ET Rate Constants

What Eqs. (10-70) and (10-71) state is this One can predict the ET rate constant, given the other parameters. Or, a missing EE rate constant can be obtained by determining if the other factors are known. We shall return to this point shortly, after considering how to obtain these equations. The original derivation from statistical mechanics is outside the scope of this book, so we shall consider two others. 

The first controversial point in this mechanism is the nature of the reaction planes where the precursor formation and the ET reaction take place. Samec assumed that the ET step occurs across an ion-free layer composed of oriented solvent molecules [1]. By contrast, Girault and Schiffrin considered a mixed solvent region where electrochemical potentials are dependent on the position of the reactants at the interface [60]. From a general perspective, the phenomenological ET rate constant can be expressed in terms of... 

In subsequent works, Marcus developed his theory further in a series of papers providing expressions for the work terms, the reorganization energy and the macroscopic ET rate constants [3 6]. Assuming a sharp liquid-liquid boundary, the solution of the mean molar volume of reactants yields an expression for of the form... 

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. 

Analysis of Zje and Z- as a function of the frequency of potential modulation (Randles plots) provides the phenomenological ET rate constant [63,74]. It should be noted that the extrapolation of Z at high frequency gives effectively the sum 7 ct + where 7 ct is the charge transfer resistance. 

The non-steady-state optical analysis introduced by Ding et al. also featured deviations from the Butler-Volmer behavior under identical conditions [43]. In this case, the large potential range accessible with these techniques allows measurements of the rate constant in the vicinity of the potential of zero charge (k j). The potential dependence of the ET rate constant normalized by as obtained from the optical analysis of the TCNQ reduction by ferrocyanide is displayed in Fig. 10(a) [43]. This dependence was analyzed in terms of the preencounter equilibrium model associated with a mixed-solvent layer type of interfacial structure [see Eqs. (14) and (16)]. The experimental results were compared to the theoretical curve obtained from Eq. (14) assuming that the potential drop between the reaction planes (A 0) is zero. The potential drop in the aqueous side was estimated by the Gouy-Chapman model. The theoretical curve underestimates the experimental trend, and the difference can be associated with the third term in Eq. (14). 

The first estimations of for photoinduced processes were reported by Dvorak et al. for the photoreaction in Eq. (40) [157,158]. In this work, the authors proposed that the impedance under illumination could be estimated from the ratio between the AC photopotential under chopped illumination and the AC photocurrent responses. Subsequently, the faradaic impedance was calculated following a treatment similar to that described in Eqs. (22) to (26), i.e., subtracting the impedance under illumination and in the dark. From this analysis, a pseudo-first-order photoinduced ET rate constant of the order of 10 to 10 ms was estimated, corresponding to a rather unrealistic ket > 10 M cms . Considering the nonactivated limit for adiabatic outer sphere heterogeneous ET at liquid-liquid interfaces given by Eq. (17) [5], the maximum bimolecular rate constant is approximately 1000 smaller than the values reported by these authors. 

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.)... 

Since the first use of SECM to study ET kinetics at a liquid-liquid interface in 1995 [47], the methodology has been proven a powerful approach for investigating the dependence of ET rate constants on the Galvani potential drop across an ITIES. 

With a fixed concentration of redox species in phase 2, the dependence of the ET rate constant, kf, on the driving force can be written as [48,70] ... 

Similar experiments were performed to measure the ET rate constants for the reaction between ZnPor in benzonitrile and aqueous reductants, Ru(CN)g, Mo(CN)g and FeEDTA (where EDTA denotes ethylenediaminetetra-acetic acid). Although the... 

The ET reaction between aqueous oxidants and decamethylferrocene (DMFc), in both DCE and NB, has been studied over a wide range of conditions and shown to be a complex process [86]. The apparent potential-dependence of the ET rate constant was contrary to Butler-Volmer theory, when the interfacial potential drop at the ITIES was adjusted via the CIO4 concentration in the aqueous phase. The highest reaction rate was observed with the smallest concentration of CIO4 in the aqueous phase, which corresponded to the lowest driving force for the oxidation process. In contrast, the ET rate increased with driving force when this was adjusted via the redox potential of the aqueous oxidant. Moreover, a Butler-Volmer trend was found when TBA was used as the potential-determining ion, with an a value of 0.38 [86]. 

A feature of these studies was the further demonstration that higher ET rate constants can be measured more readily by using a lower relative concentration of the redox reactant in the second phase. For example, Fig. 23 shows approach curves for the oxidation of DMFc by tip-generated IrClg which is characterized by a rate constant of 180cms M , with 0.1 M CIO4 in each phase, and readily distinguished from a diffu-... 

In this chapter we have shown that the dynamics and spectroscopy of the initial events taking place in bacterial photosynthetic RCs can be described by the model shown in Table I and Fig. 19. Using these physical constants we can calculate the absorption spectra, ET rate constants, and fs time-resolved spectra. It should be noted that for processes taking place in sub-ps range, it is more reasonable not to use rate constant because the concept of rate constant requires the validity of the Markoff approximation [82,88]. Instead the... 

All these theories provide the basis for using, as first approximation, the simple phenomenological equations to describe the ET rate constant k in D-B-A systems as k = k0 z d and the current flow 7 as 7 = / e /ld in molecular junctions, where d is the length of the molecule, and [1 is a decay factor. Although the decay... 

It is generally assumed that Vd, and the associated ET rate constant, ket, fall off approximately exponentially with increasing interchromophore separation, r, viz. 

In conclusion, the free energy change of an ET step is already a good indicator of the feasibility of the reaction. A highly endergonic reaction, with, say, AG° > 20 kcal mol-1, corresponds to a rather slow ET reaction that is not likely to compete with other reactions of polar nature. In the region where AG° lies between 20 and -A kcal mol-1, we need to apply the Marcus approach in order to get an approximate value of the ET rate constant, whereas at AG° < - A kcal mol-1 most intermolecular ET reactions appear to be diffusion controlled. 

The effect of anion binding on k is not nearly as great. The data in Fig. 6 show that for both metals, there is less than a 50% decrease in the thermal ET rate constant between hybrids containing neutral and anionic ligands. 

This scheme includes ET rate constants only for the d - d electron-transfer processes, in which the system conformation is conserved, and conformational and ET steps only occur sequentially. Intuitively, it might be expected that the kinetic scheme must include ET that is synchronous with a conformational change in the medium coordinate. However, we showed [10a] that it is not necessary to include the diagonal processes (e.g., A Ig) when considering stable substates. 

Our early reports of the I A reaction in [ZnCcP, Cc] complexes were interpreted within the context of the simple ET cycle. Scheme I [8b, c]. As noted above, there are two limiting cases for this scheme, i) The I A reaction is slow I(t) appears with the triplet decay rate constant kp and disappears with the thermal ET rate constant k. ii) It is rapid here I(t) appears with k and disappears with kp. We reported that ZnCcP complexes with heterologous Cc s [8d] and with aliphatic Phe-82 mutants of yeast Cc [8b] fell into the former class, while the intermediate for fungal Cc appears rapidly [8d]. 

Figure 3 Driving force (-AG°et) dependence of intramolecular ET rate constants in ZnP-Cso (CS white circles CR white squares), Fc-ZnP-Ceo (black circles), FC-H2P-C60 (black triangles), ZnP-H2P-Ceo (black squares), and Fc-ZnP-H2P-C6o (white triangles). The lines represent the best fit to the Marcus relation, [Eq. (1)] (see text). (From Ref. 47.)...

A simple diagram depicting the differences between these two complementary theories is shown in Fig. 1, which represents reactions at zero driving force. Thus, the activation energy corresponds to the intrinsic barrier. Marcus theory assumes a harmonic potential for reactants and products and, in its simplest form, assumes that the reactant and product surfaces have the same curvature (Fig. la). In his derivation of the dissociative ET theory, Saveant assumed that the reactants should be described by a Morse potential and that the products should simply be the dissociative part of this potential (Fig. Ib). Some concerns about the latter condition have been raised. " On the other hand, comparison of experimental data pertaining to alkyl halides and peroxides (Section 3) with equations (7) and (8) seems to indicate that the simple model proposed by Saveant for the nuclear factor of the ET rate constant expression satisfactorily describes concerted dissociative reductions in the condensed phase. A similar treatment was used by Wentworth and coworkers to describe dissociative electron attachment to aromatic and alkyl halides in the gas phase. ... 

The final step of the convolution analysis is the determination of the transfer coefficient a. This coefficient, sometimes called the symmetry factor, describes how variations in the reaction free energy affect the activation free energy (equation 26). The value of a does not depend on whether the reaction is a heterogeneous or a homogeneous ET (or even a different type of reaction such as a proton transfer, where a is better known as the Bronsted coefficient). Since the ET rate constant may be described by equation (4), the experimental determination of a is carried out by derivatization of the ln/Chet-AG° and thus of the experimental Inkhei- plots (AG° = F E — E°)) (equation 27). 

ET rate constants can also be determined using solution donors. Indirect reduction of peroxides and endoperoxides was accomplished by homogeneous... 

Fig. 15 Hammett plot of the logarithm of the heterogeneous ET rate constant for the electroreduction of diaryl disulfides in DMF. The solid line illustrates the experimental trend.

Electron transfer (ET) reactions play a key role in both natural (photosynthesis, metabolism) and industrial processes (photography, polymerisation, solar cells). The study of intermolecular photoinduced ET reactions in solution is complicated by diffusion. In fact, as soon as the latter is slower than the ET process, it is not anymore possible to measure km, the intrinsic ET rate constant, directly [1], One way to circumvent this problem, it is to work in a reacting solvent [2]. However, in this case, the relationship between the observed quenching rate constant and k T is not clear. Indeed, it has been suggested that several solvent molecules could act as efficient donors [3]. In this situation, the measured rate constant is the sum of the individual ksr-... 

To interpret our data, we have developed a simple model which takes into account the orientation of the n donor molecules in the first solvent shell with respect to the acceptor. The donor molecules can be sorted into three types. Donors that are in an optimal position to quench the acceptor molecules (Da), donors that have first to rotate (Z) ) and those which need translational motion (Dc). Hence, the rate constant, kq, for the fluorescence decay of an acceptor with a given solvent configuration is the sum of the individual ET rate constants with the donors of the first solvent shell... 

To test this simple model, we fitted the experimental data by means of Eq. 2 (for convenience, instead of the raw data, we used the best multiexponential fits), n, the number of donor molecules in the first solvent shell, k/t, and kj, could be estimated. The other parameters, Pa, Pb, Pc (the probability of a donor molecule of being Da, Db or Dc respectively) and ksr, the intrinsic ET rate constant, were obtained from the fit. 

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