Electromotive Force and Thermodynamics

An electrochemical cell's electromotive force (emf) — its voltage — is directly tied to the reaction's thermodynamics. Three linked equations describe that connection: ΔG=−nFEcell relates a cell's instantaneous emf to its free energy change, the Nernst equation extends the standard cell potential to nonstandard concentrations, and the E°cell–Keq relationship ties the standard emf to the reaction's equilibrium constant. Together, these three equations describe a redox reaction's spontaneity and extent from every angle.

Key Takeaways

• ΔG = −nFEcell links a cell's free energy change to its emf: positive Ecell → negative ΔG → spontaneous; negative Ecell → positive ΔG → nonspontaneous.

• Worked example: the Daniell cell's ΔG° = −(2)(96,485)(1.10) ≈ −212 kJ/mol, confirming spontaneity.

• The Nernst equation, Ecell = E°cell − (RT/nF)lnQ (or Ecell = E°cell − (0.0592/n)logQ at 25°C), extends E°cell to nonstandard concentrations using the reaction quotient Q.

• Worked example: the Daniell cell at [Cu²⁺] = 1.0 M, [Zn²⁺] = 0.010 M gives Ecell ≈ 1.16 V, above E°cell because Q < 1.

• E°cell = (RT/nF)lnKeq (or (0.0592/n)logKeq at 25°C) links standard emf to the equilibrium constant: larger Keq → more positive E°cell; smaller Keq → more negative E°cell.

• Worked example: the Daniell cell's Keq ≈ 1.5 × 10³⁷, showing the reaction runs essentially to completion.

• ΔG°=−nFE°cell=−RTlnKeq ties free energy, cell potential, and equilibrium position into one consistent picture of a redox reaction's thermodynamics.

Gibbs Free Energy and Electromotive Force

The emf of an electrochemical cell arises from the difference in reduction potentials between its two half-cells. Thermodynamically, that emf is directly related to the reaction's Gibbs free energy change (ΔG):

ΔG = −nFEcell

• ΔG is the Gibbs free energy change.

• n is the number of moles of electrons transferred in the reaction.

• F is the Faraday constant (≈ 96,485 C/mol).

• Ecell is the emf of the cell.

Because of the negative sign, this equation directly ties a cell's spontaneity to the sign of its emf:

• A positive Ecell gives a negative ΔG — the reaction is spontaneous.

• A negative Ecell gives a positive ΔG — the reaction is nonspontaneous.

This is the same spontaneity rule already used to identify galvanic cells (positive Ecell, spontaneous) and electrolytic cells (negative Ecell, requires an external power source) — ΔG=−nFEcell is simply the quantitative link between the two.

Worked Example: ΔG for the Daniell Cell

The Daniell cell (zinc anode, copper cathode) has an established E°cell of +1.10 V, and its overall reaction transfers n = 2 moles of electrons:

Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s)

Plugging into ΔG=−nFEcell:

ΔG° = −nFE°cell = −(2)(96,485 C/mol)(1.10 V) = −212,267 J/mol ≈ −212 kJ/mol

The large negative ΔG° confirms what's already known about the Daniell cell — it's strongly spontaneous, consistent with its positive E°cell and its role as a galvanic cell.

The Nernst Equation

E°cell only applies under standard conditions — 25°C (298 K), 1 atm pressure, and 1 M concentrations for every reactant and product. Real electrochemical cells rarely stay at exactly 1 M as a reaction proceeds, so a tool is needed to find the emf under nonstandard conditions. That tool is the Nernst equation:

Ecell = E°cell − (RT/nF) lnQ

• R is the gas constant, T is temperature in kelvin, n and F are the same as above.

• Q is the reaction quotient — the same ratio of product to reactant concentrations used in equilibrium expressions, evaluated at the current (nonstandard) concentrations.

At the standard temperature of 25°C, RT/F simplifies to a fixed number, and switching from a natural log to a base-10 log gives the more practical working form:

Ecell = E°cell − (0.0592 V/n) logQ

Reading the equation: as Q grows (more products relative to reactants), the logQ term grows, and Ecell drops below E°cell. As Q shrinks (more reactants relative to products), Ecell rises above E°cell. This matches Le Chatelier's principle — pushing a reaction further from equilibrium in the forward direction (low Q) makes it more thermodynamically favorable, which shows up here as a larger emf.

Worked Example: Cell Potential Under Nonstandard Conditions

Take the Daniell cell again (E°cell = +1.10 V, n = 2), but now with nonstandard concentrations: [Cu²⁺] = 1.0 M and [Zn²⁺] = 0.010 M. For the reaction Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s), the reaction quotient is:

Q = [Zn²⁺] / [Cu²⁺] = 0.010 / 1.0 = 0.010

Plugging into the 25°C form of the Nernst equation:

Ecell = E°cell − (0.0592/n) logQ = 1.10 − (0.0592/2)(log 0.010) = 1.10 − (0.0296)(−2) = 1.10 + 0.0592 ≈ 1.16 V

Lowering [Zn²⁺] relative to standard conditions makes Q smaller than 1, so logQ is negative — and subtracting a negative term raises Ecell above E°cell. The cell becomes slightly more spontaneous than it is under standard conditions, exactly as the equation predicts.

Standard Emf and the Equilibrium Constant

ΔG° can also be found a second way — from the equilibrium constant (Keq) directly:

ΔG° = −RT lnKeq

Since ΔG° also equals −nFE°cell, setting the two expressions for ΔG° equal to each other gives a direct link between standard emf and Keq:

−nFE°cell = −RT lnKeq, which rearranges to E°cell = (RT/nF) lnKeq

At 25°C, using the same base-10 simplification as the Nernst equation:

E°cell = (0.0592 V/n) logKeq

This relationship shows that E°cell is directly proportional to the natural log of Keq:

• A larger Keq (greater than 1) means the reaction strongly favors products, corresponding to a more positive E°cell.

• A smaller Keq (less than 1) means the reaction favors reactants, corresponding to a more negative E°cell.

Together, ΔG°=−nFE°cell=−RTlnKeq forms a closed loop between three ways of describing the same reaction's thermodynamics: free energy, cell potential, and equilibrium position.

Worked Example: Equilibrium Constant for the Daniell Cell

Using the Daniell cell's E°cell = +1.10 V and n = 2, solve E°cell = (0.0592/n) logKeq for Keq:

logKeq = (n × E°cell) / 0.0592 = (2 × 1.10) / 0.0592 ≈ 37.2

Keq = 10³·² ≈ 1.5 × 10³⁷

An equilibrium constant this enormous means the Daniell cell's reaction runs essentially to completion — at equilibrium, almost none of the original Cu²⁺ remains unreacted. This is the same conclusion the large negative ΔG° already pointed to: a strongly spontaneous reaction has a strongly product-favored equilibrium, and a large positive E°cell.

Common MCAT Mistakes

• Dropping the negative sign in ΔG = −nFEcell. A positive Ecell gives a negative ΔG (spontaneous) — not a positive one. The negative sign is what makes the equation match the already-known rule that galvanic cells (positive emf) run spontaneous reactions.

• Misremembering which way Ecell moves as Q changes. As Q increases (more products relative to reactants), Ecell decreases below E°cell — not the other way around — because Ecell = E°cell − (RT/nF)lnQ subtracts a growing term.

• Using the 0.0592 V shortcut at a temperature other than 25°C. That simplified constant is only valid at standard temperature (298 K); at any other temperature, the full RT/nF (or RT) form of the equation must be used instead.

• Assuming a larger Keq means a smaller (less negative) ΔG°. It's the opposite — a very large Keq corresponds to a large, very negative ΔG° and a large, positive E°cell, since all three describe the same strongly product-favored, strongly spontaneous reaction.

MCAT-Style Concept Check

Question: A galvanic cell has E°cell = 0.46 V and n = 2. At 25°C, if the reaction quotient Q = 100, what is Ecell? Use Ecell = E°cell − (0.0592/n)logQ.

• A) 0.34 V

• B) 0.40 V

• C) 0.46 V

• D) 0.52 V

Answer: B

Explanation: logQ = log(100) = 2, so (0.0592/2)(2) = 0.0592 V. Ecell = 0.46 − 0.0592 ≈ 0.40 V. Because Q > 1 (more products relative to reactants than at standard conditions), Ecell drops below E°cell, exactly as the Nernst equation predicts.

FAQ

How is ΔG related to a cell's emf?

ΔG = −nFEcell, where n is moles of electrons transferred and F is the Faraday constant. A positive Ecell always produces a negative ΔG (spontaneous), and a negative Ecell always produces a positive ΔG (nonspontaneous) — the equation is the quantitative version of the same spontaneity rule used to classify galvanic and electrolytic cells.

What does the Nernst equation let you calculate that E°cell alone can't?

E°cell only holds at standard conditions (25°C, 1 atm, 1 M concentrations). The Nernst equation, Ecell = E°cell − (RT/nF)lnQ, extends that standard value to any nonstandard concentrations by incorporating the reaction quotient Q.

What happens to a cell's emf as the reaction quotient Q increases?

Ecell decreases below E°cell as Q increases, because the Nernst equation subtracts a growing (RT/nF)lnQ term. As Q decreases below 1, Ecell rises above E°cell instead.

How is E°cell related to the equilibrium constant Keq?

E°cell = (RT/nF)lnKeq (or (0.0592/n)logKeq at 25°C). A larger Keq corresponds to a more positive E°cell, and a smaller Keq corresponds to a more negative E°cell — both describe the same reaction from different angles, since ΔG°=−nFE°cell=−RTlnKeq links all three quantities together.