Nernst Equation

Every cell you have met so far — the Daniell cell, the dry cell, the electrode-potential tables — was quoted under one restrictive condition: all ions at 1 M, all gases at 1 bar, 298 K. Real cells rarely sit there. A battery discharges, its ion concentrations drift, and its voltage sags. The Nernst equation is the tool that tracks this: it converts the fixed, tabulated $E^\circ_{cell}$ into the actual, moment-by-moment $E_{cell}$ for whatever concentrations are really present. It is the single most numerically tested idea in the electrochemistry unit — board papers lean on it for cell EMF, for concentration cells, and for reading off the equilibrium constant $K_c$ straight from a voltage. Master the log-Q bookkeeping here and the rest of the chapter's numericals fall into place.

The core insight

The Nernst equation, $E_{cell} = E^\circ_{cell} - \dfrac{RT}{nF}\ln Q$ , corrects the standard cell EMF for whatever concentrations (or pressures) are actually present, via the reaction quotient $Q$ .

The core idea

A cell's EMF is not a fixed number stamped on the cell — it depends on how far the reaction quotient $Q$ has drifted from the standard state of $Q = 1$ . The Nernst equation is the correction term that captures this drift. For a general reaction $a\,A + b\,B \rightarrow c\,C + d\,D$ occurring in a cell, the relation is

E_{cell} = E^\circ_{cell} - \frac{RT}{nF}\ln Q

where: $E^\circ_{cell}$ is the standard cell EMF (from tabulated electrode potentials), $R = 8.314\ \text{J K}^{-1}\text{mol}^{-1}$ , $T$ is the temperature in kelvin, $n$ is the number of electrons transferred in the balanced cell reaction, $F = 96{,}500\ \text{C mol}^{-1}$ , and $Q = \dfrac{[\text{C}]^c[\text{D}]^d}{[\text{A}]^a[\text{B}]^b}$ is the reaction quotient (products over reactants, pure solids and liquids omitted, exactly as in an equilibrium expression). At $T = 298\ \text{K}$ , converting from natural log to base-10 log collects the constants $\dfrac{2.303\,RT}{F}$ into a single number:

E_{cell} = E^\circ_{cell} - \frac{0.0591}{n}\log Q \quad \text{(at 298 K)}

This is the form used in nearly every board numerical. The figure below shows how $E_{cell}$ moves away from $E^\circ_{cell}$ as $Q$ moves away from 1 — the equation is nothing but a straight line in $\log Q$ .

Figure

A line graph of cell EMF against log Q showing a downward sloping straight line crossing the standard EMF value at log Q equals zero

At equilibrium, the cell can do no more net work, so $E_{cell} = 0$ and $Q$ has reached $K_c$ . Substituting directly gives a bridge between the tabulated standard EMF and the equilibrium constant:

\log K_c = \frac{nE^\circ_{cell}}{0.0591}

Link to Gibbs energy. The maximum electrical work a cell can deliver equals the fall in Gibbs free energy: $\Delta G = -nFE_{cell}$ and $\Delta G^\circ = -nFE^\circ_{cell} = -RT\ln K_c$ . This is exactly the relation used in the derivation below to build the Nernst equation from scratch.

Concentration cells apply the same equation with $E^\circ_{cell} = 0$ , since both electrodes are made of the same material. The entire EMF then comes purely from the concentration difference between the two half-cells, driving ions from the more concentrated half-cell (cathode) toward the more dilute one (anode). Notice the temperature dependence hiding inside the $0.0591/n$ constant: that value is only valid at $T = 298\ \text{K}$ . At any other temperature the correct multiplier is $2.303\,RT/F$ , recomputed for that $T$ — a detail examiners sometimes test by quoting a non-298 K temperature and expecting you to fall back on the $\ln Q$ form rather than blindly reusing $0.0591$ . Also note that $n$ , the number of electrons, is fixed by how the cell reaction is balanced: doubling every stoichiometric coefficient in a balanced equation doubles $n$ but leaves $Q$ 's value unchanged in a way that keeps $E_{cell}$ itself the same — EMF is an intensive property, independent of how the equation happens to be written.

Derivation ABuilding the Nernst equation from Gibbs free energy

The goal is to show that the Nernst equation is not an extra postulate but a direct algebraic consequence of two relations you already know from thermodynamics and electrochemistry: how Gibbs free energy tracks a reaction quotient away from equilibrium, and how Gibbs free energy connects to cell EMF; combining them and rearranging for $E_{cell}$ produces the equation used throughout this chapter.

Step 1. Start from the thermodynamic relation linking the free energy change of a reaction at any composition, $\Delta G$ , to its standard value $\Delta G^\circ$ and the reaction quotient $Q$ : $\Delta G = \Delta G^\circ + RT\ln Q$ . This says the reaction's actual driving force differs from its standard driving force by an amount that depends on how far the concentrations are from standard conditions.

Step 2. Bring in the electrochemical link between free energy and EMF: the maximum non-expansion work a cell can perform equals $-\Delta G$ , and this work is also $nFE_{cell}$ (charge $nF$ moved through potential $E_{cell}$ ). So $\Delta G = -nFE_{cell}$ , and identically at the standard state, $\Delta G^\circ = -nFE^\circ_{cell}$ .

Step 3. Substitute both EMF expressions into the Step 1 relation: $-nFE_{cell} = -nFE^\circ_{cell} + RT\ln Q$ .

Step 4. Divide through by $-nF$ to isolate $E_{cell}$ : $E_{cell} = E^\circ_{cell} - \dfrac{RT}{nF}\ln Q$ . This is exactly the Nernst equation. Converting $\ln$ to $\log_{10}$ (multiply by $2.303$ ) and inserting $R = 8.314$ , $F = 96{,}500$ , $T = 298\ \text{K}$ collapses $\dfrac{2.303RT}{F}$ to $0.0591\ \text{V}$ , giving the familiar working form $E_{cell} = E^\circ_{cell} - \dfrac{0.0591}{n}\log Q$ .

Worked example

Problem: For the cell $\text{Zn}(s)\mid \text{Zn}^{2+}(0.01\ \text{M}) \parallel \text{Cu}^{2+}(1.0\ \text{M})\mid \text{Cu}(s)$ with $E^\circ_{cell} = 1.10\ \text{V}$ , the EMF at 298 K was found earlier to be $1.159\ \text{V}$ (from $Q = 0.01$ ). Now use that same cell's standard EMF to estimate its equilibrium constant $K_c$ .

Step 1. Identify $n$ from the balanced cell reaction $\text{Zn}(s) + \text{Cu}^{2+}(aq) \rightarrow \text{Zn}^{2+}(aq) + \text{Cu}(s)$ : two electrons are transferred, so $n = 2$ .

Step 2. Apply the equilibrium relation with the standard EMF, never the instantaneous one: $\log K_c = \dfrac{nE^\circ_{cell}}{0.0591} = \dfrac{2 \times 1.10}{0.0591} = 37.23$ .

Step 3. Undo the logarithm: $K_c = 10^{37.23} \approx 1.7\times10^{37}$ .

Step 4. Interpret the size. A $K_c$ this enormous means the reaction essentially goes to completion — at equilibrium, virtually all the $\text{Cu}^{2+}$ has been reduced and virtually all the zinc oxidised, consistent with $E^\circ_{cell}$ being a healthy positive $1.10\ \text{V}$ : a strongly spontaneous cell reaction always corresponds to a very large $K_c$ . $K_c \approx 1.7\times10^{37}$ .

Common traps

Mixing $\ln$ and $\log$ without converting the constant. The $\dfrac{RT}{nF}$ form needs natural log; the $0.0591/n$ form (at 298 K) needs base-10 log — swapping them gives answers off by a factor of $2.303$ .
Writing $Q$ upside down. $Q$ is always products over reactants for the balanced overall cell reaction, not "cathode concentration over anode concentration" by rote — write the full equation first, then build $Q$ from it.
Including solids or pure liquids in $Q$ . Only aqueous ion concentrations (or gas pressures) belong in the expression; metal electrodes and solvent water are omitted.
Using the non-standard $E_{cell}$ in $\log K_c = nE^\circ_{cell}/0.0591$ . That shortcut only works with the standard EMF $E^\circ_{cell}$ , because it comes from setting $E_{cell} = 0$ at equilibrium.

Ask the tutor: want to work through a trickier multi-ion concentration cell, or see how a measured $E_{cell}$ is used to back-calculate an unknown ion concentration?

Why does the Nernst equation use log base 10 with 0.0591 instead of natural log?

Both forms are correct — they are the same equation, just with different constants. The fundamental version from thermodynamics uses natural log: $E_{cell} = E^\circ_{cell} - (RT/nF)\ln Q$ . Converting $\ln Q$ to $\log_{10}Q$ needs a factor of $2.303$ (since $\ln x = 2.303\log_{10}x$ ), and bundling $2.303 \times R \times T / F$ at $T = 298\ \text{K}$ gives exactly $0.0591\ \text{V}$ . Board papers almost always want the $0.0591/n$ form because $\log_{10}$ of round concentrations like $0.01$ or $10^{-4}$ is easy to evaluate by hand.

Why is Q written as products over reactants and not the other way round?

This mirrors the equilibrium constant expression exactly, because the Nernst equation is derived from the same thermodynamic relation, $\Delta G = \Delta G^\circ + RT\ln Q$ , that defines $Q$ for any reaction. For the cell reaction as written left to right (oxidation at the anode feeding electrons to reduction at the cathode), $Q$ is the concentrations of the species produced, raised to their stoichiometric coefficients, divided by the concentrations of the species consumed. Get the direction of the overall reaction wrong and $Q$ flips to its reciprocal, which flips the sign of the correction term and produces an EMF error.

Can the Nernst equation give a cell EMF of zero even when concentrations are non-standard?

Yes, and this is exactly what happens at equilibrium: as the reaction proceeds, reactant concentrations fall and product concentrations rise, so $Q$ climbs toward $K_c$ and the correction term $\dfrac{0.0591}{n}\log Q$ grows until it exactly cancels $E^\circ_{cell}$ . At that point $E_{cell} = 0$ , meaning the cell can no longer push a current — it is "dead," even though concentrations are non-standard and, in fact, non-standard concentrations are the entire reason it stopped. A freshly assembled cell far from equilibrium is exactly what gives a non-zero, useful EMF.

Nernst Equation

The core idea

Worked example

Common traps

Before this

Where this leads