6: Molar Mass Determination

For a dilute non-electrolyte solution, osmotic pressure is π = wRT/(MV), so M = wRT/(πV). Convert 27 °C to 300 K and 250 mL to 0.250 L. Therefore M = (0.5 × 0.082 × 300)/(0.82 × 0.250) = 12.3/0.205 = 60 g mol⁻¹. Hence option A is correct. Correct unit conversion is essential in this calculation.

For a dilute solution containing a non-volatile solute, Raoult’s law gives (p⁰ − p)/p⁰ ≈ x₂, the mole fraction of the solute. If w₁ and M₁ represent the mass and molar mass of the solvent, and w₂ and M₂ those of the solute, then x₂ ≈ (w₂/M₂)/(w₁/M₁) = w₂M₁/(w₁M₂). Thus option A is the correct mass-form relation.

For a dilute solution, osmotic pressure is π = cRT, or π = nRT/V. At the same temperature and comparable volume, higher osmotic pressure means a greater concentration of effective solute particles. With equal solute masses, the substance having a lower molar mass provides more moles because n = w/M, so it can produce the higher osmotic pressure. Dissociation could also contribute, so “may be lower” is the suitably cautious conclusion.

Colligative-property equations assume that the solute is uniformly distributed in a sufficiently dilute solution. If a sparingly soluble substance remains undissolved, the measured mass is greater than the mass actually present as dissolved particles. The calculated molality and colligative effect will then be wrong, leading to an incorrect apparent molar mass. Therefore, complete dissolution, homogeneity, and suitable dilution must be checked carefully.

The van’t Hoff factor i compares the actual number of particles in solution with the number expected if no association or dissociation occurred. When i is less than 1, separate solute molecules combine to form larger associated units, so the effective particle number decreases. An ideal dimerisation example gives i close to 0.5 when association is essentially complete. Therefore, i = 0.5 indicates association, not dissociation.

A polymer has a very large molar mass, so a given mass contains only a very small number of moles. Consequently, changes in boiling point or freezing point are often extremely small and difficult to measure accurately. Osmotic pressure can be measured for dilute polymer solutions at room temperature and is sufficiently sensitive for determining number-average molar mass. It also avoids heating that might degrade the polymer, making it the preferred colligative method.

For a dilute solution, the molar mass of the solute is calculated using M = Kf × w₂ × 1000 /(ΔTf × w₁), where w₁ is the mass of solvent. Since w₁ is in the denominator, recording it as too large makes the calculated value of M smaller than the correct value. The actual molar mass does not change; only the experimental calculation is affected.

The molar mass relation is M = Kf × w₂ × 1000 /(ΔTf × w₁). The freezing-point depression, ΔTf, occurs in the denominator. Therefore, if ΔTf is recorded smaller than its true value while the other quantities remain unchanged, the denominator becomes smaller and the calculated molar mass becomes larger. This is a mathematical measurement error, not a change in the substance.

Molar-mass determination assumes that the weighed sample contains a known amount of pure solute. An impurity changes the relationship between the measured sample mass and the number of moles of the actual solute particles. Consequently, the colligative effect may not correspond to the assumed composition, and the calculated molar mass can be inaccurate. The direction of error depends on the nature and amount of impurity.

Simple colligative equations are derived for dilute, nearly ideal solutions, where solute particles interact only weakly and the solvent behaves regularly. In a concentrated solution, solute–solute and solute–solvent interactions become important, so the measured property may not follow the simple proportional relation. Therefore, option A is correct; temperature, colour, and solvent mass do not explain this deviation.

Each colligative property responds to the effective number of solute particles. If vapour-pressure, boiling-point, freezing-point, or osmotic-pressure methods give consistent molar masses, the result is more trustworthy. A systematic difference can reveal association, dissociation, or an experimental error. Thus A is correct; the other choices are unrelated to method comparison.

A reliable solution begins by identifying the method: vapour pressure, boiling-point elevation, freezing-point depression, or osmotic pressure. Then label solute and solvent quantities, convert units consistently, use kelvin temperature where required, and decide whether the van’t Hoff factor is needed. This prevents formula and substitution errors; the other strategies deliberately create them. Thus A is correct.