Butler-Volmer equation total current density

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 ... 

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 only quantity in this equation that can be measured directly is the total current density. The partial current densities for each of the three reactions are obtained by weighing the deposit and analyzing its composition. Each of the current densities is calculated from the Butler-Volmer equation... 

Introducing the overpotential rj = E-Eo yields the Butler-Volmer equation. The total current density appears as the superposition of the anodic and the cathodic partial... 

Once the local concentration overpotential is known, the activation overpotential, ria, is obtained by subtracting Tjc from total Tj. The local activation overpotential is the actual driving force of the electrochemical reaction. It is related to the local current density at any point of the reaction zone by an electrochemical rate equation such as the Butler-Volmer equation (Eq. (10a)). Therefore, the rate equation, the Nernst equation (Eq. (37)), and the potential balance in combination couple the electric field with the species diffusion field. In addition, the energy balance applies also at the electrode level. Although this introduces another complication, a model including a temperature profile in the electrode is very useful because heat generation occurs mainly by electrochemical reaction and is localised at the reaction zone, while the... 

---