18.6: The Nernst Equation

Nonstandard Reaction Conditions

The interconnection between standard cell potentials and various thermodynamic parameters such as the standard free energy change ΔG° and equilibrium constant K has been previously explored. For example, a redox reaction involving zinc(II) and tin(II) ions at 1 M concentration with Eº_cell = +0.291 V and ΔG° = −56.2 kJ is spontaneous.

Discharge of this cell, however, results in a change in the reactant concentration and a steady decrease of the cell potential. At such a condition, however, the relationships between the cell potential and the thermodynamic parameters cannot be easily established as they only hold true at standardized conditions of concentration, temperature, and pressure (i.e., 1 M concentration, 298 K or 25 °C and a pressure of 1 atmosphere). Many redox reactions of significant scientific interest occur under nonstandard state conditions, e.g., different reactant concentrations in a galvanic cell or concentration gradients occurring across biological membranes. Hence, it becomes important to calculate the potentials of such systems.

When the concentration of zinc ions in the reaction is lesser and the concentration of the tin ions greater compared to standard conditions, the spontaneity of the redox reaction can be qualitatively predicted using Le Chatelier’s Principle. Given the higher concentration of product to reactant, the reaction has a higher tendency to proceed in the direction favoring the generation of the products. This results in a higher cell potential value or E_cell than that of the E°_cell value.

This reaction proceeds in the forward direction; however, the quantitative value of this cell potential cannot be easily determined.

Derivation of the Nernst Equation for Redox Reactions Occurring under Nonstandard Conditions

The relationship between E_cell and E°_cell values can be derived from the previously established relationship between free energy changes at standard and nonstandard conditions, which is given as follows:

ΔG is the change in free energy, ΔG° is the standard change in free energy, R is the gas constant (value = 8.314 J/mol∙K), and Q is the reaction quotient, which accounts for the change in free energy due to the difference in the reaction mixtures’ composition. The Q value is omitted if the reactants are solid.

On substitution of the equation relating free energy change to cell potential, a modified equation is obtained, known as the Nernst equation.

The Nernst equation describes the variation in the potential of a redox system (such as a galvanic cell) from its standard state value. It is dependent on the number of electrons transferred during the redox reaction, n, the temperature measured in kelvin, T, and the reaction mixture composition given as Q.

A simplified form of the Nernst equation for most work is one in which values for the fundamental constants (R and F) and a factor converting from natural log to base-10 logarithms have been included:

Under standard state conditions, the value of the reaction quotient Q is unity, whose logarithm is zero. This is due to the equal concentration of reactants and products at standard state conditions. Here, E_cell is equal to E°_cell. A Q value less than one indicates a higher concentration of reactants, shifting the reaction equilibrium to the right and, thus, yielding a higher value of cell potential. A Q value greater than one indicates a higher product concentration, driving the reaction to the left, and a lower value of cell potential. At equilibrium, the Q value is equal to K, and the cell potential becomes zero, i.e., the reaction shows no tendency to proceed in either direction. This explains why batteries “die” on continuous discharge: the decrease in reactant concentration drives the reaction towards equilibrium and its cell potential steadily decreases to zero.

This text is adapted from Openstax, Chemistry 2e, Section 17.4: Potential, Free Energy, and Equilibrium.

A zinc-copper galvanic cell at standard conditions has a cell potential of +1.10 V and a ΔG value of −212 kJ, indicating that it operates spontaneously. However, as the reactant’s concentration changes during the cell’s discharge, it leads to a gradual decrease of cell potential until the reaction stops completely. 

Conditions like these are called nonstandard. Here, the established standard values of cell potential, Gibbs free energy, and the equilibrium constant are no longer valid.

Nonstandard conditions are prevalent in many reactions ranging from redox reactions to ion gradients in neuronal membranes. But how is an accurate cell potential determined in such systems?

If the concentration of a reactant is greater, and the concentration of a product is smaller compared to standard conditions, then Le Châtelier’s Principle is used to determine the reaction’s direction qualitatively; however, it cannot be used for quantifying the deviating cell potential.

Thus, this necessitates establishing a relationship between the cell potentials for cells under standard and nonstandard conditions. Recall that the free energy changes under standard and nonstandard conditions are related. 

Substituting the equation of change in free energy with the cell potential results in a modified equation known as the Nernst equation. The Nernst equation determines how the cell potential differs from its standard value depending on the number of electrons transferred, temperature, and reaction composition. 

The reaction quotient, Q, accounts for the change in free energy due to the difference in the reaction mixtures’ composition. If reactants are solid, Q is omitted.

Under standard state conditions, the value of Q is unity and the concentration of reactants and products is equal. The logarithm of one is zero, so the cell potential equals the standard cell potential. 

A Q value less than one indicates a higher concentration of reactants compared to products, which shifts the equilibrium to the right, increasing the cell potential. 

A Q value greater than one indicates a higher product to reactant concentration, driving the reaction to the left and lowering the cell potential.

At equilibrium, the Q value is equal to K, and the cell potential becomes zero. 

The Nernst equation explains why electrochemical batteries “die” post-discharge: as the reactant concentration decreases, the cell approaches equilibrium conditions and its potential decreases to zero