Tafel equation, electrode reactions

While the form of the Tafel equation with regard to the potential dependence of i is of major general interest and has been discussed previously both in terms of the role of linear and quadratic terms in 77 " and the dependence of the form of the Tafel equation on reaction mechanisms/ the temperature dependence of Tafel slopes for various processes is of equal, if not greater general, significance, as this is a critical matter for the whole basis of ideas of activation and reorganization processes " in the kinetics of electrode reactions. 

The measurement of a from the experimental slope of the Tafel equation may help to decide between rate-determining steps in an electrode process. Thus in the reduction water to evolve H2 gas, if the slow step is the reaction of with the metal M to form surface hydrogen atoms, M—H, a is expected to be about If, on the other hand, the slow step is the surface combination of two hydrogen atoms to form H2, a second-order process, then a should be 2 (see Ref. 150). 

Overvoltage. Overvoltage (ti. ) arises from kinetic limitations or from the inherent rate (be it slow or fast) of the electrode reaction on a given substrate. The magnitude of this value can be generally expressed in the form of the Tafel equation... 

Fig. 5. Currrent—potential behavior of an electrode reaction based on equation 37. Tafel behavior is noted at high currents for two different values of d.

Equation (6.13), in fact, reflects the physical nature of the electrode process, consisting of the anode (the first term) and cathode (the second term) reactions. At equilibrium potential, E = Eq, the rates of both reactions are equal and the net current is zero, although both anode and cathode currents are nonzero and are equal to the exchange current f. With the variation of the electrode potential, the rate of one of these reactions increases, whereas that of the other decreases. At sufficiently large electrode polarization (i.e., deviation of the electrode potential from Eg), one of these processes dominates (depending on the sign of E - Eg) and the dependence of the net current on the potential is approximately exponential (Tafel equation). 

Plotting the overpotential against the decadic logarithm of the absolute value of the current density yields the Tafel plot (see Fig. 5.3). Both branches of the resultant curve approach the asymptotes for r RT/F. When this condition is fulfilled, either the first or second exponential term on the right-hand side of Eq. (5.2.28) can be neglected. The electrode reaction then becomes irreversible (cf. page 257) and the polarization curve is given by the Tafel equation... 

The above-described theory, which has been extended for the transfer of protons from an oxonium ion to the electrode (see page 353) and some more complicated reactions was applied in only a limited number of cases to interpretation of the experimental data nonetheless, it still represents a basic contribution to the understanding of electrode reactions. More frequently, the empirical values n, k° and a (Eq. 5.2.24) are the final result of the investigation, and still more often only fcconv and cm (cf. Eq. 5.2.49) or the corresponding constant of the Tafel equation (5.2.32) and the reaction order of the electrode reaction with respect to the electroactive substance (Eq. 5.2.4) are determined. 

This is the Tafel equation (5.2.32) or (5.2.36) for the rate of an irreversible electrode reaction in the absence of transport processes. Clearly, transport to and from the electrode has no effect on the rate of the overall process and on the current density. Under these conditions, the current density is termed the kinetic current density as it is controlled by the kinetics of the electrode process alone. 

Table 5.5 Constants a and b of the Tafel equation and the probable mechanism of the hydrogen evolution reaction at various electrodes with H30+ as electroactive species (aH3o+ ) (According to L. I. Krishtalik)... 

In this notation, anodic current is positive, while cathodic current is negative. As the later section on oxygen reduction will show, the Tafel slope can change with overpotential. This is because the Butler-Volmer law only applies to outer-sphere reactions. Although it can describe electrode reactions, the equation does not account for repulsive interactions of the adsorbates or changes in the reaction mechanism as potential is changed. 

The transfer currents of redox electrons and redox holes represented by Eqns. 8-63 and 8-64 are formally in agreement with the Tafel equation given by Eqn. 7-32. However, the Tafel constant (the transfer coefficient) a equals one or zero at semiconductor electrodes in contrast with metal electrodes at which a is close to 0.5. From Eqns. 8-64 and 8-65 for reaction currents, the Tafel constants is obtained as defined in Eqns. 8-66 and 8-67 ... 

Activation Polarization Activation polarization is present when the rate of an electrochemical reaction at an electrode surface is controlled by sluggish electrode kinetics. In other words, activation polarization is directly related to the rates of electrochemical reactions. There is a close similarity between electrochemical and chemical reactions in that both involve an activation barrier that must be overcome by the reacting species. In the case of an electrochemical reaction with riact> 50-100 mV, rjact is described by the general form of the Tafel equation (see Section 2.2.4) ... 

The Tafel equation implies that the overvoltage is a measure of the thermodynamic irreversibility of the electrode reaction, and it is associated with the slow step of the process. We distinguish some types of overvoltage depending on the type of slow reaction. 

Equations 1.44 and 1.45 are called Tafel equations. Figure 1.10 shows the Tafel plots for both forward and backward electrode reactions. 

The current-voltage curve of the decomposition reaction of the medium can be described by the well-known Tafel equation, E = a + b log I, where a and b are constants and I is the current density (amps/cm2) a is dependent on the electrode material, and b is determined by the mechanism of the electrode reaction. 

In many cases, the electrode reactions are not reversible and the decomposition potential is observed to be in excess of the thermodynamically calculated value. The excess voltage, referred to as an overvoltage, is found to vary with the nature and surface area (e.g., roughness) of the electrodes, impurities in the solution, and the actual current density passing through the solution. The relation between current density Id and overvoltage E was investigated by Tafel, who proposed the very successful empirical equation... 

The most reliable data are from studies of hydrogen evolution on mercury cathodes in acid solutions. This reaction has been studied most extensively over the years. The use of a renewable surface (a dropping mercury electrode, in which a new surface is formed every few seconds), our ability to purify the electrode by distillation, the long range of overpotentials over which the Tafel equation is applicable and the relatively simple mechanism of the reaction in this system all combine to give high credence to the conclusion that p = 0.5. This value has been used in almost all mechanistic studies in electrode kinetics and has led to consistent interpretations of the experimental behavior. It... 

The impedance behavior of electrode reactions is often complex but can be conveniently simulated by computer calculations, especially in the case of the method based on kinetic equations (108, 113). The forms of the frequency response represented in terms of the Z versus Z" complex-plane plots and by relations of Z or phase angle to frequency ai or log (o (Bode plots) are often characteristic of the reaction mechanism and involvement of one or more adsorbed intermediates, and they thus provide diagnostic bases for mechanism determination complementary to those based on dc, steady-state rate versus potential responses. The variations of Z versus Z" plots with dc -level potential, in controlled-potential experiments, also give rise to useful diagnostic information related to the dc Tafel behavior. 

We have remarked earlier that the treatment given above is based on an assumption for the case of that is, they are in an effective parallel combination. This is not strictly correct for a number of conditions, so the logarithmic potential-decay slopes in relation to Tafel slopes must be worked out from the full kinetic equations of Harrington and Conway (104) referred to earlier, based on the relevant mechanism of the electrode reaction. Numerical solution procedures, using computer simulation calculations, are then usually necessary for comparison with observed experimental behavior. 

The local reaction rate in each of the electrodes is described by a variant of the Tafel equation with effective parameters, that are, in first approximation, considered to be constant over each catalyst layer. 

The kinetics of this reaction on stainless steel cathode in a wide range of current densities up to very high i can be described by the Tafel equation Ec = —1.25 — 0.14 log i [27], where Ec (in volts) is referred to a standard hydrogen electrode (SHE), i is expressed in amperes per square centimeter. 

This is the so-called Tafel equation (Tafel was the first person to find empirically a linear relation between log / and rj. Meanwhile many redox reactions at metal electrodes... 

Contrary to common belief, the Tafel equation as represented by Eqs. (1) or (4), with b given by Eq. (5), virtually never represents the electrode-kinetic behavior of electrochemical processes (except probably simple ionic redox reactions that have minimal chemical coupling of one kind or another, p. 125) in particular with reference... 

A final and very important general phenomenological conclusion is that the conventional form of the Tafel equation with slope b = RT/aF with a constant is rarely observed at least for those cases where adequate and reliable 7-dependence studies of the electrode kinetics have been made (cf. Yeager ). Simple ionic redox reactions seem, however, to be an exception. ... 

Chapter 2, by B. E. Conway, deals with a curious fundamental but hitherto little-examined problem in electrode kinetics the real form of the Tafel equation with regard to the temperature dependence of the Tafel-slope parameter 6, conventionally written as fe = RT/ aF where a is a transfer coefficient. He shows, extending his 1970 paper and earlier works of others, that this form of the relation for b rarely represents the experimental behavior for a variety of reactions over any appreciable temperature range. Rather, b is of the form RT/(aH + ctsT)F or RT/a F + X, where and as are enthalpy and entropy components of the transfer coefficient (or symmetry factor for a one-step electron transfer reaction), and X is a temperature-independent parameter, the apparent limiting... 

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