The Nernst equation

Result

The Nernst equation

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Statement

How cell potential depends on concentration.

Why it matters

galvanic-cell-emf established the standard cell potential \(E^\circ\) from tabulated half-reactions under standard conditions; the Nernst equation extends this to any concentration, and in doing so ties electrochemistry directly back to the equilibrium framework of law-of-mass-action via gibbs-emf-relation's \(\Delta G=-nFE\) link to \(\Delta G^\circ=-RT\ln K\). It is the working equation behind concentration cells, the pH-measuring glass electrode, and every practical use of a battery or biological membrane potential away from standard conditions.

Hypotheses

The cell operates reversibly, so measured potentials reflect thermodynamic rather than kinetic quantities.The Nernst equation is derived from \(\Delta G\), a state function; drawing significant current introduces kinetic overpotentials at the electrodes that this purely thermodynamic equation does not capture. Species behave ideally, so activities are well approximated by molar concentrations (or partial pressures for gases).Strictly, \(Q\) should be built from thermodynamic activities; at higher ionic strength, activity coefficients deviate measurably from \(1\) (the concern addressed by Debye–Hückel-type corrections elsewhere), and the naive concentration-based equation becomes noticeably inaccurate. Temperature is fixed and known.The equation's prefactor \(RT/nF\) is explicitly temperature-dependent, so the equation must be re-evaluated, not merely reused with a fixed numeric constant, at any temperature other than the one a shortcut value was derived for.

Proof

\Delta G = -nFE

gibbs-emf-relation gives the general relation between a cell reaction's Gibbs energy and its measured potential, for any composition, not only standard conditions. A

\Delta G = \Delta G^\circ + RT\ln Q

The general thermodynamic relation for reaction Gibbs energy at arbitrary composition, with \(Q\) the reaction quotient of the overall cell reaction, built the same way as in law-of-mass-action. A

-nFE = -nFE^\circ + RT\ln Q

Substituting Step 1 into Step 2, for both the actual and standard conditions, gives an equation relating \(E\), \(E^\circ\), and \(Q\) directly. A

E = E^\circ - \frac{RT}{nF}\ln Q

Dividing through by \(-nF\) isolates the Nernst equation in its standard form. A

E = E^\circ - \frac{0.0592}{n}\log_{10}Q \quad (T=298\,\text{K})

Substituting \(R=8.314\,\text{J/mol/K}\), \(F=96485\,\text{C/mol}\), \(T=298\,\text{K}\), and converting to base-10 logarithm gives the common practical numeric shortcut, valid specifically at room temperature. A

Result

E = E^\circ - \frac{RT}{nF}\ln Q

Reading. Cell potential departs from its standard value by an amount set by how far the reaction quotient \(Q\) is from \(1\), scaled by temperature and the number of electrons transferred.

Scope. Valid for reversible, ideal (or activity-corrected) electrochemical cells; setting \(E=0\) (the cell has run down to equilibrium, no further net current flows) gives \(Q=K\), the direct link between \(E^\circ\) and the equilibrium constant via \(\ln K = nFE^\circ/RT\).

Corollaries & converses

Setting \(E=0\) gives \(\ln K = nFE^\circ/RT\), the standard route from tabulated \(E^\circ\) values to an equilibrium constant, without any separate mass-action measurement.
Concentration cells (two half-cells of the identical redox couple at different concentrations, so \(E^\circ=0\)) have their entire measured potential arise purely from the Nernst equation's \(Q\) term — the practical basis of the pH-measuring glass electrode.
electrochemical-series' ranking of \(E^\circ\) values, combined with this result, predicts actual (not merely standard) cell potentials under real, non-standard laboratory or physiological concentrations.

Fails without

Draw significant current from the cell, violating the reversibility hypothesis: the measured potential picks up a real kinetic overpotential at the electrodes and departs from the purely thermodynamic prediction of the Nernst equation.
Apply the concentration-based \(Q\) at high ionic strength (violating the ideal-behaviour hypothesis): activity coefficients deviate substantially from \(1\), and the computed \(E\) differs measurably from the value actually observed experimentally.

Common errors

Forgetting to raise concentrations to their stoichiometric-coefficient powers inside \(Q\), exactly as in any equilibrium expression.
Using the wrong value of \(n\) for the specific balanced half-reactions in question, which rescales the entire correction term.
Applying the room-temperature \(0.0592\,\text{V}\) numeric shortcut at a different temperature without rederiving the correct \(RT/F\) prefactor.
Interpreting \(E=0\) as "the cell is broken," rather than correctly recognising it as the cell having reached chemical equilibrium (\(Q=K\)), after which no further net current flows.

Discussion

Walther Nernst derived this relation around 1889; he was awarded the 1920 Nobel Prize in Chemistry substantially for his broader contributions to thermodynamics, including the third law.

Strictly, \(Q\) in the Nernst equation should be built from thermodynamic activities rather than raw molar concentrations. At low ionic strength, activities are well approximated by concentrations; at higher ionic strength, activity coefficients deviate measurably from \(1\), and the naive concentration-based Nernst equation becomes noticeably inaccurate, requiring activity-coefficient corrections to remain quantitatively reliable.

Common misconception: that a cell with positive \(E^\circ\) can never reach \(E=0\). A cell with a favourable \(E^\circ\) still runs down to \(E=0\) once reactant concentrations are depleted and product concentrations built up enough that \(Q\) has risen to equal \(K\) — running down is a statement about \(Q\) approaching \(K\), not about the sign of \(E^\circ\) itself.

Worked examples

\text{Zn}|\text{Zn}^{2+}||\text{Cu}^{2+}|\text{Cu}, \quad E^\circ=1.10\,\text{V}, \ n=2

For the Daniell cell, the overall reaction is \(\text{Zn}(s)+\text{Cu}^{2+}(aq)\to\text{Zn}^{2+}(aq)+\text{Cu}(s)\), with \(Q=[\text{Zn}^{2+}]/[\text{Cu}^{2+}]\) (pure solids omitted, as in law-of-mass-action). A

[\text{Zn}^{2+}]=0.10\,\text{M}, \ [\text{Cu}^{2+}]=1.0\times10^{-4}\,\text{M} \ \Rightarrow\ Q = \frac{0.10}{1.0\times10^{-4}}=1.0\times10^3

Substituting into the room-temperature shortcut: \(E=1.10-\frac{0.0592}{2}\log_{10}(1.0\times10^3)=1.10-\frac{0.0592}{2}(3)=1.10-0.0888=1.011\,\text{V}\). A

E \approx 1.01\,\text{V}

Reading. Depleting \(\text{Cu}^{2+}\) relative to standard concentration measurably lowers the cell potential below \(E^\circ\), exactly as the Nernst equation predicts for a product-quotient-dependent term.

Scope. The identical substitution applies for any known \(E^\circ\), \(n\), and set of concentrations, at the temperature the shortcut prefactor was derived for.

Problems

For the cell of Worked Example 1, find \(E\) if instead \([\text{Zn}^{2+}]=1.0\times10^{-4}\,\text{M}\) and \([\text{Cu}^{2+}]=0.10\,\text{M}\).
Solution
\(Q=(1.0\times10^{-4})/0.10=1.0\times10^{-3}\). \(E=1.10-\frac{0.0592}{2}\log_{10}(1.0\times10^{-3})=1.10-\frac{0.0592}{2}(-3)=1.10+0.0888=1.19\,\text{V}\), higher than \(E^\circ\) since \(Q<1\) favours the forward reaction.
A cell has \(E^\circ=0.20\,\text{V}\) and \(n=1\) at \(298\,\text{K}\). Find \(K\).
Solution
Setting \(E=0\): \(0=E^\circ-\frac{0.0592}{n}\log_{10}K \Rightarrow \log_{10}K = \frac{nE^\circ}{0.0592}=\frac{1\times0.20}{0.0592}=3.38\Rightarrow K\approx2.4\times10^3\).
Explain why a pH meter's glass electrode potential is a direct application of the Nernst equation.
Solution
The glass electrode responds selectively to \(\text{H}^+\) activity across a thin glass membrane, effectively forming a concentration-cell-like arrangement (\(E^\circ=0\) between the internal reference solution and the external sample) whose entire measured potential is set purely by the Nernst equation's \(Q\) term, here a ratio of \(\text{H}^+\) activities — making the measured potential a direct, linear function of \(\text{pH}\), exactly as concentration cells are described in the Corollaries.