4.2 The Nernst Equation and Its Applications

The Nernst equation connects cell potential to concentration and temperature in electrochemical reactions. It's crucial for understanding how these factors affect the energy available in a system, helping predict reaction spontaneity and equilibrium conditions.

This equation bridges thermodynamics and electrochemistry, allowing us to calculate cell potentials under non-standard conditions. It's a powerful tool for analyzing real-world electrochemical systems and their behavior in various environments.

Thermodynamic Principles and the Nernst Equation

Derivation of Nernst equation

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• Relates Gibbs free energy ($\Delta G$) to cell potential ($E_{cell}$) using $\Delta G = -nFE_{cell}$
  • $n$ represents number of electrons transferred in redox reaction
  • $F$ is Faraday's constant (96,485 C/mol)
• Change in Gibbs free energy also depends on standard Gibbs free energy change ($\Delta G^{\circ}$) and reaction quotient ($Q$) via $\Delta G = \Delta G^{\circ} + RT \ln Q$
  • $R$ is the gas constant (8.314 J/mol·K)
  • $T$ is the temperature in Kelvin
• Combining equations and solving for $E_{cell}$ yields Nernst equation $E_{cell} = E_{cell}^{\circ} - \frac{RT}{nF} \ln Q$
• At standard temperature (298 K), Nernst equation simplifies to $E_{cell} = E_{cell}^{\circ} - \frac{0.0592V}{n} \log Q$

Applications of the Nernst Equation

Application of Nernst equation

• Calculates cell potentials under non-standard conditions (concentrations ≠ 1 M or gas pressures ≠ 1 atm)
• Steps to calculate non-standard cell potential:
  1. Determine standard cell potential ($E_{cell}^{\circ}$) from table of standard reduction potentials
  2. Calculate reaction quotient ($Q$) based on concentrations or partial pressures of reactants and products
  3. Substitute values into Nernst equation and solve for $E_{cell}$
• Same process applies to calculate potential of individual electrodes under non-standard conditions
  • Use standard reduction potential of electrode instead of standard cell potential

Concentration effects on cell potentials

• Nernst equation reveals cell potential depends on concentrations of reactants and products
  • Increasing reactant concentration or decreasing product concentration increases cell potential
  • Decreasing reactant concentration or increasing product concentration decreases cell potential
• Magnitude of change in cell potential depends on reaction stoichiometry
  • For 1:1 stoichiometry, tenfold concentration change results in $\frac{0.0592V}{n}$ change in cell potential
  • For other stoichiometries, change in cell potential calculated using Nernst equation

Relationship between Cell Potential, Free Energy, and Equilibrium Constants

Cell potential vs free energy

• Cell potential and Gibbs free energy related by $\Delta G = -nFE_{cell}$
• Under standard conditions, becomes $\Delta G^{\circ} = -nFE_{cell}^{\circ}$
• Standard Gibbs free energy change related to equilibrium constant ($K$) by $\Delta G^{\circ} = -RT \ln K$
• Combining equations yields relationship between standard cell potential and equilibrium constant $E_{cell}^{\circ} = \frac{RT}{nF} \ln K$
• These relationships allow:
  1. Calculation of equilibrium constant from standard cell potential
  2. Calculation of standard cell potential from equilibrium constant
  3. Determination of redox reaction spontaneity based on sign of cell potential or Gibbs free energy change