Nernst Equation

A galvanic cell converts chemical energy into electrical energy through spontaneous redox reactions. The electromotive force (EMF), or cell potential E_cell, measures the 'push' driving electrons from the anode to the cathode. Under standard conditions — all solutes at 1 mol/L, all gases at 1 atm, and temperature at 25°C — we calculate a standard cell potential E°_cell using tabulated standard reduction potentials. However, in real-world cells and in living systems, concentrations and pressures rarely sit at these ideal values. The Nernst Equation, developed by the German physical chemist Walther Nernst in 1889, extends our calculation of cell potential to non-standard conditions. It shows mathematically how E_cell shifts when reactant or product concentrations change. This is not merely a theoretical refinement — pH meters, blood-gas analysers, and nerve-impulse models all depend on the Nernst relationship to translate a measured voltage into a meaningful concentration. Understanding the Nernst Equation requires combining three ideas you have already met: standard reduction potentials, the reaction quotient Q, and the thermodynamic relationship between free energy and electrical work. When you bring these together, you can predict whether a cell delivers more or less voltage than standard conditions suggest, and you can work backwards from a measured potential to determine an unknown concentration — a powerful analytical technique.

Key takeaways
  • Connecting ΔG to Cell Potential
  • Algebraic Derivation
  • Simplifying at 25 °C
  • Writing the Reaction Quotient Q
  • Interpreting the Result

Connecting ΔG to Cell Potential

Electrical work done by a cell equals the charge transferred multiplied by the potential difference. For n moles of electrons, the charge is nF (where F = 96 485 C/mol is the Faraday constant). Therefore the maximum electrical work — equal to the Gibbs free energy change — is: ΔG = −nFE_cell. Under standard conditions this becomes ΔG° = −nFE°_cell. Because ΔG = ΔG° + RT ln Q, we can substitute both expressions for ΔG.

Algebraic Derivation

Substituting ΔG = −nFE and ΔG° = −nFE° into ΔG = ΔG° + RT ln Q gives: −nFE = −nFE° + RT ln Q. Dividing every term by −nF isolates E: E = E° − (RT/nF) ln Q. This is the Nernst Equation in its general form. R is the universal gas constant (8.314 J mol⁻¹ K⁻¹), T is the absolute temperature in Kelvin, n is the number of moles of electrons transferred in the balanced equation, and Q is the reaction quotient.

Simplifying at 25 °C

At T = 298.15 K, the factor RT/F = (8.314 × 298.15)/96485 = 0.02569 V. Converting from natural logarithm to base-10 logarithm (ln Q = 2.303 log Q) gives the practical form most used in high school: E = E° − (0.0592/n) log Q. The coefficient 0.0592 V (sometimes written 0.05916 V) is valid only at 25 °C; at other temperatures you must use the full RT/nF expression.

Writing the Reaction Quotient Q

Q is written exactly like an equilibrium constant K — products over reactants, each raised to its stoichiometric coefficient, with pure solids and liquids omitted. For the general cell reaction aA + bB → cC + dD, Q = [C]^c[D]^d / ([A]^a[B]^b). For gases, use partial pressures in atm. For example, in a Zn–Cu Daniell cell, \ce{Zn(s) + Cu^{2+}(aq) -> Zn^{2+}(aq) + Cu(s)}, Q = [Zn²⁺]/[Cu²⁺] because the two solids are excluded.

Interpreting the Result

When Q < 1 (reactant concentrations exceed product concentrations), log Q is negative, so the correction term is subtracted from a negative number — E_cell increases above E°. The cell has 'extra driving force'. When Q > 1, E_cell falls below E°. At equilibrium Q = K, E_cell = 0 V (the cell is 'dead'), and the Nernst Equation reduces to ΔG° = −nFE° = −RT ln K, linking electrochemistry directly to equilibrium thermodynamics.

The Key Equation

Ecell=Ecell∘−0.0592nlog⁡QE_{\text{cell}} = E^{\circ}_{\text{cell}} - \frac{0.0592}{n} \log Q
  • E_{\text{cell}} — Cell potential (EMF) (V (volts)): The actual electromotive force of the cell under the given non-standard conditions.
  • E^{\circ}_{\text{cell}} — Standard cell potential (V (volts)): The EMF when all solutes are 1 mol/L, all gases at 1 atm, and temperature is 25 °C. Calculated as E°_cathode − E°_anode using standard reduction potentials.
  • 0.0592 — Thermal voltage factor at 25 °C (V (volts)): The combined constant RT ln(10)/F evaluated at 298 K. Equals approximately 0.05916 V; often rounded to 0.0592 V in calculations.
  • n — Number of electrons transferred (dimensionless (mol e⁻ per mol reaction)): The stoichiometric number of moles of electrons exchanged in the balanced overall cell reaction.
  • Q — Reaction quotient (dimensionless): Products over reactants at the current concentrations/pressures, written with the same form as the equilibrium constant K. Pure solids and liquids are excluded.

Worked Example

Problem. A Daniell cell operates with the overall reaction \ce{Zn(s) + Cu^{2+}(aq) -> Zn^{2+}(aq) + Cu(s)}. The standard reduction potentials are E°(Cu²⁺/Cu) = +0.34 V and E°(Zn²⁺/Zn) = −0.76 V. Calculate E_cell when [Cu²⁺] = 0.010 mol/L and [Zn²⁺] = 1.50 mol/L at 25 °C.

Ecell=1.10−0.05922log⁡(150)≈1.04 VE_{\text{cell}} = 1.10 - \frac{0.0592}{2}\log(150) \approx 1.04\text{ V}

Answer. E_cell ≈ 1.04 V

Figure
Diagram
Nernst Equation: E_cell vs. log Qlog QE_cell (V)+EQ=1, E=E°E° lineIncreasing [reactant]Q↓ → E↑ above E°Increasing [product]Q↑ → E↓ below E°−3−2−10+1+2+3Slope = −0.0592/nGraph for n=2: slope = −0.0296 V per log unit; cell is dead (E=0) at equilibrium
Origins · 1 minWalther Nernst and how this was discovered
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Walther Nernst
1864–1941 · German

Nernst derived his famous equation in 1889 while working on the thermodynamics of electrochemical cells, showing how the EMF of a cell depends on the concentrations of the species involved. He also formulated the third law of thermodynamics (the Nernst Heat Theorem, 1906), which states that the entropy of a perfect crystal approaches zero as temperature approaches absolute zero. He was awarded the Nobel Prize in Chemistry in 1920. Nernst was also an inventor — he developed the Nernst lamp, an early electric light using a ceramic rod that glowed when heated, which briefly competed with Edison's carbon-filament lamp before the tungsten bulb made both obsolete.

  • Standard Reduction Potentials — E°_cell is the starting point of every Nernst calculation; you must know how to look up and combine standard reduction potentials before applying the correction.
  • Galvanic Cells and the Daniell Cell — The Nernst Equation is applied directly to galvanic cell reactions; understanding cell construction and half-reactions is a prerequisite.
  • Reaction Quotient Q and Equilibrium Constant K — Q appears in every Nernst calculation; the relationship E_cell = 0 at equilibrium directly links electrochemistry to chemical equilibrium.
  • Gibbs Free Energy and Spontaneity — The Nernst Equation is derived from ΔG = ΔG° + RT ln Q; understanding ΔG explains why a positive E_cell means a spontaneous reaction.
  • Electrolytic Cells — Contrasting galvanic cells (spontaneous, E > 0) with electrolytic cells (non-spontaneous, requires external voltage) deepens understanding of when the Nernst correction can push a cell over or under zero volts.