Why Sea Water Freezes at a Lower Temperature Than Tap Water

Adding salt to water lowers its freezing point, raises its boiling point and draws water towards it through a membrane. Remarkably, these effects depend on how many solute particles are present, not on what they are.

This part covers relative lowering of vapour pressure, elevation of boiling point and depression of freezing point, osmotic pressure, and the van't Hoff factor.

Colligative properties depend only on the number of solute particles in a given amount of solvent, and the first of them, the relative lowering of vapour pressure, equals the mole fraction of the solute.

The four colligative properties:

- Relative lowering of vapour pressure
- Elevation of boiling point
- Depression of freezing point
- Osmotic pressure

Relative lowering of vapour pressure. From Raoult's law for a non-volatile solute:

$$\frac{p^0-p}{p^0}=x_2\approx \frac{w_2{M}_1}{M_2{w}_1}$$

for a dilute solution, where $w$ is mass and $M$ is molar mass.

Worked example. $18$ g of a non-volatile solute in $90$ g of water lowers the vapour pressure from $23.8$ to $23.32$ mm Hg:

$$\frac{p^0-p}{p^0}=\frac{0.48}{23.8}\approx 0.0202,M_2=\frac{18\times 18}{0.0202\times 90}\approx 179{\text{ g mol}}^{-1}$$

close to the molar mass of glucose.

An everyday example. Thick sugar syrup stays liquid in an open bowl far longer than plain water, because so many dissolved particles lower its vapour pressure.

The substance. Only the number of particles matters, so equal moles of glucose and urea in the same solvent give the same lowering.

**For a dilute solution, $\Delta T_b=K_{b}m$ and $\Delta T_f=K_{f}m$, where $m$ is the molality and $K_b$ and $K_f$ belong to the solvent; rearranging gives the solute's molar mass, $M_2=\frac{1000K{w}_2}{\Delta T{w}_1}$ with $w_1$ in grams.

Why boiling point rises. The solute lowers the vapour pressure, so the solution must be heated further before its vapour pressure reaches atmospheric pressure.

Why freezing point falls. The solution's vapour pressure matches that of the solid solvent only at a lower temperature.

For water:** $K_b=0.52$ K kg mol${}^{-1}$ and $K_f=1.86$ K kg mol${}^{-1}$.

Worked example 1. $18$ g of glucose ($180$ g mol${}^{-1}$) in $1$ kg of water gives $m=0.1$ mol kg${}^{-1}$:

$$\Delta T_b=0.52\times 0.1=0.052\text{ K},\Delta T_f=1.86\times 0.1=0.186\text{ K}$$

Worked example 2 — molar mass. $1$ g of a non-electrolyte in $50$ g of benzene lowers its freezing point by $0.40$ K, with $K_f=5.12$ K kg mol${}^{-1}$:

$$M_2=\frac{1000\times 5.12\times 1}{0.40\times 50}=256{\text{ g mol}}^{-1}$$

An everyday example. Traditional kulfi makers pack the moulds in ice mixed with salt — salty water freezes well below $0^{\circ }$C, so the mixture gets cold enough to set the kulfi.

The substance. Freezing-point depression is usually preferred for molar mass because $K_f$ is larger than $K_b$, giving a bigger, easier-to-measure change.

**Osmotic pressure is the extra pressure that must be applied to a solution to stop solvent flowing into it through a semipermeable membrane, with $\pi =CRT$ for dilute solutions; applying more than this pressure drives solvent out instead — reverse osmosis.

Osmosis.** Solvent passes through a semipermeable membrane from pure solvent, or a dilute solution, into a more concentrated solution.

$$\pi =CRT=\frac{n_2}{V}RT,M_2=\frac{w_{2}RT}{\pi V}$$

Comparing a solution with a cell's contents:

- Isotonic — same osmotic pressure, no net flow, like $0.9$ g of NaCl per $100$ mL for red blood cells
- Hypertonic — more concentrated outside; the cell shrinks
- Hypotonic — less concentrated outside; the cell swells

Worked example. $200$ mL of solution containing $1.26$ g of a protein has $\pi =2.57\times 10^{-3}$ bar at $300$ K. With $R=0.083$ L bar mol${}^{-1}$ K${}^{-1}$:

$$M_2=\frac{1.26\times 0.083\times 300}{2.57\times 10^{-3}\times 0.200}\approx 6.1\times 10^4{\text{ g mol}}^{-1}$$

Reverse osmosis. Pressure greater than $\pi$ on the solution side forces water out through the membrane, leaving salts behind — the basis of RO water purifiers and seawater desalination.

An everyday example. Raisins soaked in water swell up as water enters by osmosis, while salted raw mango pieces release juice as water flows out into the salt.

The substance. Osmotic pressure suits proteins and polymers because even tiny concentrations give a measurable pressure at room temperature.

**When a solute dissociates into ions or associates into larger groups, the number of particles changes, so colligative properties — and the molar masses calculated from them — become abnormal; the van't Hoff factor $i$ corrects for this.**

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

The equations become $\Delta T_f=i{K}_{f}m$, $\Delta T_b=i{K}_{b}m$ and $\pi =iCRT$.

- Dissociation — $i>1$; NaCl gives two ions, so $i\approx 2$ when fully dissociated
- Association — $i<1$; ethanoic acid forms dimers in benzene, so $i$ approaches $0.5$

Worked example. $0.1$ mol of NaCl in $1$ kg of water, fully dissociated, would give

$$\Delta T_f=2\times 1.86\times 0.1=0.372\text{ K}$$

If the measured depression is $0.34$ K, then $i=\frac{0.34}{0.186}\approx 1.83$, and the degree of dissociation is $\alpha =i-1\approx 0.83$.

An everyday example. Calcium chloride is used to de-ice roads in very cold places because each formula unit gives three ions, lowering the freezing point more than the same number of moles of sodium chloride.

The substance. **The further $i$ is from $1$, the more the solute has dissociated or associated** — colligative measurements reveal what the solute really does in solution.