The Butler-Volmer-Equation

In this chapter we treat electron-transfer reactions from a macroscopic point of view using concepts familiar from chemical kinetics. The overall rate v of an electrochemical reaction is the difference between the rates of oxidation (the anodic reaction) and reduction (the cathodic reaction) it is customary to denote the anodic reaction, and the current associated with it, as positive  

The phenomenological treatment assumes that the Gibbs energies of activation Gox and Gred depend on the electrode potential f , but that the pre-exponential factor A does not. We expand the energy of activation about the standard equilibrium potential 0o of the redox reaction keeping terms up to first order, we obtain for the anodic reaction  

The quantity a is the anodic transfer coefficient-, the factor l/F was introduced, because Fcf is the electrostatic contribution to the molar Gibbs energy, and the sign was chosen such that a is positive - obviously an increase in the electrode potential makes the anodic reaction go faster, and decreases the corresponding energy of activation. Note that a is dimensionless. For the cathodic reaction  

By differentiation we obtain for the sume of the two transfer coefficients the relation  

Since both coefficients are positive, they lie between zero and one we can generally expect a value near 1/2 unless the reaction is strongly unsymmetrical. 

Bockris Reddy (1970) describes the Butler-Volmer-equation as the central equation of electrode kinetics . In equilibrium the adsorption and desorption fluxes of charges at the interface are equal. There are common principles for the kinetics of charge exchange at the polarisable mercury/water interface and the adsorption kinetics of charged surfactants at the liquid/fluid interface. Theoretical considerations about the electrostatic retardation for the adsorption kinetics of ions were first introduced by Dukhin et al. (1973). 

P is called the symmetry factor, defined by the ratio of the distances across the DL up to the summit over the total DL thickness. The electric field at the interface is a vector. represents a characteristic equilibriiun potential difference across the interface and is charaeteristic for the reaction. 

PAOF is the magnitude at which the energy barrier for the ion-electrode transfer is lowered, and consequently, (1-P)A I F is the action for the raise of the metal-solution reaction. In conclusion we can say that in the presence of an electric field, the total free energy of activation for the electrode reaction is equal to the chemical free energy of activation. 

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The exchange current density for common redox couples (at room temperature) can range from 10-6 pAcm-2 to A cm"2. Equation (1-24) can be written in terms of the exchange current to give the Butler-Volmer equation ... 

This is the relaxation time of the polymer oxidation under electro-chemically stimulated conformational relaxation control. So features concerning both electrochemistry and polymer science are integrated in a single equation defining a temporal magnitude for electrochemical oxidation as a function of the energetic terms acting on this oxidation. A theoretical development similar to the one performed for the Butler-Volmer equation yields... 

The activation overpotential Tiac,w is due to slow charge transfer reactions at the electrode-electrolyte interface and is related to current via the Butler-Volmer equation (4.7). A slow chemical reaction (e.g. adsorption, desorption, spillover) preceding or following the charge-transfer step can also contribute to the development of activation overpotential. 

The two-step charge transfer [cf. Eqs. (7) and (8)] with formation of a significant amount of monovalent aluminum ion is indicated by experimental evidence. As early as 1857, Wholer and Buff discovered that aluminum dissolves with a current efficiency larger than 100% if calculated on the basis of three electrons per atom.22 The anomalous overall valency (between 1 and 3) is likely to result from some monovalent ions going away from the M/O interface, before they are further oxidized electrochemically, and reacting chemically with water further away in the oxide or at the O/S interface.23,24 If such a mechanism was operative with activation-controlled kinetics,25 the current-potential relationship should be given by the Butler-Volmer equation... 

This is the most commonly employed form of the Butler -Volmer equation as it does not involve the unmeasurable surface concentration terms. It must be remembered, however, that equation (1.35) is only applicable under the conditions where [0]0 [O ] and [R]0 =t- [R ]- We must now examine this equation in some detail, as its form dictates the nature of a number of electrochemical techniques for exploring reaction mechanisms. 

The Butler-Volmer equation can be employed only when O and R are chemically stable on the timescale of the experiment. Within this requirement, we can envisage two limiting cases ... 

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,... 

Providing that the potential is sufficiently negative, the kinetics of the reduction reaction in equation (2.125) can usually be rendered fast enough to tip the system into the diffusion-controlled regime, as was shown in the discussion of the Butler-Volmer equation in chapter 1. 

Thus, cyclic or linear sweep voltammetry can be used to indicate whether a reaction occurs, at what potential and may indicate, for reversible processes, the number of electrons taking part overall. In addition, for an irreversible reaction, the kinetic parameters na and (i can be obtained. However, LSV and CV are dynamic techniques and cannot give any information about the kinetics of a typical static electrochemical reaction at a given potential. This is possible in chronoamperometry and chronocoulometry over short periods by applying the Butler Volmer equations, i.e. while the reaction is still under diffusion control. However, after a very short time such factors as thermal... 

Figure 5.2 Current-potential curves according to the Butler-Volmer equation.

For small overpotentials, in the range Fr RT, the Butler-Volmer equation can be linearized by expanding the exponentials ... 

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. 

On application of an overpotential rj, the Gibbs energy of the electron-transfer step changes by eo[r) — Afa rj), where Afa(rj) is the corresponding change in the potential fa at the reaction site. Consequently, rj must be replaced by [rj — Afa r )] in the Butler-Volmer equation (5.13). 

These arguments are similar to those employed in the derivation of the Butler-Volmer equation for electron-transfer reactions in Chapter 5. However, here the reaction coordinate corresponds to the motion of the ion, while for electron transfer it describes the reorganization of the solvent. For ion transfer the Gibbs energy curves are less symmetric, and the transfer coefficient need not be close to 1/2 it may also vary somewhat with temperature since the structure of the solution changes. 

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 ... 

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. 

For simplicity we assume that the intermediate stays at the electrode surface, and does not diffuse to the bulk of the solution. Let (j>l0 and 0oo denote the standard equilibrium potentials of the two individual steps, and cred, Cint, cox the surface concentrations of the three species involved. If the two steps obey the Butler-Volmer equation the current densities j and j2 associated with the two steps are ... 

If the electrochemical reaction obeys the Butler-Volmer equation, the current density j at an electrode potential [Pg.146]

We assume that both reactions obey the Butler-Volmer equation, and denote the corresponding transfer coefficients by a and 2, the exchange current densities by jo,i and jo,2> and the equilibrium potentials by total current density is zero we have ... 

We consider a simple redox reaction obeying the Butler-Volmer equation. From Eq. (5.15), valid for small overpotentials, the charge-... 

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. 

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