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⚡ Nernst Equation
#22

§1 Interactive Simulation

Concentration Cell
pH-Dependent (Quinhydrone)
Battery Discharge
Equilibrium (E→0)
Membrane Potential
E vs log Q
E vs pH (pH electrode)
E vs T
E vs [conc]
ΔG vs Time (discharge)
E°ₒₑₗₗ
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Eₒₑₗₗ
—
Q
—
ΔG
—
K (eq.)
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RT/nF
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E° trace
Show formula
0.0592 form (25°C)

§2 The Idea, Step by Step

🔋 Why a Battery's Voltage Sags as It Runs Down

A brand-new AA battery reads a little higher on a meter than a tired one that has powered a flashlight all night, even before it dies completely. Nothing about the chemicals inside has changed identity — it is still the same metals and the same reaction. What changed is how much reactant is left versus how much product has piled up. The Nernst equation is the rule that turns that running tally of "how far along is the reaction?" into the exact voltage you would measure.

Start from the cell's headline number, the standard cell potential $E^\circ_{\text{cell}}$ — the voltage when every dissolved species sits at the textbook concentration of 1 M. Real cells almost never sit there, so we track a single bookkeeping number called the reaction quotient $Q$: pile up products and $Q$ grows, leave lots of reactants and $Q$ stays small. The working voltage drops a little for every tenfold rise in $Q$. At room temperature the simplest form is:

$$E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{0.0592}{n}\,\log Q \qquad(\text{25 }^\circ\text{C})$$ Here $n$ is how many electrons the reaction shuffles per round. For a copper concentration cell ($E^\circ_{\text{cell}}=0$, $n=2$) with the cathode at $1.0$ M and the anode at $0.1$ M, $Q = 0.1/1.0 = 0.1$, so $E = 0 - \tfrac{0.0592}{2}\log(0.1) = +0.030$ V. A pure concentration difference — no special chemistry — still pushes about $30$ mV.

The precise statement keeps the temperature visible instead of hiding it in the number $0.0592$:

$$E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF}\ln Q$$ where $R=8.314$ J·K⁻¹·mol⁻¹, $T$ is the absolute temperature, and $F=96\,485$ C/mol is Faraday's constant. The factor $0.0592$ V is just $(RT/F)\ln 10$ evaluated at $298$ K — warm the cell and that slope grows. In the sim, the preset sets $E^\circ_{\text{cell}}$, the two concentration sliders set $Q$, the $n$ slider sets the electron count, and the temperature slider scales the $RT/nF$ term you can read live in the side panel.

Try this in the sim above: (1) On the "E vs log Q" graph, drag [Cathode] down and watch the operating dot slide down a straight line whose slope is exactly $-0.0592/n$ V per decade. (2) Push the n slider from 1 to 3 and see that same line flatten — more electrons means less voltage swing per tenfold concentration change. (3) Open the Battery Discharge tab, press Play, and watch $Q$ climb toward $K$ while $E$ collapses to $0$ — that is a battery going "dead" at equilibrium, where $\Delta G = 0$.

§3 Equation Derivation

⚡ The Nernst Equation

$$\boxed{\;E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF}\ln Q\;}\qquad\text{(at 298 K: }E = E^\circ - \frac{0.0592}{n}\log Q\text{)}$$

Symbol Definitions

SymbolMeaningSI Unit
$E_{\text{cell}}$Cell potential at non-standard conditionsV
$E^\circ_{\text{cell}}$Standard cell potential ([all]=1 M, P=1 bar)V
$R$Universal gas constant8.314 J·K⁻¹·mol⁻¹
$T$Absolute temperatureK
$n$Number of electrons in balanced cell reactionmol
$F$Faraday's constant96 485 C/mol
$Q$Reaction quotientdimensionless

Step-by-Step Derivation from First Principles

1Start with Gibbs free energy under non-standard conditions. For a general reaction at any composition: $$\Delta G = \Delta G^\circ + RT\ln Q$$ where $Q$ is the reaction quotient (same form as $K$ but with current activities/concentrations).
2Express $\Delta G$ in electrical terms. The maximum non-PV work from a reversible cell is: $$\Delta G = -nFE_{\text{cell}},\qquad \Delta G^\circ = -nFE^\circ_{\text{cell}}$$ where the negative sign reflects that spontaneous reactions ($\Delta G < 0$) give positive $E$.
3Substitute and rearrange. $$-nFE_{\text{cell}} = -nFE^\circ_{\text{cell}} + RT\ln Q$$ Divide both sides by $-nF$: $$\boxed{E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF}\ln Q}$$
4Convert to base-10 log at 298 K. Using $\ln Q = 2.303\log Q$, $R = 8.314$, $F = 96485$, $T = 298$: $$\frac{RT}{nF}\cdot 2.303 = \frac{(8.314)(298)(2.303)}{n\cdot 96485} = \frac{0.0592}{n}\text{ V}$$ Hence the famous engineering form: $$E = E^\circ - \frac{0.0592}{n}\log Q\qquad\text{(at 298 K)}$$
5Equilibrium limit. When $E_{\text{cell}} \to 0$, the reaction reaches equilibrium and $Q \to K$: $$0 = E^\circ - \frac{RT}{nF}\ln K\;\Longrightarrow\;\ln K = \frac{nFE^\circ}{RT}$$ A 1.10 V Daniell cell has $\log K = 37$ — essentially complete reaction.
6pH-dependent reactions. If H⁺ is involved, e.g. MnO₄⁻ + 8H⁺ + 5e⁻ → Mn²⁺ + 4H₂O: $$E = E^\circ - \frac{0.0592}{5}\log\frac{[\text{Mn}^{2+}]}{[\text{MnO}_4^-][\text{H}^+]^8} = E^\circ - 0.0947\,\text{pH}\;\text{(at standard [Mn},\text{MnO}_4^-])$$ This is why MnO₄⁻ is a stronger oxidizer in acid than in base.

Worked Example — Concentration Cell

Problem: Cu | Cu²⁺ (0.01 M) ‖ Cu²⁺ (1.00 M) | Cu — both half-cells use Cu electrodes.
$E^\circ_{\text{cell}} = 0$ (same metal!), but the cell still produces voltage due to concentration gradient.
Cell reaction: Cu²⁺(1.00 M, cathode) → Cu²⁺(0.01 M, anode); $n = 2$. $$Q = \frac{[\text{Cu}^{2+}]_{\text{anode}}}{[\text{Cu}^{2+}]_{\text{cathode}}} = \frac{0.01}{1.00} = 0.01$$ $$E = 0 - \frac{0.0592}{2}\log(0.01) = -0.0296\times(-2) = +0.0592\text{ V}$$ A small but measurable voltage — this is the basis of pH meters and ion-selective electrodes.

📚 Atkins & de Paula — Physical Chemistry, 11th Ed., §6E.2: "The Nernst equation" | Levine — Physical Chemistry, 6th Ed., §13.4 | Skoog et al. — Fundamentals of Analytical Chemistry, 9th Ed., Ch. 18.4

§4 Frequently Asked Questions

📚 LibreTexts Chemistry — "Nernst Equation" (chem.libretexts.org) | Khan Academy — Nernst equation | MIT OCW 5.111

§5 Common Misconceptions

📚 Sanger & Greenbowe — J. Chem. Educ. 74, 819 (1997) "Common student misconceptions in electrochemistry" | Ahtee & Varjola — Int. J. Sci. Educ. 20, 305 (1998) | Taber — Chemical Misconceptions (RSC, 2002)