Validity of the Butler-Volmer equation

1 shows the slope of their Tafel plots, d(lni)/dr), as a function of the inverse temperature 1/T. The Butler-Volmer equation predicts a straight line of slope aeo/k, which is indeed observed. Over the investigated temperature range both the transfer coefficient and the energy of activation are constant a = 0.425 0.01 and Eact = 0.59 0.01 eV at equilibrium, confirming the validity of the Butler-Volmer equation in the region of low overpotentials, from which the Tafel slopes were obtained. 

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Since the proton is transferred from a position right in front of the electrode, the assumptions made in the phenomenological derivation of the Butler-Volmer equation may not be valid furthermore, a proton can tunnel through a potential energy barrier in the reaction path. Nevertheless, an empirical law of the form ... 

Equation (7.144) is the most general form of the Butler-Volmer equation it is valid for a multistep overall electrodic reaction in which there may be electron transfers in steps other than the rds and in which the rds may have to occur V times per occurrence of the overall reaction. This generalized equation is seen to be of the same form as the simple Butler-Volmer equation for a one-step, single-electron transfer reaction ... 

Equations (30) and (31) are known as Tafle equations and they are approximations of the Butler-Volmer equation, which is valid for irreversible reactions. Equation (30) is valid for irreversible anodic reactions, while Eq. (31) is valid for irreversible cathodic reactions. 

The usual procedure for measuring the exchange current density ig is then to measure t] as a function of I and to plot In I vs. (Tafel plot). Such plots are shown in Figure 13.3 for Pt and Ag electrodes deposited on YSZ. From the slopes of the linear part of these plots ( ri > 200 mV, in which case Equation 13.15 are valid) one obtains the transfer coefficients and a. By extrapolating the linear part of the plot to r] = 0, one obtains io. One can then plot i vs. T] and use the low field approximation of the Butler-Volmer equation which is valid for < 10 mV, i.e.. 

This treatment remains valid for two other possible reaction sequences these are sequences in which there are (a) chemical, i.e., noncharge-transfer, steps before and after a charge-transfer rds and (b) charge-transfer steps before and after a chemical rds. In the latter case, where no charge transfer occurs in the rds, the number of electrons transferred after the rds will be n — y. There will be no effect of potential on the rate of the rds except that arising from previous charge-transfer steps thus, the Butler-Volmer equation for a chemical rds is given as... 

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. 

Therefore, the current density depends on the exchange current density ( o), transfer coefficient ( p), overpotential r ), and temperature (r). Fig. 7 represents typical current-overpotential curves based on Eq. (39). The net current is the result of the combined effects of the forward (anodic) and reverse (cathodic) currents. Although the Butler-Volmer equation for an electrochemical reaction in PEMFC is valid over the full potential range, simpler approximate equations may often be used for limited conditions. Thus, for the common value dp = 1/2, Eq. (39) becomes... 

A theoretical current-potential curve (/7/q vs. fj) is given in Fig. 7.3 for r] = 0.5. It should be emphasized here that Eq. (7.11) is only valid in this simple form if the current is really kinetically controlled, i.e. if diffusion of the redox species toward the electrode surface is sufficiently fast. According to the Butler-Volmer equation (Eq. 7.11) the current increases exponentially with potential in both directions. In this aspect charge transfer processes at metal electrodes differ completely from those at semiconductors. When the overpotential is sufficiently large, erj/kT 1. one of the exponential terms in Eq. (7.11) can be neglected compared to the other. In this case we have either... 

Following Bard and Faulkner (2001), for the reaction with the RDS involving single-electron transfer, parameter a is 0.5. Measurements (Holtappels et al., 1999) gave a 0.7 however, to simplify calculations we will adopt the value 0 = 1 — a = 0.5. Note that this choice of a is important only in the region of small currents, when both exponents in the Butler-Volmer equation contribute to the activation polarization. At larger currents the second exponent in Eq. (4.153) is small and the results below are valid for arbitrary a. 

In order to reach these expressions, we remove an exponential term from the Butler-Volmer equations. The approximate expression for inversion of the equations is valid only if the current density is sufficiently high. This approximation is, of course, seen again after inversion of that expression equations [1.55] are not correct for overly low values of current density and are not defined in 0. In order to avoid numerical instabiUty, it is usual to add a term of internal current into the expression of the activation overvoltage ... 

Current voltage measurements of ET at lES sometimes conform to the Butler-Volmer equation and sometimes not. This is not surprising because some of the assumptions on which this equation is based may fail at the liquid/liquid interface. These assumptions include (1) The potential drop across the interface is close to that imposed on the electrodes, or if not, a correction is included to properly account for the potential carried by the diffuse layers of ions at the interface (2) The current is due to ET alone, and if not, a correction due to ion transfer must be included (3) Marcus theory is valid. 

In the Tafel regime, we consider the Butler-Volmer equation in terms of two subregions, one for high anodic polarization (rjs aAp/RT), the second for high cathodic polarization -rjs cicF/RT). It can be shown that these approximations are valid when / avg io- The mathematical representations for the anodic and cathodic Tafel regions are similar ... 

In spite of the above justification for the kinetic approach to the estimate of l, this has a number of drawbacks. First of all, there is no point in using a kinetic approach to determine a thermodynamic equilibrium quantity such as l. The justification of the validity ofEqs. (42) and (45) by the resulting equilibrium condition of Eq. (46) is far from rigorous, just as is the justification of the empirical Butler-Volmer equation by the thermodynamic Nernst equation. Moreover, the kinetic expressions of Eq. (41) involve a number of arbitrary assumptions. Thus, considering the adsorption step of Eq. (38a) in quasi-equilibrium under kinetic conditions cannot be taken for granted a heterogeneous chemical step, such as a deformation of the solvation shell of the... 

Though the limiting cases of Butler-Volmer equations are easy to use, one should be careful about the range of activation overpotential for which these equations are valid. Chan et al. [76] reported the lower limit of activation overpotential for which the Tafel equation can be used as > 0.28 V, and the upper limit for linear current-potential relation ship as 7act < 0.1 V. 

It is worth noting that Eq. (1.25) is valid if the whole cell siu face is run in a current-generating (fuel cell) mode. If part of the cell experiences strong oxygen depletion, the respective domain may turn into the electrolysis mode (Kulikovsky et al., 2006). In that case, the exact Butler-Volmer equation in the form (1.24) has to be used. 

Simplified Butler-Volmer Equation 3 Butler-Volmer Equation with Identical Charge Transfer Coefficients-sinh Simplification A very nice simplification can be made to the BV model if the anodic and cathodic charge transfer coefficients at the electrode are equivalent (i.e., Uc = a a). In this case, no approximation is needed, and a new form explicit in r] and mathematically equivalent to the original BV model can be written. This model is valid over all regions of the electrode polarization, as shown in Figure 4.25. 

This is the famous Butler-Volmer (B-V) equation, the central equation of phenomenological electrode kinetics, valid under conditions where there is a plentiful supply of reactant (e.g., the Ag+ ions) by easy diffusion to and from electrodes in the solution, so that the rate of the reaction is indeed controlled by the electric charge transfer at the interface, and not by transport of ions to the electrode or away from it. 

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