Problem 54 - Chemistry: The Central Science

Question

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

Step-by-Step Solution

Answer

The standard Gibbs free energy change (ΔG°) for the given redox reaction is -171,410.3 J/mol, and the reduction potential (E_red) is 1.776 V.

Step 1: Calculate the standard Gibbs free energy change (ΔG°)

Given the equilibrium constant, K = 8.7 × 10^4, temperature T = 298 K, and the gas constant R = 8.314 J/K/mol. We can now calculate the standard Gibbs free energy change, ΔG° using the formula: \( ΔG° = -RT \ln{K} \) Substitute the given values into the equation: \( ΔG° = -(8.314 \frac{J}{K \cdot mol})(298 K) \ln{(8.7 \times 10^4)} \) After calculation, we find that: \( ΔG° = -171,410.3 \frac{J}{mol} \)

Step 2: Calculate the reduction potential (E_red)

We will use the Nernst equation to find the reduction potential E_red, considering the fact that it's a one-electron redox reaction (n = 1): \( E_red = E° - \frac{RT}{nF} \ln{Q} \) Since we need E_red, let's reorganize the equation to find it: \( E_red = \frac{ΔG°}{-nF} \) In this case, n = 1 (one-electron), and the Faraday's constant F = 96,485 C/mol. The ΔG° value we found in the first step is -171,410.3 J/mol. Plug all these values into the equation: \( E_red = \frac{-171,410.3 \frac{J}{mol}}{-1 \times 96,485 \frac{C}{mol}} \) After calculation, we find that: \( E_red = 1.776 V (rounded to three decimal places) \) In conclusion, the standard Gibbs free energy change, ΔG°, is -171,410.3 J/mol, and the reduction potential, E_red, is 1.776 V.

Key Concepts

Equilibrium Constant

In the world of chemistry, the equilibrium constant, often denoted as \( K \), is a significant value that indicates the extent of a reaction under a given set of conditions. It expresses the ratio of the concentrations of products to reactants at equilibrium. In essence, a high equilibrium constant, like \( 8.7 \times 10^4 \), suggests that at equilibrium, the products are overwhelmingly more abundant than the reactants. This value helps chemists understand the favorability of the reaction towards the products or reactants, thereby guiding various experimental and industrial processes.

• When \( K \) is greater than one, the reaction favors products.
• If \( K \) is less than one, it favors reactants.
• \( K \) is dimensionless, as it's simply a ratio.

Gibbs Free Energy

Gibbs free energy, denoted as \( \Delta G \), is a thermodynamic property that can help predict the feasibility of a process. If \( \Delta G \) is negative, the reaction tends to be spontaneous, meaning it can proceed without any input of energy. The measure tells whether a chemical change is energetically favorable. The formula to find the standard Gibbs free energy change \( (\Delta G^{\circ}) \) involves the equilibrium constant:\[ \Delta G^{\circ} = -RT \ln{K} \].
Here:

• \( R \) is the universal gas constant, which is \(8.314 \, \text{J/mol K}\).
• \( T \) is the temperature in Kelvin.

By inserting the known values into this equation, you can find \( \Delta G^{\circ} \), which represents the change in Gibbs free energy under standard conditions.

Reduction Potential

Reduction potential, often represented as \( E_{\text{red}} \), is a measure of the propensity of a chemical species to acquire electrons and thus be reduced. Higher values of reduction potential indicate a greater tendency for reduction, signaling the ease with which a species can gain electrons. In electrochemistry, it is crucial as it gives insight into the relative strengths of different reducers.
To find the reduction potential, we can use the equation:\[ E_{\text{red}} = \frac{\Delta G^{\circ}}{-nF} \]where:

• \( \Delta G^{\circ} \) is the standard Gibbs free energy change.
• \( n \) is the number of moles of electrons transferred in the reaction (in this case, 1).
• \( F \) is Faraday's constant \( (96,485 \, \text{C/mol}) \).

This formula helps calculate \( E_{\text{red}} \) using the Gibbs free energy value obtained earlier.

Nernst Equation

The Nernst equation is a pivotal equation in electrochemistry, used to determine the cell potential of an electrochemical cell under non-standard conditions. The equation provides a relationship between the reduction potential at any conditions and the reduction potential at standard conditions. It's expressed as:\[ E = E^{\circ} - \frac{RT}{nF} \ln{Q} \].
In this expression:

• \( E \) is the electrode potential.
• \( E^{\circ} \) is the standard electrode potential.
• \( R \) is the universal gas constant \( (8.314 \, \text{J/mol K}) \).
• \( n \) is the number of electrons participating in the redox reaction.
• \( F \) is Faraday's constant \( (96,485 \, \text{C/mol}) \).
• \( Q \) is the reaction quotient, representing the ratio of the concentration of products to reactants at any point during the reaction.

The Nernst equation elegantly shows how the cell potential varies with changes in concentration, temperature, and reaction progress.