Phase Equilibria

A phase transition occurs when a substance changes between solid, liquid, and gas. At the transition, two phases coexist at equilibrium.

The Clapeyron equation describes the pressure-temperature dependence of the phase boundary: $$\dfrac{dP}{dT} = \dfrac{\Delta S}{\Delta V} = \dfrac{\Delta H}{T\Delta V}$$ For liquid-vapor transitions (assuming ideal gas and $\Delta V \approx V_{gas}$), the Clausius-Clapeyron equation applies: $$\ln\!\left(\dfrac{P_2}{P_1}\right) = -\dfrac{\Delta H_{vap}}{R}\left(\dfrac{1}{T_2} - \dfrac{1}{T_1}\right)$$

A phase diagram plots $P$ vs. $T$ and shows the regions of stability for each phase. Key points:

• Triple point: three phases coexist
• Critical point: above $T_c$ and $P_c$ the liquid and gas phases become indistinguishable (supercritical fluid)

Examples: $\mathrm{H_2O}$ has a negative solid-liquid slope (ice less dense than water). $\mathrm{CO_2}$ has a positive solid-liquid slope and $T_c = 31$°C.

Raoult's law states that the partial vapor pressure of a component above an ideal solution is proportional to its mole fraction: $$P_i = \chi_i P_i^\circ$$ where $P_i^\circ$ is the vapor pressure of the pure component.

Positive deviation ($P > \chi P^\circ$): weaker A-B interactions than A-A/B-B (e.g. ethanol + water → azeotrope). Negative deviation ($P < \chi P^\circ$): stronger A-B interactions (e.g. chloroform + acetone).

Properties that depend only on the number of solute particles (not their identity):

• Ebullioscopy (boiling point elevation): $\Delta T_b = K_b \cdot m$
• Cryoscopy (freezing point depression): $\Delta T_f = K_f \cdot m$
• Osmotic pressure: $\Pi = iMRT$ (where $i$ is the van't Hoff factor)

For electrolytes, $i$ accounts for dissociation (e.g. NaCl $i=2$, CaCl₂ $i=3$).