Electron transfer rate coefficient

Recently five monometallic (Au, Pd, Pt, Ru, Rh) nanoparticles were investigated as electron mediators together with four core/shell bimetallic (Au/Pd, Au/Pt, Au/Rh, Pt/ Ru) nanoparticles [53,194-196]. The linear relationship was observed between the electron transfer rate coefficients and the hydrogen generation rate coefficient as shown in Figure 15. 

Probably the most important outcome of the Marcus theory is the relationship between homogeneous homonuclear (fcs) and heterogeneous electron transfer rate coefficients (feei) after corrections for electrostatic work terms... 

Note that assumptions (2) and (3) are about timescales. Denoting by x, and tlz the characteristic times (inverse rates) of the electron transfer reaction, the solvent relaxation, and the Landau-Zener transition, respectively, (the latter is the duration of a single curve-crossing event) we are assuming that the inequalities Tr A Ts tlz hold. The validity of this assumption has to be addressed, but for now let us consider its consequences. When assumptions (1)—(3) are satisfied we can invoke the extended transition-state theory of Section 14.3.5 that leads to an expression for the electron transfer rate coefficient of the form (cf. Eq. 14.32)... 

In its simplest form the Marcus expression for the electron-transfer rate coefficient is given by [15-20] ... 

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... 

Examination of the peak potential locations and transfer coefficient values in a series of 16 cyclic and acyclic dibromides according to the procedures detailed in Chapter 3 points to a first dissociative electron transfer rate-determining step (Scheme 4.1). It is followed by another dissociative electron transfer step, leading directly to the olefin. Intrinsic barriers for the first, rate-determining step range from 0.6 to 0.8 eV, consisting mostly of the bond dissociation contribution (one-fourth of the bond dissociation energy). 

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. 

Fig. 9. Schematic representation of the variation of individual charge transfer rate coefficients, kf, and the observed rate coefficients (solid lines) with potential for a two-step electron transfer reaction [eqn. (122)] according to ref. 13.

Here, k° is the standard heterogeneous electron transfer rate constant and a is the electrochemical transfer coefficient [33], which corresponds in electrochemistry to the Bronsted coefficient in organic chemistry. It is seen from Equations 6.10 and 6.11 that kTsei and k°x are both equal to k° at E = E°. 

In this section, microdisc electrodes will be discussed since the disc is the most important geometry for microelectrodes (see Sect. 2.7). Note that discs are not uniformly accessible electrodes so the mass flux is not the same at different points of the electrode surface. For non-reversible processes, the applied potential controls the rate constant but not the surface concentrations, since these are defined by the local balance of electron transfer rates and mass transport rates at each point of the surface. This local balance is characteristic of a particular electrode geometry and will evolve along the voltammetric response. For this reason, it is difficult (if not impossible) to find analytical rigorous expressions for the current analogous to that presented above for spherical electrodes. To deal with this complex situation, different numerical or semi-analytical approaches have been followed [19-25]. The expression most employed for analyzing stationary responses at disc microelectrodes was derived by Oldham [20], and takes the following form when equal diffusion coefficients are assumed ... 

The higher the barrier, the larger the exponential coefficient p and the more dramatically the electron transfer rate decays with distance. By a fortunate coincidence of units, the P in " is approximated by the square root of the barrier height in eV. Thus for typical biological redox centers that must overcome a barrier of about 8eV to be ionized in a vacuum, we can estimate the P for exponential decay of electron transfer in vacuum to be about 2.8 Much less of a barrier is presented by a surrounding organic... 

Figure 36. The dependency of the homogeneous electron-transfer rate constant, A et on the potential difference, Erx - a- In the logarithmic plot, three asymptotes are noted, with slopes of 0, -1/118 mV, and -1/59 mV , representing diffusion-controlled, activation-controlled, and counter-diflfusion-controlled electron-transfer reactions, respectively [125]. The transfer coefficient...

Electrocatalytic reactions often involve several elementary steps some of which are not necessarily electrochemical. The transfer coefficient is, then, related to the symmetry factor of an electron transfer rate-determining step (rds) through (74)... 

The current-potential curves discussed so far can be used to measure concentrations, mass-transfer coefficients, and standard potentials. Under conditions where the electron-transfer rate at the interface is rate-determining, they can be employed to measure heterogeneous kinetic parameters as well (see Chapters 3 and 9). Often, however, one is interested in using electrochemical methods to find equilibrium constants and rate constants of homogeneous reactions that are coupled to the electron-transfer step. This section provides a brief introduction to these applications. 

Figure 18.2.6 Schematic representation of the variation of electron-transfer rate, and transfer coefficient, a, with electrode potential for an ideal semiconductor electrode. The current is is equivalent to that defined in (18.2.9) or (18.2.10). At sufficiently extreme potentials (not shown) mass transfer would lead to a limiting current on the right side of the diagram. [Reprinted with permission from B. R. Horrocks, M. V. Mirkin, and A. J. Bard, 7. Phys. Chem., 98, 9106 (1994). Copyright 1994, American Chemical Society.]...

A schematic representation of the ideal electron-transfer rate and transfer coefficient as functions of potential for a semiconductor electrode is shown in Figure 18.2.6. Although there have been numerous studies with semiconductor electrodes, such ideal behavior is rarely seen (45, 47, 49, 57-59). Difficulties in such measurements include the presence of processes in parallel with the electron-transfer reaction involving dissolved reactant at the semiconductor surface, such as corrosion of the semiconductor material, effects of the resistance of the electrode material, and charge-transfer reactions that occur via surface states. 

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