Mass vs. mole fraction — An aqueous solution has a methanol mass (weight) fraction of 0.64. Which statement about the methanol mole fraction XM is correct? (Molar masses: methanol ≈ 32 g/mol, water ≈ 18 g/mol.)

Introduction:
Converting between mass fraction and mole fraction is routine in chemical engineering material balances. Because different species have different molar masses, a given mass fraction rarely equals the mole fraction.

Given Data / Assumptions:

• Methanol mass fraction w_MeOH = 0.64; water mass fraction w_H2O = 0.36.
• Molar masses: M_MeOH ≈ 32 g/mol, M_H2O ≈ 18 g/mol.
• Binary mixture only (methanol + water).

Concept / Approach:
Mole fraction XM = n_MeOH / (n_MeOH + n_H2O), where n_i = mass_i / M_i. Choose a convenient basis of 1 kg total solution to convert mass fractions into component masses, then into moles.

Step-by-Step Solution:
Take basis: 1 kg solution → m_MeOH = 0.64 kg, m_H2O = 0.36 kg.Compute moles: n_MeOH = 0.64 kg / 0.032 kg/mol = 20 mol/kg × 0.032? Simplify using grams: 0.64 kg = 640 g → n_MeOH = 640/32 = 20 mol.Compute moles: n_H2O = 0.36 kg = 360 g → n_H2O = 360/18 = 20 mol.Total moles = 20 + 20 = 40 mol → XM = 20/40 = 0.5.

Verification / Alternative check:
Use a fraction form directly: XM = (w_MeOH/M_MeOH) / [(w_MeOH/M_MeOH) + (w_H2O/M_H2O)] = (0.64/32)/[(0.64/32) + (0.36/18)] = 0.02/(0.02+0.02) = 0.5.

Why Other Options Are Wrong:

• (a) and (c) contradict the exact calculation.
• (d) Misinterprets mass fraction as mole fraction; due to lighter water, equal moles occur at these masses.
• (e) 0.36 is the mass fraction of water, not the mole fraction of methanol.

Common Pitfalls:
Failing to use molar masses; assuming mass fraction equals mole fraction; arithmetic slips when converting units.

Final Answer:
XM = 0.5