Entropy, Free Energy, and Electrochemical Work

Driving question

How can the same thermodynamic accounting predict both reaction direction and electrical work?

Entropy and dispersal

Entropy tracks the number of energetically accessible arrangements, not simply “disorder.” Processes commonly increase system entropy when particles spread into more volume, more gas particles form, or thermal energy becomes distributed among more accessible states. The surroundings matter too: at constant pressure, heat released by the system increases the surroundings' entropy by approximately $-\Delta H_\mathrm{sys}/T$.

Gibbs free energy

At constant temperature and pressure, $\Delta G=\Delta H-T\Delta S.$ A process is thermodynamically favorable in the forward direction when $\Delta G<0$, at equilibrium when $\Delta G=0$, and unfavorable when $\Delta G>0$. Favorability does not imply a fast rate. Under nonstandard conditions, $\Delta G=\Delta G^\circ+RT\ln Q,$ and at equilibrium, $\Delta G^\circ=-RT\ln K.$

Electrochemical work

For a balanced redox reaction transferring $n$ moles of electrons, $\Delta G^\circ=-nFE^\circ_\mathrm{cell}.$ A positive $E^\circ_\mathrm{cell}$ corresponds to a negative $\Delta G^\circ$ for the reaction as written. Oxidation occurs at the anode and reduction at the cathode in both galvanic and electrolytic cells. Electrons travel through the external circuit from anode to cathode; ions migrate through the electrolyte to maintain electrical neutrality.

Worked connection with units

For a balanced standard-state cell reaction with $n=2$ and $E^\circ_\mathrm{cell}=+1.10\ \mathrm{V}$, $\Delta G^\circ=-(2\ \mathrm{mol\ e^-})(96485\ \mathrm{C\ mol^{-1}\ e^-})(1.10\ \mathrm{J\ C^{-1}})=-2.12\times10^5\ \mathrm{J\ mol^{-1}}.$ To three significant figures, $\Delta G^\circ=-212\ \mathrm{kJ\ mol^{-1}}$ for the balanced reaction as written. The negative sign is consistent with a favorable forward standard-state reaction. Reversing the reaction reverses the signs of both $E^\circ_\mathrm{cell}$ and $\Delta G^\circ$.

Nonstandard cells

Combining $\Delta G=\Delta G^\circ+RT\ln Q$ with electrical work gives the Nernst equation, $E=E^\circ-\frac{RT}{nF}\ln Q.$ At $298\ \mathrm{K}$ this is often written $E=E^\circ-(0.05916\ \mathrm{V}/n)\log Q$. A cell reaches equilibrium when $E=0$ and $Q=K$.

Media prompts

• Before: Predict the sign of $\Delta G^\circ$ for a galvanic cell with $E^\circ=+1.10\ \mathrm{V}$.
• During: Track electrons, cations, and anions separately.
• After: Explain why a favorable redox reaction can still be slow.
• Non-video fallback: Use the Gibbs-energy sign map and solve Quiz Question 3.

Laboratory connection

Complete the supervised microscale electrochemical-cell investigation in lab.md. Online simulation is an accessibility and prelab option, not a substitute for supervised AP laboratory time.

Sources

• College Board, AP Chemistry Course and Exam Description, effective Fall 2024, Unit 9.
• NIST, Standard Electrode Potentials and Temperature Coefficients in Water at 298.15 K, including $\Delta G^\circ=-nFE^\circ$ and the Nernst relationship.
• IUPAC/NIST, ThermoML Recommendations, for thermodynamic quantity and unit conventions.