Nernst Equation
Reference for E = E0 - (RT/nF) x ln(Q) Nernst equation.
Calculates cell potential at non-standard conditions for batteries, corrosion, and biochemistry.
The Formula
E = E° - (RT / nF) × ln(Q)
At 25°C: E = E° - (0.0592 / n) × log₁₀(Q)
At 25°C: E = E° - (0.0592 / n) × log₁₀(Q)
The Nernst equation adjusts the standard cell potential for real-world conditions. When concentrations are not 1 M or pressures are not 1 atm, this formula gives the actual voltage.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| E | Cell potential under actual conditions | Volts (V) |
| E° | Standard cell potential | Volts (V) |
| R | Gas constant (8.314 J/mol·K) | J/(mol·K) |
| T | Temperature | Kelvin (K) |
| n | Number of electrons transferred | (unitless) |
| F | Faraday's constant (96,485 C/mol) | C/mol |
| Q | Reaction quotient [products]/[reactants] | (unitless) |
Example 1
Zn-Cu cell (E° = 1.10 V, n = 2) with [Cu²⁺] = 0.01 M and [Zn²⁺] = 1.0 M at 25°C
Q = [Zn²⁺] / [Cu²⁺] = 1.0 / 0.01 = 100
E = 1.10 - (0.0592/2) × log₁₀(100)
= 1.10 - 0.0296 × 2
= 1.04 V
Example 2
At equilibrium (E = 0), find the equilibrium constant K for the Zn-Cu cell
0 = 1.10 - (0.0592/2) × log₁₀(K)
log₁₀(K) = 1.10 / 0.0296 = 37.16
K = 10³⁷·¹⁶ ≈ 1.45 × 10³⁷
When to Use It
Use the Nernst equation when:
- Calculating actual battery voltage at non-standard concentrations
- Predicting cell potential during discharge (as concentrations change)
- Relating cell potential to the equilibrium constant
- Designing concentration cells and pH meters
Key Notes
- Formula: E = E° − (RT/nF) ln Q: E° is the standard cell potential, R = 8.314 J/(mol·K), T is temperature (K), n is moles of electrons transferred, F = 96,485 C/mol (Faraday's constant), and Q is the reaction quotient.
- Simplified form at 25°C: At 298 K, RT/F ≈ 0.02569 V. Using log₁₀: E = E° − (0.05916/n) log Q. This is the most common form used in electrochemistry problems.
- Reaction quotient Q: Q = [products]/[reactants] (same form as the equilibrium constant K, but not at equilibrium). When Q < 1 (products low), E > E° — the cell produces more voltage than under standard conditions.
- At equilibrium, E = 0: When Q = K_eq, the cell is at equilibrium and produces no net voltage. This gives the relationship: ln K = nFE°/RT — connecting thermodynamics and electrochemistry.
- Applications: The Nernst equation is used to calculate cell voltage at non-standard concentrations, model biological membrane potentials (Nernst equation for ion channels), and design pH electrodes and ion-selective sensors.