Colligative Properties — Freezing Point Depression

Theoretical Background

Colligative properties depend only on the number of solute particles present in solution, not on their chemical identity.
Typical colligative properties: vapor-pressure lowering, boiling-point elevation, freezing-point depression, osmotic pressure.
For freezing point depression:
$\Delta T_f = i K_f \, m$ where:
- $\Delta T_f$ = freezing point lowering
- $i$ = van ’t Hoff factor (number of effective particles per solute unit)
- $K_f$ = cryoscopic constant of the solvent
- $m$ = molality of solute
Molality is defined as:
$m = \frac{n_{\text{solute}}}{m_{\text{solvent}}(kg)}$

A solution is prepared by dissolving 10.0 g of NaCl in 200 g of water.

Step 1. Moles of solute
Molar mass NaCl = $58.44\, g\,mol^{-1}$
$n = \frac{10.0}{58.44} = 0.171\, mol$

Step 2. Molality
Mass of solvent = $200 g = 0.200 kg$
$m = \frac{0.171}{0.200} = 0.855\, mol\,kg^{-1}$

Step 3. van ’t Hoff factor
NaCl dissociates ideally into 2 ions ($Na^+, Cl^-$), so:
$i = 2$

Effective molality:
$m_{\text{eff}} = i m = 2 \times 0.855 = 1.71\, mol\,kg^{-1}$

Step 4. Freezing point depression
$\Delta T_f = K_f m_{\text{eff}} = (1.86)(1.71) = 3.18\, K$

So the new freezing point is:
$T_f = 273.15 - 3.18 = 269.97\,K \;\approx -3.2^\circ C$

The assumption of complete dissociation is an idealization; in real solutions the van ’t Hoff factor $i$ is slightly less than 2 due to ion pairing.
Colligative properties provide a powerful experimental tool to determine molar masses of solutes or to estimate their degree of dissociation.
This case illustrates why adding salt lowers the freezing point of water — the scientific basis of road de-icing in winter and of antifreeze mixtures in car engines.