Electrochemical Free Energy

Core Concept

The electrical work a galvanic cell can perform is tied directly to its free-energy change by ΔG° = −nFE°cell, where n is the number of moles of electrons transferred in the balanced equation and F is the Faraday constant (96,485 C/mol e⁻). A positive E°cell gives a negative ΔG° (spontaneous); a negative E°cell gives a positive ΔG° (nonspontaneous). This equation is the quantitative bridge between the electrical world (volts) and the thermodynamic world (joules), and it explains why batteries store chemical energy as useful electrical energy.

Key Ideas

• Work = charge × voltage. ΔG° = −(nF)·E°cell is just "maximum electrical work = total charge moved × potential difference," with the minus sign matching the ΔG sign convention.
• n counts electrons, not coefficients blindly. n is the number of electrons transferred per reaction as balanced — the same value in both half-reactions.
• Faraday constant F = 96,485 C/mol e⁻. The charge of one mole of electrons.
• Signs coupled. E°cell > 0 ⇔ ΔG° < 0; E°cell < 0 ⇔ ΔG° > 0.
• Unit hygiene. nF·E°cell comes out in joules (C × V = J); convert to kJ/mol as needed.

Equations and Variables

ΔG° = −nFE°cell

How It Works

1. Write the balanced net redox equation and identify n (electrons transferred).
2. Compute E°cell from reduction potentials.
3. Multiply n × F × E°cell to get the electrical work (in joules), then attach the minus sign for ΔG°.
4. Interpret the sign: negative ΔG° = spontaneous/product-favored.

Worked Example

Find ΔG° for Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s).

• n = 2 (two electrons transferred: Zn → Zn²⁺ + 2 e⁻ and Cu²⁺ + 2 e⁻ → Cu).
• E°cell = +1.10 V (from the earlier note).
• ΔG° = −nFE°cell = −(2)(96,485 C/mol)(1.10 V)
• = −212,267 J/mol = −212.3 kJ/mol

The large negative value confirms the reaction is strongly spontaneous — exactly the energy released by a Daniell cell per mole of reaction.

How it works

1. Write the balanced net redox equation and identify n (electrons transferred).
2. Compute E°cell from reduction potentials.
3. Multiply n × F × E°cell to get the electrical work (in joules), then attach the minus sign for ΔG°.
4. Interpret the sign: negative ΔG° = spontaneous/product-favored.

Common confusions

• "n is the largest coefficient in the equation." — Wrong. n is the number of electrons transferred, equalized between the half-reactions (e.g., 2 for Zn/Cu, 6 for Cr₂O₇²⁻ + Fe²⁺).
• "E°cell must be multiplied by n." — Wrong. n multiplies F to give total charge; E°cell is a per-electron potential and is never scaled.
• "A negative E°cell means ΔG° is negative." — Wrong. The minus sign flips it: negative E°cell ⇒ positive ΔG°.
• "F is the gas constant." — Wrong. F is the Faraday constant (96,485 C/mol); R is the gas constant (8.314 J/(mol·K)).
• "Units come out in kJ automatically." — Wrong. nF·E°cell gives joules; divide by 1000 for kJ.

Quick review

• ΔG° = −nFE°cell.
• n = electrons transferred; F = 96,485 C/mol e⁻.
• Positive E°cell ⇒ negative ΔG° (spontaneous).
• E°cell is intensive; ΔG° scales with n.
• Work in joules, then convert to kJ.