Butler-Volmer equation anodic

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.)...

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) ... 

Large Cathodic Current We have seen from Figure 6.7 that for the large negative values of overpotential r], the partial cathodic current density i approaches i, i i. For these conditions the Butler-Volmer equation (6.45) can be simplified. Analysis of Eq. (6.45) shows that when rj becomes more negative, the first exponential term in the equation (corresponding to the anodic partial current) decreases, whereas the second exponential term (corresponding to the cathodic partial reaction) increases. Thus, under these conditions. 

Butler-Volmer equation An anodic current at an clcctrodc/elcctrolytc interface was recorded as function of overpotential T and the results tabulated as follows ... 

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 ... 

Here the subscripts s and m denote solid (electrodes and current collectors) and membrane (electrolyte) respectively. Note that these two equations can be treated as only one equation with variable a and source terms. The R s are the volumetric transfer currents due to electrochemical reaction which are non-zero only in the catalyst layers and can be calculated from the Butler-Volmer equation for anode and cathode sides as ... 

Most industrial processes are operated at current densities of more than 50 mA/cm2. In this range the overpotential is relatively high, and one of the terms in the Butler-Volmer equation can be neglected. By convention the anodic overpotential is positive, and the cathodic overpotential is negative. If the anodic overpotential is high, then the second term of the Butler-Volmer equation can be neglected ... 

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)... 

When the rate is controlled by the - charge transfer step according to the - Butler-Volmer equation the anodic partial current density ( a) can be expressed as follows ... 

Corrosion current density — Anodic metal dissolution is compensated electronically by a cathodic process, like cathodic hydrogen evolution or oxygen reduction. These processes follow the exponential current density-potential relationship of the - Butler-Volmer equation in case of their charge transfer control or they may be transport controlled (- diffusion or - migration). At the -> rest potential Er both - current densities have the same value with opposite sign and compensate each other with a zero current density in the outer electronic circuit. In this case the rest potential is a -> mixed potential. This metal dissolution is related to the corro-... 

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 Butler-Volmer equation relates the effect of anodic or cathodic overpotential to net anodic or cathodic current density for an electrode reaction under activation control that is, free from mass transport and concentration effects. 

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),... 

The relationship between overpotential and current density of a single, activation-controlled electrochemical reaction is the Butler-Volmer equation, Eq. (10). As the equation shows, the rates of anodic and cathodic partial reactions are exponentially dependent on the overpotential. The net current is the sum of the anodic and cathodic partial currents. 

Fig. 10 Fit of the Sorensen and Kjelstrup equations (Butler-Volmer equations applied to two supposed reactions) to the anode polarization data of Solli [12].

Equations (30) and (31) are known as Tafle equations and they are approximations of the Butler-Volmer equation, which is valid for irreversible reactions. Equation (30) is valid for irreversible anodic reactions, while Eq. (31) is valid for irreversible cathodic reactions. 

Ni molar flux of species i, moles per square meter second a anodic transfer coefficients for Butler-Volmer equation,... 

Physically, the reason for the dramatic difference between performances of cathode and anode active layers is the exchange current density ia at the anode the latter is 10 orders of magnitude higher than at the cathode [6]. Due to the large ia, the electrode potential r]a is small. The anode of PEFC, hence, operates in the linear regime, when both exponential terms in the Butler-Volmer equation can be expanded [178]. This leads to exponential variation of rja across the catalyst layer with the characteristic length (in the exponent)... 

Therefore, the current density depends on the exchange current density ( o), transfer coefficient ( p), overpotential r ), and temperature (r). Fig. 7 represents typical current-overpotential curves based on Eq. (39). The net current is the result of the combined effects of the forward (anodic) and reverse (cathodic) currents. Although the Butler-Volmer equation for an electrochemical reaction in PEMFC is valid over the full potential range, simpler approximate equations may often be used for limited conditions. Thus, for the common value dp = 1/2, Eq. (39) becomes... 

The treatment by Pritzker and Fahidy [6] also involves anodic dissolution of the metal by a generalized expression of the activation overpotential by a mass transport modified Butler-Volmer equation. In both cases, anodic dissolution and cathodic deposition and stability and instability are possible. Some other studies have been presented with fewer terms that contribute to the surface stability such as those in Refs. [1,10] but with some erroneous conclusions. Anyhow, the limitations of Fahidy s work [6] arise from the fact that only a linear perturbation is considered, and so it can be applied only to the early stages of the instability. 

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 a sufficiently large value of anodic polarization from the reversible potential (overpotential ja > 50 mV), the first term on the right side of Eq. (18) dominates the second term. Therefore, at large overpotentials, the Butler-Volmer equation simplifies to ... 

The Butler-Volmer equation can be simplified for the anodic and cathodic reactions to ... 

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