Frequencies electron transfer

The expressions for ket in equations (29) and (30) are the products of two factors (1) an exponential term which, assuming a common force constant, gives the fractional population of reactants at temperature T with vibrational distributions at the intersection regions for each of the trapping vibrations, and (2) a pre-exponential term ve, the electron transfer frequency ve is defined in equation (31). 

Reversibility is not a problem in the limit that ve < v n. If the electron transfer frequency is slow relative to the rate of vibrational redistribution, the vibrational distribution at the intersection region will change before back electron transfer can occur. 

Estimate the electron transfer frequency for a reaction occurring in water at 25°C for which is 314kJmol assuming that is 0.1 kJmol Repeat the calculation for the case that is 5kJmol Compare these estimates with the value obtained for an adiabatic electron transfer. 

Since ya is very small, the electron transfer frequency Vet is equal to the diabatic frequency, that is, to 1.0 x 10 s-. ... 

When is 5000 JmoP, the corresponding value of Vj is 2.50 x 10 s. The adiabacity parameter also increases to 6.3. According to equation (7.8.40) the electron transfer frequency is now... 

The discussion thus far in this chapter has been centred on classical mechanics. However, in many systems, an explicit quantum treatment is required (not to mention the fact that it is the correct law of physics). This statement is particularly true for proton and electron transfer reactions in chemistry, as well as for reactions involving high-frequency vibrations. 

Much of tills chapter concerns ET reactions in solution. However, gas phase ET processes are well known too. See figure C3.2.1. The Tiarjioon mechanism by which halogens oxidize alkali metals is fundamentally an electron transfer reaction [2]. One might guess, from tliis simple reaction, some of tlie stmctural parameters tliat control ET rates relative electron affinities of reactants, reactant separation distance, bond lengtli changes upon oxidation/reduction, vibrational frequencies, etc. 

Early studies showed tliat tire rates of ET are limited by solvation rates for certain barrierless electron transfer reactions. However, more recent studies showed tliat electron-transfer rates can far exceed tire rates of diffusional solvation, which indicate critical roles for intramolecular (high frequency) vibrational mode couplings and inertial solvation. The interiDlay between inter- and intramolecular degrees of freedom is particularly significant in tire Marcus inverted regime [45] (figure C3.2.12)). 

Thus far we have discussed the direct mechanism of dissipation, when the reaction coordinate is coupled directly to the continuous spectrum of the bath degrees of freedom. For chemical reactions this situation is rather rare, since low-frequency acoustic phonon modes have much larger wavelengths than the size of the reaction complex, and so they cannot cause a considerable relative displacement of the reactants. The direct mechanism may play an essential role in long-distance electron transfer in dielectric media, when the reorganization energy is created by displacement of equilibrium positions of low-frequency polarization phonons. Another cause of friction may be anharmonicity of solids which leads to multiphonon processes. In particular, the Raman processes may provide small energy losses. 

While being very similar in the general description, the RLT and electron-transfer processes differ in the vibration types they involve. In the first case, those are the high-frequency intramolecular modes, while in the second case the major role is played by the continuous spectrum of polarization phonons in condensed 3D media [Dogonadze and Kuznetsov 1975]. The localization effects mentioned in the previous section, connected with the low-frequency part of the phonon spectrum, still do not show up in electron-transfer reactions because of the asymmetry of the potential. 

Nuclear, electronic and frequency factors in electron transfer reactions. N. Sutin, Acc. Chem. Res., 1982,15, 275-282 (78). 

From an analysis of data for polypyrrole, Albery and Mount concluded that the high-frequency semicircle was indeed due to the electron-transfer resistance.203 We have confirmed this using a polystyrene sulfonate-doped polypyrrole with known ion and electron-transport resistances.145 The charge-transfer resistance was found to decrease exponentially with increasing potential, in parallel with the decreasing electronic resistance. The slope of 60 mV/decade indicates a Nemstian response at low doping levels. 

The present approach has been applied to the experiment done by Nelsen et ah, [112], which is a measurement of the intramolecular electron transfer of 2,7-dinitronaphthalene in three kinds of solvents. Since the solvent dynamics effect is supposed to be unimportant in these cases, we can use the present theory within the effective ID model approach. The basic parameters are taken from the above reference except for the effective frequency. The results are shown in Fig. 26, which shows an excellent agreement with the experiment. The electronic couphng is quite strong and the perturbative treatment cannot work. The effective frequencies used are 1200, 950, and 800 cm for CH3CN, dimethylformamide (DMF), and PrCN [113]. 

For simple outer-sphere electron transfer reactions, the effective frequency co is determined by the properties of the slow polarization of the medium. For a liquid like water, where the temporal relaxation of the slow polarization as a response to the external field is single exponential, tfie effective frequency is equal to... 

Schmickler W. 1996. Interfacial Electrochemistry. New York Oxford University Press. Schmickler W, Koper MTM. 1999. Adiabahc electrochemical electron-transfer reactions involving frequency changes of iimer-sphete modes. Electrochem Comm 1 402-405. Schmickler W, Mohr J. 2002. The rate of electrochemical electron-transfer reachons. J Chem Phys 117 2867-2872. 

Depending on pH, increasing the acidity of the solution either makes the potential required to yield a fixed turnover frequency more oxidizing by 60 mV/pH or does not affect it. This pH dependence is in most cases the same as that of the Fe /n (.Q jpjg jjj jjjg absence of a substrate. These identical pH dependences suggest a pre-equilibrium between the ferric and ferrous forms of the catalyst followed, by a kinetically irreversible step that does not involve proton or electron transfer (e.g., O2 binding). 

Only three steps of the proposed mechanism (Fig. 18.20) could not be carried out individually under stoichiometric conditions. At pH 7 and the potential-dependent part of the catalytic wave (>150 mV vs. NHE), the —30 mV/pH dependence of the turnover frequency was observed for both Ee/Cu and Cu-free (Fe-only) forms of catalysts 2, and therefore it requires two reversible electron transfer steps prior to the turnover-determining step (TDS) and one proton transfer step either prior to the TDS or as the TDS. Under these conditions, the resting state of the catalyst was determined to be ferric-aqua/Cu which was in a rapid equilibrium with the fully reduced ferrous-aqua/Cu form (the Fe - and potentials were measured to be within < 20 mV of each other, as they are in cytochrome c oxidase, resulting in a two-electron redox equilibrium). This first redox equilibrium is biased toward the catalytically inactive fully oxidized state at potentials >0.1 V, and therefore it controls the molar fraction of the catalytically active metalloporphyrin. The fully reduced ferrous-aqua/Cu form is also in a rapid equilibrium with the catalytically active 5-coordinate ferrous porphyrin. As a result of these two equilibria, at 150 mV (vs. NHE), only <0.1%... 

The discrepancy may also be caused by the approximations in the calculation of the EEDF. This EEDF is obtained by solving the two-term Boltzmann equation, assuming full relaxation during one RF period. When the RF frequency becomes comparable to the energy loss frequencies of the electrons, it is not correct to use the time-independent Boltzmann equation to calculate the EEDF [253]. The saturation of the growth rate in the model is not caused by the fact that the RF frequency approaches the momentum transfer frequency Ume [254]. That would lead to less effective power dissipation by the electrons at higher RF frequencies and thus to a smaller deposition rate at high frequencies than at lower frequencies. 

A more quantitative description of the photocurrent responses, taking into account the contributions from back electron transfer and RuQi attenuation, was achieved by IMPS measurements [83]. Considering the mechanism in Fig. 11, excluding the supersensitization step, and the equivalent circuit in Fig. 18, the frequency-dependent photocurrent for a perturbation as in Eq. (42) is given by... 

A complex representation of IMPS data obtained for the heterogeneous quenching of ZnTPPC -diferrocenylethane is displayed in Fig. 21(a). The semicircular response in the first quadrant corresponds to the competition between product separation and back electron transfer, while the lower quadrant is determined by the i uQi constant. The i uQi attenuation limited the frequency range to less than 1 kHz. Equation (45) describes the experimental spectra at various Galvani potential differences [solid lines in Fig. 21(a)],... 

It is clear that one of the major limitations of this analysis is the assumption of constant excited-state coverage. Deviations from the behavior described by Eq. (45) in the low frequency range have been observed at photocurrent densities higher than 10 Acm [50]. These deviations are expected to be connected to excited-state diffusion profiles similar to those considered by Dryfe et al. [see Eq. (38)] [127]. A more general expression for IMPS responses is undoubtedly required for a better understanding of the dynamics involved in back electron transfer as well as separation of the photoproducts. 

As described in the introduction, certain cosurfactants appear able to drive percolation transitions. Variations in the cosurfactant chemical potential, RT n (where is cosurfactant concentration or activity), holding other compositional features constant, provide the driving force for these percolation transitions. A water, toluene, and AOT microemulsion system using acrylamide as cosurfactant exhibited percolation type behavior for a variety of redox electron-transfer processes. The corresponding low-frequency electrical conductivity data for such a system is illustrated in Fig. 8, where the water, toluene, and AOT mole ratio (11.2 19.2 1.00) is held approximately constant, and the acrylamide concentration, is varied from 0 to 6% (w/w). At about = 1.2%, the arrow labeled in Fig. 8 indicates the onset of percolation in electrical conductivity. 

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