Nernst Equation - Battery Design

The Nernst equation relates the equilibrium potential of an electrochemical half-reaction (or a full cell) to its standard potential and the activities, effectively concentrations, of the species taking part. It’s the thermodynamic link between how much reactant or product is present and what voltage that reaction produces: the same link that, for a battery, connects lithium content in an electrode to the cell’s open circuit voltage.

E = E_0 – \frac{RT}{nF}\ln(Q)

where:

E is the electrode (or cell) potential under the actual, non-standard conditions (Volts)
E⁰ is the standard electrode (or cell) potential (Volts)
R is the universal gas constant (8.314 J·mol^-1·K^-1)
T is the absolute temperature (Kelvin)
n is the number of electrons transferred in the reaction
F is the Faraday constant (96,485 C·mol^-1)
Q is the reaction quotient – the ratio of the activities of the products to the reactants

At 25°C, and switching to base-10 logarithms, this is often written in the simplified form E = E₀ – (0.0592/n) × log₁₀(Q).

Why it matters for battery design

The textbook form above assumes two chemically distinct species trading electrons, which suits a classic redox couple but doesn’t quite match what happens inside a lithium-ion electrode. There, lithium ions insert into and are extracted from a single host lattice, so what varies is not “concentration of species A versus species B” but the fraction of lattice sites already occupied by lithium. For an intercalation electrode this gives a specific form of the Nernst equation:

\mu = E_0 – kT\ln\left[\frac{x}{1-x}\right]

where x is the ratio of occupied to available intercalation sites in the host material, k is the Boltzmann constant, and E₀ is the standard redox potential of that electrode material [1]. This is part of why a lithium-ion cell’s Open Circuit Voltage traces a smooth curve against State of Charge rather than sitting at one fixed value: as x moves from empty towards full, the ln[x/(1-x)] term runs from strongly negative, through zero at half-occupancy, to strongly positive. A full cell’s OCV is then the difference between two such intercalation potentials, one per electrode, each following its own version of this relationship over its own occupancy range.

The same physical relationship also underpins an empirical curve-fitting model sometimes used in Battery Management Systems, which approximates the measured OCV-SOC curve as OCV = k₀ + k₁·ln(s) + k₂·ln(1–s), with s the state of charge and k₀, k₁, k₂ fitted to test data rather than derived from first-principles material properties [1].

References

P. Pillai, J. Nguyen, B. Balasingam, Performance Analysis of Empirical Open-Circuit Voltage Modeling in Lithium Ion Batteries, Part-1: Performance Measures, arXiv:2306.16542, 2023