By Allan Robinson
Mar 27, 2023 11:26 pm EST

Dragan Smiljkovic/E+/GettyImages

The Nernst equation, named after the German chemist Walther Nernst, quantifies the relationship between an electrochemical cell’s potential and the concentrations of its reactants and products. Think of a ball rolling down a hill: a steep slope (high cell potential) drives the reaction forward, while a flat slope (zero potential) marks equilibrium. The term “cell potential” is synonymous with electromotive force (EMF) and is expressed in volts, equivalent to joules per coulomb.

TL;DR

Cell potential (in volts) indicates how far an electrochemical reaction has progressed toward equilibrium.

The General Form

The most versatile expression of the Nernst equation applies to any temperature and incorporates the ideal gas constant (R = 8.3145 J mol⁻¹ K⁻¹), Faraday’s constant (F = 96 485 C mol⁻¹), the number of electrons transferred (n), absolute temperature (T in Kelvin), and the standard reduction potential (E°) of the cell:

\(E_{cell}=E^{\circ}_{cell}-\frac{RT}{nF}\ln Q\)

Here, the reaction quotient Q is the ratio of product to reactant concentrations at a given instant:

\(Q=\frac{[\text{products}]}{[\text{reactants}]}\)

Because E° is a known constant for each half‑cell, this equation can predict the instantaneous cell potential under non‑standard conditions, making it indispensable for experimental design and analysis.

Standard Conditions

When the temperature is fixed at 298 K (25 °C) and concentrations are expressed in molarity, the general formula simplifies dramatically. Substituting R, F, and T and converting the natural logarithm to a base‑10 logarithm yields:

\(E_{cell}=E^{\circ}_{cell}-\frac{0.0592\,\text{V}}{n}\log Q\)

Example: Zn–Cu Redox Couple

Consider the classic galvanic reaction:

\(\text{Zn}_{(s)} + \text{Cu}^{2+}_{(aq)} \rightleftharpoons \text{Zn}^{2+}_{(aq)} + \text{Cu}_{(s)}\)

Given E°_{cell} = +1.10 V and, after one minute, concentrations of \([\text{Cu}^{2+}] = 0.05\,\text{M}\) and \([\text{Zn}^{2+}] = 1.95\,\text{M}\), we compute:

\(n = 2\) (two electrons are transferred)

\(Q = \frac{[\text{Zn}^{2+}]}{[\text{Cu}^{2+}]} = \frac{1.95}{0.05} = 39\)

Plugging into the simplified equation:

\(E = 1.10\,\text{V} - \frac{0.0592\,\text{V}}{2}\log 39 = \mathbf{1.053\,V}\)

What Does This Mean?

The reaction quotient Q directly influences the cell potential because it reflects the system’s deviation from equilibrium. The relationship mirrors that between Gibbs free energy (ΔG) and reaction progress:

• Q = 1 → E = E° (the reaction proceeds as expected)
• Q < 1 → E > E° (the reaction is more favorable than the standard state)
• Q > 1 → E < E° (the reaction is approaching equilibrium but remains favorable)
• Q = K → E = 0 (true equilibrium; no net cell potential)

Here, K is the equilibrium constant that marks the concentrations at which the reaction ceases to advance. This framework allows chemists to predict reaction direction, calculate energy changes, and design efficient electrochemical devices.

Other Applications

Beyond batteries, the Nernst equation underpins biological membrane potentials. The Goldman equation, for example, expands on Nernst by incorporating ion permeability and membrane geometry to predict the resting potential of cells. In biotechnology, accurate modeling of membrane voltage is essential for nerve impulse propagation, muscle contraction, and cellular signaling.

Galvanic cells—often called electrochemical batteries—rely on the principle that a voltage difference between two electrodes converts stored chemical energy into electrical work. Mastery of the Nernst equation enables engineers to optimize electrode materials, electrolyte composition, and operating conditions for maximum performance.