The standard electrode potentials and cell potentials (E°_cell) are valid only under standard conditions (1.0 mol dm⁻³ concentration for solutions, 100 kPa for gases, 298 K). In non-standard conditions, the cell potential will differ. The Nernst equation quantifies this relationship.

The Nernst equation for a half-cell or a full cell is:
E = E° - (RT / nF) ln Q
Where:
● E = Cell potential (or electrode potential) under non-standard conditions (V)
● E° = Standard cell potential (or standard electrode potential) (V)
● R = Ideal gas constant (8.314 J K−1 mol−1)
● T = Absolute temperature (K)
● n = Number of moles of electrons transferred in the balanced reaction (or half-reaction)
● F = Faraday constant (96485 C mol−1)
● ln Q = Natural logarithm of the reaction quotient (Q)

The Reaction Quotient (Q): Q has the same form as the equilibrium constant (K), but it uses current (non-equilibrium) concentrations or partial pressures. For the general reaction aA + bB ⇌ cC + dD:
Q = ([C]ᶜ [D]ᵈ) / ([A]ᵃ [B]ᵇ)
● If Q < K, the reaction will proceed forward to reach equilibrium.
● If Q > K, the reaction will proceed in reverse to reach equilibrium.
● If Q = K, the system is at equilibrium, and E = 0 (no net cell potential).

Simplified Nernst Equation (at 298 K): At 298 K (25 °C), the term (RT/F) is constant: RT/F = (8.314 J K⁻¹ mol⁻¹) × (298 K) / (96485 C mol⁻¹) ≈ 0.02569 V
The natural logarithm (ln) can also be converted to log₁₀ by multiplying by 2.303: ln x = 2.303 log₁₀ x
So, at 298 K, the Nernst equation can be simplified to:
E = E° - (0.0592 / n) log₁₀ Q
This simplified form is often used for quick calculations at standard laboratory temperature.

Applications of the Nernst Equation:
● Calculating cell potential at non-standard concentrations: Used to predict the voltage of a battery as it discharges and reactant concentrations change.
● Calculating equilibrium constant (K) from E°: At equilibrium, E = 0 and Q = K. So, 0 = E° - (RT/nF) ln K, which means E° = (RT/nF) ln K. This gives ln K = nFE° / RT, and K = e^(nFE°/RT). This equation provides an alternative way to calculate K if E° is known, linking electrochemistry to thermodynamics.
● Concentration cells: Cells where the driving force is solely due to a difference in concentration of the same species in two half-cells. The E° for such a cell is 0, and the potential arises entirely from the Nernst term.

Example: Calculate the cell potential of the Daniell cell at 298 K if [Zn²⁺] = 0.10 M and [Cu²⁺] = 2.0 M. (From earlier, E°_cell = +1.10 V and n = 2 electrons transferred)
1. Write the overall balanced reaction: Zn(s) + Cu²⁺(aq) ⇌ Zn²⁺(aq) + Cu(s)
2. Determine the reaction quotient (Q): Q = [Zn²⁺] / [Cu²⁺] (Pure solids are not included) Q = 0.10 / 2.0 = 0.050
3. Apply the simplified Nernst equation: E = E° - (0.0592 / n) log₁₀ Q E = 1.10 - (0.0592 / 2) log₁₀ (0.050)
E = 1.10 - (0.0296) × (-1.301) E = 1.10 + 0.0385 E = 1.1385 V
The cell potential is slightly higher than the standard potential because the concentration of reactants (Cu²⁺) is higher than standard, and products (Zn²⁺) is lower than standard, shifting the reaction to favour the forward direction.