Transfer coefficient potential dependence

One important question in the light of current electron transfer theories [85-87] is that of the transition between stepwise (electron transfer and bond cleavage as separate elementary steps) or concerted (dissociative electron transfer [88]) mechanisms. For the two extremes, one expects largely different activation parameters for the electron transfer at an electrode. In particular, in contrast to the simple Butler-Vohner relationship (Eq. 18) with a constant transfer coefficient, potential dependent a values become evident. The experimentally accessible apparent transfer coefficient... 

The above analysis shows that in the simple case of one adsorbed intermediate (according to Langmuirian adsorption), various complex plane plots may be obtained, depending on the relative values of the system parameters. These plots are described by various equivalent circuits, which are only the electrical representations of the interfacial phenomena. In fact, there are no real capacitances, inductances, or resistances in the circuit (faradaic process). These parameters originate from the behavior of the kinetic equations and are functions of the rate constants, transfer coefficients, potential, diffusion coefficients, concentrations, etc. In addition, all these parameters are highly nonlinear, that is, they depend on the electrode potential. It seems that the electrical representation of the faradaic impedance, however useful it may sound, is not necessary in the description of the system. The systen may be described in a simpler way directly by the equations describing impedances or admittances (see also Section IV). In... 

Figure 6.23. Effect of partial charge transfer coefficient XD on catalyst performance for fixed X.A depending on dimensionless potential n, (a) electrophobic, (b) electrophilic, (c) volcano-type, (d) inverted volcano-type.

Both the frequency of the well and its depth cancel, so that the free energy of activation is determined by the height of the maximum in the potential of mean force. The height of this maximum varies with the applied overpotential (see Fig. 13). To a first approximation this dependence is linear, and a Butler-Volmer type relation should hold over a limited range of potentials. Explicit model calculation gives transfer coefficients between zero and unity there is no reason why they should be close to 1/2. For large overpotentials the barrier disappears, and the rate will then be determined by ion transport. 

The constants characterizing the electrode reaction can be found from this type of polarization curve in the following manner. The quantity k"e is determined directly from the half-wave potential value (Eq. 5.4.27) if E0r is known and the mass transfer coefficient kQx is determined from the limiting current density (Eq. 5.4.20). The charge transfer coefficient oc is determined from the slope of the dependence of In [(yd —/)//] on E. 

A closer scrutiny of Figure 6 reveals the persistence of small, but consistent curvature in all of the plots. In order to verify the curvature, the transfer coefficient 8 was also determined independently from the width of the CV wave, as described by Nicholson and Shain (10). The potential dependence of 8 obtained in this manner is shown in Figure 7. The slopes 88/3E represent the unmistakable presence of curvature in Figure 6. 

Figure 7. Dependence of the heterogeneous transfer coefficient (5 with the applied electrode potential for the representative class of organometals I-IV.

The transfer coefficient a has a dual role (1) It determines the dependence of the current on the electrode potential. (2) It gives the variation of the Gibbs energy of activation with potential, and hence affects the temperature dependence of the current. If an experimental value for a is obtained from current-potential curves, its value should be independent of temperature. A small temperature dependence may arise from quantum effects (not treated here), but a strong dependence is not compatible with an outer-sphere mechanism. 

This potential-energy surface will change when the electrode potential is varied consequently the energy of activation will change, too. These changes will depend on the structure of the double layer, so we cannot predict the value of the transfer coefficient a unless we have a detailed model for the distribution of the potential in the double layer. There is, however, no particular reason why a should be close to 1/2. Also, a temperature dependence of the transfer coefficient is not surprising since the structure of the double layer changes with temperature. 

Hydrogen evolution, the other reaction studied, is a classical reaction for electrochemical kinetic studies. It was this reaction that led Tafel (24) to formulate his semi-logarithmic relation between potential and current which is named for him and that later resulted in the derivation of the equation that today is called "Butler-Volmer-equation" (25,26). The influence of the electrode potential is considered to modify the activation barrier for the charge transfer step of the reaction at the interface. This results in an exponential dependence of the reaction rate on the electrode potential, the extent of which is given by the transfer coefficient, a. 

This result is quite in contrast to the common expectation that the electrode potential changes the activation barrier at the interface which would result in a temperature independent transfer coefficient a. Following Agar s discussion (30), such a behavior indicates a potential dependence of the entropy of activation rather than the enthalpy of activation. Such "anomalous" behavior in which the transfer coefficient depends on the temperature seems to be rather common as recently reviewed by Conway (31). 

FIGURE 2.6. EC reaction scheme in cyclic voltammetry. Mixed kinetic control by an electron transfer obeying a MHL kinetic law (Xt — 0.7 eV, koo — 4 x 103 cms-1, implying that kg = 0.69 cms-1) and an irreversible follow-up reaction (from bottom to top, k+ = 103, 105, 107, 109s 1). Temperature, 25°C. a Potential-dependent rate constant derived from convolutive manipulation of the cyclic voltammetric data (see the text), b Variation with potential of the apparent transfer coefficient (see the text) obtained from differentiation of the curves in part a. 

As anticipated, for an irreversible process the forward peak is located at potentials more negative than if it were reversible (i.e. compared to its standard thermodynamic potential). Moreover, since the above relationship shows that the peak current depends on the square root of the transfer coefficient a, under equivalent conditions the height of the irreversible peak equals 78.6% of the reversible peak given that, as often happens, ot = 0.5... 

In addition, the split peaks can be used for estimation of electron-transfer coefficient as well as for precise determination of the formal potential of the surface electrode reaction. The potential separation between split peaks is insensitive to the electron-transfer coefficient. However, the relative ratio of the heights of the split peaks depends on the electron-transfer coefficient according to the following function ... 

The physical meaning of the kinetic parameter m is identical as for surface electrode reaction (Chap. 2.5.1). The electrochemical reversibility is primarily controlled by 03 (Fig. 2.71). The reaction is totally irreversible for log(m) < —3 and electrochemically reversible for log(fo) > 1. Between these intervals, the reaction appears quasireversible, attributed with a quasireversible maximum. Though the absolute net peak current value depends on the adsorption parameter. Fig. 2.71 reveals that the quasireversible interval, together with the position of the maximum, is independent of the adsorption strength. Similar to the surface reactions, the position of the maximum varies with the electron transfer coefficient and the amphtude of the potential modrrlation [92]. 

Similar to the surface electrode processes (Chap. 2.5.1) the peak current ratio of the split peaks ( fp,c/ lp,a) is a function of the electron transfer coefficient o. Note that the anodic and the cathodic peak is located at the more negative and more positive potentials, respectively. This type of dependence is given in Fig. 2.98. Over the interval 0.3 < < 0.7 the dependence vs. is hnear, associated with the... 

The potential-dependent part of the activation energy AGp can be estimated by introduction of the transfer coefficient a, which was introduced in 1930 by Erdey-Gruz and Volmer (3). The potential-dependent contribution AgJ to the free energy of activation... 

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