The Tafel Equation

For most values of overvoltage its value is given by the equation 

This equation is known as the Tafel equation. It can be expressed in many forms. One simple variation is to use natural logarithms instead of base 10, which is preferred since 

The constant A is higher for an electrochemical reaction that is slow. The constant t o is higher if the reaction is faster. The current density t o can be considered as the current density at which the overvoltage begins to move from zero. It is important to remember that the Tafel equation only holds true when i io. This current density io is usually called the exchange current density, as we shall see in Section 3.4.2. 

Electrochemistry deals principally with chemical and physical phenomena at interfaces between an electronic conductor (typically a metal or a semiconductor) and an ionic conductor such as electrolyte solution, as affected by the electric potential of the solid conductor. An electrochemical reaction involves a current going through the interface, hence the passing of either electrons or ions. But even if the current is carried through by ions, an electron-transfer step often occurs within the ions, which must be either generated or discharged by an exchange of electrons. 

The driving force for these heterogeneous reactions can be achieved by varying the electrode potential according to 

The rate of the electronic transfer is associated with the intensity, i, of the electric current that goes through the electrode, where i represents the number of electrons which react with the active species per unit of time 

Since electrode reactions are heterogeneous process, the rates are defined by mol sec per unit area. 

Here n is the number of electrons transferred in each elementary process and A the active surface of the electrode. 

---

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... 

In order to evaluate from the Tafel equation it must be expressed in terms of E. By definition... 

Initially, the curve conforms to the Tafel equation and curve AB which is referred to as the active region, corresponds with the reaction Fe- Fe (aq). At B there is a departure from linearity that b omes more pronounced ns the potential is increased, and at a potential C the current decreases to a very small value. The current density and potential at which the transition occurs are referred to as the critical current density, and the passivation potential Fpp, respectively. In this connection it should be noted that whereas is determined from the active to passive transition, the Flade potential Ef is determined from the passive to active transition... 

This is commonly known as the high field equation. It is of similar form to the Tafel equation for activation controlled electrochemical reactions with... 

As the corrosion rate, inclusive of local-cell corrosion, of a metal is related to electrode potential, usually by means of the Tafel equation and, of course, Faraday s second law of electrolysis, a necessary precursor to corrosion rate calculation is the assessment of electrode potential distribution on each metal in a system. In the absence of significant concentration variations in the electrolyte, a condition certainly satisfied in most practical sea-water systems, the exact prediction of electrode potential distribution at a given time involves the solution of the Laplace equation for the electrostatic potential (P) in the electrolyte at the position given by the three spatial coordinates (x, y, z). 

Referring to Fig. 11.5b, the initial rise in current corresponds to simple metal dissolution, expressed quantitatively through the Tafel equation relating potential and current logarithmically, and for multi-grained metals... 

Conway, B. E. The Temperature and Potential Dependence of Electrochemical Reaction Rates, and the Real Form of the Tafel Equation 16... 

It follows when Eqs. (6.5) and (6.12) are compared that the value of the empirical constant a in the Tafel equation is given by... 

Thus, in the region of very high anodic or cathodic polarization, the RDS is always the first step in the reaction path. The transfer coefficient of the full reaction which is equal to that of this step is always smaller than unity (for a one-electron RDS), while slope i in the Tafel equation is always larger than 0.06 V. When the potential is outside the region of low polarization, a section will appear in the polarization curve at intermediate values of anodic or cathodic polarization where the transfer coefficient is larger than unity and b is smaller than 0.06 V. This indicates that in this region the step that is second in the reaction path is rate determining. 

It can be seen from Fig. 15.2 that in semilogarithmic plots of AE vs. log/, the polarization characteristics are linear [i.e., obey the Tafel equation (6.3)]. Slopes b practically coincide for most metals and have values of 0.11 to 0.13 V. However, the absolute values of polarization recorded for a given current density (CD) vary within... 

Whereas is relatively easy to determine from the calculated binding energies, it is not easy to measure experimentally, since the measured potentials are always related to a specific current. Therefore, in order to compare directly with experiment, we have to calculate polarization curves, i.e., the current. The link between Gqrr and the current is the Tafel equation. 

Applying the Tafel equation with Uq, we obtain the polarization curves for Pt and PtsNi (Fig. 3.10). The experimental polarization curves fall off at the transport limiting current since the model only deals with the surface catalysis, this part of the polarization curve is not included in the theoretical curves. Looking at the low current limit, the model actually predicts the relative activity semiquantitatively. We call it semiquantitative since the absolute value for the prefactor on Pt is really a fitting parameter. 

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)... 

The anodic evolution of oxygen takes place at platinum and other noble metal electrodes at high overpotentials. The polarization curve obeys the Tafel equation in the potential range from 1.2 to 2.0 V with a b value between 0.10 and 0.13. Under these conditions, the rate-controlling process is probably the oxidation of hydroxide ions or water molecules on the surface of the electrode covered with surface oxide ... 

In addition to hydrocarbons, other products have also been found, especially in the reactions of the higher fatty acids. In steady state, the current density obeys the Tafel equation with a high value of constant b 0.5. At a constant potential the current usually does not depend very much on the sort of acid. The fact that the evolution of oxygen ceases in the... 

In Moscow Power Engineering Institute (TU) portable air aluminum batteries with saline electrolyte were developed [7, 18, and 20], In our devices, the air electrodes consist of two layers. Diffusion layer contains PTFE, carbon black and metal screen active layer consists of activated carbon and PTFE. At 293 K and the range of current density 2-25 mA/ cm2 dependence of cathode potential E (in H-scale) upon current density J (Figure 2) may by written by the Tafel equation (12). 

These reaction currents given by Eqns. 7-32 and 7-33 are formally in agreement with the Tafel equation of Eqn. 7-19 obtained by experimental observations. Note that the rate equations in Eqns. 7-32 and 7-33 apply to the forward reaction only and disregard the backward reaction rate. 

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 ... 

In the range of potential away from the equilibrium potential, the backward reaction current can be disregarded, and the anodic and the cathodic reaction currents are expressed, respectively, by the Tafel equations described in Eqn. 9-11 ... 

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) ... 

Activation Polarization It is customary to express the voltage drop due to activation polarization by a semi-empirical equation, called the Tafel equation (2). The equation for activation polarization is shown by Equation (2-38) ... 

The usual form of the Tafel equation that can be easily expressed by a Tafel Plot is... 

Care The Tafel equation is different to all the other equations we have discussed so far in this chapter, because in this case I is not a limiting current. 

The current is zero at equilibrium. Indeed, = 0 is one definition of equilibrium (see Chapter 2). As the potential is shifted away from V equilibrium so the electrode is polarized (cf Section 6.1). We recall that the deviation of the potential from its equilibrium value is termed the overpotential q (as defined by equation (6.1)). The portion of the Tafel graph at extreme overpotentials represents insufficient flux at the electrode in effect, the potential is so extreme that extra charge could flow if sufficient flux were available but, because of solvent viscosity, rate of solution stirring, etc., the flux is simply not large enough for the behaviour to follow the Tafel equation. 

---