Marcus theory for

FIG. 21 Plot of log ki2 vs. AEi/2 showing the dependence of ET rate on the driving force for the reaction between ZnPor and four aqueous reductants. The difference between the half-wave potentials for an aqueous redox species and ZnPor, AE-i/2 = AE° + A°0, where AE° is the difference in the formal potentials of the aqueous redox species and ZnPor and A° is the potential drop across the ITIES. The solid line is the expected behavior based on Marcus theory for X = 0.55 eV and a maximum rate constant of 50 cm s M . (Reprinted from Ref. 49. Copyright 1999 American Chemical Society.)... 

In order to account for the foregoing kinetic behavior, we rely on the Marcus theory for outer-sphere electron transfer to provide the quantitative basis for establishing the free energy relationship (8), i.e.,... 

Fig. 4 Predictions of Marcus Theory for the variation of forward (fcf)and reverse (kr) rate coefficients with ApA" for reactions with different intrinsic barriers (X/4)...

A very brief introduction to the important topic of bioinorganic electron transfer mechanisms has been included in Section 1.8 (Electron Transfer) of Chapter 1. Discussions of Marcus theory for protein-protein electron transfer and electron or nuclear tunneling are included in the texts mentioned in Chapter 1 (references 3-7). A definitive explanation of the underlying theory is found in the article entitled Electron-Transfer in Chemistry and Biology, written by R. A. Marcus and N. Sutin and published in Biochem. Biophys. Acta, 1985, 811, 265-322. 

Fig. 12 Test of the Marcus theory for methyl transfers in H,0. The graph compares the values of G(29) calculated by equation (29) from experimental free energies with C,7) calculated by equation (27) from the Cx x for the symmetrical reactions...

To fully understand the relaxation pathways for photoinduced charge-transfer reactions in solutions we need to take solvent effects into account. For that reason it is necessary to recall some basic principles of the classical Marcus Theory for electron-transfer reactions in solution. 

Figure 5.2 Tafel plots of In k versus overpotential for a mixed self-assembled monolayer containing HS(CH2)i600C-ferrocene and HS(CH2)isCH3 in 1.0 M HCIO4 at three different temperatures V, 1 °C O/ 25 °C , 47°C. The solid lines are the predictions of the Marcus theory for a standard heterogeneous electron transfer rate constant of 1.25 s-1 at 25 °C, and a reorganization energy of 0.85 eV (= 54.8 kj moh1). Reprinted with permission from C. E. D Chidsey, Free energy and temperature dependence of electron transfer at the metal-electrolyte interface, Science, 251, 919-922 (1991). Copyright (1991) American Association for the Advancement of Science...

These tunneling concepts can be compared with the Marcus theory for electron transfer (Closs and Miller 1988 Kang et al. 1990 Kim and Hynes 1990a,b Marcus and Sutin 1985 McLendon 1988 Minaga et al. 1991 Sutin 1986). In the normal regime, solvent fluctuation is required to equilibrate the reactant and... 

Reactions with Co(III)aq and other Co(III) species often do not conform to the Marcus theory (for a summary, see Pennington, 1978) and reactions nos. 28 and 32 thus constitute expected cases of deviation. The reason for this discrepancy is not known. 

We have now surveyed the use of Marcus theory for organic processes of many different kinds. With a few notable exceptions, the agreement between... 

The use of the energy-gap reaction coordinate allows us to calculate solvent reorganization energies in a way analogous to that in the Marcus theory for electron transfer reactions.19 The major difference here is that the diabatic states for electron transfer reactions are well-defined, whereas for chemical reactions, the definition of the effective diabatic states is not straightforward. The Marcus theory predicts that... 

Fig. 1. The classical (a) and the semi-classical (b-d) representations of the Marcus theory for X — 1.0 eV at T = 300 K. In the classical expression (Eq. 1), X determines both the position of the maximum and the breadth of the parabola. The maximum keI is determined by the frequency factor (Z, here taken as 6 x 10 1 s ) in the Eyring expression (ket = KZexp( — AGlJkbT) where k is the transmission coefficient, usually taken to be unity). In the semi-classical approach the reorganization energy is explicitly divided into Xh (here equal 0.2 eV) and 2a (0.8 eV). The value of V is chosen to...

Therefore, if the Marcus theory describes properly the effect of solvents of k, a linear correlation between In and ( op -fis ) should be observed in the experimental results. Before turning to the experimental studies, the (Sop - s ) parameter for various solvents used in electrochemical work is presented in Table 1. Inspection of these data reveals that the largest difference of the (Cop -Ss ) parameter for the listed solvents amounts to 0.263. Thus, on the basis of the Marcus theory for the outer-sphere electrode reactions, the largest change of the reaction rate for different solvents should amount to exp (const 0.263). In this estimation any double-layer effect on the rate constant was neglected. 

A quasi-linear correlation of log/c or AG with the ionization potentials of the electron donors as observed in the FeL3 - + reactions is predicted by Marcus theory for outer-sphere electron transfers. Accordingly, the free-energy dependence of AG can be satisfactorily simulated with the Marcus equation (Eq. 90), taking a (constant) value of A = 41 kcal mol as reorganization energy for all tetraalkyltin compounds (see Figure 19A) [32]. 

Figure 11.20. Linear free energy relation for the oxygenation of metal ions. The slope of unity is predicted by Marcus theory for endergonic outer-sphere electron transfer steps. (From Wehrli, 1990.)...

A refinement of the mechanism was searched for in the more recent literature when a differentiation was made between what was called inner-sphere ET and outer-sphere ET. It was assumed that, in the reaction of a Grignard reagent with a ketone (i.e., benzophenone), the electron transfer was rate-limiting [44] furthermore, for a series of Grignard reagents, a correlation had been found between the reaction rates and their oxidation potentials [21], according to the Marcus theory for outer-sphere ET [55]. Nevertheless, it seemed questionable [56] whether the electron transfer was an independent step (steps l->2->3-+4 in Scheme 19), or whether it was concerted with the transfer of the magnesium atom (steps l->3->4). 

The approach used to obtain the EVB free-energy functionals (the Ag of Equation (7)) has been originally developed in Ref. 25 in order to provide the microscopic equivalent of the Marcus theory for electron transfer (ET) reactions.38 This approach allows one to explore the validity of the Marcus formula and the underlying linear response approximation (LRA) on a microscopic molecular level.39 While this point is now widely accepted by the ET community,40 the validity of the EVB as perhaps the most general tool in microscopic LFER studies is less appreciated. This issue will be addressed below. 

If there is a strong electronic coupling between P and R in the transition state, one commonly speaks of an inner-sphere mechanism, and, conversely, if the interaction is weak, one uses the term outer-sphere mechanism. There are various theoretical approaches for quantifying the rates of redox reactions, including the so-called Marcus theory. For a description of these approaches, we refer to the literature (e.g., Eberson, 1987). For our discussion here, we content ourselves with trying to identify the factors that determine the rate at which a given organic pollutant is reduced or oxidized in the environment. 

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