Why Kulfi Sellers Pack Salt Into the Ice Around Their Moulds

Add salt or sugar to water and the water boils a little hotter, freezes a little colder and pulls more water through membranes. These changes depend only on the number of solute particles, not on what the particles are — which lets chemists use them to find molar masses.

This lesson covers the four colligative properties and their calculations, and the van't Hoff factor for solutes that dissociate or associate.

Colligative properties — relative lowering of vapour pressure, elevation in boiling point, depression in freezing point and osmotic pressure — depend only on the number of solute particles in a given amount of solvent, not on their nature.

The four properties, for a non-volatile solute:

$$\frac{p^{\circ }-p}{p^{\circ }}=x_{\text{solute}}\Delta T_b=K_{b}m\Delta T_f=K_{f}m\pi =CRT$$

- $K_b$ and $K_f$ are the molal elevation and depression constants; for water, $K_b=0.52$ and $K_f=1.86$ K kg mol${}^{-1}$
- Osmotic pressure is the pressure needed to stop solvent flowing through a semipermeable membrane into the solution

Worked example 1 — boiling and freezing points. 18 g of glucose ($M=180$ g mol${}^{-1}$) is dissolved in 500 g of water:

$$m=\frac{0.10}{0.500}=0.20{\text{ mol kg}}^{-1}\Delta T_b=0.52\times 0.20=0.104\text{ K}\Delta T_f=1.86\times 0.20=0.372\text{ K}$$

So the solution boils about 0.10 K higher and freezes about 0.37 K lower than pure water.

Worked example 2 — osmotic pressure. A 0.010 M protein solution at 300 K, with $R=0.0831$ L bar mol${}^{-1}$ K${}^{-1}$:

$$\pi =0.010\times 0.0831\times 300=0.249\text{ bar}$$

Worked example 3 — molar mass. 1.00 g of a non-electrolyte in 50.0 g of water lowers the freezing point by 0.372 K:

$$M_2=\frac{K_f\times w_2\times 1000}{\Delta T_f\times w_1}=\frac{1.86\times 1.00\times 1000}{0.372\times 50.0}=100{\text{ g mol}}^{-1}$$

Osmosis terms. Solutions with equal osmotic pressure are isotonic; one with higher osmotic pressure is hypertonic and one with lower pressure hypotonic. Applying a pressure greater than the osmotic pressure drives reverse osmosis.

An everyday example. Household RO water purifiers push tap water through a semipermeable membrane at a pressure above its osmotic pressure, leaving dissolved salts behind.

The substance. Osmotic pressure is the preferred way to find the molar mass of proteins and polymers — it gives a measurable value at room temperature even in very dilute solutions, where freezing point changes would be too small to detect.

The van't Hoff factor, i, is the ratio of the observed colligative property to the value expected with no dissociation or association; it is greater than 1 for solutes that split into ions and less than 1 for solutes whose molecules join together.

$$i=\frac{\text{observed colligative property}}{\text{calculated colligative property}}=\frac{\text{normal molar mass}}{\text{observed molar mass}}$$

Modified equations:

$$\Delta T_b=i{K}_{b}m\Delta T_f=i{K}_{f}m\pi =iCRT$$

Degree of dissociation. For a solute that splits into n ions with degree of dissociation $\alpha$:

$$i=1+(n-1)\alpha \alpha =\frac{i-1}{n-1}$$

Degree of association. When n molecules join into one, as ethanoic acid dimerises in benzene, $i=1-\left(1-\frac{1}{n}\right)\alpha$.

Worked example 1 — dissociation. A 0.10 mol kg${}^{-1}$ KCl solution freezes at $-0.35$ °C, while the expected depression is $1.86\times 0.10=0.186$ K:

$$i=\frac{0.35}{0.186}=1.88\alpha =\frac{1.88-1}{2-1}=0.88$$

So potassium chloride is about 88 per cent dissociated.

Worked example 2 — association. Ethanoic acid in benzene shows an observed molar mass of 114 g mol${}^{-1}$ against a normal value of 60 g mol${}^{-1}$:

$$i=\frac{60}{114}=0.526\alpha =\frac{1-0.526}{1-\frac{1}{2}}=0.95$$

So about 95 per cent of the acid exists as dimers.

Typical values of i. Glucose and urea give 1; NaCl and KCl give close to 2; $\mathrm{CaC}{\mathrm{l}}_2$ and ${\mathrm{K}}_2\mathrm{S}{\mathrm{O}}_4$ give close to 3; ethanoic acid in benzene gives close to 0.5.

An everyday example. Road crews spreading salt on snowbound Himalayan passes melt ice more effectively than the same number of moles of sugar would, because each unit of sodium chloride supplies two ions.

The substance. Measured values of i for strong electrolytes fall a little short of the ideal whole numbers — oppositely charged ions attract one another, so they do not behave as fully independent particles.