Colligative Properties and Relative Lowering of Vapour Pressure

Colligative properties are those properties of a solution that depend solely on the number of solute particles dissolved in a given quantity of solvent, and not at all on the chemical nature of those particles. Four colligative properties are studied in this chapter: the relative lowering of vapour pressure, the elevation of boiling point, the depression of freezing point, and osmotic pressure. All four arise from the same underlying cause — dissolving a non-volatile solute in a solvent lowers the solvent's vapour pressure, because a fraction of the liquid surface that would otherwise be occupied by escaping solvent molecules is instead occupied by non-volatile solute particles that do not contribute to the vapour at all.

Consider a solution formed by dissolving a non-volatile solute (component 2) in a volatile solvent (component 1). Since the solute itself contributes essentially no vapour pressure, applying Raoult's law to the solvent alone gives the vapour pressure of the solution as $p_1 = x_1 p_1^{\circ}$, where $p_1^{\circ}$ is the vapour pressure of the pure solvent and $x_1$ is the solvent's mole fraction in the solution. The lowering of vapour pressure is therefore $\Delta p = p_1^{\circ} - p_1 = p_1^{\circ}(1 - x_1) = p_1^{\circ} x_2$ (using $x_1 + x_2 = 1$, since only two components are present). Dividing through by $p_1^{\circ}$ gives the relative lowering of vapour pressure:

$\frac{p_1^{\circ} - p_1}{p_1^{\circ}} = x_2$ …