Henry's Law

$p = K_H \times x$

• $p$ = partial pressure of gas (atm or bar)
• $K_H$ = Henry's law constant (atm or bar)
• $x$ = mole fraction of dissolved gas in solution (dimensionless)
• Condition: Dilute solutions; gas does not react with solvent; applicable at constant temperature

Raoult's Law — Volatile Binary

$P_{total} = x_A \cdot P^\circ_A + x_B \cdot P^\circ_B$

• $P^\circ_A, P^\circ_B$ = vapour pressures of pure components A and B (mmHg or atm)
• $x_A, x_B$ = mole fractions in liquid phase (x_A + x_B = 1)

Raoult's Law — Non-Volatile Solute

$\frac{\Delta P}{P^\circ} = x_{solute} = \frac{n_2}{n_1 + n_2}$

Mole Fraction of Vapour Phase (Dalton's Law combined)

$y_A = \frac{P_A}{P_{total}} = \frac{x_A \cdot P^\circ_A}{x_A \cdot P^\circ_A + x_B \cdot P^\circ_B}$

Boiling Point Elevation

$\Delta T_b = i \cdot K_b \cdot m$

$K_b = \frac{R \cdot T_b^2 \cdot M_1}{1000 \cdot \Delta H_{vap}}$

• $K_b$ (water) = 0.52 K·kg/mol; $K_b$ (benzene) = 2.53; $K_b$ (acetic acid) = 3.07
• $m$ = molality (mol/kg of solvent)
• $i$ = van't Hoff factor

Freezing Point Depression

$\Delta T_f = i \cdot K_f \cdot m$

$K_f = \frac{R \cdot T_f^2 \cdot M_1}{1000 \cdot \Delta H_{fus}}$

• $K_f$ (water) = 1.86 K·kg/mol; $K_f$ (benzene) = 5.12; $K_f$ (camphor) = 40.0
• Note: $K_f > K_b$ for water because $\Delta H_{fus} < \Delta H_{vap}$

Osmotic Pressure

$\pi = i \cdot C \cdot R \cdot T$

• $C$ = molarity (mol/L) — NOT molality
• $R$ = 0.0821 L·atm/(mol·K)
• $T$ = temperature in Kelvin (always)

Molar Mass Determination

$M_2 = \frac{K_b \times w_2 \times 1000}{\Delta T_b \times w_1}$

$M_2 = \frac{K_f \times w_2 \times 1000}{\Delta T_f \times w_1}$

$M_2 = \frac{w_2 \cdot R \cdot T}{\pi \cdot V}$

• $w_2$ = mass of solute (g), $w_1$ = mass of solvent (g), $V$ = volume of solution (L)

Van't Hoff Factor — Dissociation

$i = 1 + (n-1)\alpha$

• $n$ = number of ions per formula unit
• $\alpha$ = degree of dissociation (0 to 1)
• Example: NaCl (n=2, α=1) → i = 1 + (2−1)(1) = 2

Van't Hoff Factor — Association

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

For dimerization (n=2): $i = 1 - \frac{\alpha}{2}$

• Example: $CH_{3}COOH$ in benzene (n=2, α=1) → i = 1 − 1/2 = 0.5

Van't Hoff Factor Definition

$i = \frac{\text{observed colligative property}}{\text{theoretical colligative property (for non-electrolyte)}}$

$i = \frac{M_{calculated}}{M_{actual}} \text{ (for association: } i < 1 \Rightarrow M_{calc} > M_{actual}\text{)}$