The Nernst Equation & Thermodynamics

Beyond Standard Conditions

Standard electrode potentials are measured assuming the concentration of all species is unity (1 M). But what happens in real-world batteries where concentrations change over time?

The Nernst Equation allows us to calculate the electrode potential at any concentration. For a generic reduction half-reaction $M^{n+}(aq) + ne^- \rightarrow M(s)$, the potential is given by:

$E_{(M^{n+}/M)} = E^\circ_{(M^{n+}/M)} - \frac{RT}{nF} \ln \frac{1}{[M^{n+}]}$

Example 2.1: Non-Standard Cell Potential

Problem. Represent the cell in which the following reaction takes place:

$Mg(s) + 2Ag^+(0.0001M) \rightarrow Mg^{2+}(0.130M) + 2Ag(s)$

Calculate its $E_{(cell)}$ if $E^\circ_{(cell)} = 3.17$ V.

Equilibrium and Gibbs Energy

If you leave a circuit closed, the battery will eventually die. As the chemical reaction proceeds, the reactant concentration drops, product concentration rises, and the cell voltage decreases.

When the system reaches equilibrium, the reaction stops and the voltmeter reads exactly zero ($E_{(cell)} = 0$). The reaction quotient $Q$ becomes the equilibrium constant $K_c$.

Example 2.2: Calculating Equilibrium Constant

Problem. Calculate the equilibrium constant of the reaction:

$Cu(s) + 2Ag^+(aq) \rightarrow Cu^{2+}(aq) + 2Ag(s)$

Given: $E^\circ_{(cell)} = 0.46$ V

Example 2.3: Calculating Standard Gibbs Energy

Problem. The standard electrode potential for the Daniell cell is $1.1$ V. Calculate the standard Gibbs energy for the reaction:

$Zn(s) + Cu^{2+}(aq) \rightarrow Zn^{2+}(aq) + Cu(s)$