Butler-Volmer equation, rates

Standard rate coefficient Butler-Volmer equation... 

As tire reaction leading to tire complex involves electron transfer it is clear that tire activation energy AG" for complex fonnation can be lowered or raised by an applied potential (A). Of course, botlr tire forward (oxidation) and well as tire reverse (reduction) reaction are influenced by A4>. If one expresses tire reaction rate as a current flow (/ ), tire above equation C2.8.11 can be expressed in tenns of tire Butler-Volmer equation (for a more detailed... 

The effect of the phospholipids on the rate of ion transfer has been controversial over the last years. While the early studies found a retardation effect [6-8], more recent ones reported that the rate of ion transfer is either not retarded [9,10] or even enhanced due to the presence of the monolayer [11 14]. Furthermore, the theoretical efforts to explain this effect were unsatisfactory. The retardation observed in the early studies was explained in terms of the blocking of the interfacial area by the phospholipids, and therefore was related to the size of the transferring ion and the state of the monolayer [8,15]. The enhancement observed in the following years was attributed to electrical double layer effects, but a Frumkin-type correction to the Butler Volmer (BV) equation was found unsuitable to explain the observations [11,16]. Recently, Manzanares et al. showed that the enhancement can be described by an electrical double layer correction provided that an accurate picture of the electrical double layer structure is used [17]. This theoretical approach will be the subject of Section III.C. 

In practice, the Butler- Volmer equation is only obeyed in any case for potentials close to Et, or, more generally, for small currents, not through any neglect of factors such as anharmonicity but rather because the rate of transport of the ions to the electrode becomes rate-limiting, a problem we turn to next. 

The values of the parameters derived from the best fit can be related to the fundamental physical constants, such as the electrochemical rate constants, by explicit calculation. From the Butler- Volmer equation,... 

Inner-sphere electron-transfer reactions are not expected to obey the Butler-Volmer equation. In these reactions the breaking or formation of a bond, or an adsorption step, may be rate determining. When the reactant is adsorbed on the metal surface, the electrostatic potential that it experiences must change appreciably when the electrode potential is varied. 

We assume that k and /c i are independent of the coverage and the electrode potential. We further assume that the rate of the electron-transfer step obeys a Butler-Volmer equation of the form ... 

Ao L the two terms involving L/Xo cancel, surface diffusion is fast, the deposition of adatoms is rate determining, and Eq. (10.6) reduces to the Butler-Volmer equation. 

Hydrogen evolution, the other reaction studied, is a classical reaction for electrochemical kinetic studies. It was this reaction that led Tafel (24) to formulate his semi-logarithmic relation between potential and current which is named for him and that later resulted in the derivation of the equation that today is called "Butler-Volmer-equation" (25,26). The influence of the electrode potential is considered to modify the activation barrier for the charge transfer step of the reaction at the interface. This results in an exponential dependence of the reaction rate on the electrode potential, the extent of which is given by the transfer coefficient, a. 

Butler27 and Volmer28 advanced Tafel s equation by relating overpotentials to activation barriers. The quantitative relationship between current and overpotential is called the Butler-Volmer equation (eqn (32)), and is valid for electrochemical reactions that are rate limited by charge transfer. 

The dimensionless time (t), potential ( ), and current (i/0 are all as defined in equations (1.4). The exact characteristics of the voltammograms depend on the rate law. In the case of Butler-Volmer kinetics,... 

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 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 ... 

It is now time to define some terms. The exchange current (/o) is best thought of as the rate constant of electron transfer at zero overpotential. This current is commonly expressed as a form of current density, Iq/A (cf. equation (1.1)), in which case it is called the exchange current density, io- (Incidentally, this also explains why the Butler-Volmer equation does not include an area term. This follows since both / et and /q are functions of area, thus causing the two area terms to cancel out.)... 

Occasionally, the analyst is required to determine the rate of electron transfer, ket, and can then use the Butler-Volmer equation (equation (7.16)) to determine 7o, from which ket is readily calculated by using equation (7.17). The preferred method of obtaining the exchange currents in such cases is under conditions of infinite rotation speed i.e. via a Koutecky-Levich plot. 

Activation polarization arises from kinetics hindrances of the charge-transfer reaction taking place at the electrode/electrolyte interface. This type of kinetics is best understood using the absolute reaction rate theory or the transition state theory. In these treatments, the path followed by the reaction proceeds by a route involving an activated complex, where the rate-limiting step is the dissociation of the activated complex. The rate, current flow, i (/ = HA and lo = lolA, where A is the electrode surface area), of a charge-transfer-controlled battery reaction can be given by the Butler—Volmer equation as... 

Figure 5. Measurement and analysis of steady-state i— V characteristics, (a) Following subtraction of ohmic losses (determined from impedance or current-interrupt measurements), the electrode overpotential rj is plotted vs ln(i). For systems governed by classic electrochemical kinetics, the slope at high overpotential yields anodic and cathodic transfer coefficients (Ua and aj while the intercept yields the exchange current density (i o). These parameters can be used in an empirical rate expression for the kinetics (Butler—Volmer equation) or related to more specific parameters associated with individual reaction steps.(b) Example of Mn(IV) reduction to Mn(III) at a Pt electrode in 7.5 M H2SO4 solution at 25 Below limiting current the system obeys Tafel kinetics with Ua 1/4. Data are from ref 363. (Reprinted with permission from ref 362. Copyright 2001 John Wiley Sons.)...

Consider a system in which a potential difference AV, in general different from the equilibrium potential between the two phases A 0, is applied from an external source to the phase boundary between two immiscible electrolyte solutions. Then an electric current is passed, which in the simplest case corresponds to the transfer of a single kind of ion across the phase boundary. Assume that the Butler-Volmer equation for the rate of an electrode reaction (see p. 255 of [18]) can also be used for charge transfer across the phase boundary between two electrolytes (cf. [16, 19]). It is mostly assumed (in the framework of the Frumkin correction) that only the potential difference in the compact part of the double layer affects the actual charge transfer, so that it follows for the current density in our system that... 

Thus, we have expressed the rate constant as a function of potential difference A(/>. This was the aim of the second part of the derivation of the Butler-Volmer equation. 

The relationship between q and I, the reaction rate of an electrode reaction, is expressed by the Butler-Volmer equation, whose model describes a linear variation of the activation energy with the applied overpotential [22]. Hence,... 

Equation (25) is general in that it does not depend on the electrochemical method employed to obtain the i-E data. Moreover, unlike conventional electrochemical methods such as cyclic or linear scan voltammetry, all of the experimental i-E data are used in kinetic analysis (as opposed to using limited information such as the peak potentials and half-widths when using cyclic voltammetry). Finally, and of particular importance, the convolution analysis has the great advantage that the heterogeneous ET kinetics can be analyzed without the need of defining a priori the ET rate law. By contrast, in conventional voltammetric analyses, a specific ET rate law (as a rule, the Butler-Volmer rate law) must be used to extract the relevant kinetic information. 

Both activation and concentration polarization typically occur at the same electrode, although activation polarization is predominant at low reaction rates (small cnrrent densities) and concentration polarization controls at higher reaction rates (see Fignre 3.10). The combined effect of activation and concentration polarization on the cnrrent density can be obtained by adding the contribntions from each [Eqs. (3.26) and (3.28)], with appropriate signs for a redaction process only to obtain the Butler-Volmer equation ... 

The Butler-Volmer (BV) approximation is the simplest approach to model and capture the essential features of the empirical Tafel equation. It considers an electrochemical half-cell reaction as an activated process, with the forward and backward reaction rates following an Arrhenius type law according to... 

Where k0 denotes the standard rate constant. The overall Butler-Volmer equation assuming the reactant and product concentrations to be the same, cA = cB = c, can be formulated as... 

Derivation of the Butler-Volmer equation in terms of electrode reaction rate constants is given in most electrochemical texts.1,3 7 15... 

Butler- Volmer equation and, 1217 controlled reaction rates, 1213, 1218 -convective mechanism. 1229 flux-equality equation, 1213 heat flow and, similarities, 1215 interfacial response at constant current 1216, 1218... 

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. 

If this is the case, the interest in the phenomenology of /—r) in terms of the Butler Volmer equation (no diffusion control, all electron transfer at the interface) is lessened. It will be acceptable to use equilibrium concepts at the interface for many purposes, and concentrate on the rate-determining transport process outside the interfacial region. 

It is clear that these questions cluster around some basic quantities that appear in the Butler—Volmer electrodic equation, such as P, the interfacial concentrations of electron acceptors and donors, and the potential difference that affects the reaction rate. Some attempts must be made to answer these questions. 

There is a good economic reason for this. Look back at the Butler Volmer equation (Eq. 7.24) the larger the ifl (Le., the better the catalysis), the smaller the overpotential needed to get a given rate of reaction. However, the smaller the overpotential, the less the total cell potential, and hence the kilowatt hours, to produce a given amount of a substance in an electrochemical reactor. 

---