Butler-Volmer law

The current-potential relationship predieted by Eqs. (49) and (50) differs strongly from the Butler-Volmer law. For y 1 the eurrent density is proportional to the eleetro-static driving force. Further, the shape of the eurrent-potential curves depends on the ratio C1/C2 the curve is symmetrical only when the two bulk concentrations are equal (see Fig. 19), otherwise it can be quite unsymmetrieal, so that the interface can have rectifying properties. Obviously, these current-potential eurves are quite different from those obtained from the lattice-gas model. 

The first attempt to describe the dynamics of dissociative electron transfer started with the derivation from existing thermochemical data of the standard potential for the dissociative electron transfer reaction, rx r.+x-,12 14 with application of the Butler-Volmer law for electrochemical reactions12 and of the Marcus quadratic equation for a series of homogeneous reactions.1314 Application of the Marcus-Hush model to dissociative electron transfers had little basis in electron transfer theory (the same is true for applications to proton transfer or SN2 reactions). Thus, there was no real justification for the application of the Marcus equation and the contribution of bond breaking to the intrinsic barrier was not established. 

According to the Butler-Volmer law, the rates of simple electron-transfer reactions follow a particularly simple law. Both the anodic... 

Both contributions to the current obey the Butler-Volmer law. The current flowing through the conduction band has a vanishing anodic transfer coefficient, ac = 0, and a cathodic coefficient of unity, /3C — 1. Conversely, the current through the valence band has av — 1 and j3v = 0. Real systems do not always show this perfect behavior. There can be various reasons for this we list a few of the more common ones ... 

As an example, Fig. 12.6 shows Tafel plots for the exchange of the acetylcholine ion between an aqueous solution and 1,2-DCE. The two branches were obtained under conditions in which the ion was initially present in one phase only. This reaction obeys the Butler-Volmer law surprisingly well, even though a microscopic interpretation faces the same difficulty that we have discussed for electron-transfer reactions. 

The principle of this method is quite simple The electrode is kept at the equilibrium potential at times t < 0 at t = 0 a potential step of magnitude r) is applied with the aid of a potentiostat (a device that keeps the potential constant at a preset value), and the current transient is recorded. Since the surface concentrations of the reactants change as the reaction proceeds, the current varies with time, and will generally decrease. Transport to and from the electrode is by diffusion. In the case of a simple redox reaction obeying the Butler-Volmer law, the diffusion equation can be solved explicitly, and the transient of the current density j(t) is (see Fig. 13.1) ... 

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. 

When the slopes of the straight lines are the same (a = 0.5), half of the excess driving force is employed to accelerate the forward reaction and half to slow down the reverse reaction. Overall, the Butler-Volmer law may thus be expressed as... 

Although extremely useful in practice, the Butler-Volmer law is entirely empirical, with no justification of its linear character and no prediction of how the rate constants could be related to the molecular structure of the... 

The cyclic voltammetric responses depend on the manner in which the rate constants are related to the electrode potential. We start with cases in which the Butler-Volmer law applies ... 

FIGURE 1.17. Cyclic voltammetry of slow electron transfer involving immobilized reactants and obeying a Butler Volmer law. Normalized current-potential curves as a function of the kinetic parameter (the number on each curve is the value of log A ) for a. — 0.5. Insert irreversible dimensionless response (applies whatever the value of a). 

FIGURE 2.5. EC reaction scheme in cyclic voltammetry. Mixed kinetic control by an electron transfer obeying the Butler-Volmer law (with a = 0.5) and an irreversible follow-up reaction, a Variation of the peak potential with the scan rate, b Variation of the peak width with scan rate. Dots represent examples of experimental data points obtained over a six-order-of-magnitude variation of the scan rate. 

If the kinetics of electron transfer does not obey the Butler-Volmer law, as when it follows a quadratic or quasi-quadratic law of the MHL type, convolution (Sections 1.3.2 and 1.4.3) offers the most convenient treatment of the kinetic data. A potential-dependent apparent rate constant, kap(E), may indeed be obtained derived from a dimensioned version of equation (2.10) ... 

The Butler-Volmer law may be applied within the potential range of each wave with standard potentials E and E2, transfer coefficients standard rate constants and kc f2. The simulations shown in Figure 2.3527 were carried out as depicted in Section 6.2.6 and led determination of the following parameters ... 

As compared to the Nemstian case, the plateau is the same but the wave is shifted toward more negative potentials, the more so the slower the electrode electron transfer. An illustration is given in Figure 4.13 for a value of the kinetic parameter where the catalytic plateau is under mixed kinetic control, in between catalytic reaction and substrate diffusion control. For the kjet(E) function, rather than the classical Butler-Volmer law [equation (1.26)], we have chosen the nonlinear MHL law [equation (1.37)]. 

The two successive electron transfer reactions are assumed to obey the Butler-Volmer law with the values of standard potentials, transfer coefficient, and standard rate constants indicated in Scheme 6.1. It is also assumed, matching the examples dealt with in Sections 2.5.2 and 2.6.1, that the reduction product, D, of the intermediate C, is converted rapidly into other products at such a rate that the reduction of B is irreversible. With the same dimensionless variables and parameters as in Section 6.2.4, the following system of partial derivative equations, and initial and boundary conditions, is obtained ... 

Redox systems in solution may behave reversibly, and polarization T of a polarizable electrode in contact with the solution usually follows the Butler-Volmer law ... 

Figure 27. Dependence of specific rate constant for pre-filamentary center growth upon bromide ion concentration, showing conformity with the Butler-Volmer law of electrode kinetics. Developing agent, 10 m ascorbic acid [107].

It is easily seen from this simple representation that as soon as the rate of the chemical removal of P is larger than that of the back electron transfer, the rate-determining step of the overall process is the forward electron transfer. Thus, independently of the intrinsic value of k°, the Butler-Volmer law in Eq. (109) simplifies to that in Eq. (117), because (P)x=o is made negligible. As a result, the R/P electron transfer presents all the kinetic characteristics of a slow electron transfer [85]. However, since this behavior is not related to the intrinsic value of k°, the current-potential curve may be observed in the close vicinity of E° or even positive to E° for a reduction (negative to E° for an oxidation) [81]... 

Obviously, when K = kf/kb 1, the equilibrium lies toward P and does not affect the R/P electron transfer. In the converse situation, and when kf and kb are larger than the mass transfer rate, a rapid equilibrium displaced toward Z establishes, and (P)x=0 (Z)x=o/K. Introduction of this relation in the current rate law [Eq. (109)] then yields Eq. (123), which is identical in its formulation to a Butler-Volmer law but for a... 

Electrochemical electron and ion transfer reactions are commonly interpreted by the phenomenological Butler-Volmer law, according to which the rate constant k for... 

For outer sphere electron transfer reactions the Butler-Volmer law rests on solid experimental and theoretical evidence. An outer sphere electron transfer reaction is the simplest possible case of an electron transfer reaction, a reaction where only an electron is exchanged, no bonds are broken, the reactants are not specifically adsorbed, and catalysts play no role (see, e.g.. Ref. 2). Experimental investigations such as those by Curtiss et al. [206] have shown that the transfer coefficient of simple electron transfer reactions is independent of temperature. The theoretical basis is given by the theories of Marcus [207] and of Levich and Dogonadze [208] these theories also predict deviations at high overpotentials which were experimentally confirmed [209, 210]. 

Contrary to outer sphere electron transfer reactions, the validity of the Butler-Volmer law for ion transfer reactions is doubtful. Conway and coworkers [225] have collected data for a number of proton and ion transfer reactions and find a pronounced dependence of the transfer coefficient on temperature in all cases. These findings were supported by experiments conducted in liquid and frozen aqueous electrolytes over a large temperature range [226, 227]. On the other hand, Tsionskii et al. [228] have claimed that any apparent dependence of the transfer coefficient on temperature is caused by double layer effects, a statement which is difficult to validate because double layer corrections, in particular their temperature dependence, depend on an exact knowledge of the distribution of the electrostatic potential at the interface, which is not available experimentally. Here, computer simulations may be helpful in the future. Theoretical treatments of ion transfer reactions are few they are generally based on variants of electron transfer theory, which is surprising in view of the different nature of the elementary act [229]. 

In Table 4, the boundary conditions at the electrode surface involve the Butler-Volmer law describing the kinetics of electron transfer between the electrode and the cosubstrate rather than the Nernst law describing its thermodynamics (as in Table 2), because it is interesting... 

In spite of its simplicity (or rather because of it), Eq. (4.152) raises several questions. To a good approximation, the kinetics of electrochemical reactions on both sides of the cell follow the Butler-Volmer law, which establishes exponential dependence of cell current on the respective halfcell overpotential. Why then is the resulting polarization curve linear Does this mean that the resistive losses in SOFC dominate and the contribution of activation polarizations to the overall voltage loss is small ... 

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