To solve this problem, we need to find the molality of an aqueous solution given that the mole fraction of the solvent (water) is 0.8. Molality (m) is defined as the number of moles of solute per kilogram of solvent.

Step 1: Define the variables

Let:

• Mole fraction of solvent (water), χ_solvent = 0.8
• Mole fraction of solute, χ_solute = 1 - χ_solvent = 1 - 0.8 = 0.2
• Let n_solute be the moles of solute
• Let n_solvent be the moles of solvent (water)
• Mass of solvent in kg = (n_solvent × M_solvent) / 1000, where M_solvent is the molar mass of water = 18 g/mol

Step 2: Write the formula for mole fraction

The mole fraction of solute is given by:

So, 0.2 = n_solute / (n_solute + n_solvent)

Step 3: Express n_solute in terms of n_solvent

From the equation above:

Cross-multiplying:

0.2 n_solute + 0.2 n_solvent = n_solute

0.2 n_solvent = n_solute - 0.2 n_solute

0.2 n_solvent = 0.8 n_solute

So, n_solute = (0.2 / 0.8) n_solvent = (1/4) n_solvent = 0.25 n_solvent

Step 4: Write the formula for molality

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

Mass of solvent = n_solvent × M_water = n_solvent × 18 grams = (n_solvent × 18) / 1000 kg

So, m = n_solute / (mass of solvent in kg) = n_solute / [ (n_solvent × 18) / 1000 ] = (n_solute × 1000) / (n_solvent × 18)

Step 5: Substitute n_solute = 0.25 n_solvent

m = (0.25 n_solvent × 1000) / (n_solvent × 18) = (0.25 × 1000) / 18 = 250 / 18 ≈ 13.8889 mol/kg

So, the molality is approximately 13.89 mol/kg, which matches the option 13.88 (considering rounding).

Final Answer: 13.88 mol kg^–1

Related Topics and Formulae

Mole Fraction (χ): The mole fraction of a component in a solution is the ratio of the number of moles of that component to the total number of moles of all components in the solution. For a binary solution with solute and solvent:

And χ_solute + χ_solvent = 1.

Molality (m): Molality is defined as the number of moles of solute per kilogram of solvent. It is independent of temperature and is given by:

It is a useful concentration unit in colligative properties.

Relation between Mole Fraction and Molality: For a dilute solution, where n_solute << n_solvent, the mole fraction of solute is approximately proportional to molality. The exact relation is:

where M_solvent is the molar mass of the solvent in g/mol. For water, M = 18 g/mol.