The Nernst equation expresses electrode potential at nonstandard conditions in terms of the standard electrode potential, the temperature, the number of electrons transferred, and the reaction quotient \(Q\).
Core form of the Nernst equation
For a half-reaction or an overall cell reaction written in the forward direction, the Nernst equation is
\[ E = E^\circ - \frac{RT}{nF}\ln Q \]
Here \(E\) is the electrode potential (or cell potential), \(E^\circ\) is the standard potential, \(R\) is the gas constant, \(T\) is the absolute temperature (K), \(n\) is the number of electrons transferred in the balanced redox equation, \(F\) is the Faraday constant, and \(Q\) is the reaction quotient built from activities.
The base-10 form is obtained from \(\ln Q = 2.303 \log_{10} Q\): \[ E = E^\circ - \frac{2.303RT}{nF}\log_{10} Q \] At \(T = 298.15\ \mathrm{K}\) (25 °C), \[ E = E^\circ - \frac{0.05916}{n}\log_{10} Q \] with \(E\) in volts.
Reaction quotient and activities
The reaction quotient \(Q\) has the same algebraic form as an equilibrium constant, except it uses the current activities rather than equilibrium activities. For a reaction \[ \alpha A + \beta B \rightleftharpoons \gamma C + \delta D \] the quotient is \[ Q = \frac{a_C^{\gamma}a_D^{\delta}}{a_A^{\alpha}a_B^{\beta}} \] where \(a_i\) denotes activity.
| Species type | Activity contribution in \(Q\) | Common approximation |
|---|---|---|
| Pure solids, pure liquids | Activity treated as 1, omitted from \(Q\) | Always 1 in dilute-solution electrochemistry |
| Aqueous ions and solutes | Use activity \(a_i\) | \(a_i \approx [i]\) for sufficiently dilute solutions |
| Gases | Use activity based on fugacity/partial pressure | \(a \approx \frac{P}{P^\circ}\) when ideal-gas behavior is reasonable |
Cell potential from half-cells
For a galvanic cell written as a net cell reaction, the same Nernst equation applies to the overall cell potential:
\[ E_{\text{cell}} = E_{\text{cell}}^\circ - \frac{RT}{nF}\ln Q \]
The electron count \(n\) corresponds to the electrons transferred in the balanced net redox reaction (not the sum of stoichiometric coefficients).
Worked example with a standard Zn/Cu cell
The reaction \[ \mathrm{Zn}(s) + \mathrm{Cu}^{2+}(aq) \rightarrow \mathrm{Zn}^{2+}(aq) + \mathrm{Cu}(s) \] has \(n = 2\) and a typical standard potential \(E^\circ_{\text{cell}} = 1.10\ \mathrm{V}\).
With \([\mathrm{Zn}^{2+}] = 0.10\ \mathrm{M}\) and \([\mathrm{Cu}^{2+}] = 1.0 \times 10^{-3}\ \mathrm{M}\), solids omitted: \[ Q = \frac{a(\mathrm{Zn}^{2+})}{a(\mathrm{Cu}^{2+})} \approx \frac{[\mathrm{Zn}^{2+}]}{[\mathrm{Cu}^{2+}]} = \frac{0.10}{1.0 \times 10^{-3}} = 100 \] \[ \log_{10} Q = 2 \] At 25 °C, \[ E_{\text{cell}} = 1.10 - \frac{0.05916}{2}\cdot 2 = 1.10 - 0.05916 = 1.04084\ \mathrm{V} \] The positive value remains consistent with a spontaneous galvanic direction under these concentrations.
Visualization of \(E\) versus \(\log_{10}Q\) at 25 °C
Common pitfalls and consistency checks
- Logarithm base consistency: \(\ln\) pairs with \(\frac{RT}{nF}\), while \(\log_{10}\) pairs with \(\frac{2.303RT}{nF}\).
- Electron count \(n\): the transferred electrons in the balanced net redox reaction, not a coefficient sum.
- Activity versus concentration: concentrated electrolytes and high ionic strength reduce the accuracy of \(a \approx [\,]\).
- Omission rules: pure solids and pure liquids omitted from \(Q\); dissolved ions and gases retained with appropriate activity models.
- Reaction direction: reversing the written cell reaction replaces \(Q\) by \(1/Q\) and changes the sign of \(\ln Q\) in the expression for \(E\).
Concentration cells as a limiting case
When both electrodes involve the same redox couple, \(E^\circ_{\text{cell}} = 0\) and the potential arises entirely from a concentration ratio inside \(Q\). This makes the Nernst equation the defining relation for concentration-cell voltages under nonstandard conditions.