Marcus equation, which relates the free energy of ET determined electrochemically and the free energy of activation determined by kinetics, revealed that the barriers for the ET mechanism should be unacceptably high. They concluded that the reaction proceeds via the polar mechanism . Consistent with the polar mechanism, cyclizable probe experiments were negative . The mechanistic discrepancy between the reactions of benzaldehyde and benzophenone was later solved by carbon kinetic isotope effect study vide infraf. ... 

Many reactions exhibit effects of thermodynamics on reaction rates. Embodied in the Bell-Evans-Polanyi principle and extended and modified by many critical chemists in a variety of interesting ways, the idea can be expressed quantitatively in its simplest form as the Marcus theory (15-18). Murdoch (19) showed some time ago how the Marcus equation can be derived from simple concepts based on the Hammond-Leffler postulate (20-22). Further, in this context, the equation is expected to be applicable to a wide range of reactions rather than only the electron-transfer processes for which it was originally developed and is generally used. Other more elaborate theories may be more correct (for instance, in terms of the physical aspects of the assumptions involving continuity). For the present, our discussion is in terms of Marcus theory, in part because of its simplicity and clear presentation of concepts and in part because our data are not sufficiently reliable to choose anything else. We do have sufficient data to show that Marcus theory cannot explain all of the results, but we view these deviations as fairly minor. 

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