The Nernst equation

The Nernst equation is named after the German physical chemist Walther Nernst who first formulated it. The Nernst equation links the actual reversible potential of an electrode (measured in volts), E, to the standard reversible potential of the electrode couple, E0 which is a thermodynamic value.

Lets consider the origin of the Nernst equation. The rate of an electrochemical reaction in terms of oxidation and reduction reactions, the concentration of the reacting species, the electrode potentials and the current densities can all be related quantitatively according to equation (1):eq. 1

in which i is the net current density, i and i are the partial current densities of the oxidation and reduction respectively, Cred and Cox are the concentrations of the reducing and oxidizing agents, respectively, k and k are the corresponding rate constants, while aa and ac are the so-called transfer coefficients - that is, specific constants giving a proper influence factor to the exponential dependence of the rate on the potential, E. In the case of a simple one-electron transfer, these factors are termed symmetry factors, for they, in a way, reflect the symmetry of the energy barrier. It can be proved that aa + ac = n, the number of electrons exchanged in a single act of an electrode reaction.

For a particular value of E the two partial current densities must become equal. This value of potential is the reversible electrode potential. From equation (1) one can deduce equation (2):eq. 2

This equation is known as the Nernst equation; Eš is the standard electrode potential (at Cox = Cred = 1) characteristic of the given redox couple.

The final and the most fundamental form the Nernst equation is written as:
eq. 3

where R is the universal gas constant, T is the absolute temperature in degrees Kelvin, z is the charge number of the electrode reaction (which is the number of moles of electrons involved in the reaction as written), and F is the Faraday constant (96,500 C mole-1). The notation ared represents the chemical activities of all of the species which appear on the reduced side of the electrode reaction and the notation aox represents the chemical activities of all of the species which appear on the oxidized side of the electrode reaction.

Some of the species which take part in electrode reactions are pure solid compounds and pure liquid compounds. In dilute aqueous solutions, water can be treated as a pure liquid because the amount of water is so much greater than the amount of any other species. For pure solid compounds or pure liquid compounds, activities are constant, so in the Nernst equation, as elsewhere in chemical thermodynamics, their values are considered to be one. The activities of gases are usually taken as their partial pressures and the activities of solutes such as ions are usually taken as their molar concentrations. Since ions are present in electrode reactions far more often than are gases or pure solids, the Nernst equation is often written with the gas and pure solid or liquid activities understood, in the form:
eq. 4

It is understood, however, that the activities of pure solid compounds and pure liquid compounds are still taken as equal to one and that the activities of gases are still taken as equal to their partial pressures.

At 298.15 K (25 oC), the numeric values of the constants and of the conversions from logarithms of base e (ln) to logarithms of base ten (log) can be combined to give a simpler form of the Nernst equation:
eq. 5

The standard electrode potential on the hydrogen scale is related to the thermodynamics of the electrode process. It reflects the standard free energy change of the redox reaction between the electron and the given redox couple, relative to the free energy change that takes place in the hydrogen electrode process.