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Nernst Equation | Electrochemistry | Derivation, Formula, Applications & Calculator

Nernst Equation

Predicting electrode potentials under non‑standard conditions – the cornerstone of electrochemistry

1. What is the Nernst Equation?

The Nernst equation relates the electrode potential (or cell potential) of an electrochemical reaction to the standard potential, temperature, and the activities (concentrations) of the chemical species involved. It is used to calculate the voltage of a galvanic cell or half‑cell when the concentrations are not at standard conditions (1 M, 1 atm, 25°C).

E = E° – (RT / nF) × ln(Q)

Where:

  • E = observed cell potential (V)
  • E° = standard cell potential (V)
  • R = gas constant (8.314 J·mol⁻¹·K⁻¹)
  • T = absolute temperature (K)
  • n = number of electrons transferred in the balanced reaction
  • F = Faraday constant (96485 C·mol⁻¹)
  • Q = reaction quotient (ratio of product activities to reactant activities)

For aqueous solutions at 298 K (25°C), using log₁₀, the equation simplifies to:

E = E° – (0.0591 / n) × log₁₀(Q)

2. Derivation of the Nernst Equation

The Nernst equation is derived from the relationship between Gibbs free energy and cell potential.

ΔG = ΔG° + RT ln(Q)

Standard Gibbs free energy change is related to standard cell potential:

ΔG° = – nFE°

Substituting:

– nFE = – nFE° + RT ln(Q)

Dividing by – nF gives the Nernst equation:

E = E° – (RT / nF) ln(Q)

At equilibrium, ΔG = 0, so E = 0 and Q = K (equilibrium constant):

0 = E° – (RT / nF) ln(K) ⇒ ln(K) = nFE° / RT

This allows calculation of the equilibrium constant from standard cell potential.

3. Interactive Nernst Equation Calculator

E = — V

Example: For the Daniell cell (Zn + Cu²⁺ → Zn²⁺ + Cu), E° = 1.10 V, n=2, Q = [Zn²⁺]/[Cu²⁺]. At Q=0.1, E ≈ 1.13 V.

4. Half‑Cell Potential (Reduction potential)

For a reduction half‑reaction: Mⁿ⁺ + n e⁻ → M(s)

E = E° – (RT / nF) ln(1 / [Mⁿ⁺]) = E° + (RT / nF) ln[Mⁿ⁺]

At 298 K, using log₁₀:

E = E° + (0.0591 / n) log₁₀[Mⁿ⁺]
Example: Calculate the reduction potential of Cu²⁺/Cu half‑cell at 25°C when [Cu²⁺] = 0.01 M. E°(Cu²⁺/Cu) = +0.34 V.
E = 0.34 + (0.0591/2) log(0.01) = 0.34 + 0.02955 × (-2) = 0.34 – 0.0591 = 0.2809 V.

5. Cell Potential under Non‑Standard Conditions

For a spontaneous redox reaction: aA + bB → cC + dD

Q = [C]^c [D]^d / ([A]^a [B]^b)
E_cell = E°_cell – (RT / nF) ln(Q)

As a reaction proceeds, the concentrations change, Q increases, and E_cell decreases until equilibrium (Q=K, E=0).

Example: For the Daniell cell (Zn | Zn²⁺ || Cu²⁺ | Cu), E° = 1.10 V, n=2. If [Zn²⁺] = 0.1 M and [Cu²⁺] = 1.0 M, Q = 0.1/1.0 = 0.1.
E = 1.10 – (0.0591/2) log(0.1) = 1.10 – 0.02955 × (-1) = 1.1296 V.

6. Equilibrium Constant from Nernst Equation

At equilibrium, E_cell = 0, Q = K. Thus:

0 = E° – (RT / nF) ln(K) ⇒ ln(K) = nFE° / RT
log₁₀(K) = nE° / 0.0591 (at 298 K)
Example: For the reaction 2Fe³⁺ + 2I⁻ → 2Fe²⁺ + I₂, E° = 0.23 V, n=2.
log K = (2 × 0.23) / 0.0591 = 7.78, so K ≈ 6.0 × 10⁷ (strongly product‑favored).

7. Applications of the Nernst Equation

Electrochemistry & Batteries
Predicts voltage of batteries under load; explains why battery voltage drops as it discharges (concentration changes).
Biochemistry
Calculates membrane potentials (e.g., resting potential of neurons) using ion concentrations inside/outside cells.
Analytical Chemistry
Used in potentiometry (pH meters, ion‑selective electrodes) to determine ion concentrations from measured potentials.
Corrosion Science
Evaluates whether a metal will corrode under given environmental conditions.
Environmental Monitoring
Measures dissolved oxygen or heavy metal ions in water using electrochemical sensors.
Fuel Cells & Electrolysis
Optimizes operating conditions by relating potential to reactant/product concentrations.

8. Limitations of the Nernst Equation

  • Ideal conditions assumed: The equation assumes dilute solutions where activity ≈ concentration. At high concentrations, ionic interactions cause deviations.
  • Temperature and pressure restrictions: Only valid at constant T and P; real systems may have temperature gradients.
  • Kinetic limitations ignored: Does not account for overpotentials, activation barriers, or electrode polarization.
  • Complex reactions: For multi‑step or complicated mechanisms, the simple Nernst equation may not apply directly.
  • Non‑ideal behaviour: Activity coefficients must be used for accurate results in concentrated solutions.

Despite these limitations, the Nernst equation remains indispensable for understanding and predicting electrochemical behaviour.

9. Summary & Key Takeaways

  • The Nernst equation allows calculation of electrode/cell potentials under non‑standard conditions.
  • It is derived from the Gibbs free energy–potential relationship: ΔG = –nFE.
  • At 298 K, E = E° – (0.0591/n) log₁₀(Q).
  • When Q < K, E > 0; when Q > K, E < 0; at equilibrium E=0 and Q=K.
  • Applications range from batteries and pH meters to biology and corrosion monitoring.
  • Limitations include ideal solution assumptions and neglect of kinetic effects.
The Nernst equation is the bridge between thermodynamics and electrochemistry.

10. Video Lectures

Complete Lectures

Detailed explanations of the Nernst equation, derivation, and applications.

Complete guide to the Nernst Equation – all content original, with interactive calculator, examples, and video lectures.

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