A galvanic cell delivers a voltage that depends on both the redox couples involved and the concentrations of species in solution. The Nernst equation quantifies this dependence.
Standard potential
Each redox couple Ox / Red has a standard electrode potential E°(Ox/Red), measured by convention against the standard hydrogen electrode (SHE, E° = 0.000 V). Examples:
- E°(F₂/F⁻) = +2.87 V (strong oxidant)
- E°(Cl₂/Cl⁻) = +1.36 V
- E°(Cu²⁺/Cu) = +0.34 V
- E°(2H⁺/H₂) = 0.00 V (reference)
- E°(Zn²⁺/Zn) = −0.76 V (strong reductant)
- E°(Li⁺/Li) = −3.04 V
The higher E°, the stronger the oxidant. A cell combines two couples: cathode (reduction, higher potential) and anode (oxidation, lower potential). Its standard emf is:
E°cell = E°cathode − E°anode
The Nernst equation
When concentrations differ from standard (1 mol/L for solutes, 1 bar for gases), the half-cell potential becomes:
E(Ox/Red) = E°(Ox/Red) + (R·T / n·F) · ln([Ox] / [Red])
with: - R = 8.314 J·mol⁻¹·K⁻¹ - T: absolute temperature (K) - n: number of electrons exchanged - F = 96,485 C/mol (Faraday constant)
At 25 °C (298 K), the factor R·T/F · ln(10) ≈ 0.0592 V, giving the practical form:
E = E° + (0.0592 / n) · log([Ox] / [Red])
Full-cell application
For a cell with non-standard emf U, combining the two half-equations:
U = E°cell + (0.0592 / n) · log(Q⁻¹)
where Q is the reaction quotient. At equilibrium, U = 0, yielding the fundamental relation:
E°cell = (0.0592 / n) · log(K)
linking standard emf to the overall equilibrium constant.
Example: Daniell cell at non-standard concentrations
Couples: Cu²⁺/Cu and Zn²⁺/Zn. At 25 °C, n = 2 and:
E°cell = E°(Cu²⁺/Cu) − E°(Zn²⁺/Zn) = 0.34 − (−0.76) = +1.10 V
With [Cu²⁺] = 0.01 mol/L and [Zn²⁺] = 1 mol/L:
U = 1.10 + (0.0592 / 2) · log(0.01 / 1) = 1.10 − 0.0592 = 1.04 V
Lower [Cu²⁺] drags the emf down, as expected (less reactant on the cathode side).
Useful special cases
pH measurement: for H⁺/H₂ at P(H₂) = 1 bar:
E = 0 − 0.0592 · pH
At 25 °C, the hydrogen electrode potential is exactly E = −0.0592 · pH. This is the basis of pH metering: measure a cell voltage to read off the pH.
Concentration cell: two identical half-cells at different concentrations — emf comes entirely from the log term and goes to zero as concentrations equalize (equilibrium).
Why it matters
Nernst connects thermodynamics (E°, K, ΔG°) to experimental electrochemistry (measured voltages at known concentrations). It is the central quantitative tool for:
- designing a battery (pick couples, predict emf).
- predicting corrosion (potentials in acid/base media).
- interpreting a pH measurement or a redox titration.
- understanding mitochondrial respiratory chains.