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Cell potential, nonstandard

Standard potentials assume a tidy 1 M, 1 atm world. Real cells are messier — and a running battery, or a concentration cell built from one metal, gets its voltage from exactly that departure from standard.

§1

The Nernst equation.

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When concentrations differ from standard, the cell potential shifts from E° according to the Nernst equation. At 25 °C, E = E° − (0.05916/n) log Q, where Q is the reaction quotient.

So an operating cell is not pinned to E°: as reactants are consumed, Q rises, and E falls below E°. A concentration cell — the same species at different concentrations on each side — has E° = 0 but a nonzero E, driven entirely by the concentration difference.

The potential reaches zero only at equilibrium (when Q = K). A working cell delivering current is not at equilibrium, so its potential is nonzero.

UNIT 9 TOPIC 9.10 • CELL POTENTIAL UNDER NONSTANDARD CONDITIONS NERNST EQUATION THE NERNST EQUATION E = E° − RT nF ln Q at 25 °C: E = E° − (0.05916/n) log Q WHAT EACH TERM MEANS E = cell potential (nonstandard) E° = standard cell potential R = 8.314 J·mol⁻¹·K⁻¹ T = temperature (K) n = moles e⁻ transferred F = 96,485 C·mol⁻¹ Q = reaction quotient Concentration cell Ecell e⁻ 0.10 M anode (−) 1.0 M cathode (+) Worked example — copper(II) concentration cell Cu(s) │ Cu²⁺ (0.10 M) ‖ Cu²⁺ (1.0 M) │ Cu(s) Overall cell reaction (net ionic): Cu²⁺(aq, 1.0 M) → Cu²⁺(aq, 0.10 M) At 25 °C, for Cu²⁺ + 2e⁻ → Cu(s): E° = +0.340 V Both electrodes are identical, so E°cell = 0 V (E°cathode = E°anode) Reaction quotient: Q = [Cu²⁺]anode / [Cu²⁺]cathode = 0.10 / 1.0 = 0.10 Nernst equation (n = 2, 25 °C): Ecell = 0 − (0.05916/2) · log(0.10) Ecell = +0.0296 V Ecell > 0 → spontaneous as written Q < K shifts forward · E_cell > 0 Q = K equilibrium · E_cell = 0 Q > K shifts reverse · E_cell < 0 this cell: Q = 0.10 < K = 1 CED ANCHOR Cu²⁺ is consumed in the concentrated half-cell and produced in the dilute half-cell until the concentrations approach equilibrium Quantify electrochemical systems with Q and Ecell (Nernst) for both direction and magnitude of the driving force — not vague labels like “favored.” AP Chemistry · Unit 9 · Applications of Thermodynamics
Fig. 9.10.1 The Nernst equation, E = E° − (0.05916/n) log Q at 25 °C, gives the cell potential under nonstandard conditions. An operating cell and a concentration cell both have nonzero potentials; E reaches zero only at equilibrium.
§2

Applying the Nernst equation.

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Shift E° by the Q term.

  1. Start from E°. The standard potential is the baseline.
  2. Form Q. The reaction quotient from the actual (nonstandard) concentrations.
  3. Apply the shift. E = E° − (0.05916/n) log Q at 25 °C; E moves away from E° as Q changes.
  4. Locate equilibrium. E = 0 only when Q = K; a working cell is not there yet.
§3

The pieces you'll meet.

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Nonstandard potentials.

Nernst
Nernst equation
E = E° − (0.05916/n) log Q at 25 °C.
Q
Reaction quotient
From the actual concentrations.
operate
Operating cell
E differs from E° as Q changes.
conc cell
Concentration cell
E° = 0 but E ≠ 0.
equil
Equilibrium
E = 0 only when Q = K.
n
n
Moles of electrons in the balanced reaction.
§4

Worked example: shifting from standard.

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Setup. A cell has E° = +1.10 V and transfers n = 2 electrons at 25 °C.

Q > 1. If product ions have built up so Q = 10, then log Q = 1 and E = 1.10 − (0.05916/2)(1) = 1.10 − 0.0296 ≈ +1.07 V — slightly below E°.

Concentration cell. If both half-cells used the same couple (E° = 0) with different concentrations, E would be nonzero, set entirely by the −(0.05916/n) log Q term.

Equilibrium. Only when Q reaches K does E fall to 0 and the cell die.

§5

Mistakes that cost real points.

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Pitfall · 01

"The cell potential stays at E° regardless of concentration."

Under nonstandard conditions the potential shifts from E° according to the Nernst equation, E = E° − (0.05916/n) log Q. As concentrations change, so does E. Assuming E is fixed at E° ignores how real, operating cells behave.

Fix. Use the Nernst equation: E moves away from E° as Q departs from 1.

Pitfall · 02

"A concentration cell has zero voltage because both sides are the same substance."

A concentration cell has E° = 0 but a nonzero E, because the two half-cells are at different concentrations — the Nernst term drives a real potential. It produces voltage until the concentrations equalize.

Fix. Recognize a concentration cell runs on the concentration difference (E° = 0, E ≠ 0).

Pitfall · 03

"A cell delivering current is already at equilibrium."

A cell that is producing current is away from equilibrium (Q ≠ K); its potential is nonzero. The potential reaches zero only at equilibrium, when the cell is dead. A working cell has not gotten there yet.

Fix. Treat an operating cell as not at equilibrium; E = 0 only when Q = K.

§6

Skill Check.

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