Electrode reaction rate: Butler-Volmer

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

As ksh in this instance is very small, then according to the Butler-Volmer formulation (eqn. 3.5) the reaction rate of the forward reaction, K — 8,he "F(E 0)/flr, even at E = E°, is also very low. Hence Etppl. must be appreciably more negative to reach the half-wave situation than for a reversible electrode process. Therefore, in the case of irreversibility, the polarographic curve is not only shifted to a more negative potential, but also the value of its slope is considerably less than in the case of reversibility (see Fig. 3.21). In... 

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. 

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. 

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.)...

Like all cathodes, early electrochemical kinetic studies of LSM focused heavily on steady-state d.c. characteristics, attempting to extract mechanistic information from the Tand F02 dependence of linear and Tafel parameters.As recently as 1997, some workers have continued to support a view that LSM is limited entirely by electrochemical kinetics at the LSM/electrolyte Interface based on this type of analysis. However, as we have seen for other materials (including Pt), the fact that an electrode obeys Butler—Volmer kinetics means little in terms of identifying rate-limiting phenomena or in determining how close the reaction occurs to the TPB. To understand LSM at a nonempirical level, we must examine other techniques and results. 

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... 

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,... 

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 ... 

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. 

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. 

In electrodics, the reaction rate is expressed in terms of current density i (Section 7.2.1). Thus one would expect, by analogy, the electrochemical order of the reaction to be given by an expression similar to (7.145) which should result from the Butler-Volmer expression ... 

The introduction of 0 in the equations for current density need by no means refer only to the adsorbed intermediates in the electrode reaction. What of other entities that may he adsorbed on the surface For example, suppose one adds to the solution an oiganic substance (e.g., aniline) and this becomes adsorbed on the electrode surface. Then, the 0 for the adsorbed organic substance must also be allowed for in the electrode kinetic equations. So, in Eq. (7.149), the value of 0 would really have to become a 0, where the summation is over all the entities that remain upon the surface and block off sites for the discharging entities. Many practical aspects of electrodics arise from this aspect of the Butler-Volmer equation. For example, the action of organic corrosion inhibitors partly arises in this way (adsorption and blocking of the surface of the electrode and hence reduction of the rate of the corrosion reaction per apparent unit area).67... 

When the symmetry factor was introduced by Volmer and Erdey-Gruz in 1930, it was thought to be a simple matter of the fraction of the potential that helps or hinders the transfer of an ion to or from the electrode (Section 7.2). A more molecularly oriented version of the effect of P upon reaction rate was introduced by Butler, who was the first to apply Morse-curve-type thinking to the dependence of theenergy-dis -tance relation in respect to nonsolvent and metal—hydrogen bonds. 

It is an experimental fact that whenever mass transfer limitations are excluded, the rate of charge transfer for a given electrochemical reaction varies exponentially with the so-called overpotential rj, which is the potential difference between the equilibrium potential F0 and the actual electrode potential E (t) = E — Ed). Since for the electrode reaction Eq. (1) there exists a forward and back reaction, both of which are changed by the applied overpotential in exponential fashion but in an opposite sense, one obtains as the effective total current density the difference between anodic and cathodic partial current densities according to the generalized Butler-Volmer equation ... 

Exchange current density — When an electrode reaction is in equilibrium, the reaction rate in the anodic direction is equal to that in the cathodic direction. Even though the net current is zero at equilibrium, we still envisage that there is the anodic current component (If) balanced with the cathodic one (Ic). The current value /() = Ja = IC is called the exchange current . The corresponding value of current density jo = Io/A (A, the electrode area) is called the exchange current density . If the rate constants for an electrode reaction obey the Butler-Volmer equation, jo is given by... 

Volmer turned his attention to processes at - nonpo-larizable electrodes [iv], and in 1930 followed the famous publication (together with - Erdey-Gruz) on the theory of hydrogen - overpotential [v], which today forms the background of phenomenological kinetics of electrochemistry, and which resulted in the famous - Butler-Volmer equation that describes the dependence of the electrochemical rate constant on applied overpotential. His major work, Kinetics of Phase Formation , was published in 1939 [v]. See also the Volmer reaction (- hydrogen), and the Volmer biography with selected papers [vi]. 

The Nemst equation is a thermodynamic expression of the equilibrium state of an electrochemical reaction. It can give the value of the thermodynamic electrode potential for electrochemical reactions as well as point out the reaction direction. However, it cannot show the reaction rate. To connect the reaction rate and the electrode potential, one needs to use the Butler-Volmer equation. 

The free energy barrier for the flow of ionic charge across the oxide electrode-electrolyte interface has an electrical contribution and consequently the reaction rate can be formally described by Butler-Volmer-type equations [38]. The cation current density corresponding to the process... 

Many of the electrode theories have assumed that the anodic reaction is rate-limiting and that the cathodic reduction of silver ions from silver halide is not rate-limiting and might not present any limitations to the process of development. Hamano et al. [112] contend that there are instances where the cathodic process does influence development. They use the Butler-Volmer equation as the basis for their development rate model and derive Eq. (83),... 

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