Derive an expression relating the depression of freezing point with the amount of solute present in the solution.

Answer:

the freezing point of a substance may be defined as the temperature at which the vapour pressure of the substance in its liquid phase is equal to its vapour pressure in the solid phase.
According to Raoult’s law, when a non-volatile solid is added to the solvent its vapour pressure decreases and now it would become equal to that of solid solvent at lower temperature. Thus, the freezing point of the solvent decreases.

Let ${\mathrm{T}}_{\mathrm{f}}^0$ be the freezing point of pure solvent and T_f be its freezing point when non-volatile solute is dissolved in it. The decrease in freezing point.
$∆{\mathrm{T}}_{\mathrm{f}} = {\mathrm{T}}_{\mathrm{f}}^0 - {\mathrm{T}}_{\mathrm{f}}$

Similar to elevation of boiling point, depression of freezing point (ΔT_f) for dilute solution (ideal solution) is directly proportional to molality,
m of the solution. Thus,
ΔT_f ∝ m

or
ΔT_f= K_fm

The proportionality constant, K_f, which depends on the nature of the solvent is known as Freezing Point Depression Constant or Molal Depression Constant or Cryoscopic Constant. The unit of Kf is K kg mol^-1

If w₂ gram of the solute having molar mass as M₂, present in w₁ gram of solvent, produces the depression in freezing point ΔT_f of the
solvent then molality of the solute is given by the equation:
$$\begin{gathered} \mathrm{m}= \frac{{\displaystyle \raisebox{1ex}{${\mathrm{w}}_2$}\!\left/ \!\raisebox{-1ex}{${\mathrm{M}}_2$}\right.}}{{\displaystyle \raisebox{1ex}{${\mathrm{w}}_1$}\!\left/ \!\raisebox{-1ex}{$1000$}\right.}} \\ \mathrm{substituting} \mathrm{this} \mathrm{value} \mathrm{in} ∆{\mathrm{T}}_{\mathrm{f}} = {\mathrm{K}}_{\mathrm{f}}\mathrm{m} \\ ∆{\mathrm{T}}_{\mathrm{f}} = \frac{{\mathrm{K}}_{\mathrm{f}} \times {\mathrm{w}}_2 \times 1000}{{\mathrm{M}}_2\times {\mathrm{w}}_1} \\ {\mathrm{M}}_2 = \frac{{\mathrm{K}}_{\mathrm{f}} \times {\mathrm{w}}_2 \times 1000}{∆{\mathrm{T}}_{\mathrm{f}}\times {\mathrm{w}}_1} \end{gathered}$$

Thus for determining the molar mass of the solute we should know the quantities w₁, w₂, ΔT_f, along with the molal freezing point depression constant.