The Nernst Equation & Galvanic Cells

Where the standard potential comes from

A galvanic cell pairs an oxidation half-reaction at the anode with a reduction half-reaction at the cathode, and electrons flow spontaneously from anode to cathode when the reaction is thermodynamically favourable. The standard cell potential, written E°, is measured under fixed reference conditions — 1 M concentrations, 1 atm partial pressure, 25°C — and can be looked up as the difference of two tabulated standard reduction potentials, E°cell = E°cathode - E°anode. A positive E°cell corresponds to a negative Gibbs free energy change, delta-G° = -nFE°, which is the thermodynamic reason the reaction proceeds on its own.

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The Nernst equation itself

Real cells rarely sit at exactly 1 M and 25°C. Combining delta-G = delta-G° + RT ln Q (the free energy of a reaction away from standard conditions, where Q is the reaction quotient) with delta-G = -nFE gives the Nernst equation, named for Walther Nernst, who derived it in 1889.

E = E deg - (RT / nF) * ln(Q)

at 25 degC (298 K), switching to log base 10:
E = E deg - (0.0592 V / n) * log10(Q)

Q = reaction quotient = [products] / [reactants]  (activities)

Reading the equation: concentration, temperature, and n

Each term has a direct physical reading. Raising product concentration pushes Q up, which lowers E because the cell is being driven closer to equilibrium; raising reactant concentration does the opposite. Higher temperature amplifies the RT/nF prefactor, making the cell's voltage more sensitive to concentration changes. And n, the number of electrons transferred per reaction as written, sits in the denominator: a one-electron process (n = 1) swings its voltage far more for a given change in Q than a two-electron process (n = 2) does for the identical change.

At equilibrium: E = 0 and the link to K

When a cell fully discharges there is no more net driving force left, so E = 0 and the reaction quotient has reached the equilibrium constant, Q = K. Substituting those two facts into the Nernst equation gives a direct algebraic bridge between a purely thermodynamic quantity, E°, and a purely chemical-equilibrium quantity, K.

at equilibrium:  0 = E deg - (RT/nF) * ln(K)
              =>  ln(K) = n F E deg / (R T)

Real-world Nernst potentials: batteries and biology

In a discharging battery, reactants deplete and products accumulate, which is exactly the Nernst-driven voltage sag that separates a real, finite-capacity battery from an idealized constant-voltage source. The same equation, applied separately to each ion species, gives the Nernst potential across a biological cell membrane — the equilibrium voltage for a single ion given its concentration difference between the inside and outside of a cell. Those per-ion Nernst potentials, particularly for potassium and sodium, are the building blocks used to compute a neuron's resting membrane potential and to model how an action potential fires, tying this piece of 19th-century electrochemistry directly to modern neuroscience.

Frequently asked questions

Why does a battery's voltage sag as it discharges?

As a battery discharges, reactant concentrations fall and product concentrations rise, which increases the reaction quotient Q. The Nernst equation shows that cell potential decreases as ln Q increases, so the measured voltage sags below the fresh, standard-condition value even though the underlying chemistry hasn't changed — it's simply responding to the changing concentrations inside the cell.

What happens to the cell potential at equilibrium?

At equilibrium the reaction quotient Q equals the equilibrium constant K, and the cell potential E drops to exactly zero — a fully discharged battery has no remaining driving force to push electrons through an external circuit. Setting E = 0 in the Nernst equation gives a direct algebraic relationship between the standard potential E deg and K, ln K = nFE deg / RT.

How does the Nernst equation apply to neurons?

Applied separately to each ion species, the Nernst equation gives the equilibrium (Nernst) potential across a cell membrane for that ion, based on its concentration difference between the inside and outside of the cell. These per-ion Nernst potentials for potassium, sodium and other ions are the building blocks used to calculate a neuron's resting membrane potential and to model how action potentials are generated.

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The Nernst Equation: Cell Potential Beyond Standard Conditions