Marcus theory dependence

On this basis Cr(V), not Cr(IV), is the kinetically important intermediate such that k = 3 k4 and k = k Jk. The hydrogen-ion dependence of the reaction rate has been discussed. Furthermore, comparisons are drawn with the rate of the Cr(VI)+Fe(phen)3 reaction, and Sullivan has speculated on the intimate nature of both mechanisms in the light of Marcus theory... 

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.)... 

According to the Marcus theory [9], the electron transfer rate depends upon the reaction enthalpy (AG), the electronic coupling (V) and the reorganization energy (A). By changing the electron donor and the bridge we measured the influence of these parameters on the charge transfer rate. The re-... 

In summary, to apply the Marcus theory of electron transfer, it is necessary to see if the temperature dependence of the electron transfer rate constant can be described by a function of the Arrhenius form. When this is valid, one can then determine the activation energy AEa only under this condition can we use AEa to determine if the parabolic dependence on AG/ is valid and if the reaction coordinate is defined. 

Figure 6.24 Marcus theory predictions of the dependence of the electron-transfer rate on the thermodynamic driving force...

If the interaction between the donor and acceptor in the encounter pair (D. .. A) is weak (Scheme 4.2), the rate constant kET can be estimated by the Marcus theory. This theory predicts a quadratic dependence of the activation free energy AG versus AG° (standard free energy of the reaction). 

Q vibration is not directly coupled to the bath of harmonic oscillators. This assumption is similar to the approach employed by Silbey and Suarez who used a tunneling splitting that depends on the oscillating transfer distance Q in their spin-boson Hamiltonian. Borgis and Hynes, too, have made this assumption in the context of Marcus theory. 

Take a mean value of 80 (i.e., 0.83 eV). Numerical calculations show that T] < 0.2 V is the condition up to which 9.38 yields the experimental version of Tafel s law (of course, the value depends on the Fs chosen and the allowed T, for the applicability of 9.38 will be roughly halved at the lower limit and doubled at the higher one. In any case, this harmonic approximation, which is involved in the Weiss—Marcus theory, cannot be applied to the experimental current-potential data, which in reality extend over 0.2 V and even 1.0 V (for hydrogen and oxygen evolution). 

Marcus theory (15) has been applied to the study of the reductions of the jU,2-superoxo complexes [Co2(NH3)8(/u.2-02)(/i2-NH2)]4+ and [Co2(NH3)10(ju.2-O2)]6+ with the well-characterized outer-sphere reagents [Co(bipy)3]2+, [Co(phen)3]2+, and [Co(terpy)2]2+, where bipy = 2,2 -bipyridine, phen = 1,10-phenanthroline, and terpy = 2,2 6, 2"-terpyridine (16a). The kinetics of these reactions could be adequately described using a simple outer-sphere pathway, as predicted by Marcus theory. However, the differences in reactivity between the mono-bridged and di-bridged systems do not appear to be explicable in purely structural terms. Rather, the reactivity differences appear to be caused by charge-dependent effects during the formation of the precursor complex. Some of the values for reduction potentials reported earlier for these species (16a) have been revised and corrected by later work (16b). 

Radicals can be either reduced (to anions or organometallics) or oxidized to cations by formal single electron transfer (Scheme 11).50 Such redox reactions can be conducted either chemically or electro-chemically51 and the rates of electron transfer are usually analyzed by the Marcus theory and related treatments.50 These rates depend (in part) on the difference in reduction potential between the radical and the reductant (or oxidant). Thus a species such as an a-amino radical with high-lying singly occupied molecular orbital (SOMO) is more readily oxidized, while a species such as the malonyl radical with a low-lying SOMO is more readily reduced. The inherent difference in reduction potential of substituted radicals is an important control element in several kinds of reactions. 

Inner-sphere electron transfers are characterized by (a) temperature-independent rate constants that are greatly higher and rather poorly correlated by Marcus theory (b) weak dependence on solvent polarity (c) low sensitivity to kinetic salt effects. This type of electron transfer does not produce ion radicals as observable species but deals with the preequilibrium formation of encountered complexes with the charge-transfer (inner-sphere) nature (see also Rosokha Kochi 2001). 

---