The Nernst Equation

Every electrochemical system has stored energy tied up in the redox reactions happening within it. The Nernst equation connects that stored energy to a measurable parameter, voltage. More specifically, it shows how voltage evolves as a reaction proceeds and the system's composition changes.

What Drives the Reaction?

Redox reactions involve the transfer of electrons between chemical species. Whether that transfer happens spontaneously or needs a push depends on which direction lowers the system's free energy (i.e., is thermodynamically favourable). This is captured by the Gibbs free energy change, ΔG.

• In a spontaneous reaction (like a battery discharging), electrons naturally move from a higher-energy state to a lower-energy state, releasing energy in the process.

• In a non-spontaneous reaction (like a battery charging), an external voltage is applied to move electrons against their natural direction, from a lower energy state to a higher energy state.

The energy difference between the reacting species is the driving force for the reaction. This determines how much energy can be harnessed as electrical energy in spontaneous systems, or how much electrical energy needs to be supplied in non-spontaneous systems.

Connecting Thermodynamics to Voltage

The reaction driving force can be expressed in two ways: as free energy or as voltage. These two terms are related through the equation:

Where: ΔG = Gibbs free energy change, n = number of electrons transferred in the reaction, F = Faraday constant, E = cell potential (V)

This relationship shows that voltage is simply a measure of how much free energy per charge is available or needs to be supplied to drive the reaction.

Why the Driving Force Changes Over Time

ΔG is not constant. It shifts as the reaction progresses. Along the reaction, the activities of the reactants decrease, and the activities of the products increase. That means the composition of the system is constantly changing. Since ΔG depends on composition, the driving force also changes, and so does the voltage.

The more negative ΔG is, the stronger the driving force for the forward reaction to proceed, and the higher the voltage. The less negative ΔG, the weaker the driving force, and the lower the voltage.

The Nernst Equation

The Nernst equation captures exactly how voltage changes with composition. It translates the free energy change at a specific reaction composition into a voltage, not just at the start or under idealized conditions. The Nernst equation is written as:

Where: E = cell potential, E° = standard potential (measured under standard-state conditions), R = universal gas constant, T = temperature, F = Faraday constant, n = number of electrons transferred in the reaction, Q = the reaction quotient

The Reaction Quotient, Q

The reaction quotient describes the composition of the system at any given moment. It is the ratio of the activities of products to reactants, each raised to their stoichiometric coefficients. In simpler terms, it shows how much product there is relative to reactant.

Here's how Q affects voltage:

• A larger reaction quotient (Q>1) indicates a higher relative amount of products compared to reactants. In this situation, ΔG becomes less negative, the driving force for the forward reaction becomes smaller, and the cell voltage is lower.

• A smaller reaction quotient (Q < 1) indicates a higher relative amount of reactants compared to products. In this situation, ΔG becomes more negative, the driving force for the forward reaction becomes stronger, and the cell voltage is higher.

• When the cell is at equilibrium (Q = K, where K is the equilibrium constant), it means the rate of the forward reaction is equal to that of the reverse reaction. In this situation, the Gibbs free energy change is zero, there is no overall push causing electrons to flow, and the voltage is zero.

Conclusion

The Nernst equation isn’t just a thermodynamics formula, it’s a window into what’s happening inside any electrochemical system. In batteries, by linking composition to voltage, it explains why voltage changes during charge and discharge, helps track how state of charge evolves, and provides the thermodynamic baseline for modeling and diagnostics. It's hard to optimize what you don’t understand, and the Nernst equation is one of the clearest lenses to see what’s going on beneath the surface of the battery.