Chemistry 2e · Electrochemistry
Potential, Free Energy, and Equilibrium
On this page 9 sections
In 30 seconds
A working galvanic cell converts a spontaneous redox reaction into useful electrical work — but how much work, and under what conditions? The answer ties together three of chemistry's most important quantities: the cell potential E, the Gibbs free energy change ΔG, and the equilibrium constant K. This topic develops the exact equations connecting them — ΔG°= -nFE°cell, the Nernst equation Formula correcting E° for nonstandard concentrations Full entry →, and lnK = nFE°cellRT — so you can predict spontaneity, calculate maximum work, and find equilibrium constants that no direct experiment could measure.
Why this matters
These relationships are the quantitative engine of electrochemistry. Battery ratings, fuel-cell efficiencies, electroplating cost estimates, and even the way cells couple nonspontaneous reactions to ATP hydrolysis all rest on ΔG = -nFE. The equations let you predict a cell's voltage at any concentration (not just the 1 M standard state), decide whether a proposed redox pair will react at all, and compute equilibrium constants like K ≈ 1037 for the zinc–copper cell — a number far too large for ordinary titration but obtained instantly from a single table of standard potentials.
The college version
Core Concepts
The master equation: ΔG°= -nFE°cell
The standard cell potential E°cell (in volts) measures the driving force per electron transferred. The symbol n is the number of moles of electrons in the balanced reaction, and the Faraday constant converts moles of electrons into charge:
F = 96,485 C/mol e-
Because 1 V = 1 J/C, the product nFE° has energy units: mol × Cmol × JC = J. A spontaneous reaction has E°cell > 0 and ΔG°< 0. Note the intensive/extensive distinction: E° is intensive — doubling every coefficient in the reaction leaves E° unchanged, while both n and ΔG° double.
The Nernst equation: voltage at real concentrations
Standard potentials apply only at 1 M solutes, 1 atm gases, and 25 °C. Real cells rarely sit at standard conditions, and the Nernst equation corrects for that:
E = E°cell - RTnFlnQ
where Q is the reaction quotient built exactly as for any equilibrium. At 298 K the constants fold into a convenient base-10 form: E = E°cell - 0.0592nlog10Q (since RTFln10 ≈ 0.0592 V). For the zinc–copper cell, Q = [Zn2+][Cu2+]. As a cell discharges, Q grows and E falls toward zero. A Concentration cell Cell with identical electrodes but different ion concentrations Full entry → — the same redox couple at two different concentrations — shows the Nernst equation at its purest: E°cell = 0, yet the cell still produces voltage because Q ≠ 1.
Equilibrium: when the cell is "dead"
At equilibrium the cell can do no more work: E = 0, ΔG = 0, and Q = K. Substituting E = 0 and Q = K into the Nernst equation gives:
lnK = nFE°cellRT
At 298 K this becomes log10K = nE°cell0.0592. A modest potential of +0.30 V with n = 2 corresponds to K ≈ 1010 — the reaction is overwhelmingly product-favored, which is why redox titrations and batteries behave so decisively.
The spontaneity triangle
Three statements are equivalent: E°cell > 0, ΔG°< 0, and K > 1. Given any one of them you can rank reactions and predict direction. For nonstandard conditions, use the actual E from the Nernst equation in ΔG = -nFE — the same formula, just with the real driving force instead of the standard one.
Common Confusions
| Do Not Confuse | With | Difference |
|---|---|---|
| Standard potential E° | Actual potential E | E° is fixed at standard conditions; E changes with Q |
| ΔG° | ΔG | ΔG° comes from E°; ΔG comes from the real E |
| n = moles of substance | n = moles of electrons | Read n from the balanced half-reactions, not the overall equation's coefficients |
| lnQ in the Nernst equation | log10 Q | Same equation, different bases: 0.0592 = (RT/F) × ln 10 |
| "Cell is dead" = no reactants left | "Cell is dead" = at equilibrium | At equilibrium Q = K and E = 0, but both reactants and products remain |
Eli explains
The same idea, in plain words
Explain it like I’m 10
Think of a battery as a water slide. The voltage is how high the water starts — higher water means more speed at the bottom. As the ride goes on, the water level drops (concentrations change) and the slide slows down. When the water is level at both ends, the ride stops completely — that's equilibrium. Free energy is just a way of asking how much fun (work) the slide can still deliver.
Worked example
Worked Example 1 — Maximum work and K from E°. For the zinc–copper cell, Zn(s) + Cu2+(aq) → Zn2+(aq) + Cu(s), with E°cell = +1.10 V and n = 2:
ΔG°= -nFE°= -(2 mol e-)(96,485 C/mol e-)(1.10 V) = -2.12 × 105 J = -212 kJ
Unit check: mol × Cmol × JC = J. The negative sign confirms spontaneity. Now find K:
lnK = nFE°RT = (2)(96,485)(1.10)(8.314)(298) = 212,2672,478 = 85.7
so K = e85.7 ≈ 1.5 × 1037 — the reaction runs essentially to completion.
Worked Example 2 — Nernst equation at nonstandard concentrations. Suppose [Zn2+] = 0.10 M and [Cu2+] = 2.0 M. Then Q = 0.102.0 = 0.050, and at 298 K:
E = E°- 0.0592nlog10Q = 1.10 V - 0.05922log10(0.050) = 1.10 - 0.0296(-1.30) = 1.14 V
The cell delivers slightly more than its standard 1.10 V because the reactant side ([Cu2+]) is concentrated and the product side ([Zn2+]) is dilute — Le Châtelier's principle in electrochemical form.
Key takeaways
- Memorize the trio: ΔG°= -nFE°cell, E = E°- RTnFlnQ, and lnK = nFE°RT.
- F = 96,485 C/mol e-; at 298 K, RTFln10 ≈ 0.0592 V.
- n = moles of electrons transferred — read it from the balanced half-reactions, and make sure both half-reactions involve the same number of electrons before adding.
- E°> 0 ⇔ ΔG°< 0 ⇔ K > 1: all three say "spontaneous."
- At equilibrium E = 0, even though the reaction is not necessarily complete.
- Doubling a reaction leaves E° unchanged but doubles n and ΔG°.
- A concentration cell runs on E°= 0 but Q ≠ 1; the higher-concentration side is the cathode.
Check yourself
5 review questions from the chapter. Try each one, then open the answer.
A cell has E°cell = +1.10 V and n = 2. What is ΔG°, and is the reaction spontaneous?
Show answer
ΔG°= -(2)(96,485)(1.10) = -2.12 × 105 J = -212 kJ. Negative ΔG° means spontaneous.
Estimate K for a reaction with E°= +0.30 V and n = 2 at 298 K.
Show answer
log10K = (2)(0.30)0.0592 ≈ 10.1, so K ≈ 1 × 1010 — strongly product-favored.
If you double every coefficient in a cell reaction, what happens to E°, n, and ΔG°?
Show answer
E° is unchanged (intensive); n doubles; ΔG° doubles.
Why does a battery's voltage drop as it discharges, even though E° is constant?
Show answer
Discharging builds up products and depletes reactants, so Q increases and E = E°- 0.0592nlogQ falls toward zero.
A copper concentration cell has [Cu2+] = 1.0 M in one compartment and 0.0010 M in the other. Which side is the cathode, and what is E at 298 K?
Show answer
The 1.0 M side is the cathode (reduction of Cu²⁺). E = 0.05922log10(1.00.0010) = 0.0296 × 3.0 = 0.089 V.
Study tools & related lessonsKey vocabulary · Related
Key vocabulary
- Cell potential E
- Voltage a cell actually produces under its real conditions
- Standard cell potential E°cell
- Voltage when all solutes are 1 M, gases 1 atm, at 25 °C
- Gibbs free energy Δ G
- Energy available to do work at constant pressure and temperature
- Faraday constant F
- Charge carried by one mole of electrons, 96,485 C/mol
- Nernst equation
- Formula correcting E° for nonstandard concentrations
- Reaction quotient Q
- Concentration ratio like K, but at any moment, not just equilibrium
- Equilibrium constant K
- The value of Q when the reaction stops changing
- Concentration cell
- Cell with identical electrodes but different ion concentrations
- equilibrium constant (K)
- A temperature-dependent ratio of product-to-reactant concentrations (or pressures) at equilibrium
- Faraday constant, F
- Charge of one mole of electrons, 96,485 C/mol.
- reaction quotient (Q)
- The same ratio as K, but with current, possibly non-equilibrium concentrations
Sources & references
This lesson was adapted from the open educational references above; their licenses and attributions are preserved. See Copyright & Licensing.
Educational content only. It is not medical, legal or professional advice. Found an error? Tell us.