What are \(E_{\text {cell }}^{\circ}\) and \(\Delta G^{\circ}\) of a redox reaction at \(25^{\circ} \mathrm{C}\) for which \(n=2\) and \(K=0.065 ?\)

The Nernst equation is crucial when it comes to understanding electrochemical cells. In simple terms, it relates the reduction potential of a half-cell in an electrochemical cell to the standard electrode potential, temperature, activity, and the reaction quotient. At standard conditions, the Nernst equation for calculating the cell potential is:

\[ E_{\text {cell }}^{\text {o}} = \frac{RT}{nF} \times \text{ln}(K) \]

Here’s a quick guide to the important components:

• \text{R} = 8.314 J/(mol·K) is the universal gas constant.
• \text{T} is the temperature in Kelvins; for standard conditions, it's 298 K (25°C).
• \text{n} is the number of moles of electrons transferred in the reaction.
• \text{F} is Faraday's constant (96485 C/mol).
• \text{K} is the equilibrium constant.

This equation helps us find \( E_{\text {cell }}^{\text {o}} \) when we're given \( K \), translating the equilibrium state information into the cell potential. This relationship shows us how the spontaneity of a reaction changes with different species concentrations.

The standard Gibbs free energy change (ΔG°) indicates the spontaneity of a reaction under standard conditions. The formula linking ΔG° to the standard electrode potential is:

\[ \triangle G^{\text{o}} = -nFE_{\text {cell }}^{\text{o}} \]

Let’s break down what this means:

• \text{n} is the number of moles of electrons transferred.
• \text{F} is Faraday's constant (96485 C/mol).
• \text{E_{\text {cell }}^{\text{o}}} is the standard electrode potential found using the Nernst equation.

ΔG° tells us if a reaction is spontaneous (ΔG° < 0) or non-spontaneous (ΔG° > 0) under standard conditions. If ΔG° is zero, the system is at equilibrium. In essence, this relationship connects electrochemistry with thermodynamics.
In our exercise, the calculated ΔG° is approximately +6.847 kJ/mol, indicating that the reaction is non-spontaneous at standard conditions.

The equilibrium constant (K) is a reflection of the ratio of products to reactants at equilibrium for a particular reaction. A larger K (>1) generally means a reaction favors the formation of products, while a smaller K (<1) suggests that reactants are preferred.

In electrochemical cells, the equilibrium constant impacts the standard electrode potential (\text{E_{\text {cell }}^{\text{o}}}). Remember, the Nernst equation:

\[ E_{\text {cell }}^{\text{o}} = \frac{RT}{nF} \times \text{ln}(K) \]

This equation helps to link the equilibrium constant with the measurable electrode potential of the cell. By calculating the natural logarithm of K and incorporating temperature, number of electrons, and constants like \text{R} and \text{F}, we can find \text{E_{\text {cell }}^{\text{o}}}.
In our given problem, K was 0.065, indicating the reaction heavily favors reactants over products at equilibrium. This is reflected in a small and slightly negative \text{E_{\text {cell }}^{\text{o}}}.