Intermolecular Forces

• Dispersion forces < dipole-dipole < hydrogen bonding (increasing strength)
• Stronger forces → higher boiling point, lower vapour pressure, higher surface tension, higher viscosity
• Only $H_{2}O$, HF, $NH_{3}$ form significant hydrogen bonds among common molecules

Gas Laws — One-Line Versions

• Boyle: $PV = k$ at constant $T, n$ — pressure and volume inversely proportional
• Charles: $V/T = k$ at constant $P, n$ — volume and temperature directly proportional (use Kelvin always)
• Gay-Lussac: $P/T = k$ at constant $V, n$ — pressure and temperature directly proportional
• Avogadro: $V \propto n$ at constant $T, P$ — molar volume of ideal gas = 22.4 L at STP
• Ideal gas: $PV = nRT$; $R = 0.0821\ \text{L·atm/(mol·K)}$ or $8.314\ \text{J/(mol·K)}$

Graham's Law

• $r_1/r_2 = \sqrt{M_2/M_1}$ — rate inversely proportional to $\sqrt{M}$
• Lighter gas (smaller $M$) always diffuses faster
• Effusion time ratio: $t_1/t_2 = \sqrt{M_1/M_2}$ (inverse of rate ratio)

Molecular Speeds

• $v_{mp} = \sqrt{2RT/M}$; $v_{avg} = \sqrt{8RT/\pi M}$; $v_{rms} = \sqrt{3RT/M}$
• Order: $v_{mp} < v_{avg} < v_{rms}$; ratio ≈ $1 : 1.128 : 1.224$
• All increase with temperature; all decrease with molar mass
• Only $v_{rms}$ relates directly to kinetic energy: $KE = \frac{1}{2}Mv_{rms}^2 = \frac{3}{2}RT$

Real Gases

• Deviation from ideal behavior: high pressure, low temperature
• Van der Waals: $(P + an^2/V^2)(V - nb) = nRT$; $a$ = attraction, $b$ = volume
• $Z = PV/(nRT)$: ideal $Z = 1$; $Z < 1$ (attraction); $Z > 1$ (repulsion)
• $H_{2}$ and He: $Z > 1$ at all pressures (very small $a$)
• Boyle temperature: $T_B = a/(Rb)$ — real gas behaves ideally
• Critical constants: $T_c = 8a/(27Rb)$; $P_c = a/(27b^2)$; $V_c = 3b$

Liquid State

• Vapour pressure ↑ with temperature (more molecules escape)
• Surface tension ↓ with temperature (weaker cohesive forces)
• Viscosity of liquids ↓ with temperature (more KE overcomes intermolecular forces)
• Boiling point = temperature at which vapour pressure = external pressure
• Joule-Thomson effect: real gas cools on adiabatic expansion (below inversion temperature)

Dalton's Law

• $P_{total} = \sum p_i$; partial pressure $p_i = x_i \times P_{total}$
• Gas collected over water: $P_{gas} = P_{total} - P_{H_2O}$