Thermodynamics of Cells

Gibbs free energy (ΔG) predicts spontaneity: if ΔG < 0, the reaction is spontaneous; if ΔG > 0, it is non-spontaneous; if ΔG = 0, the system is at equilibrium. In a galvanic cell, the spontaneous redox reaction drives electrons through an external circuit, doing electrical work. The maximum work a system can do at constant temperature and pressure equals −ΔG.

Electrical work (w_elec) is defined as w_elec = charge × potential difference = (nF) × E_cell, where n is the number of moles of electrons transferred and F is the Faraday constant (96,485 C mol⁻¹). Since the maximum electrical work equals −ΔG, we write: ΔG = −nFE_cell. A positive E_cell means ΔG is negative — the cell reaction is spontaneous. For example, in a Daniell cell ($\ce{Zn | Zn^{2+} || Cu^{2+} | Cu}$), E°_cell = +1.10 V, so ΔG° = −2 × 96485 × 1.10 = −212,267 J mol⁻¹ ≈ −212 kJ mol⁻¹.

Under standard conditions (all solutes at 1 mol L⁻¹, all gases at 1 atm, temperature 298 K), the equation becomes ΔG° = −nFE°_cell. The standard cell potential E°_cell is calculated from standard reduction potentials: E°_cell = E°_cathode − E°_anode. For the Daniell cell: E°(Cu²⁺/Cu) = +0.34 V, E°(Zn²⁺/Zn) = −0.76 V, so E°_cell = 0.34 − (−0.76) = +1.10 V.

From thermodynamics: ΔG° = −RT ln K, and from electrochemistry: ΔG° = −nFE°_cell. Setting these equal: −nFE°_cell = −RT ln K, which rearranges to ln K = nFE°_cell / RT. At T = 298 K, using R = 8.314 J mol⁻¹ K⁻¹, this simplifies in base-10 logarithm form to: log K = nE°_cell / 0.05916. A large positive E°_cell gives a very large K, confirming the reaction strongly favours products. For the Daniell cell: log K = (2 × 1.10) / 0.05916 ≈ 37.2, so K ≈ 10^37.2, an enormous equilibrium constant showing the reaction is essentially complete.

When concentrations differ from standard, the actual cell potential is found using the Nernst equation: E_cell = E°_cell − (RT/nF) ln Q, or at 298 K: E_cell = E°_cell − (0.05916/n) log Q, where Q is the reaction quotient. As the cell discharges, reactant concentrations fall and product concentrations rise, causing Q to increase and E_cell to decrease. At equilibrium, E_cell = 0 and Q = K, consistent with ΔG = 0.

• \Delta G — Gibbs free energy change (J mol⁻¹): Energy available to do useful work at constant T and P; negative value indicates spontaneous reaction.
• n — Moles of electrons transferred (mol): The stoichiometric number of electrons exchanged in the balanced redox half-reactions.
• F — Faraday constant (C mol⁻¹): Charge of one mole of electrons; F = 96,485 C mol⁻¹.
• E_{\text{cell}} — Cell potential (EMF) (V (volts)): The voltage produced by the galvanic cell; positive for a spontaneous cell reaction.

Problem. A galvanic cell is constructed using a silver half-cell ($\ce{Ag+}$/Ag, E° = +0.80 V) and a nickel half-cell ($\ce{Ni^{2+}}$/Ni, E° = −0.25 V). (a) Write the overall cell reaction. (b) Calculate ΔG° for the reaction. (c) Calculate the equilibrium constant K at 25°C.

Answer. E°_cell = +1.05 V; ΔG° = −203 kJ mol⁻¹; K ≈ 3.2 × 10³⁵

• Standard Electrode Potentials — E°_cell is calculated from standard electrode potentials (E°_cathode − E°_anode), which is the direct input to ΔG° = −nFE°_cell.
• The Nernst Equation — The Nernst equation extends ΔG = −nFE to non-standard conditions by incorporating the reaction quotient Q, directly derived from ΔG = ΔG° + RT ln Q.
• Chemical Equilibrium and Gibbs Energy — ΔG° = −RT ln K is a thermodynamic principle that connects standard free energy to equilibrium, forming one half of the triangle linking K, ΔG°, and E°_cell.
• Electrolysis and Faraday's Laws — The Faraday constant (F = 96,485 C mol⁻¹) appears in both ΔG = −nFE and Faraday's laws of electrolysis, showing the unity of thermodynamic and quantitative electrochemistry.