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Gibbs Free Energy from Cell Potential (ΔG° = −nFE°)

Also known as dG from E cell · free energy of a galvanic cell · electrical work of a cell · nFE

ΔG∘=−nFE∘\Delta G^{\circ} = -n F E^{\circ}

Worked example: Daniell cell, E = 1.10 V with n = 2 → dG = -212.27 kJ/mol — press Try an example to run it live, then adjust anything.

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Constant used — built into this formula, no need to enter
F=96,485.33212331001 C/molF = 96,485.33212331001\ \text{C/mol}Faraday constant · exact
See also
Standard Cell Potential from Half-Cells — Where E° comes from: cathode minus anode, off the standard reduction table.Gibbs Free Energy and the Equilibrium Constant — The third corner of the triangle — ΔG° = −RT ln K, so a cell voltage also fixes the equilibrium constant.Nernst Equation — What happens once the cell is no longer at standard conditions.Gibbs Free Energy Change (ΔG = ΔH − TΔS) — The same ΔG° from enthalpy and entropy instead of from a voltmeter.

Learning zone

A volt is a joule per coulomb. That single fact is the whole derivation: multiply a cell's potential by the charge that moves through the circuit per mole of reaction, and you have an energy per mole. The charge is n·F, where n counts electrons transferred as the balanced half-equations are written and F is the Faraday constant, 96 485 C/mol. The minus sign converts "work the cell can do on the outside world" into "the system's free energy change", which falls when the cell discharges.

Worked: the Daniell cell, zinc against copper, E° = +1.10 V with n = 2. ΔG° = −2 × 96 485 × 1.10 = −212 kJ/mol. Negative, so the reaction runs on its own and will drive a load — which is what a battery is. Run it the other way: water's formation from its elements has ΔG° = −237.13 kJ/mol, so an ideal hydrogen fuel cell with n = 2 offers E° = 237 130/(2 × 96 485) = 1.229 V. Every hydrogen fuel cell in the world is a commentary on that number.

The minus sign is not decoration. Drop it and every galvanic cell reads as non-spontaneous, which is a conclusion the existence of batteries makes difficult to defend. A positive E° goes with a negative ΔG°; a negative E° goes with a positive ΔG° and means the reaction must be driven — that is electrolysis, and |E°| is the thermodynamic minimum voltage before any overpotential or resistive loss is added.

The value of n is the other place answers go wrong. It is the number of electrons in the balanced overall reaction, after the two half-equations have been scaled to match. Copper deposition, Cu²⁺ + 2e⁻ → Cu, uses two; silver, Ag⁺ + e⁻ → Ag, uses one. Note also that E° does not scale when you double the equation but ΔG° does — potential is intensive, free energy is extensive — and n absorbs the difference.

This equation is one corner of a triangle. ΔG° = −RT ln K is another, and setting the two equal gives E° = (RT/nF) ln K, which means a voltmeter reading is an equilibrium constant measurement. The Nernst equation is the same relation once the cell leaves standard conditions.

Gibbs Free Energy from Cell Potential (ΔG° = −nFE°)
ΔG∘=−nFE∘\Delta G^{\circ} = -n F E^{\circ}
Where
  • ΔG∘\Delta G^{\circ}= Standard Gibbs free energy change (kJ/mol)
  • nn= Electrons transferred
  • E∘E^{\circ}= Standard cell potential (V)
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