# The Nernst Equation

This equation relates the zero-current cell potential to the activities of the reactants and products in the cell reaction. The Gibbs Energy of reaction, ΔGr , can always be written in the following way:

where ΔGºr is the Standard Gibbs Energy of Reaction, and Q is the reaction quotient, dependant upon the activities of the reactants and products, and thus also dependant upon the composition of the reaction mixture. Division of both sides of this equation by -νF gives:

we replace the term  -ΔGºr / νF with the symbol Eº , which is called the Standard Cell Potential.

This gives the expression

which is the Nernst Equation, relating cell potential to the composition of the cell. The dependence of cell potential upon composition that it predicts is as follows:

 (E – Eº) F RT The graph has been plotted for three  different values of ν ; ν = 1, ν = 2 and ν = 3. lnQ

The above graph was plotted using this rearranged version of the Nernst Equation;

Note that when lnQ = 0 (which occurs when Q = 1, i.e. when all reactants and products are in their standard states and thus have an activity of one) , the standard cell potential, Eº, is equal to the zero-current cell potential E.

Eº is thus sometimes formally defined as the zero-current cell potential when all reactants and products are in their standard states.

The Nernst Equation leads to another important result, for an electrochemical cell at equilibrium. At equilibrium, Q = K, the equilibrium constant for the reaction, and E = 0, as an electrochemical cell at equilibrium can do no work. Substituting these results into the Nernst Equation, we obtain

which rearranges to

This allows prediction of equilibrium constants from measured standard cell potentials. Note that substitution of the definition  Eº = – ΔGºr / νF followed by a rearrangement gives another very useful relation,

which may be applied quite generally, outside the field of electrochemistry.