Colligative Properties — Notes

Colligative properties of dilute solutions of non-volatile solutes depend on the number of solute particles, not their nature: relative lowering of vapour pressure, elevation of boiling point, depression of freezing point and osmotic pressure.

Relative lowering of vapour pressure (from Raoult’s law, $p = p^\circ x_1$): $\dfrac{p^\circ - p}{p^\circ} = x_2$. For dilute solutions $x_2 \approx \dfrac{n_2}{n_1} = \dfrac{w_2M_1}{M_2w_1}$, giving $M_2 = \dfrac{w_2M_1p^\circ}{w_1(p^\circ - p)}$.

A liquid boils when its vapour pressure equals the external pressure; a solution’s vapour pressure is lower, so it boils at a higher temperature: $\Delta T_\text{b} = T_\text{b} - T^\circ_\text{b}$.

Derivation: near $T^\circ_\text{b}$ the vapour pressure curves of solvent and solution are nearly parallel straight lines, so $\Delta T_\text{b} \propto \Delta p$. With $p^\circ$ constant, $\Delta T_\text{b} \propto \Delta p/p^\circ = x_2 \approx n_2/n_1 = mM_1/1000$. Hence $\Delta T_\text{b} = K_\text{b}m$, with $K_\text{b}$ the ebullioscopic (molal elevation) constant, in K kg mol⁻¹ (water 0.52, benzene 2.53). Thermodynamically $K_\text{b} = \dfrac{R(T^\circ_\text{b})^2M_1}{1000\,\Delta_\text{vap}H}$.

Molar mass: $M_2 = \dfrac{1000K_\text{b}w_2}{\Delta T_\text{b}w_1}$.

At the freezing point the solid and liquid solvent have equal vapour pressures. The solution’s vapour pressure is lower, so it equals that of the solid at a lower temperature: $\Delta T_\text{f} = T^\circ_\text{f} - T_\text{f}$.

Derivation: as for boiling point, $\Delta T_\text{f} \propto \Delta p \propto x_2 \propto m$, so $\Delta T_\text{f} = K_\text{f}m$, with $K_\text{f}$ the cryoscopic (molal depression) constant (water 1.86, benzene 5.12, camphor 37.7 K kg mol⁻¹); $K_\text{f} = \dfrac{R(T^\circ_\text{f})^2M_1}{1000\,\Delta_\text{fus}H}$.

Molar mass: $M_2 = \dfrac{1000K_\text{f}w_2}{\Delta T_\text{f}w_1}$. Uses: antifreeze, salting roads, camphor (Rast) method.

Diffusion: spreading of particles from high to low concentration, no membrane. Osmosis: flow of solvent only through a semipermeable membrane (parchment, cellophane, copper ferrocyanide, animal bladder) from pure solvent or dilute solution into a more concentrated solution. Osmotic pressure $\pi$: the excess pressure on the solution that just stops osmosis.

van’t Hoff laws: Boyle–van’t Hoff ($\pi \propto C$ at constant $T$); Charles–van’t Hoff ($\pi \propto T$ at constant $C$); Avogadro–van’t Hoff (equimolar solutions at the same $T$ have equal $\pi$). Combined: $\pi = CRT$, $\pi V = nRT$, $M_2 = \dfrac{w_2RT}{\pi V}$ — best for proteins and polymers (room temperature, large values).

Isotonic solutions have equal $\pi$ (0.9% NaCl and blood); a cell shrinks in a hypertonic solution and swells in a hypotonic one. Reverse osmosis: pressure greater than $\pi$ forces solvent out of the solution; used for desalination of sea water.

Dissociation increases the number of particles (molar mass appears low); association decreases it (molar mass appears high, e.g. carboxylic acids dimerise in benzene).

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

Modified equations: $\Delta p/p^\circ = ix_2$; $\Delta T_\text{b} = iK_\text{b}m$; $\Delta T_\text{f} = iK_\text{f}m$; $\pi = iCRT$.

• Dissociation into $n$ ions: $i = 1 + (n-1)\alpha$, so $\alpha = \dfrac{i-1}{n-1}$.
• Association of $n$ molecules: $i = 1 - \alpha + \dfrac{\alpha}{n}$, so $\alpha = \dfrac{1-i}{1-1/n}$.