Tafel relation

The polarization curve (polarization current i, versus polarization potential E) of a corroding metallic electrode can be measured by polarizing the electrode in the anodic and cathodic directions. In the range of electrode potential a short distance away from the corrosion potential, the polarization curve follows the Tafel relation as shown in Fig. 11-6. Here, the polarization current, ip, in the anodic direction equals the dissolution current of the metal i and the polarization current, ip, in the cathodic direction equals the reduction current of the oxidant i. In the range of potential near the corrosion potential, however, the polarization current, ip, is the difference between the anodic dissolution current of the metal... 

Fig. 7.19. The existence of the region of diffuse charge in the electrified interface has an effect on the Tafel relation, making it change the slope and deviate from lineality.

The great importance of the Tafel relation—because it is too widely observed to be applicable in electrode kinetics—does not seem to have been appreciated during the time (about 1960-1980) in which Gaussian concepts were frequently used to present a quantal approach to electrode kinetics. Supporting a theoretical view that does not yield what is in effect the first law of electrode kinetics is similar to supporting a theory of gas reactions that does not lead to the exponential dependence of rate on temperature. It represents a remarkable historical aberration in the field. Thus the... 

Fig. 9.25. (e) The hydrogen evolution reaction overthe Tafel relation which is linear over eleven orders of magnitude (experimental points). Curved line Marcus expression with assumption of harmonic oscillators. (Reprinted from J. O M. Bockris and S. U. M. Khan, Quantum Electrochemistry, Plenum 1979, p. 228.)... 

The corrosion current density icoa is evaluated by the electrochemical polarization resistance method assuming that both the anodic and the cathodic partial currents obey the Tafel relation ... 

It is often claimed that electrocatalysts in fuel cells are dependent on the exchange current density, i0 of the slowest reaction in the cell, (a) Make Tafel plots for i0 = 10 I,10 6, and 10-3 A cm-2 and bTaM = 0.12. (b) Then draw plots of the same type and the same i0, but with b values of 0.12, 0.05, 0.038, and 0.029 (T = 298 K). (c) Write out your conclusions concerning the interplay of /0 and b in the Tafel relation (B = RT/aF). (d) How does this relate to the choice of electrocatalytic surfaces for optimal fuel cell performance (Bockris)... 

Experimentally, rates of electrode reactions are measured as the current passed, to which they are directly proportional. The dependence of current, /, on potential is exponential, suggesting a linear relation between lg I and potential—this is the Tafel relation. However, the rate (product of rate constant and reagent concentration) cannot rise indefinitely because the supply of reactants begins to diminish and becomes transport-limited. 

This is a form of the Tafel relation (Section 6.6) so long as the current is proportional to the rate constant. It is an example of a linear free energy relationship (a kinetic parameter, In A , varies linearly with a thermodynamic parameter, E). Substituting AG we get... 

The reaction mechanism is the same as that shown in Section 6.2.1.1. According to Jarvi and Stuve [50], an exponent of 5 should be used for Reaction 1 in the case of there being one site for CO and one for H. Assuming the Tafel relation holds for all three steps (see the previous section), the reaction rate can be expressed as... 

The first factor determines the tendency for dissolution to occur while the second and third, which are closely related, determine the rate of dissolution. The use of the standard electrode potentials as a measure of nobility is well known. The recognition that the exchange current density is a measure of the reversibility of a process and therefore a quantity characteristic of the reactivity of the system is more recent (13,32). As indicated by the Tafel relations, the exchange current density is a direct measure of the rate of the electrode reaction for any given value of the activation overvoltage (33). The values of iG may then be taken as a criterion for the electrochemical activity of a system. 

The Butler-Volmer equations simplify in the case where one of the exponential terms dominate, which will happen if the loss potential in question is large. In this case the electrode potential is linearly related to the logarithm of the corresponding current, and if the losses occur equally at both electrodes, then the total potential becomes linearly related to log(f). This is called the Tafel relation, and the slope of the line is called the "Tafel slope". The following sections will give many examples of potential-current relationships, and except for the interval of smallest currents, the Tafel approximation is often a valid one. 

The current ip at any V(t) during the transient is then assumed to be equal to the value the steady-state current would have at the same V as determined by the Tafel relation for the electrode process. If the steady-state current, i , obeys the Tafel equation, and C is assumed to be independent of potential, Eq. (41) may be integrated to give Eq. (43) thus. 

Fig. 17. Tafel relations of logi plotted versus overpotential for bulk Ni (5) and bulk Ni 80%-Mo 20% (6) in comparison with electroplated Ni (80%)-Mo (19%)-Cd (1%) composites at six temperatures, (1) to (4), in 0.2 M aqueous NaOH. (Original plots based on 200 points.) (From Ref. 75.)...

Fig. 25. Comparison of logarithmic plots of C o versus overpotential For the 0 R at Pt in acid (curve a) and alkaline solution (curve b) derived from the TaFel relations and potential-relaxation transients (257, 2S8).

Fig. 34. Curved Tafel relations [overpotential versus log(current density)] for Clj evolution on Pt. Curves (a)-(g) represent various cr concentrations in water at 298 K in the range 0.1-4.8 M (points are shown in the original reference). (From Ref. 341.)...

We will not go into details of a microscopic modeling of these reactions here but rather parameterize the reaction rates by standard approximations. We adopt the Tafel relation, given at the cathode side by... 

During the reaction current passes through the pores of the electrode. According to Ohm s law this leads to a potential gradient in the electrolyte. In simple cases, the current density j depends on the overpotential t] (the difference between actual potential and equilibrium potential) according to the Tafel relation... 

Figure 4. Tafel relations for the h.e.r. at Ni in methanolic HCl over a wide range of temperature (from Ref. 2). Note that upper-region slopes increase with increasing r, while lower-region slopes decrease for the same electrode process.

The two linear Tafel regions at Ni in methanolic HCl vary with r in a continuous and complementary way one has a slope that increases with T while the slope of the other simultaneously decreases with T (Fig. 4), so there is a singular temperature at which the Tafel relation is one line over the whole c.d. range. The directions of change of the slopes of the two Tafel lines at each temperature, other than at the singular temperature, correspond apparently to reaction mechanisms that are consecutive or parallel ). However, we believe, based on new data... 

Figure 6. Tafel relations for the h.e.r. at Ni-Mo electroplated electrocatalysts at several temperature curves (1-4), 341, 319, 298, and 278 K, compared with behavior of metallic nickel (curve 5) (Real apparent area factor for Ni-Mo coated electrode 450 X ) electrolyte is 1.0 M aqueous KOH (from Ref. 46, see also Fig. 19).

From the form of the Fermi function, it is seen that when it is used for an electrode process where Ep is modulated according to Ep = E°p Ve, the Ve term does not simply factorize from the Fermi equation giving a normal Tafel relation even with an empirically included j8 factor (cf. Gurney ), since... 

More detailed calculations of these effects were given later by Christov and Conway, who calculated proton tunneling probabilities through an Eckart barrier, the height of which was varied with potential. This gave a Tafel relation, as shown in Fig. 13, for proton transfer at a cathode for the case of complete tunneling control. In practice, both classical and nonclassical transfer occur in parallel " to relative extents dependent on temperature. 

Figure 13. Tafel relation arising from complete proton tunneling control in the h.e.r. (from Conway, ).

The expression derived by Despic and Bockris gives a potential dependence of especially when initial- and final-state profiles cross asymmetrically (see Fig. 15), i.e., when AFq is either a large exothermic or endothermic quantity, so that the transition state is, respectively, either close in configuration to the final state or to the initial state. The Tafel relation is consequently nonlinear simulated Tafel relations were calculated showing this behavior. 

From Eq. (34), it is seen that a pp is a function of A in the case where quadratic terms in the Tafel relation [Eq. (33)] are significant. It is interesting that this situation could evidently give... 

Reasonably linear Tafel relations were observed at all temperatures (down to low T) and the b values corresponded closely to those obtained for the high c.d. region at Ni in the alcohol + HCl solutions. The (6, T) plot (Fig. 5) shows that b decreases linearly with T down to the lowest temperature attained i.e., -89°C). This behavior is to be contrasted with that shown in Fig. 4 for the chloride solutions. 

Figure 19. Comparison of Tafel relations for the h.e.r. at various Ni electrode preparations, and Fe, having varying real to apparent area ratios T = 343 K (from Tilak ).

Figure 22. (a) Tafel relations for a process with a finite heat of activation (log Iq dependent on T) and a b value linear in T (conventional case, schematic) and (b) as in part a but for a proportional to T b independent of T) (schematic). 

For the case where the conventional form of the Lefat slope applies = aF/RT with a independent of T, it is easy to show that a family of Tafel relations obtained (Fig. 22a) at various temperatures intersect at a common overpotential 17 given by... 

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