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Cell potential and free energy

A cell's voltage is a verdict on favorability wearing different units. One equation, ΔG° = −nFE°, translates between the volts you measure and the free energy that decides whether the reaction runs.

§1

ΔG° = −nFE°cell.

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Cell potential and free energy are linked by ΔG° = −nFE°cell, where n is the moles of electrons transferred and F is Faraday's constant. The minus sign means a positive E° gives a negative ΔG° — a favorable (spontaneous) reaction.

The standard cell potential is E°cell = E°cathode − E°anode (both as reduction potentials). A positive E°cell marks a galvanic, spontaneous cell.

Crucially, cell potential is intensive: multiplying the balanced equation by a factor does not change E°. Free energy, by contrast, is extensive — scaling the equation scales ΔG (and n).

UNIT 9 TOPIC 9.9 • CELL POTENTIAL AND FREE ENERGY ΔG° = −nFE°cell WHAT THE SYMBOLS MEAN ΔG° standard Gibbs free energy change (J·mol⁻¹) n moles of electrons transferred F Faraday constant = 96,485 C·mol⁻¹ E°cell standard cell potential (V) GALVANIC CELL (EXAMPLE) V e⁻ salt bridge Zn²⁺ Cu²⁺ Zn(s) anode oxidation (−) Cu(s) cathode reduction (+) Zn(s) | Zn²⁺(aq) ‖ Cu²⁺(aq) | Cu(s) KEY IDEA Cell potential and Gibbs free energy change are directly related. A positive E°cell gives a negative ΔG°, so the reaction is spontaneous under standard conditions. BUILT FROM HALF-CELLS E°cell = E°cathode − E°anode Both from standard reduction potentials. ΔG° = −nFE°cell n = moles of e⁻ transferred RULE OF THUMB E°cell > 0 → ΔG° < 0 spontaneous E°cell = 0 → ΔG° = 0 equilibrium E°cell < 0 → ΔG° > 0 nonspontaneous CONNECTING THE THREE E°cell ΔG° spontaneity > 0 < 0 spontaneous = 0 = 0 equilibrium < 0 > 0 nonspont. WORKED EXAMPLE Galvanic Zn/Cu cell: E°cell = +1.10 V, n = 2 ΔG° = −nFE°cell = −(2)(96,485 C·mol⁻¹)(1.10 V) = −212,267 J·mol⁻¹ ≈ −212 kJ·mol⁻¹ ΔG° < 0 → spontaneous The cell reaction is thermodynamically favorable. AP Chemistry · Unit 9 · Applications of Thermodynamics
Fig. 9.9.1 ΔG° = −nFE°cell links cell potential to free energy. A positive E°cell means a favorable (spontaneous) reaction and a negative ΔG°. E°cell = E°cathode − E°anode, and the potential is intensive — scaling the equation does not change it.
§2

From potentials to favorability.

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Get E°, then translate to ΔG°.

  1. Find E°cell. E°cell = E°cathode − E°anode, using reduction potentials.
  2. Read favorability. A positive E°cell means favorable (galvanic); negative means it needs driving.
  3. Count n. The moles of electrons transferred in the balanced reaction.
  4. Compute ΔG°. ΔG° = −nFE°cell — a positive E° gives a negative ΔG°.
§3

The pieces you'll meet.

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Potential, electrons, and free energy.

E°cell
Standard cell potential
E°cathode − E°anode.
ΔG°
Free energy
= −nFE°cell.
n
Electrons (n)
Moles of electrons transferred.
F
Faraday's constant
Charge per mole of electrons (96,485 C).
intensive
Intensive
E° does not change when the equation is scaled.
sign
Sign rule
Positive E° → negative ΔG° → favorable.
§4

Worked example: potential, favorability, free energy.

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E°cell. A cell has E°cathode = +0.34 V and E°anode = −0.76 V, so E°cell = 0.34 − (−0.76) = +1.10 V.

Favorable. E°cell > 0, so the reaction is spontaneous (galvanic) and ΔG° < 0.

Scaling. Doubling the balanced equation leaves E° at +1.10 V (intensive), but doubles n, so ΔG° = −nFE° doubles.

Key point. Voltage does not scale with amount; free energy does.

§5

Mistakes that cost real points.

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Pitfall · 01

"If you double the reaction, the cell potential doubles."

Cell potential is intensive — it does not depend on how much reaction is written. Doubling the coefficients leaves E° unchanged (though it doubles n and therefore ΔG°). Scaling voltage with the equation is a mistake.

Fix. Keep E° fixed when scaling the equation; only n and ΔG° scale.

Pitfall · 02

"A more negative cell potential is more favorable."

It is the reverse: a more positive E°cell is more favorable (more negative ΔG°), because ΔG° = −nFE°. Reading favorability with the wrong sign inverts every conclusion. Positive E° means spontaneous.

Fix. Read a more positive E°cell as more favorable (ΔG° more negative).

Pitfall · 03

"E°cell is the sum of the two reduction potentials."

E°cell = E°cathode − E°anode, a difference of reduction potentials, not their sum. Adding them (or forgetting to flip the anode) gives the wrong voltage and can flip the predicted spontaneity.

Fix. Use E°cell = E°cathode − E°anode; subtract, do not add.

§6

Skill Check.

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