Tafel approximation

By analogy with the case of single-step reactions at [ 171 > 0.1 the Tafel approximation is obtained, with slopes... 

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. 

At high anodic or cathodic overpotentials the plots are linear, in accordance with the Tafel approximation, and the intercept of the anodic and cathodic Tafel lines on the Pj = 0 axis gives the valne of log i . Additional details of the experimental setup and procednres for measuring kinetic parameters are fonnd in many electrochemistry texts [2, 3, 12, 13]. 

Equation (3.30) is termed the Tafel approximation for the German scientist who found that the current equivalent to the rate for a single reaction on a metal surface is related to the potential of the metal [38]. Substituting for ac=2.3 RT/acnF)logi° and bc = —2.3 RT/acnF) ... 

The adsorption of OH ions on the Pt(lll) plane has been studied by Drazic and coworkers [102] using fast cyclic voltammetry in sodium hydroxide solutions (0.03 to 1 M), under quasi-equilibrium and Tafel approximation conditions. It was shown that the OH ion adsorption is an electrosorption process with one electron exchanged... 

However, if the reaction is irreversible and can be described by the Tafel approximation, then... 

The Tafel approximation underestimates the leakage currents (by 6% in the middle compartment) except at the end electrodes. This effect is lessened as the Iq increases, other parameters remaining the same. 

Often, the exponential dependence of the dark current at semiconductor-electrolyte contacts is interpreted as Tafel behavior [49], since the Tafel approximation of the Butler-Volmer equation [50] also shows an exponential increase of the current with applied potential. One should, however, be aware of the fundamental differences of the situation at the metal-electrolyte versus the semiconductor-electrolyte contact. In the former, applied potentials result in an energetic change of the activated complex [51] that resides between the metal surface and the outer Helmholtz plane. The supply of electrons from the Fermi level of the metal is not the limiting factor rather, the exponential behavior results from the Arrhenius-type voltage dependence of the reaction rate that contains the Gibbs free energy in the expraient It is therefore somewhat misleading to refer to Tafel behavior at semiconductor-electrolyte contacts. 

Regardless of fuel cell type, in a certain range of polarization voltages this rate is well approximated by the Butler-Volmer or Tafel equations. A better (non-Tafel) approximation of the reaction rate for DMFC anode is considered in Section 2.7. 

If the charge transfer step is rate-controlling, conditions at the electrode interface will be far from equilibrium except at very low current densities, i.e., i 1+ and therefore as i. For this Tafel approximation we have ... 

The complexity of Eq. (5.25) depends on the order of the electrochemical reaction and the spatial dependence of the concentration terms. For a first-order reaction you can use a form of the Butler-Volmer Eq. (3.83). We shall use the Tafel approximation, assuming that the reaction is cathodic see Eq. (3.84) ... 

The Tafel approximation is valid provided that > 2. In that case, sinhiji cosh rji (exp i)/2. On the other hand, the argument of the tan function in Equations 4.148 and 4.151 must be less than tt/2. Setting = 2 in this argument gives a minimal e for which the parametric form of the polarization curve (4.148) and (4.151) can be used ... 

From Table 5.2, we can see that the error between the Tafel approximation of BV and the full BV equations is less than 1% for q larger than 118 mV. Thus, the Tafel form of the BV equation is a good approximation for q larger than 118 mV at 25°C. 

Find the values of overpotential as a function of temperature from 25°C to 800°C for which the linear approximation and Tafel approximation of BV equations are valid. (Hint see Examples 5.2 and 5.3.)... 

Equation (8.21) can be applied to different charge transfer processes both at the cathode and the anode by fitting different parameter values for a and I, to these processes. The approximation in Equation (8.21) for the Butler-Volmer equatiotf is much better than the Tafel approximation. The reason is that it is exact both for large cirrrents (in one of the directions) and for zero current. The equation can be used as an approximation for the Butler-Volmer equation in both current directions if a different pair of parameters values a and Iq ate used for each direction. 

SOLUTION From the last example we know Tafel kinetics will be valid when x = a,FrjlRuT > 1.2, which occurs for r] > 0.07 V in this case. With the Tafel approximation, we have to evaluate... 

These polarization curves were deduced using Tafel kinetics at a single electrode only and no other losses. Note that the Tafel expression results in zero losses (mathematically, it will actually predict negative polarization, which is physically unrealistic) until i > io. The exact solution shown for io = 0.3 A/cm shows the error associated with the Tafel approximation with a higher exchange current density. 

Using the Tafel approximation from the BV equation (4.35), we can show that... 

Figure 4.23), since the opposing branch of the BV equation can no longer be neglected, and the Tafel approximation is not valid. At very high current density, deviation is observed experimentally due to concentration polarization effects. An experimental Tafel slope for the ORR in a humidified H2/O2 fuel cell is shown in Figure 4.24. The data in Figure 4.24 were extracted from the polarization curves of fully humidified PEFCs operating on pure...

High polarization, Tafel approximation (explicit rj expression) ... 

It can be shown that the Tafel approximation of the cathodic branches of the two forms of the modified Butler-Volmer equation, ((18) and (20)) are identical when... 

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