Cell potential, Gibbs energy and K

As a Daniell cell runs, [Zn²⁺] climbs, [Cu²⁺] falls and the voltmeter reading drops. When the reading reaches zero the concentrations stop changing: the reaction is at equilibrium.

Putting E_cell = 0 and Q = Kc into Nernst gives E°_cell = (2.303RT/nF) log Kc, which at 298 K is E° = (0.059/n) log Kc.

Daniell cell: log Kc = 2 × 1.1/0.059 = 37.288, so Kc ≈ 2 × 10³⁷. Cu + 2Ag⁺ → Cu²⁺ + 2Ag with E° = 0.46 V: log Kc = 15.6, Kc = 3.92 × 10¹⁵. Equilibrium constants too large to measure directly come from E°.

The maximum (reversible) electrical work a cell can do equals the fall in its Gibbs energy: ΔrG = −nFE_cell; under standard conditions ΔrG° = −nFE°_cell.

E_cell is intensive, ΔrG is extensive. Doubling the equation doubles n and ΔrG (−2FE becomes −4FE) but leaves E unchanged.

Daniell cell: ΔrG° = −2 × 96487 × 1.1 = −212.27 kJ mol⁻¹. From ΔrG° = −RT ln K the equilibrium constant follows.

Sign check: E° > 0 ⇔ ΔrG° < 0 ⇔ K > 1, a spontaneous reaction.