Question

If the equilibrium constant for a one-electron redox reaction at \(298 \mathrm{~K}\) is \(8.7 \times 10^{4}\), calculate the corresponding \(\Delta G^{\circ}\) and \(E_{\text {cell }}^{0}\)

Short answer

The standard Gibbs free energy change (ΔG°) for the redox reaction is -44.39 kJ/mol, and the standard cell potential (E°cell) is 0.44 V.

Step-by-step solution

Find ΔG°

: First, we'll use the formula 𝚫G° = -RT lnK to find the Gibbs free energy change (ΔG°): ΔG° = -(8.314 J/mol·K) × (298 K) × ln(8.7 × 10^4) ΔG° ≈ -44387.6 J/mol Since the result is typically expressed in kJ/mol, we'll convert it: ΔG° ≈ -44.39 kJ/mol

Calculate E°cell

: Now, we'll use the formula E°cell = -ΔG° / nF to find the standard cell potential: E°cell = -(-44.39 kJ/mol) / (1 × 96,485 C/mol) E°cell ≈ 0.44 V So, the standard Gibbs free energy change (ΔG°) for the redox reaction is -44.39 kJ/mol, and the standard cell potential (E°cell) is 0.44 V.

Key concepts

Gibbs Free Energy

Gibbs free energy, often denoted as \( \Delta G \), is a thermodynamic quantity that helps predict the spontaneity of a chemical reaction.

When calculating \( \Delta G^{\circ} \), the formula used is:

• \( \Delta G^{\circ} = -RT \ln K \)
• \( R \) is the universal gas constant (8.314 J/mol·K)
• \( T \) is the temperature in Kelvin
• \( K \) is the equilibrium constant of the reaction

This formula shows the relationship between Gibbs free energy and the equilibrium constant. A negative \( \Delta G^{\circ} \) indicates a spontaneous reaction under standard conditions, meaning the reaction will proceed without external energy input.

In our case, substituting the values gives \( \Delta G^{\circ} \approx -44.39 \, \text{kJ/mol} \), indicating the reaction is spontaneous.

Redox Reaction

Redox, short for reduction-oxidation, reactions are chemical reactions where electrons are transferred between substances. These reactions are crucial in many natural processes and technological applications.

Each redox reaction consists of two half-reactions:

• Oxidation: Loss of electrons
• Reduction: Gain of electrons

These processes must occur simultaneously. The electrons lost in the oxidation process are gained by another species in the reduction process.

In the provided exercise, the reaction involves a one-electron redox process. The equilibrium constant helps us understand how far the reaction proceeds, while the associated Gibbs free energy tells us about spontaneity.

Standard Cell Potential

Standard cell potential \( (E_{\text{cell}}^{0}) \) is a measure of the voltage or electric potential difference of a cell under standard conditions. It reflects the cell's ability to do electrical work.

To find \( E_{\text{cell}}^{0} \), we use the formula:

• \( E_{\text{cell}}^{0} = -\frac{\Delta G^{\circ}}{nF} \)
• \( n \) is the number of moles of electrons transferred in the reaction
• \( F \) is Faraday’s constant (96,485 C/mol)

In this scenario, since \( \Delta G^{\circ} \approx -44.39 \, \text{kJ/mol} \) and \( n = 1 \), we calculate \( E_{\text{cell}}^{0} \approx 0.44 \, \text{V} \).

A positive standard cell potential indicates that the cell reaction is spontaneous and can produce electrical energy. This aligns with our Gibbs free energy calculation, confirming the reaction's spontaneity.