Electromotive Force and Thermodynamics

ΔG = −nFEcell links emf to free energy: positive Ecell → negative ΔG (spontaneous); negative Ecell → positive ΔG (nonspontaneous).
Nernst equation extends E°cell to nonstandard concentrations: Ecell = E°cell − (RT/nF)lnQ, or at 25°C, Ecell = E°cell − (0.0592/n)logQ.
As Q grows (more products), Ecell drops below E°cell; as Q shrinks (more reactants), Ecell rises above E°cell — matches Le Chatelier's principle.
E°cell = (RT/nF)lnKeq, or at 25°C, E°cell = (0.0592/n)logKeq — links standard emf directly to the equilibrium constant.
Larger Keq (>1, product-favored) → more positive E°cell; smaller Keq (<1, reactant-favored) → more negative E°cell.
ΔG° = −nFE°cell = −RTlnKeq ties free energy, cell potential, and equilibrium position into one closed loop.

Nernst equation: Extends standard cell potential to nonstandard concentrations using the reaction quotient Q: Ecell = E°cell − (RT/nF)lnQ.
Reaction quotient (Q): Ratio of product to reactant concentrations at the current (nonstandard) conditions, same form as an equilibrium expression.
Standard conditions: 25°C (298 K), 1 atm pressure, 1 M concentration for all reactants and products — where E°cell applies exactly.

ΔG = −nFEcell

Positive Ecell → negative ΔG → spontaneous (matches galvanic cells); negative Ecell → positive ΔG → nonspontaneous (matches electrolytic cells).
Daniell cell worked example: ΔG° = −(2)(96,485)(1.10) ≈ −212,267 J/mol ≈ −212 kJ/mol — strongly spontaneous.

Ecell = E°cell − (0.0592 V/n) logQ

Worked example: Daniell cell, [Cu²⁺]=1.0 M, [Zn²⁺]=0.010 M → Q=0.010/1.0=0.010. Ecell=1.10−(0.0296)(log 0.010)=1.10−(0.0296)(−2)=1.10+0.0592≈1.16 V.
Lower [Zn²⁺] than standard makes Q<1, logQ negative, so Ecell rises above E°cell — more spontaneous than standard conditions.

E°cell = (0.0592 V/n) logKeq

Derived by setting ΔG°=−nFE°cell equal to ΔG°=−RTlnKeq.
Worked example, Daniell cell: logKeq=(2×1.10)/0.0592≈37.2 → Keq=10³⁷·²≈1.5×10³⁷ — reaction runs essentially to completion.

Three linked equations (ΔG=−nFEcell, Nernst equation, E°cell=(0.0592/n)logKeq) describe the same reaction's thermodynamics from three angles: free energy, cell potential, and equilibrium position.
A huge Keq (like the Daniell cell's ~10³⁷) means the reaction goes essentially to completion — barely any reactant remains at equilibrium.

The Nernst equation's sign logic: MORE products relative to reactants (higher Q) LOWERS Ecell below E°cell — don't assume more product formation always raises voltage.
n (moles of electrons) DOES appear in ΔG=−nFEcell and in the Nernst/Keq equations — this is different from the E°cell=E°cathode−E°anode formula, where n never appears. Don't conflate the two.

If Ecell is positive, is ΔG positive or negative, and is the reaction spontaneous?

What does the Nernst equation let you calculate that E°cell alone cannot?

A reaction has a very large Keq. Is E°cell more positive or more negative?

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