Calculate the depression in the freezing point of water when 10 g of CH3CH2CHClCOOH is added to 250 g of water. Ka = 1.4 × 10−3, Kf = 1.86 K kg mol−1.
Molar mass of $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CHClCOOH}=15+14+13+35.5+12+16+16+1$
$=122.5 \mathrm{~g} \mathrm{~mol}^{-1}$
$\therefore$ No. of moles present in $10 \mathrm{~g}$ of $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CHClCOOH}=\frac{10 \mathrm{~g}}{122.5 \mathrm{~g} \mathrm{~mol}^{-1}}$
$=0.0816 \mathrm{~mol}$
It is given that $10 \mathrm{~g}$ of $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CHClCOOH}$ is added to $250 \mathrm{~g}$ of water.
$\therefore$ Molality of the solution, $=\frac{0.0186}{250} \times 1000$
$=0.3264 \mathrm{~mol} \mathrm{~kg}^{-1}$
Let $a$ be the degree of dissociation of $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CHClCOOH}$.
$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CHClCOOH}$ undergoes dissociation according to the following equation:
$\therefore K_{a}=\frac{C \alpha \cdot C \alpha}{C(1-\alpha)}$
$=\frac{C \alpha^{2}}{1-\alpha}$
Since α is very small with respect to 1, 1 − α ≈ 1
Now, $K_{a}=\frac{C \alpha^{2}}{1}$
$\Rightarrow \alpha=\sqrt{\frac{K_{a}}{C}}$
$=\sqrt{\frac{1.4 \times 10^{-3}}{0.3264}} \quad\left(\because K_{a}=1.4 \times 10^{-3}\right)$
$=0.0655$
Again,
Total moles of equilibrium = 1 − α + α + α
= 1 + α
$\therefore i=\frac{1+\alpha}{1}$
$=1+\alpha$
$=1+0.0655$
$=1.0655$
Hence, the depression in the freezing point of water is given as:
$\Delta T_{f}=i . K_{f} m$
$=1.0655 \times 1.86 \mathrm{Kkg} \mathrm{mol}^{-1} \times 0.3264 \mathrm{~mol} \mathrm{~kg}^{-1}$
$=0.65 \mathrm{~K}$