Nernst equation, K and Gibbs energy

E° is measured at standard conditions: every solution at 1 M. Real cells use other concentrations, and then the cell potential changes. The Nernst equation gives it. At 298 K:

E(cell) = E°(cell) – (0.059 ÷ n) log Q

Two things to find from the cell reaction:

• n is the number of electrons transferred. In Zn + Cu²⁺ → Zn²⁺ + Cu, zinc goes from 0 to +2, so n = 2.
• Q is the reaction quotient, written from the ions only: product ions on top, reactant ions underneath, each raised to the power of its coefficient. Solid metals are left out. For the Daniell cell, Q = [Zn²⁺] ÷ [Cu²⁺].

As a cell runs, the reactants are used up, Q grows, and E falls. When E reaches zero the cell is 'flat': the reaction is at equilibrium, and Q has become K. Putting E = 0 into the Nernst equation:

E°(cell) = (0.059 ÷ n) log K

So the standard cell potential tells you how far a reaction goes. A large positive E° means a huge K.

The cell potential also gives the Gibbs energy change, which measures the useful work a reaction can do:

ΔG = –nFE(cell) and, at standard conditions, ΔG° = –nFE°(cell) = –RT ln K

F is the Faraday constant, the charge on one mole of electrons: 96487 C mol⁻¹, usually rounded to 96500. Since there is a minus sign, a positive E° gives a negative ΔG°, which means the reaction is spontaneous.

One special case: a hydrogen electrode in a solution of pH x has E = –0.059x V.