Butler-Volmer equation equilibrium

Equations such as V-96 are known as Butler-Volmer equations [150]. At equilibrium, there will be equal and opposite currents in both directions, =... 

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 ... 

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 ... 

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... 

Potential Difference A< Departs from Equilibrium Butler-Volmer Equation, When the interphase is not in equihbrium, a net current density i flows through the electrode (the double layer). It is given by the difference between the anodic partial current density i (a positive quantity) and the cathodic partial current density i (a negative quantity) ... 

Butler Volmer, equation (coni.) multistep reaction. 1176. 1179 non-equilibrium, 1191 theory of diffusion, 1217 Butyl compounds, adsorption. 979... 

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. 

This general equation covers charge transfer at electrified interfaces under conditions both of zero excess field, low excess fields, and high excess fields, and of the corresponding overpotentials. Thus the Butler-Volmer equation spans a large range of potentials. At equilibrium, it settles down into the Nernst equation. Near equilibrium it reduces to a linear / vs. T) (Ohm slaw for interfaces), whereas, if T) > (RT/fiF) (i.e., one is 50 mV or more from equilibrium at room temperature), it becomes an exponential /vs. T) relation, the logarithmic version ofwhich is called Tafel s equation. 

The procedure for correcting forthe departure from equilibrium to nonequilibrium surface coverage consists in (1) writing down the actual concentration in the Butler-Volmer equation or its relevant special case and (2) transforming this expression into one involving the equilibrium exchange-current density Iq, which contains the bulk concentration. 

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 ... 

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 ... 

If there is no other contributions to the - overpotential (tj = E - eq, where Eeq is the - equilibrium potential), i.e., q = qac, taking into account the appropriate form of the -> Butler-Volmer equation for high enough anodic -> polarization (t] RT/nF)... 

Butler-Volmer equation — The Butler-Volmer or -> Erdey-Gruz-Volmer or Butler-Erdey-Gruz-Volmer equation is the fundamental equation of -> electrode kinetics that describes the exponential relationship between the -> current density and the -> electrode potential. Based on this model the -> equilibrium electrode potential (or the reversible electrode potential) can also be interpreted. 

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... 

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. 

In spite of the above justification for the kinetic approach to the estimate of l, this has a number of drawbacks. First of all, there is no point in using a kinetic approach to determine a thermodynamic equilibrium quantity such as l. The justification of the validity ofEqs. (42) and (45) by the resulting equilibrium condition of Eq. (46) is far from rigorous, just as is the justification of the empirical Butler-Volmer equation by the thermodynamic Nernst equation. Moreover, the kinetic expressions of Eq. (41) involve a number of arbitrary assumptions. Thus, considering the adsorption step of Eq. (38a) in quasi-equilibrium under kinetic conditions cannot be taken for granted a heterogeneous chemical step, such as a deformation of the solvation shell of the... 

E and the reversible equilibrium potential of the electrochemical reaction Thus the driving force for the electrochemical reaction is not the absolute potential it is the activation overpotential riaof This relationship between the current density and activation overpotential has been further developed and resulted in the Butler-Volmer equation ... 

In this equation, and represent the surface concentrations of the oxidized and reduced forms of the electroactive species, respectively k° is the standard rate constant for the heterogeneous electron transfer process at the standard potential (cm/sec) and oc is the symmetry factor, a parameter characterizing the symmetry of the energy barrier that has to be surpassed during charge transfer. In Equation (1.2), E represents the applied potential and E° is the formal electrode potential, usually close to the standard electrode potential. The difference E-E° represents the overvoltage, a measure of the extra energy imparted to the electrode beyond the equilibrium potential for the reaction. Note that the Butler-Volmer equation reduces to the Nernst equation when the current is equal to zero (i.e., under equilibrium conditions) and when the reaction is very fast (i.e., when k° tends to approach oo). The latter is the condition of reversibility (Oldham and Myland, 1994 Rolison, 1995). 

As described above, if this electrochemical reaction is driven out of equilibrium with the thermodynamic force ry, a current / will flow. One of the great successes of electrochemistry is its ability to provide a quantitative description of the charge transfer current characteristics. Max Volmer (1885—1965) and John Alfred Valentine Butler (1899—1977) proposed the following relation, which is known today as the Butler—Volmer equation, between the over potential 7j and the current density j (Fig. 3.4) ... 

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). 

It is obvious that this reaction can only lead to metal dissolution, if the metal electrode potential is negative from the hydrogen electrode potential. This is the reason for the classification of metals into noble metals (the equilibrium potential is more positive than the standard hydrogen potential) and non-noble metals (the equilibrium potential is more negative than the standard hydrogen potential). The kinetic of the total process can be described by the Butler-Volmer equation for the two partial reactions. 

In the given form, the Butler-Volmer equation is applicable rather broadly, for flat model electrodes, as well as for heterogeneous fuel cell electrodes. In the latter case, concentrations in Eq. (2.13) are local concentrations, established by mass transport and reaction in the random composite structure. At equilibrium,/f = 0, concentrations are uniform. These externally controlled equilibrium concentrations serve as the reference (superscript ref) for defining the equilibrium electrode potential via the Nernst equation. 

The Butler-Volmer Equation The Rate of an Electrochemical Reaction at a Given Degree of Displacement from Equilibrium... 

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