This problem involves relating Gibbs free energy change of a reaction to the electrochemical oxidation of a species. Let's break it down step by step.

Step 1: Understand the Given Data

The reaction X → Y has a standard Gibbs free energy change, Δ_rG⁰ = -193 kJ/mol. This energy is used to oxidize M⁺ to M³⁺ via the half-reaction:

M⁺ → M³⁺ + 2e^-, with a standard electrode potential E⁰ = -0.25 V.

We are to find how many moles of M⁺ are oxidized when 1 mole of X is converted to Y.

Step 2: Relate Gibbs Free Energy to Electrochemical Work

The energy released by the reaction X → Y (which is -Δ_rG⁰) is used to do electrochemical work for oxidizing M⁺. The work done in an electrochemical cell is related to the Gibbs free energy change of the cell reaction.

For the oxidation half-reaction: M⁺ → M³⁺ + 2e^-, the standard Gibbs free energy change is given by:

$\mathrm{\Delta G}{}^0=-nF{E}^0$

where n is the number of electrons transferred (which is 2 for this reaction), F is the Faraday constant (96500 C/mol), and E⁰ is the standard electrode potential.

Substituting the values:

$\mathrm{\Delta G}{}^0=-2\times 96500\times (-0.25)$

First, calculate the numerical value:

2 × 96500 = 193000

193000 × 0.25 = 48250

So, ΔG⁰ = -(-48250) = +48250 J/mol = +48.25 kJ/mol (since 1000 J = 1 kJ)

This positive value indicates that the oxidation of M⁺ is non-spontaneous under standard conditions and requires energy input.

Step 3: Equate the Energy Released to Energy Used

The energy released by the conversion of 1 mole of X to Y is |Δ_rG⁰| = 193 kJ. This energy is used to drive the oxidation of M⁺.

Let the number of moles of M⁺ oxidized be N. The total energy required for oxidizing N moles of M⁺ is N × |ΔG⁰ for oxidation|.

Since the energy released equals the energy used:

$193=N\times 48.25$

Step 4: Solve for N

$N=\frac{193}{48.25}=4$

So, 4 moles of M⁺ are oxidized when 1 mole of X is converted to Y.

Final Answer

The number of moles of M⁺ oxidized is 4.

Related Topics and Formulae

Gibbs Free Energy and Electrochemistry

The relationship between Gibbs free energy change and cell potential is fundamental in electrochemistry:

$\mathrm{\Delta G}{}^0=-nF{E}^0$

where ΔG⁰ is the standard Gibbs free energy change, n is the number of electrons transferred in the reaction, F is the Faraday constant (96500 C/mol), and E⁰ is the standard cell potential.

A negative ΔG⁰ indicates a spontaneous reaction, while a positive ΔG⁰ indicates non-spontaneity.

Energy Conservation

In processes where energy from one reaction is used to drive another, the principle of energy conservation applies. The useful work obtained from a spontaneous reaction can be used to perform non-spontaneous work, as in this problem.