Question

Derive an equation that directly relates the standard emf of a redox reaction to its equilibrium constant.

Short answer

The equation that directly relates the standard emf of a redox reaction (E°) to its equilibrium constant (K_eq) can be derived by combining the Nernst Equation with the relationships between standard Gibbs free energy change (ΔG°) and equilibrium constants. The final equation is: \(E° = -\frac{RT}{nF} * ln(K_{eq})\) where R is the gas constant, T is temperature in Kelvin, n is the number of electrons transferred in the redox reaction, and F is Faraday's constant.

Step-by-step solution

Recall the Nernst Equation

The Nernst Equation is used to calculate the emf of an electrochemical cell under non-standard conditions. For a redox reaction, it is given as: \(E = E° - \frac{RT}{nF} * ln(Q)\) where: - E is the emf of the reaction under non-standard conditions - E° is the standard emf - R is the gas constant - T is temperature (in Kelvin) - n is the number of electrons transferred in the redox reaction - F is Faraday's constant - Q is the reaction quotient

Recall the relationship between ΔG° and K_eq

The standard Gibbs free energy change (ΔG°) of a reaction is related to its equilibrium constant (K_eq) by the equation: \(ΔG° = -RT * ln(K_{eq})\)

Recall the relationship between ΔG and E

The relationship between Gibbs free energy change (ΔG) and emf (E) is given by: \(ΔG = -nFE\) The relationship between standard Gibbs free energy change (ΔG°) and standard emf (E°) is given by: \(ΔG° = -nFE°\)

Derive the equation relating E° and K_eq

Using the relationships from Step 2 and Step 3, we have: \(ΔG° = -nFE° = -RT * ln(K_{eq})\) Now we can solve for E°: \(E° = -\frac{RT}{nF} * ln(K_{eq})\) This equation directly relates the standard emf of a redox reaction to its equilibrium constant.