A disproportionation reaction involves a substance that acts as both an oxidizing and a reducing agent, producing higher and lower oxidation states of the same element in the products. Which of the following disproportionation reactions are spontaneous under standard conditions? Calculate $\Delta G^{\circ}\( and \)K\( at \)25^{\circ} \mathrm{C}$ for those reactions that are spontaneous under standard conditions. a. $2 \mathrm{Cu}^{+}(a q) \longrightarrow \mathrm{Cu}^{2+}(a q)+\mathrm{Cu}(s)$ b. $3 \mathrm{Fe}^{2+}(a q) \longrightarrow 2 \mathrm{Fe}^{3+}(a q)+\mathrm{Fe}(s)$ c. $\mathrm{HClO}_{2}(a q) \longrightarrow \mathrm{ClO}_{3}^{-}(a q)+\mathrm{HClO}(a q) \quad$ (unbalanced) Use the half-reactions: $\mathrm{ClO}_{3}^{-}+3 \mathrm{H}^{+}+2 \mathrm{e}^{-} \longrightarrow \mathrm{HClO}_{2}+\mathrm{H}_{2} \mathrm{O} \quad \mathscr{E}^{\circ}=1.21 \mathrm{V}$ $\mathrm{HClO}_{2}+2 \mathrm{H}^{+}+2 \mathrm{e}^{-} \longrightarrow \mathrm{HClO}+\mathrm{H}_{2} \mathrm{O} \quad \mathscr{E}^{\circ}=1.65 \mathrm{V}$

Step by step solution

01

Identify the half-reactions for reaction a

Reaction a: \(2 \mathrm{Cu}^{+}(a q) \longrightarrow \mathrm{Cu}^{2+}(a q)+\mathrm{Cu}(s)\) Half-reaction 1: \(\mathrm{Cu}^{+}(a q) \longrightarrow \mathrm{Cu}^{2+}(a q) + 1 e^-\) Half-reaction 2: \(\mathrm{Cu}^{+}(a q) + 1 e^- \longrightarrow \mathrm{Cu}(s)\)

02

Identify the half-reactions for reaction b

Reaction b: \(3 \mathrm{Fe}^{2+}(a q) \longrightarrow 2 \mathrm{Fe}^{3+}(a q)+\mathrm{Fe}(s)\) Half-reaction 1: \(2 \mathrm{Fe}^{2+}(a q) \longrightarrow 2 \mathrm{Fe}^{3+}(a q) + 2 e^-\) Half-reaction 2: \(\mathrm{Fe}^{2+}(a q) + 2 e^- \longrightarrow \mathrm{Fe}(s)\) ##Step 2: Calculate the standard cell potentials and Gibbs free energy changes for reactions a and b## The equation to convert the standard cell potentials, \(\mathscr{E}^{\circ}\), to Gibbs free energy change, \(\Delta G^{\circ}\), is: \(\Delta G^{\circ} = -nFE^{\circ}\), where n is the number of moles of electrons transferred, F is the Faraday's constant (96485 C/mol).

03

Calculate \(\Delta G^{\circ}\) for reaction a

\(\Delta G^{\circ}_{a} = -1(96485)(\mathscr{E}^{\circ}_{Cu^{2+}/Cu}-\mathscr{E}^{\circ}_{Cu^{+}/Cu})\)

04

Calculate \(\Delta G^{\circ}\) for reaction b

\(\Delta G^{\circ}_{b} = -n(96485)(\mathscr{E}^{\circ}_{Fe^{3+}/Fe^2+}-\mathscr{E}^{\circ}_{Fe^{2+}/Fe})\) ##Step 3: Determine the spontaneity of reactions a and b## A reaction is spontaneous if \(\Delta G^{\circ} < 0\).

05

Determine spontaneity for reaction a

If \(\Delta G^{\circ}_{a} < 0\), reaction a is spontaneous.

06

Determine spontaneity for reaction b

If \(\Delta G^{\circ}_{b} < 0\), reaction b is spontaneous. ##Step 4: Calculate \(K\) for the spontaneous reactions## The equation relating Gibbs free energy change and the equilibrium constant is: \(\Delta G^{\circ}=-RT \ln K\) Where R is the gas constant (8.314 J/mol K) and T is the temperature (298 K).

07

Calculate \(K\) for reaction a (if spontaneous)

If reaction a is spontaneous, use \(\Delta G^{\circ}_{a}\) to calculate \(K_{a}\): \(K_{a} = e^{(-\Delta G^{\circ}_{a})/ (RT)}\)

08

Calculate \(K\) for reaction b (if spontaneous)

If reaction b is spontaneous, use \(\Delta G^{\circ}_{b}\) to calculate \(K_{b}\): \(K_{b} = e^{(-\Delta G^{\circ}_{b})/ (RT)}\) ##Step 5: Determine the disproportionation reaction for HClO2## For reaction c with HClO2, the half-reactions are provided. First, we should balance the overall redox reaction.

09

Balance the redox reaction for HClO2

Balanced disproportionation reaction for HClO2: \(2 \mathrm{HClO}_{2}(a q) \longrightarrow \mathrm{ClO}_{3}^{-}(a q) + \mathrm{HClO}(a q) + \mathrm{H}_{2}\mathrm{O}\) ##Step 6: Calculate the \(\Delta G^{\circ}\) and \(K\) for HClO2 disproportionation## Calculate the standard Gibbs free energy change and the equilibrium constant for the HClO2 disproportionation using the provided half-reaction potentials.

10

Calculate \(\Delta G^{\circ}\) for HClO2 disproportionation

\(\Delta G^{\circ}_{c} = -n(96485)(\mathscr{E}^{\circ}_{ClO_{3}^{-}/HClO_{2}}-\mathscr{E}^{\circ}_{HClO_{2}/HClO})\)

11

Calculate \(K\) for HClO2 disproportionation (if spontaneous)

If the HClO2 disproportionation is spontaneous, use \(\Delta G^{\circ}_{c}\) to calculate \(K_{c}\): \(K_{c} = e^{(-\Delta G^{\circ}_{c})/ (RT)}\) With these calculations, we can determine the spontaneous disproportionation reactions under standard conditions from the given examples and their respective equilibrium constants.

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