Calculate ∆G°_rxn and K for each reaction.
a. The disproportionation of Mn²⁺(aq) to Mn(s) and MnO₂(s) in acid solution at 25 °C.
b. The disproportionation of MnO₂(s) to Mn²⁺(aq) and MnO₄^–(aq) in acid solution at 25 °C.

Verified step by step guidance

Step 1: Identify the half-reactions involved in the disproportionation of Mn²⁺(aq) to Mn(s) and MnO₂(s). The two half-reactions are: Mn²⁺(aq) → Mn(s) and Mn²⁺(aq) → MnO₂(s).

Step 2: Write the balanced half-reactions and determine the standard reduction potentials (E°) for each half-reaction from a standard reduction potential table.

Step 3: Calculate the standard cell potential (E°_cell) for the overall reaction using the formula: E°_cell = E°_cathode - E°_anode.

Step 4: Use the Nernst equation to calculate the standard Gibbs free energy change (∆G°_rxn) for the reaction: ∆G°_rxn = -nFE°_cell, where n is the number of moles of electrons transferred and F is the Faraday constant.

Step 5: Calculate the equilibrium constant (K) for the reaction using the relationship: ∆G°_rxn = -RTlnK, where R is the universal gas constant and T is the temperature in Kelvin.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Gibbs Free Energy (∆Gr°)

Gibbs Free Energy (∆Gr°) is a thermodynamic potential that measures the maximum reversible work obtainable from a thermodynamic process at constant temperature and pressure. It indicates the spontaneity of a reaction; a negative ∆Gr° suggests that the reaction can occur spontaneously, while a positive value indicates non-spontaneity. The standard Gibbs free energy change can be calculated using standard enthalpy and entropy values.

Equilibrium Constant (K)

The equilibrium constant (K) is a dimensionless number that expresses the ratio of the concentrations of products to reactants at equilibrium for a given reaction at a specific temperature. It provides insight into the extent of a reaction; a large K value indicates that products are favored, while a small K suggests that reactants are favored. The relationship between K and Gibbs free energy change is given by the equation ∆Gr° = -RT ln(K).

Disproportionation Reaction

A disproportionation reaction is a specific type of redox reaction in which a single substance is simultaneously oxidized and reduced, resulting in two different products. In the case of Mn2+(aq), it is transformed into elemental manganese (Mn) and manganese dioxide (MnO2) under acidic conditions. Understanding the half-reactions involved is crucial for calculating the overall Gibbs free energy change and equilibrium constant for the reaction.

Calculate ∆G°rxn and K for each reaction.a. The disproportionation of Mn2+(aq) to Mn(s) and MnO2(s) in acid solution at 25 °C.b. The disproportionation of MnO2(s) to Mn2+(aq) and MnO4–(aq) in acid solution at 25 °C.