Nernst Equation Derivation

Introduction

The standard potential E° is a reference value, while actual cell potential depends on composition. As products and reactants change, their activity ratio Q changes and so does the maximum reversible voltage. The Nernst equation makes this relationship quantitative and follows directly from the Gibbs-energy equation for a nonstandard chemical reaction.

Core explanation

At constant temperature and pressure, a reaction has ΔG = ΔG° + RT ln Q, where Q is a dimensionless quotient formed from current activities for the balanced reaction. For a reversible electrochemical cell, ΔG = −nFE and ΔG° = −nFE°. Substitute these into the Gibbs relation: −nFE = −nFE° + RT ln Q. Divide by −nF to obtain E = E° − (RT/nF)ln Q. Here R is the gas constant, T is absolute temperature, F is Faraday's constant, and n is electrons transferred per mole of the reaction as written.

If Q=1, ln Q=0 and E=E° even if individual species are not all separately at their standard values, provided the reaction activity ratio equals one. If Q<K, the reaction as written has E>0 because E° = (RT/nF)ln K. At equilibrium Q=K and E=0. If Q increases as products accumulate, the subtraction term grows and the forward cell voltage falls. A decreasing Q raises forward driving voltage. These conclusions depend on Q being written for the same balanced reaction direction as E°.

For Zn(s) + Cu²⁺(aq) → Zn²⁺(aq) + Cu(s), pure solids have activity one, so Q = a(Zn²⁺)/a(Cu²⁺). Raising zinc-ion activity or lowering copper-ion activity raises Q and lowers the forward cell potential. Dilute textbook problems may replace activities by concentration ratios, but the rigorous equation uses dimensionless activities. For a gaseous reactant, its activity depends on pressure relative to a standard pressure; omitting a gas term can reverse a predicted trend.

The equation is a thermodynamic relation for an overall cell reaction. It does not account directly for resistive losses when current flows or electrode overpotentials. An observed loaded-cell voltage can be lower than its Nernst open-circuit value. Reaction kinetics also determine how quickly the composition changes. In calculations, write the full balanced equation, identify n, construct Q with correct exponents and omit pure solids and liquids, then substitute T in kelvin. Avoid using a memorized base-ten factor at the wrong temperature.

Changing the reaction equation changes E° and Q consistently. Reverse it: E° changes sign and Q becomes 1/Q. Multiply it by two: n doubles and Q becomes Q² while E° remains unchanged. Substitution into Nernst then gives the same physical voltage. This invariance is a strong algebraic check on a derived result.

Step-by-step reasoning

1. Write ΔG = ΔG° + RT ln Q for the cell reaction. 2. Replace ΔG and ΔG° with −nFE and −nFE°. 3. Rearrange to E = E° − (RT/nF)ln Q. 4. Build Q from the same balanced reaction and interpret its value.

Visual explanation

Draw a descending voltage-versus-ln Q line with intercept E° at ln Q=0 and a zero crossing at ln K. Mark product buildup toward the right.

Real-world analogy

A storage reservoir delivers less pressure difference as the receiving side fills. Product accumulation likewise reduces the reversible driving force for the forward cell reaction.

Real-world example

As a zinc-copper cell operates, Zn²⁺ can accumulate and Cu²⁺ can be consumed. Their quotient rises, so the thermodynamic forward voltage decreases from its initial value.

Why?

Why does a larger Q lower forward E? Product-rich composition raises ΔG for additional forward reaction, leaving less Gibbs energy available as electrical work per charge.

Common misconception

“E° changes when reactant concentration changes.” E° remains the standard reference at fixed temperature; actual E changes through the Q term.

Worked example

For Zn + Cu²⁺ → Zn²⁺ + Cu, let E° = 1.10 V and n=2. If the activity ratio a(Zn²⁺)/a(Cu²⁺) increases from 1 to 100 at 25 °C, the Nernst correction is (0.05916 V/2)log₁₀(100) = 0.05916 V. Thus E falls from 1.10 V to about 1.04 V. The solids do not enter Q, and the sign makes sense because products have become relatively more abundant.

Quick check

1. What is E when Q=K for the overall cell reaction? Answer: Zero reversible cell potential at equilibrium.

Exam focus

Derive or recall the minus sign and build Q from the exact reaction. Use activities for rigor and concentration approximations only when the problem permits them.

Advanced insight

The Nernst equation connects electrochemical potential to chemical activities. Measuring E can therefore infer an unknown ion activity when other cell quantities and calibration are known.

Summary

Combining Gibbs energy with reversible electrical work yields E = E° − (RT/nF)ln Q. Product buildup raises Q and generally lowers forward cell voltage at a fixed temperature.

Practice questions

1. If Q=1, how does E compare with E°? Answer: They are equal in the Nernst equation. 2. Do pure metal electrodes appear in Q for Zn/Cu reaction? Answer: No. Pure solid activities are one. 3. Why does n appear in the denominator of the Nernst correction? Answer: Voltage is energy per transferred charge, and nF is the charge for one mole of the written reaction.