Nernst Equation - Pedagogy Zone

Nernst equation can be used to calculate the emf of the cells when the concentration of reactants and products are known.

Let us consider a general redox reaction

Mⁿ⁺ + ne^– $\rightleftharpoons$ M.

For any reversible reaction, the change in free energy $ΔG$ and equilibrium constant $K$ can be related using van’t Hoff’s isotherm as

eqn … (1)

$\begin{aligned} & \Delta \mathrm{G}=-\mathrm{RT} \cdot \ln \cdot \mathrm{K}+\mathrm{RT} \ln \frac{[\text { Product }]}{[\text { Reactant }]} \\ & \Delta \mathrm{G}=\Delta \mathrm{G}^{\mathrm{o}}+\mathrm{RT} \ln \frac{[\text { Product }]}{[\text { Reactant }]}\left[\Delta \mathrm{G}^{\mathrm{o}}=-\mathrm{RT} \ln \cdot \mathrm{K}\right] \end{aligned}$

where ΔG° is standard free energy change, when the concentrations of the reactants and products are unity.

We know, in a reversible reaction, the electrical energy is produced at the expense of the free energy decrease.

eqn … (2)

ΔG = -nFE and ΔG° = -nFE°

If $n =$ number of electrons transferred, $F =$ faraday (96500 coulombs), $E=$ electrode potential and $E° =$ standard electrode potential.

Thus, by substituting Eq. (2) in Eq. (1), we get,

$-\mathrm{nFE}=-\mathrm{nFE}^{\circ}+\mathrm{RT} \ln \frac{[\mathrm{M}]}{\left[\mathrm{M}^{\mathrm{n}+}\right]}$

$\begin{aligned} & \text { Divide Eq. (3) by nF. } \\ & -\mathrm{E}=-\mathrm{E}^{\circ}+\frac{\mathrm{RT}}{\mathrm{nF}} \ln \frac{[\mathrm{M}]}{\left[\mathrm{M}^{\mathrm{n}+}\right]} \end{aligned}$

As the concentration of metal is unity, Eq. (4) becomes

$\mathrm{E}=\mathrm{E}^{\circ}-\frac{\mathrm{RT}}{\mathrm{nF}} \ln \frac{1}{\left[\mathrm{M}^{\mathrm{n}+}\right]}$

$\mathrm{E}=\mathrm{E}^{\circ}+\frac{2.303 \mathrm{RT}}{\mathrm{nF}} \log \left[\mathrm{M}^{\mathrm{n}+}\right]$

The equation derived above is known as Nernst equation for electrode potential.

At $25°C$ , this equation can be simply expressed as

$E=E^{\circ}+\frac{0.0592}{n} \log \left[M^{n+}\right]$

(since T = 298°K, F = 96,500 C, R = 8.314°JK^-1 mol^-1)

Application of Nernst equations:

It is used to calculate emf of a cell.
It is used to calculate the single electrode potential.
It is used to calculate equilibrium constant of an electrochemical reaction.
It is used to predict activity of metals in acid solutions.
Corrosion tendency of metals in a given set of environmental conditions can be predicted.