Nernst equation

The essentials

Standard potentials apply only when all concentrations are 1 M (and gases at 1 bar). The Nernst equation gives the potential at other concentrations. For a reduction Mⁿ⁺ + ne⁻ → M at 298 K: E = E° − (0.059/n) log (1/[Mⁿ⁺]). For a whole cell reaction: E_cell = E°cell − (0.059/n) log Q, where Q is the reaction quotient.

Key points

• General form: E = E° − (2.303RT/nF) log Q; at 298 K, 2.303RT/F ≈ 0.059 V.
• For the Daniell cell, E_cell = E°cell − (0.059/2) log ([Zn²⁺]/[Cu²⁺]).
• At equilibrium E_cell = 0, giving log K_c = nE°cell ÷ 0.059 at 298 K.
• ΔG° = −nFE°cell, where F = 96487 C mol⁻¹ (often taken as 96500). ΔG° = −2.303RT log K_c.
• A positive E_cell corresponds to a negative ΔG, so the reaction is spontaneous.
• For the hydrogen electrode, E = −0.059 pH at 298 K (with H₂ at 1 bar).

Exam tips

• Example: ΔG° for the Daniell cell = −2 × 96500 × 1.10 = −212300 J mol⁻¹ ≈ −212.3 kJ mol⁻¹.
• Example: if n = 2 and E°cell = 0.2955 V, log K = 2 × 0.2955 ÷ 0.059 ≈ 10, so K ≈ 10¹⁰.
• Lowering the concentration of the product ions or raising that of the reactant ions increases E_cell.

Common mistakes

• Putting the concentration of a pure solid in Q; solids are omitted.
• Using the wrong n; it is the number of electrons transferred in the balanced cell reaction.