5: Colligative Properties

For a dilute solution, the osmotic-pressure equation is π = CRT, where π is osmotic pressure, C is the molar concentration of the solute, R is the gas constant and T is the absolute temperature. Thus C is measured in moles per litre, or molarity. It does not represent solvent mass, colour or melting point, so option A is correct.

For a dilute solution, osmotic pressure is given by the van’t Hoff equation π = CRT, where C is molar concentration, R is the gas constant, and T is absolute temperature. When temperature and R remain constant, π is directly proportional to C. Therefore, doubling the molar concentration doubles the osmotic pressure, so option B is correct.

For a non-dissociating solute, the van’t Hoff factor is i = 1, so the freezing-point depression equation is ΔTf = Kf m. Rearranging gives m = ΔTf/Kf. Substitution gives m = 3.72/1.86 = 2.0 mol kg⁻¹. Therefore, option C is correct. The units of Kf and the calculated molality are consistent, and no dissociation or association correction is required.

For a non-dissociating solute, i = 1 and the elevation in boiling point is given by ΔTb = Kb m. Hence, m = ΔTb/Kb. Using the given values, m = 0.78/0.52 = 1.5 mol kg⁻¹. Thus option B is correct. The boiling-point constant Kb must be used here because the question gives boiling-point elevation, not freezing-point depression.

One formula unit of calcium chloride, CaCl₂, separates completely in water according to CaCl₂ → Ca²⁺ + 2Cl⁻. This produces one calcium ion and two chloride ions, giving a total of three solute particles. For complete dissociation under ideal conditions, the van’t Hoff factor equals the number of particles produced from one formula unit. Therefore, i = 3 and option C is correct.

Freezing requires solvent molecules to arrange themselves into an ordered solid structure. When a non-volatile solute is dissolved, its particles become distributed among the solvent molecules and disturb this orderly arrangement. Consequently, the solution must be cooled to a temperature lower than the normal freezing point of pure water before the solvent can crystallize. This lowering is called depression of freezing point and is a colligative property.

Isotonic solutions have equal osmotic pressures at the same temperature. Solvent molecules may still cross the semipermeable membrane in both directions, but the rates of movement are equal in opposite directions. Consequently, the two-way movements cancel and the net solvent flow is zero. Therefore, isotonicity means absence of net osmosis, not that molecular movement completely stops.

At the same temperature, osmotic pressure is related to the effective concentration of solute particles. A solution with greater osmotic pressure has a greater effective solute-particle concentration than the reference solution. It is therefore described as hypertonic relative to that solution. Hypotonic means lower osmotic pressure, while isotonic means equal osmotic pressure. Thus the correct answer is hypertonic.

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. Here, the solute amount is 0.1 mol and the solvent mass is 1 kg. Therefore, m = 0.1/1 = 0.1 mol kg⁻¹. The fact that the solute does not dissociate means no particle-factor correction is needed.

For a non-dissociated solute, the van’t Hoff factor is i = 1, so the freezing-point depression is ΔT_f = K_fm. If the solute produces twice as many effective particles and i becomes 2, the equation becomes ΔT_f = 2K_fm. Thus, at the same K_f and molality, the depression is twice the non-dissociated value.

A van’t Hoff factor of i = 1 means that the effective number of particles is equal to the number expected from the solute formula. In the simple idealized interpretation, there is no significant dissociation that would make i greater than 1 and no association that would make i less than 1. Thus, the solute is treated as neither dissociating nor associating.

Boiling-point elevation is defined as the increase in boiling point caused by adding a non-volatile solute. Therefore, the boiling point of the solution is T_b = T_b° + ΔT_b. Substituting the values gives T_b = 373 + 0.6 = 373.6. The value must be added, not subtracted, so option C is correct.

A colligative property depends primarily on the number of dissolved solute particles, rather than on the chemical identity of those particles. Important examples are relative lowering of vapour pressure, elevation of boiling point, depression of freezing point, and osmotic pressure. Viscosity, surface tension, and refractive index generally depend on the nature of the substances as well.

For a non-volatile solute, Raoult’s law gives the solvent vapour pressure as p = x_solvent p°. Here, the solvent mole fraction is 0.9 and the pure-solvent vapour pressure p° is 100. Therefore, p = 0.9 × 100 = 90. The vapour pressure is lowered from 100 because some of the liquid particles are solute particles.

Starting from πV = nRT, divide both sides by V to obtain π = nRT/V. If V and T are fixed, and R is the universal gas constant, then every factor except n remains constant. Consequently, π increases directly with the number of solute moles: doubling n doubles π under the stated conditions. Therefore, option A is correct.

For a dilute solution, osmotic pressure is given by π = CRT, where C is the molar concentration and T is absolute temperature. At the same temperature, π is directly proportional to C. Since 0.2 mol L⁻¹ is twice 0.1 mol L⁻¹, the 0.2 mol L⁻¹ solution has the greater osmotic pressure, approximately twice as large if the solutes have the same van’t Hoff factor. Hence B is correct.

These four effects are colligative properties. For an ideal dilute solution, their magnitude depends primarily on the number of independently moving solute particles relative to the amount of solvent, not on the chemical identity of those particles. Dissociation increases the effective particle count, whereas association decreases it. Therefore, the common basis is the number of effective solute particles, making A correct.

A binary solution contains two components: the solute and the solvent. If their mole fractions are represented by x_solute and x_solvent, the total number of moles is the sum of the moles of both components. Therefore, x_solute + x_solvent = 1. Mole fraction is dimensionless, and the sum of the mole fractions of all components in any solution is always unity. Hence, option C, (1), is correct.

For a dilute solution containing a non-volatile solute, freezing-point depression is given by ΔTf = iKf m and boiling-point elevation is given by ΔTb = iKb m. In both equations, m denotes molality, which is the number of moles of solute per kilogram of solvent. Molality is preferred because it does not change with temperature-related volume changes.

For a dilute solution, osmotic pressure is expressed as π = iCRT. When temperature and the nature of the solute remain unchanged, π is directly proportional to concentration C. Therefore, tripling the concentration triples the osmotic pressure: πnew = 3 × 2 = 6 units. Hence option D is correct.

The symbol ΔTb denotes the elevation in boiling point, so its relation is ΔTb = iKb m. The subscript b refers to boiling, whereas Kf belongs to freezing-point depression, ΔTf = iKf m. Kb and Kf are solvent-specific constants and are not generally equal. Therefore, using Kf for ΔTb is the error.

Colligative properties are properties of dilute solutions that depend primarily on the number of dissolved solute particles, not on their chemical identity. The four main examples are relative lowering of vapour pressure, elevation of boiling point, depression of freezing point, and osmotic pressure. Dissociation and association change the effective particle number through the van’t Hoff factor.

The van’t Hoff factor is defined as the ratio of the actual number of solute particles in solution to the number expected if no dissociation or association occurred. Here, i = moles of effective particles / initial moles of NaCl = 0.2/0.1 = 2. This agrees with complete dissociation of NaCl into Na⁺ and Cl⁻ ions.

The van’t Hoff factor compares the number of particles actually present after association or dissociation with the number initially expected. When two particles combine to form one particle, the effective number becomes half of the original number. Therefore, i = actual particles/initial particles = 1/2 = 0.5. This is the ideal value for complete pairwise association.

Viscosity is not a colligative property. It depends on the nature of both solute and solvent, intermolecular forces, temperature, and the shape or size of dissolved particles. In contrast, elevation of boiling point, depression of freezing point, osmotic pressure, and relative lowering of vapour pressure are the standard colligative properties because they depend mainly on the number of solute particles.