Understanding the Problem

We are given a 5.2 molal aqueous solution of methyl alcohol (CH₃OH). Molality (m) is defined as the number of moles of solute per kilogram of solvent. Here, the solute is methyl alcohol and the solvent is water. We need to find the mole fraction of methyl alcohol in this solution.

Step 1: Recall the Definitions

Molality (m) = moles of solute / mass of solvent in kg

Given molality = 5.2 m, which means there are 5.2 moles of CH₃OH in 1 kg of water.

Mole fraction (X) of a component = moles of that component / total moles in solution

So, for methyl alcohol, $X_{C{H}_{3}OH}=\frac{n_{C{H}_{3}OH}}{n_{C{H}_{3}OH}+n_{H_{2}O}}$

Step 2: Assign Known Values from Molality

From the molality (5.2 m):

Moles of CH₃OH, $n_{C{H}_{3}OH}=5.2\mathrm{mol}$

Mass of water (solvent) = 1 kg = 1000 g

Step 3: Find Moles of Water (Solvent)

Molar mass of water (H₂O) = 18 g/mol

Moles of H₂O, $n_{H_{2}O}=\frac{mass}{molarmass}=\frac{1000}{18}\approx 55.56\mathrm{mol}$

Step 4: Calculate Total Moles in Solution

Total moles = $n_{C{H}_{3}OH}+n_{H_{2}O}=5.2+55.56=60.76\mathrm{mol}$

Step 5: Calculate Mole Fraction of Methyl Alcohol

$X_{C{H}_{3}OH}=\frac{n_{C{H}_{3}OH}}{totalmoles}=\frac{5.2}{60.76}\approx 0.0856$

Rounding to three decimal places (as per the options), $X_{C{H}_{3}OH}\approx 0.086$

Final Answer

The mole fraction of methyl alcohol in the solution is 0.086.

Related Topics & Formulae

Key Concepts

• Molality (m): A concentration unit defined as moles of solute per kilogram of solvent. It is temperature-independent because it uses mass, not volume.

  $m=\frac{molesofsolute}{massofsolvent(inkg)}$

• Mole Fraction (X): A dimensionless concentration unit representing the ratio of the number of moles of one component to the total number of moles in the mixture.

  For a component A: $X_A=\frac{n_A}{n_A+n_B+...}$

  The sum of all mole fractions in a solution is always 1.

• Molarity vs. Molality: Molarity (M) is moles of solute per liter of solution. It changes with temperature because volume changes. Molality (m) does not change with temperature.

Connecting Molality to Mole Fraction

For a two-component solution (solute + solvent), the mole fraction of the solute can be derived from its molality (m) and the molar mass of the solvent (M_solvent):

$X_{solute}=\frac{m}{m+\frac{1000}{M_{solvent}}}$

This formula is useful for quickly solving such problems without going through all the intermediate steps.