Using the standard reduction potentials listed in Appendix E, calculate the equilibrium constant for each of the following reactions at 298 K:
(a) Fe(s) + Ni²⁺(aq) → Fe²⁺(aq) + Ni(s)
(b) Co(s) + 2 H⁺(aq) → Co²⁺(aq) + H₂(g)
(c) 10 Br^-(aq) + 2 MnO₄^-(aq) + 16 H⁺(aq) → 2 Mn²⁺(aq) + 8 H₂O(l) + 5 Br₂(l)

Verified step by step guidance

1. Identify the half-reactions involved in the redox process. For the given reaction, the half-reactions are: \( \text{Br}^- \rightarrow \text{Br}_2 \) and \( \text{MnO}_4^- \rightarrow \text{Mn}^{2+} \).
2. Look up the standard reduction potentials (E°) for each half-reaction from Appendix E. The standard reduction potential for \( \text{Br}_2 + 2e^- \rightarrow 2\text{Br}^- \) and \( \text{MnO}_4^- + 8H^+ + 5e^- \rightarrow \text{Mn}^{2+} + 4H_2O \) are needed.
3. Calculate the standard cell potential (E°_cell) using the formula: \( E°_\text{cell} = E°_\text{cathode} - E°_\text{anode} \). Identify which half-reaction is the reduction (cathode) and which is the oxidation (anode).
4. Use the Nernst equation to relate the standard cell potential to the equilibrium constant (K): \( E°_\text{cell} = \frac{RT}{nF} \ln K \), where \( R \) is the gas constant, \( T \) is the temperature in Kelvin, \( n \) is the number of moles of electrons transferred, and \( F \) is Faraday's constant.
5. Rearrange the Nernst equation to solve for the equilibrium constant \( K \): \( \ln K = \frac{nFE°_\text{cell}}{RT} \). Substitute the known values to find \( K \).

Key Concepts

Standard Reduction Potentials

Standard reduction potentials are measured voltages that indicate the tendency of a chemical species to gain electrons and be reduced. Each half-reaction has a specific potential, and these values are typically listed in tables. The more positive the potential, the greater the species' ability to be reduced. These values are essential for calculating the overall cell potential and determining the direction of electron flow in redox reactions.

Nernst Equation

The Nernst equation relates the cell potential to the concentrations of the reactants and products at non-standard conditions. It is expressed as E = E° - (RT/nF)ln(Q), where E° is the standard cell potential, R is the gas constant, T is the temperature in Kelvin, n is the number of moles of electrons transferred, F is Faraday's constant, and Q is the reaction quotient. This equation is crucial for calculating the equilibrium constant from the standard potentials.

Equilibrium Constant (K)

The equilibrium constant (K) quantifies the ratio of the concentrations of products to reactants at equilibrium for a given reaction at a specific temperature. It is derived from the standard Gibbs free energy change and can be calculated using the relationship K = e^(nFE°/RT). Understanding K is vital for predicting the extent of a reaction and how changes in conditions affect the position of equilibrium.