Butler-Volmer equation, rates reactions

The current-voltage curves corresponding to these processes are depicted in Fig. 5.1. As the net current across the metal/solution interface is zero the potential Ep assumed by the particle under stationary conditions is given by the point of intersection of the two i(E) curves. At this potential the anodic and cathodic currents are equal and their value corresponds to iR. The latter defines the overall reaction rate. Both the mixed potential Ep and the reaction current iR may be evaluated from electrokinetic theory. Application of the Butler-Volmer equation to reaction (5.2) gives for the reaction rate V the expression... 

For many reactions, the charge transfer is only one elementary step in a sequence of many others. Some substances break chemical bonds and form new ones. Oxygen reduction is a relatively complicated process with several intermediate species corresponding to a sequence of reaction steps. Nevertheless, a corrosion reaction is often ruled by the Butler-Volmer equation, although reaction steps other than the charge transfer may be rate determining as well. 

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... 

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. 

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. 

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... 

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 ... 

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... 

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. 

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 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 (zFVf/i0) —> °°, that is, when the specific rate of the surface diffusion is much faster dian that of the transfer reaction, Eq. (7.267) tends to become identical with the Butler-Volmer equation that is, the current density becomes uniform on the surface. With increasing y values, the second term of Eq. (7.268) tends toward zero, and the y-T fl relation becomes linear with a slope of (1 + P)zF/4.6/ 7. It is possible to employ Eq. (7.262) to calculate the concentration profile between the growth sites. Some results are shown in Fig. 7.138. 

Our chapter has two broad themes. In the first, we will consider some aspects of quantum states relevant to electrochemical systems. In the second, the theme will be the penetration of the barrier and the relation of the current density (the electrochemical reaction rate) to the electric potential across the interface. This concerns a quantum mechanical interpretation of Talel s experimental work of 1905, which led (1924-1930) to the Butler-Volmer equation. 

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 ... 

Since the electrochemical reactions are supposed to take place at the electrodeelectrolyte interface, then the Butler-Volmer equation, regulating the electrochemical kinetics, sets the boundary condition, whilst j (production rate) in Equation (3.37) is replaced with J (current density produced), as explained in detail in Section 3.7.2. 

The first exponential term in both equations is independent of the applied potential and is designated as k and A(L for the forward and backward processes, respectively. These represent the rate constants for the reaction at equilibrium, e.g. for a monolayer containing equal concentrations of both oxidized and reduced forms. However, the system is at equilibrium at E0/ and the products of the rate constant and the bulk concentration are equal for the forward and backward reactions, i.e. k must equal Therefore, the standard heterogeneous electron transfer rate constant is designated simply as k°. Substitution into Equations (2.19) and (2.20) then yields the Butler-Volmer equations as follows ... 

The interfacial potential drop at the nonpolarizable ITIES was controlled by varying the concentration of either the cation or the anion of the ionic liquid in the aqueous phase. The kinetics of interfacial ET followed the Butler-Volmer equation, and the measured bimolecular rate constant was much larger than that obtained at the water-1,2-dichloroethane interface. In the second publication, Laforge et al. [112] developed a new method for separating the contributions from the interfacial ET reaction and solute partitioning to the SECM feedback. 

In many cases Eq. (6) is called the Butler-Volmer equation since it describes the case when the charge-transfer step exclusively determines the rate of the reaction (current), i.e., the rate of mass transport is very high in comparison with the rate of the charge transfer. 

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 net rate of the elementary reaction is obtained by deducting the backward reaction from the forward reaction. This is called the Butler-Volmer equation ... 

HOR and the ORR involve two and four electrons, respectively. Since the Butler-Volmer equation is important for expressing the relationship between the current density of an electrochemical reaction and the overpotential, the rate-determining step (RDS) of a multi-electron reaction can be simplified as a pseudo-elementary reaction involving multiple electrons. The Butler-Volmer equation for this reaction is usually written as follows ... 

Assuming the Butler-Volmer equation and Langmuir adsorption isotherm hold for this reaction, the charge-transfer rate in the presence of CO adsorption is... 

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