6: Molar Mass Determination

Freezing-point depression is defined as ΔT_f = T_f° − T_f, where T_f° is the freezing point of the pure solvent and T_f is the freezing point of the solution. Thus, ΔT_f = 0 − (−0.4) = 0.4 °C. Since a temperature difference has the same numerical value in Celsius and kelvin, the depression is 0.4 K. Therefore, option A is correct.

The mass of solvent is given as 250 g. To convert grams to kilograms, divide by 1000 because 1000 g = 1 kg. Therefore, mass of solvent = 250/1000 = 0.25 kg. The solute mass is not used for this particular conversion. Hence, 0.25 kg is the correct answer, and option A is unambiguous.

Molality is defined as the number of moles of solute divided by the mass of solvent in kilograms: m = moles of solute / kilograms of solvent. Substituting the data gives m = 0.1/0.5 = 0.2 mol kg⁻¹. The mass of the solution is not used; only the mass of the solvent is required. Therefore, option A is the correct answer.

In the usual notation for a binary solution, A denotes the solvent and B denotes the solute. Therefore, M_B means the molar mass of component B, which is the solute. This quantity is used in formulas involving mole fraction, molality, and colligative-property methods for determining molar mass. The other options describe unrelated physical quantities.

The number of moles is calculated using n = m/M, where m is the given mass and M is the molar mass. When the mass is kept constant, increasing M makes n smaller because moles and molar mass are inversely proportional. Thus, a solute with greater molar mass contributes fewer moles in the same mass. Zero moles would occur only if the mass were zero.

For a fixed mass, the number of moles is n = w/M. A lower molar mass therefore gives a larger number of moles and hence more dissolved particles. Colligative properties depend on the number of solute particles, not on their colour, smell, or chemical identity alone. Consequently, the lower-molar-mass solute produces the greater effect when both solutions are otherwise comparable.

For a solution containing a non-volatile solute, Raoult’s law gives the relative lowering of vapour pressure as (p° − p)/p° = x_B, where x_B is the solute mole fraction in the dilute-solution approximation. This measurable pressure change helps determine the amount of solute and therefore its molar mass. Colour, hardness, and container volume are not the relevant quantities.

Colligative properties depend on the number of particles actually present in solution. If some solute remains undissolved, the measured solution contains fewer dissolved particles than expected from the weighed mass. The observed temperature or vapour-pressure change will therefore not correspond to the assumed amount, producing an erroneous calculated molar mass. Complete dissolution and proper mixing are essential experimental precautions.

Colligative methods such as elevation of boiling point and depression of freezing point use a temperature difference: ΔT_b = T_b(solution) − T_b° or ΔT_f = T_f° − T_f(solution). The pure-solvent temperature provides the reference value against which the solution temperature is compared. Without this reference, the magnitude of the colligative change and the subsequent molar-mass calculation cannot be obtained accurately.

Freezing-point depression is defined as ΔT_f = T_f° − T_f, where T_f° is the freezing point of the pure solvent and T_f is the freezing point of the solution. If both temperatures are equal, their difference is zero. Thus, ΔT_f = 0 K. A value of −1 K would imply that the two temperatures differ by one kelvin, which is not stated.

A non-volatile solute lowers the vapour pressure of a solvent. Consequently, the solution must be heated to a higher temperature to reach the external pressure required for boiling. The increase in boiling temperature is called elevation in boiling point and is represented by ΔT_b. It is a colligative property because its magnitude depends primarily on the number of dissolved solute particles.

When a suitable non-volatile solute is dissolved, the chemical potential of the liquid solvent decreases, so the solution freezes at a lower temperature than the pure solvent. This decrease is called depression in freezing point and is written as ΔT_f = T_f° − T_f. It is a colligative property and can be used, through the appropriate formula, to determine the solute’s molar mass.

The definition of molar mass is M = w/n, where w is the mass of the substance and n is its amount in moles. Rearranging this expression gives n = w/M. Therefore, the number of moles is obtained by dividing mass by molar mass. If w is in grams and M is in grams per mole, the resulting unit is mole, confirming that option A is dimensionally correct.

Molar mass is calculated from the amount of substance and its mass, using the relation M = m/n. Therefore, measuring the mass of the solute accurately is an essential first step in an experimental determination. The colour of the solute, the colour of the container, and the laboratory name do not enter the calculation and cannot help determine molar mass. Thus, option A is correct.

All colligative properties depend on the effective number of dissolved solute particles. During molar-mass determination, dissociation or association can change this number and may produce an abnormal observed molar mass. Therefore, the effective particle count must be considered carefully, often through the van’t Hoff factor. Colour, container shape, and smell are not controlling factors. Option A is correct.

The freezing-point depression is represented by ΔTf = Tf° − Tf, where Tf° is the freezing point of the pure solvent and Tf is the freezing point of the solution. Using ΔTf = Kf m for a nonelectrolyte, the solution molality can be calculated; with the known solute mass and solvent mass, its molar mass follows. Hence option A is required.

For a solution containing a non-volatile solute, the solvent vapour pressure is lowered. The solution must therefore be heated to a higher temperature before its vapour pressure equals the external pressure. The boiling-point elevation is ΔTb = Tb(solution) − Tb(pure solvent), a positive difference. Thus option B is correct, while option A describes the opposite effect.

High-molar-mass substances such as proteins and polymers often dissolve only in very dilute solutions and may decompose when heated. Osmotic pressure follows π = CRT and can be measured at or near room temperature, so even a small concentration produces a measurable property. This avoids heating and is suitable for macromolecules. Therefore, option A is correct.

The number of moles is calculated by n = m/M, where m is the given mass and M is the molar mass. When the mass remains fixed, increasing M makes the quotient smaller. Thus a high-molar-mass substance contributes fewer moles than a low-molar-mass substance of the same mass. The number of moles is not unchanged and cannot be equal to molar mass. Option A is correct.

For a nonelectrolyte, the depression in freezing point is related to molality by ΔTf = Kf m, where Kf is the cryoscopic constant of the solvent. Rearranging this equation by dividing both sides by Kf gives m = ΔTf/Kf. The other options use incorrect multiplication, powers, or addition and are dimensionally inconsistent. Therefore, option A is correct.

For a dilute solution containing a nonelectrolyte, the elevation in boiling point is directly proportional to the molality of the solution. The relation is ΔTb = Kb m, where Kb is the ebullioscopic constant of the solvent and m is molality. It is not a quotient, sum, or difference. Hence option A is the correct relation used in calculations.

The Rast method is a cryoscopic method used to determine the molar mass of a solute. In this method, the solute is dissolved in a suitable solvent, commonly camphor in the classical procedure, and the depression in the solvent’s freezing point is measured. Using the freezing-point relation, the molar mass is calculated. Therefore, option A is correct.

When the freezing-point depression, ΔTf, is very small, it is difficult to measure the temperature difference with high precision. A small absolute error then represents a large relative error. Since the calculated molar mass depends on ΔTf, this uncertainty can significantly affect the final result. Therefore, the main problem is increased measurement error, making option A correct.

For a nonelectrolyte solution, the freezing-point depression relation is ΔTf = Kf m, where Kf is the cryoscopic constant and m is molality. Rearranging gives m = ΔTf / Kf. Thus, the given values directly provide the molality first; quantities such as moles and molar mass may be calculated afterward if additional data are supplied. Hence, option A is correct.

For a dilute nonelectrolyte solution, the elevation in boiling point is given by ΔTb = Kb m, where Kb is the ebullioscopic constant and m is the molality. Dividing both sides by Kb gives m = ΔTb / Kb. Therefore, the boiling-point data first yield molality through option A; the other expressions do not follow from the colligative-property equation.