Rate constants electron-transfer

The first type of interaction, associated with the overlap of wavefunctions localized at different centers in the initial and final states, determines the electron-transfer rate constant. The other two are crucial for vibronic relaxation of excited electronic states. The rate constant in the first order of the perturbation theory in the unaccounted interaction is described by the statistically averaged Fermi golden-rule formula... 

In the strong-coupling limit at high temperatures the electron transfer rate constant is given by the Marcus formula [Marcus 1964]... 

As an illustration of these considerations, the Arrhenius plot of the electron-transfer rate constant, observed by DeVault and Chance [1966] (see also DeVault [1984]), is shown in fig. 13. 

The exponent in this formula is readily obtained by calculating the difference of quasiclassical actions between the turning and crossing points for each term. The most remarkable difference between (2.65) and (2.66) is that the electron-transfer rate constant grows with increasing AE, while the RLT rate constant decreases. This exponential dependence k AE) [Siebrand 1967] known as the energy gap law, is exemplified in fig. 14 for ST conversion. 

EPR studies on electron transfer systems where neighboring centers are coupled by spin-spin interactions can yield useful data for analyzing the electron transfer kinetics. In the framework of the Condon approximation, the electron transfer rate constant predicted by electron transfer theories can be expressed as the product of an electronic factor Tab by a nuclear factor that depends explicitly on temperature (258). On the one hand, since iron-sulfur clusters are spatially extended redox centers, the electronic factor strongly depends on how the various sites of the cluster are affected by the variation in the electronic structure between the oxidized and reduced forms. Theoret-... 

Figure 15. Relationship between electron transfer rate constant ke and hydrogen generation rate constant. (Reprinted from Ref [194], 2000, with permission from lUPAC.)...

Early studies of ET dynamics at externally biased interfaces were based on conventional cyclic voltammetry employing four-electrode potentiostats [62,67 70,79]. The formal pseudo-first-order electron-transfer rate constants [ket(cms )] were measured on the basis of the Nicholson method [99] and convolution potential sweep voltammetry [79,100] in the presence of an excess of one of the reactant species. The constant composition approximation allows expression of the ET rate constant with the same units as in heterogeneous reaction on solid electrodes. However, any comparison with the expression described in Section II.B requires the transformation to bimolecular units, i.e., M cms . Values of of the order of 1-2 x lO cms (0.05 to O.IM cms ) were reported for Fe(CN)g in the aqueous phase and the redox species Lu(PC)2, Sn(PC)2, TCNQ, and RuTPP(Py)2 in DCE [62,70]. Despite the fact that large potential perturbations across the interface introduce interferences in kinetic analysis [101], these early estimations allowed some preliminary comparisons to established ET models in heterogeneous media. 

FIG. 10 Potential dependence of the electron-transfer rate constant k i) normalized to the value at the potential of zero charge TCNQ reduction by hexacyanoferrate at the water-DCE... 

FIG. 21 Complex IMPS spectra obtained for the photo-oxidation of DFcET by ZnYPPC" at the water-DCE interface (a). The opposite potential dependencies of the phenomenological ET rate constant and the porph5rin coverage (b) are responsible for the maximum on the flux of electron injection obtained from IMPS responses for DFcET and Fc (c). The potential dependence of the back electron-transfer rate constant is also shown in (d). (From Ref. 83. Reproduced by permission of The Royal Society of Chemistry.)... 

Fig. 5. Plot of apparent electron self exchange rate constants kf P, derived from polymer De values for films containing the indicated metals, mixed valent states, and ligands, all in acetonitrile, using Equation 2, vs. literature heterogeneous electron transfer rate constants k° for the corresponding monomers in nitrile solvents. See Ref. 6 for details. (Reproduced from Ref. 6. Copyright 1987 American Chemical Society.)...

The donor-acceptor complexes [Ir(/r-dmpz)(CO)(PPh2 0(CH2)2R )]2 exhibit photo-induced electron-transfer rate constants of 1012s—1 and charge recombination rates slower than 2 x 10los-1 when R = pyridine and 4-phenylpyridine.534 Further studies on these complexes revealed that recombination reactions were temperature dependent and slower for the deuterated acceptors.535... 

Thus, a plot of p c vs. loger gives (l from the slope and from the intercept. Using E° = -2.21 V, n = 1 and D = lO cm s-1, the authors calculated a transfer coefficient of 0.4 and a standard electron transfer rate constant of... 

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 4. Correlation of the ionization potentials of alkylmetal donors with the electron-transfer rate constant (log kFe) for Fe(phen)s3+ (%), Fe(bpy)s3+ (O), and Fe(Cl-phen)s3+ ((D), (left). The figure on the right is the same as the left figure for Fe(phen)s3+ except for the inclusion of electron-transfer rates for some alkyl radicals as identified, (Note the expanded scale,)...

One striking prediction of the energy gap law and eq. 11 and 14 is that in the inverted region, the electron transfer rate constant (kjjj. = ket) should decrease as the reaction becomes more favorable (lnknr -AE). Some evidence has been obtained for a fall-off in rate constants with increasing -AE (or -AG) for intermolecular reactions (21). Perhaps most notable is the pulse radiolysis data of Beitz and Miller (22). Nonetheless, the applicability of the energy gap law to intermolecular electron transfer in a detailed way has yet to be proven. 

It has been recognized that sulfur donors aid the stabilization of Cu(i) in aqueous solution (Patterson Holm, 1975). In a substantial study, the Cu(ii)/Cu(i) potentials and self-exchange electron transfer rate constants have been investigated for a number of copper complexes of cyclic poly-thioether ligands (Rorabacher et al., 1983). In all cases, these macrocycles produced the expected stabilization of the Cu(i) ion in aqueous solution. For a range of macrocyclic S4-donor complexes of type... 

In one series of experiments the cytochrome c oxidase mutations replaced acidic residues by neutral ones, and some of them were thus expected to alter the nature of binding of the protein to cytochrome c. From the pattern of dependence of the heme c to Cua electron-transfer rate constant on these mutations it was deduced that the binding of cytochrome c to cytochrome c oxidase is mediated by electrostatic interactions between four specific acidic residues on cytochrome c oxidase and lysines on cytochrome c. In another series of experiments, tryptophan 143 of cytochrome c oxidase was mutated to Phe or Ala. These mutations had an insignificant effect on the binding of the two proteins, but they dramatically reduced the rate constant for electron transfer from heme c to Cua- It was concluded that electron transfer from... 

For the cerium pz sandwich complex, only data for the analogous porphyrin sandwich is available. For the porphyrin, not only are one ring reduction and two ring oxidations observed, a metal-based Ce(IV)/Ce(III) reduction is observed, at a lower potential than the ring oxidation. The electron-transfer rate constant for the Ce(IV)/Ce(III) process was found to be very slow, 2.2 1.0 X 10 cm 1 because the/orbital of the cerium ion is buried between the two macrocycles making it difficult for the incoming electron to access it. For the Ce IVl pz, a similar Ce(IV/III) reduction is observed, at a lower potential than that observed for the porphyrin. 

A general difficulty encountered in kinetic studies of outer-sphere electron-transfer processes concerns the separation of the precursor formation constant (K) and the electron-transfer rate constant (kKT) in the reactions outlined above. In the majority of cases, precursor formation is a diffusion controlled step, followed by rate-determining electron transfer. In the presence of an excess of Red, the rate expression is given by... 

The method consists of plotting the forward electron transfer rate constant against the standard potential of a series of redox catalysts as illustrated by Figure 2.29. Three regions appear on the resulting Bronsted plot, which correspond to the following reaction scheme (Scheme 2.14). The... 

Aromatic anion radicals may be used as outer-sphere electron donors, thus giving rise to redox catalysis. The ensuing variations of the electron transfer rate constant with the driving force are shown in Figure 3.2b for... 

D0 and DR are the respective diffusion coefficients k° and a are known as the standard (electron transfer) rate constant and electron transfer coefficient respectively, and both are kinetic parameters characterizing the feasibility of the electron transfer x is the distance away from the electrode surface. 

From the intercepts and slopes Vg, k2 and Kg could, be calculated. For the analog DMBA the values of Vs and k2 were directly taken from the constant maximum rate as shown in Figure 5. Most of our results are gathered in Table I ( 1 5,2J, 2 2 ), which demonstrates that the observed increase in rate with a is governed by an increase of the electron transfer rate constant, k2, whereas the Michaelis constant, Kg, changed in the wrong way considering eq. (4). ... 

Figure 9. Compensation plot of activation parameters for the electron transfer rate constant k2 taken from Table I...

---