Nernst kinetic derivation

The Nernst equation defines the equilibrium potential of an electrode. A simplified thermodynamic derivation of this equation is given in the Sections 5.3 to 5.5. Here we will give the kinetic derivation of this equation. 

Diffusion overpotential. When high current densities j exist at electrodes (at the boundary to the electrolyte), an impoverishment of the reacting substances is possible. In this case the reaction kinetics are determined only by diffusion processes through this zone, the so-called Nernst layer. Without dealing with the derivation in detail, the following formula is obtained for the diffusion overpotential that occurs (with as the maximum current density) ... 

Williams (1964) derived the relation T = kBTrQV3De2, where T is the recombination time for a geminate e-ion pair at an initial separation of rg, is the dielectric constant of the medium, and the other symbols have their usual meanings. This r-cubed rule is based on the use of the Nernst-Einstein relation in a coulom-bic field with the assumption of instantaneous limiting velocity. Mozumder (1968) criticized the rule, as it connects initial distance and recombination time uniquely without allowance for diffusional broadening and without allowing for an escape probability. Nevertheless, the r-cubed rule was used extensively in earlier studies of geminate ion recombination kinetics. 

The integral relationships above are valid for any transient technique other than cyclic voltammetry, since at this stage of the derivation, the fact that the potential is a linear function of time has not yet been introduced. It is also valid in the case where charge transfer is not fast and together with diffusion, kinetically governs the electrochemical response. In the present case, the linear relationship between potential and time comes into play through Nernst s law, leading to... 

The potential of this electrode is defined (Section 5.2) as the voltage of the cell Pt H2(l atm) H+(<2 = 1) MZ+ M, where the left-hand electrode, Et = 0, is the normal hydrogen reference electrode (described in Section 5.6). We will derive the Nernst equation on the basis of the electrochemical kinetics in Chapter 6. Here we will use a simplified approach and consider that Eq. (5.9) can be used to determine the potential E of the M/Mz+ electrode as a function of the activity of the products and reactants in the equilibrium equation (5.10). Since in reaction (5.10) there are two reactants, Mz+ and e, and only one product of reaction, M, Eq. (5.9) yields... 

This is the Nernst equation defined from the electrode kinetics considerations. Later, we derive the same relationship on purely thermodynamic grounds. 

The Warburg and Nernst impedances were derived under the assumption that the potential obeys the Nernst equation. The more realistic Randles model takes into account the kinetics of charge transfer as described by the Butler-Volmer equation. For the electrode reaction (5.147) this is written as... 

The dynamic model of a PEMFC can be realized in MATLAB and Simulink software for implementation in power systems [10]. Beginning with hydrogen flow, the three significant factors are input, output, and reaction flows during operahon [11]. The thermodynamic potential of the chemical energy that can be converted into electrical energy is derived from Nernst s law and is dependent on the partial pressures of the reactants and temperature. For reaction kinetic considerahon, overpotentials at both anode and cathode essentially constitute the energy required to drive a reaction beyond the state of thermodynamic reversibility. 

---