Question

The depression of freezing point of water (in K) for 0.1 molal solutions of NaCl and Na$_2$SO$_4$ are $\Delta T_1$ and $\Delta T_2$, respectively. Assuming ideal solutions, find the ratio $\Delta T_1/\Delta T_2$ (rounded off to two decimal places).

Hint:

For colligative properties in ideal solutions, the solute identity matters only via the van't Hoff factor \(i\): count ions to get ratios quickly.

Answer 1

For ideal solutions, $\Delta T_f = iK_f m$, where $i$ is the van't Hoff factor.
NaCl $\to$ $i_1 = 2$ (Na$^+$, Cl$^-$); Na$_2$SO$_4$ $\to$ $i_2 = 3$ (2 Na$^+$, SO$_4^{2-}$).
Hence, \[ \frac{\Delta T_1}{\Delta T_2}=\frac{i_1}{i_2}=\frac{2}{3}=0.666\ldots \approx 0.67. \]