Electromotive Force, Gibbs Free Energy & Equilibria in Electrochemical Cells

Gibbs Free Energy (ΔG) and Electromotive Force (EMF)

• Thermodynamic criterion for spontaneity
  • Spontaneous if $\Delta G < 0$; non-spontaneous if $\Delta G > 0$.
  • $\Delta G$ represents the maximum non-PV work obtainable from a chemical system.
• Electrochemical work analogy to physics
  • Physics work formula: $w = q\,\Delta V$ (charge moved through a potential).
  • In cells, total charge moved = $nF$; potential difference = $E_{\text{cell}}$.
• Fundamental relationship under standard conditions
  • $\Delta G^\circ = -n F E^\circ_{\text{cell}}$.
  • Variables:
  – $n$ = mol e⁻ transferred (integer from balanced equation).
  – $F = 96\,485\;\text{C\,mol}^{-1}$ (Faraday constant).
  • Units: if $F$ in coulombs, express $\Delta G$ in joules, not kJ.

Sign Conventions & Cell Types

• Opposite signs: $\Delta G^\circ$ and $E^\circ_{\text{cell}}$ always have opposite algebraic signs due to the negative sign in their relation.
• Galvanic (Voltaic) cells
  • Spontaneous: $\Delta G^\circ < 0$, $E^\circ_{\text{cell}} > 0$.
• Electrolytic cells
  • Non-spontaneous: \Delta G^\circ > 0, E^\circ_{\text{cell}} < 0.
• Concentration cells
  • Special galvanic cells; EMF arises solely from ion-concentration gradient even though electrodes are the same metal.
  • When concentrations equalize, $E_{\text{cell}} \to 0$.

Worked Example 1: Standard ΔG° of an Electrolytic Cell

• Given overall reaction
  $2\,\text{Fe}^{3+}<em>{(aq)} + 2\,\text{Cl}^-</em>{(aq)} \rightarrow 2\,\text{Fe}^{2+}<em>{(aq)} + \text{Cl}</em>2_{(g)}$
• Standard reduction potentials
  • $E^\circ<em>{\text{red}}(\text{Fe}^{3+}/\text{Fe}^{2+}) = +0.77\;\text{V}$ (cathode). • $E^\circ</em>{\text{red}}(\text{Cl}_2/\text{Cl}^-) = +1.36\;\text{V}$.
• Determine anode/cathode
  • Despite larger $E^\circ_{\text{red}}$, Cl⁻ is forced to oxidize (serves as anode) → reaction non-spontaneous.
• Calculate EMF
  $E^\circ<em>{\text{cell}} = E^\circ</em>{\text{red,cathode}} - E^\circ_{\text{red,anode}} = 0.77 - 1.36 = -0.59\;\text{V}$.
• Calculate $\Delta G^\circ$
  $n = 2$ e⁻.
  $\Delta G^\circ = -nF E^\circ_{\text{cell}} = -2(96\,485)(-0.59) \approx +1.14 \times 10^5\;\text{J} \,(\approx +120\,\text{kJ})$.
  • Positive value confirms non-spontaneity.

Reaction Quotient (Q) & the Nernst Equation

• Need for non-standard calculations
  • Real cells seldom have 1 M ion concentrations; concentration cells require unequal concentrations to work.
• General Nernst form
  $E<em>{\text{cell}} = E^\circ</em>{\text{cell}} - \frac{RT}{nF} \ln Q$.
  • $R = 8.314\;\text{J\,mol}^{-1}\,\text{K}^{-1}$, $T$ in kelvin.
  • $Q$ = reaction quotient, same form as equilibrium constant but instantaneous concentrations.
  • Only aqueous (or gaseous) species appear; pure solids/liquids omitted.
• Simplified (25 °C = 298 K)
  $E<em>{\text{cell}} = E^\circ</em>{\text{cell}} - \frac{0.0592}{n} \log Q$ (base-10 log).

Worked Example 2: EMF Under Non-Standard Conditions

• Half-reactions & potentials
  • $\text{Fe}^{2+} + 2e^- \rightarrow \text{Fe}$ $E^\circ<em>{\text{red}} = -0.44\;\text{V}$. • $\text{Cl}</em>2 + 2e^- \rightarrow 2\,\text{Cl}^-$ $E^\circ_{\text{red}} = +1.36\;\text{V}$.
• Identify electrodes
  • Higher $E^\circ_{\text{red}}$ → cathode (Cl₂/Cl⁻).
  • Iron acts as anode (is oxidized).
• Standard cell EMF
  $E^\circ_{\text{cell}} = 1.36 - (-0.44) = +1.80\;\text{V}$.
• Cell concentrations
  • $[\text{Fe}^{2+}] = 0.01\;\text{M}$, $[\text{Cl}^-] = 0.10\;\text{M}$.
• Net ionic equation
  $\text{Fe} + \text{Cl}_2 \rightarrow \text{Fe}^{2+} + 2\,\text{Cl}^-$.
• Reaction quotient
  $Q = \frac{[\text{Fe}^{2+}][\text{Cl}^-]^2}{\text{(Fe,Cl}_2\text{ solids/gases omitted)}} = 0.01 \times (0.10)^2 = 1\times10^{-4}$.
• EMF at 25 °C
  $E_{\text{cell}} = 1.80 - \frac{0.0592}{2} \log(1\times10^{-4})$
  $= 1.80 - 0.0296 \times (-4)$
  $= 1.80 + 0.118 \approx 1.92\;\text{V}$.
  • Cell delivers higher voltage than standard due to favorable concentration difference.

Equilibria & Standard Potentials

• Free energy via equilibrium constant
  $\Delta G^\circ = -RT \ln K_{eq}$.
• Combining with EMF relation
  $nF E^\circ<em>{\text{cell}} = RT \ln K</em>{eq}$.
• Qualitative insights
  • $K<em>{eq} > 1$ → products favored, \ln K{eq} > 0 → $E^\circ<em>{\text{cell}} > 0$ (galvanic). • K{eq} < 1 → reactants favored, $E^\circ<em>{\text{cell}} < 0$ (electrolytic). • $K</em>{eq} = 1$ → $E^\circ_{\text{cell}} = 0$ (e.g., concentration cell at equilibrium).
• Non-standard free energy
  $\Delta G = \Delta G^\circ + RT \ln Q$ (parallels Nernst equation).

Practical Measurement Tools

• Voltmeter: measures EMF while allowing current to flow.
• Potentiometer: a high-impedance voltmeter that draws negligible current → more accurate EMF at true open-circuit potential.

Key Equations (Quick Reference)

• $\Delta G = \Delta G^\circ + RT \ln Q$.
• $\Delta G^\circ = -n F E^\circ_{\text{cell}}$.
• $\Delta G^\circ = -RT \ln K_{eq}$.
• $n F E^\circ<em>{\text{cell}} = RT \ln K</em>{eq}$.
• $E<em>{\text{cell}} = E^\circ</em>{\text{cell}} - \dfrac{RT}{nF} \ln Q$.
• $E<em>{\text{cell}} = E^\circ</em>{\text{cell}} - \dfrac{0.0592}{n} \log Q$ (298 K).

Biological & Real-World Connections

• Heart: functions as a self-paced electrochemical (galvanic-like) cell generating action potentials.
• Neurons: operate as rechargeable concentration cells; ion gradients across membranes produce electrical signaling.
• Mitochondria: proton-motive force across inner membrane analogous to an electrochemical cell, driving ATP synthesis.
• Medicine: understanding electrochemistry underpins cardiology (EKG), neurophysiology, bioenergetics, and medical devices (defibrillators, pacemakers).

Terminology & Definitions

• EMF (E): voltage produced by a cell; potential difference under open-circuit conditions.
• Faraday constant (F): charge of 1 mol electrons, $96\,485\;\text{C}$.
• Standard state: 1 M solutes, 1 atm gases, 25 °C (298 K).
• Reaction quotient (Q): product-to-reactant ratio at a given moment; same form as $K_{eq}$ but not necessarily at equilibrium.
• Standard reduction potential ($E^\circ_{\text{red}}$): intrinsic tendency of a species to gain electrons under standard conditions relative to SHE.

MCAT Strategy & Tips

• Memorize sign conventions: “GALVANIC: G
• For EMF computations:
  • Identify cathode (higher $E^\circ<em>{\text{red}}$) and anode. • Use subtraction: $E^\circ</em>{\text{cell}} = E^\circ<em>{\text{red,cathode}} - E^\circ</em>{\text{red,anode}}$.
• Quick logs: $\log(10^{-x}) = -x$; handy for Nernst approximations.
• Always confirm $n$ from balanced NET ionic equation.

Summary / Take-Home Messages

• $\Delta G$, EMF, and ion concentrations are interlinked; altering any alters the others.
• The Nernst equation predicts the real-time voltage output of a cell under varying conditions.
• At equilibrium, cell voltage drops to zero, but concentration differences (or external voltage) can maintain or drive reactions.
• Electrochemical principles bridge inorganic chemistry, physiology, and medical technology.