Potential under real conditions
Tables give standard potentials E° measured with every species at 1 mol/L (or 1 bar for gases) and 25 °C. Real cells rarely work under those conditions. The Nernst equation, published by Walther Nernst in 1889, corrects the potential for actual concentrations and temperature.
The formulas
- Nernst equation: E = E° − (RT ÷ nF) × ln Q
- At 25 °C: E = E° − (0.05916 ÷ n) × log₁₀ Q
- Gibbs free energy: ΔG = −nFE
- Equilibrium constant: K = exp(nFE° ÷ RT)
R = 8.314 J/(mol·K), F = 96,485 C/mol, n is the number of electrons transferred and Q the reaction quotient: products over reactants, each raised to its coefficient, with solids and pure liquids counting as 1.
Worked example: Daniell cell
Zn(s) + Cu²⁺ → Zn²⁺ + Cu(s) has E° = 1.10 V and n = 2. With [Zn²⁺] = 0.10 mol/L and [Cu²⁺] = 1.0 mol/L, Q = 0.1 at 25 °C:
- RT/nF = 8.314 × 298.15 ÷ (2 × 96,485) = 0.01285 V.
- ln Q = ln 0.1 = −2.303.
- E = 1.10 + 0.01285 × 2.303 = 1.1296 V.
- ΔG = −2 × 96,485 × 1.1296 = −218 kJ/mol.
- log₁₀ K = 2 × 96,485 × 1.10 ÷ (8.314 × 298.15 × 2.303) = 37.19, so K ≈ 1.5 × 10³⁷.
Lowering the product concentration raises the cell potential, as Le Chatelier’s principle predicts.
The 59 mV rule
At 25 °C, each tenfold change in Q shifts the potential by 59.2 mV divided by n. For one-electron processes, that is the famous 59 mV per decade; a pH electrode, which responds to H⁺, changes by 59 mV per pH unit at 25 °C.
When a battery dies
As a cell discharges, products build up and reactants run down, so Q rises and E falls. When Q equals K, E reaches zero and no more useful work can be extracted: the battery is flat.
Applications
- pH meters and ion-selective electrodes convert a measured voltage into a concentration using the Nernst slope.
- Corrosion: the Nernst equation predicts when metals oxidize, the basis of Pourbaix diagrams.
- Neuroscience: the equilibrium potential of potassium across a nerve membrane, about −90 mV, follows from the ratio of K⁺ inside and outside the cell.
- Concentration cells: two identical electrodes in solutions of different concentration generate a voltage purely from the Nernst term.
Limits
The equation uses concentrations as stand-ins for activities, which is accurate only in dilute solutions. In concentrated electrolytes, activity coefficients well below 1 make the real potential differ noticeably.