The Nernst Equation

In electrochemistry, the Nernst equation can be used, in conjunction with other information, to determine the reduction potential of a half-cell in an electrochemical cell. It can also be used to determine the total voltage, or electromotive force, for a full electrochemical cell. It is named after the German physical chemist who first formulated it, Walther Nernst.

Electrochemical cell

Schematic of an electrochemical cell.

The Nernst equation gives a formula that relates the electromotive force of a nonstandard cell to the concentrations of species in solution:

$E = E^o - \frac {0.0257}{n} ln\ Q$

In this equation:

• E is the electromotive force of the non-standard cell
• E^o is the electromotive force of the standard cell
• n is the number of moles of electrons transferred in the reaction

ln Q is the natural log of $\frac{C^cD^d}{A^aB^b}$, where the uppercase letters are concentrations, and the lowercase letters are stoichiometric coefficients for the reaction: $aA + bB \rightarrow cC + dD$

Example

Find the cell potential of a galvanic cell based on the following reduction half-reactions where [Ni²⁺] = 0.030 M and [Pb²⁺] = 0.300 M.

Ni²⁺ + 2 e- → Ni, E⁰ = -0.25 V

Pb²⁺ + 2 e- → Pb, E⁰ = -0.13 V

First, find the electromotive force for the standard cell, which assumes concentrations of 1 M.

In order for this reaction to run spontaneously (positive E^o cell) the nickel must be oxidized and therefore its reaction needs to be reversed. The added half-reactions with the adjusted E⁰ cell are:

$Pb^{2+} + Ni \rightarrow Ni^{2+} + Pb, \ \ \ E^o = 0.12\ V$

The number of moles of electrons transferred is 2 and Q is $\frac{[Ni^{2+}][Pb]}{[Pb^{2+}][Ni]}$, where Pb and Ni are pure solids whose concentrations remain constant, so they are dropped from the equation.

$E = E^o - \frac {0.0257}{n} ln\ Q$

$E = 0.12V - \frac {0.0257}{2} ln\frac{0.030}{0.300}$

$E = 0.15\ V$