Nernst Equation

From standard potential to actual voltage

The standard cell potential E°_cell is a single tabulated number that assumes a very specific set of conditions: every dissolved species at exactly 1 M, every gas at 1 atm, temperature held at 25 °C. Real cells almost never live there. A car battery's H₂SO₄ thins as it discharges. A pH electrode is built around a 10⁻⁷ M H⁺ solution. A neuron's membrane potential depends on potassium twenty-eight times more concentrated inside than out. To predict voltage in any of those situations, you need the Nernst equation.

The equation falls out of substituting Q for K in ΔG = −nFE combined with ΔG = ΔG° + RT·ln(Q):

−nFE  = −nFE°  +  RT·ln(Q)

Divide both sides by −nF:

E = E° − (RT/nF) · ln(Q)

Q has the same form as K but is evaluated at actual concentrations, not equilibrium ones. For Zn(s) + Cu²⁺ → Zn²⁺ + Cu(s), Q = [Zn²⁺] / [Cu²⁺]. Solids have unit activity and don't appear. As the cell discharges (Zn²⁺ accumulates, Cu²⁺ depletes), Q grows, ln(Q) grows, and E falls below E°. When E reaches zero, Q = K — chemical equilibrium, and the cell is dead.

Worked example: Daniell cell at non-standard concentrations

Suppose a Daniell cell is set up with [Cu²⁺] = 0.010 M and [Zn²⁺] = 1.0 M at 25 °C. At standard conditions, E° = +1.10 V; the question is what voltmeter reading to expect.

n = 2 (electrons transferred per Cu²⁺ reduced)
Q = [Zn²⁺] / [Cu²⁺] = 1.0 / 0.010 = 100
log10(Q) = 2

E = E° − (0.0592 / n) · log10(Q)
  = 1.10 V − (0.0592 / 2) · 2
  = 1.10 V − 0.0592
  = 1.041 V

The cell delivers about 60 mV less than standard EMF — small but measurable, which is why a freshly-built voltaic cell rarely matches the textbook value to better than ±50 mV.

Push further. Drop [Cu²⁺] to 10⁻⁶ M, keep [Zn²⁺] at 1.0 M; then Q = 10⁶ and the voltage falls to 1.10 − 0.0296·6 = 0.92 V. To stop the cell entirely (E = 0), 1.10 = 0.0296 · log Q gives log Q = 37.2 and K = 10³⁷ — the equilibrium constant for the Daniell reaction, computed directly from the cell potential.

Six everyday applications of the Nernst equation

Application	What is measured	Nernst expression	Slope
pH meter	[H⁺] of sample	ΔE = (0.0592 / 1) · log([H⁺]inside / [H⁺]outside)	59.2 mV per pH unit
Ion-selective electrode (Na⁺, K⁺, Ca²⁺)	Activity of the chosen ion	E = E° + (RT/zF) · ln(aion)	59.2 mV (z=1), 29.6 mV (z=2)
Neuron membrane (resting)	Membrane potential from [K⁺] gradient	EK = (RT/F) · ln([K⁺]out / [K⁺]in)	61.5 mV per decade at 37 °C
Lithium-ion discharge	Voltage drop with state of charge	E = E° − (RT/nF)·ln(Qcell)	Plateau-shaped, with intercalation steps
Corrosion polarisation	Local pitting risk on iron	EFe = E° + (RT/2F)·ln([Fe²⁺])	29.6 mV per decade
Concentration cell	Ratio of two activities of same species	E = (RT/nF) · ln(a1 / a2)	59.2 mV per decade (n=1)

The same equation governs every row. The differences are in the value of n, the relevant Q, and whether the temperature is 25 °C or something else (37 °C raises the slope from 59.2 to 61.5 mV per decade — small but non-trivial in pharmaceutical work).

Worked example: a pH meter applied to a sample

A glass-membrane pH electrode behaves as a concentration cell selective to H⁺. Inside the bulb, a fixed [H⁺] of around 10⁻⁷ M is maintained by a buffered Ag/AgCl reference. The bulb's outer face contacts the sample. The membrane potential, ΔE, follows the Nernst equation for a one-electron process:

ΔE = (0.0592 V) · log10([H⁺]_inside / [H⁺]_outside)
   = 0.0592 · (pH_outside − pH_inside)
   = 0.0592 · ΔpH        (at 25 °C)

If the sample is pH 4 and inside reference is pH 7, ΔpH = −3 and ΔE = −0.178 V — a 178 mV signal. Two-point calibration with pH 4 and pH 7 buffers sets offset and slope (modern membranes deliver 96-99 percent of theoretical Nernstian slope). That is why a working pH meter requires recalibration each session.

Standard E° vs Nernst E — when each applies

	Standard E°	Nernst E
Concentrations	All 1 M	Any value, including very dilute
Gas pressures	1 atm	Any value
Temperature	25 °C	Any T (RT changes accordingly)
What it predicts	Reference voltage at standard state	Actual voltmeter reading
Direct measurement?	Only at carefully prepared standard conditions	What real cells deliver in practice
Used for	Tabulating reduction potentials, ΔG° calculations	pH meters, batteries, corrosion, biology
Reduces to E° when	Always (it is the constant)	Q = 1, i.e. all species at unit activity

Real-world specifications

• Commercial pH meters. Resolution typically ±0.01 pH unit (≈0.6 mV), within the Nernstian limit. High-end meters log temperature and apply (RT/F) correction so a 37 °C blood-gas measurement uses the right slope.
• Li-ion fuel-gauge ICs. Chips like BQ34Z100 sample pack voltage during rest and apply Nernstian cathode models to estimate state-of-charge within 1 percent. Without correction, voltage-only inference is off by 10-20 percent.
• Glucose strips. Coulometric Nernst cells where glucose oxidase generates H₂O₂, oxidised at a Pt anode. Current scales with [glucose]; integrated charge gives the reading in ~5 seconds.
• ECG electrodes. Cardiac action potentials are fundamentally Nernstian — the 30 mV ventricular depolarisation is Na⁺'s Nernst pull; the −80 mV resting baseline is K⁺'s.

Variants and refinements

• Goldman-Hodgkin-Katz (GHK). Generalises Nernst to a membrane permeable to multiple ions weighted by permeability P. Used for resting potentials when K⁺, Na⁺, and Cl⁻ all contribute.
• Nernst-Einstein. Links the diffusion potential to mobility μ and diffusion constant D: D = (RT/zF)·μ. The same R, T, z, F govern transport as govern equilibrium.
• Tafel approximation. When current flows, overpotential develops on top of the Nernstian voltage. η = a + b·log(i) at high currents — layered on top of Nernst.
• Activity-corrected form. Replace concentrations with activities a = γ·c. At seawater ionic strength (~0.7 M), γ for Cl⁻ ≈ 0.66 — uncorrected Nernst overestimates Cl⁻ activity by 50 percent.

Common pitfalls

• Solids and pure liquids have activity 1. They don't appear in Q. Inserting [Zn(s)] or [H₂O(l)] makes Q meaningless.
• Mixing natural and common log. 0.0592/n uses log₁₀; RT/nF uses ln. Substituting without converting underestimates by 2.303.
• Wrong sign for ion-selective electrodes. Half-reaction written as reduction: increasing [oxidised] raises E. Reverse the half-reaction and the slope flips. Anchor at standard conditions to check.
• 25 °C constants at 37 °C. Body-temperature work needs 0.0617 V instead of 0.0592 V — a 4 percent error that matters in clinical sensors.
• Treating Q as K. Q is the same expression at current concentrations; Q = K only at equilibrium, where E = 0. Confusing them predicts zero voltage from a working cell.
• Ignoring junction and offset potentials. Real electrodes have mV-scale junction potentials and surface offsets. Two-point calibration absorbs both.