States of Matter

Matter exists in three principal states — solid, liquid, and gas — governed by the balance between kinetic energy and intermolecular forces. Intermolecular interactions increase in strength: dispersion (London) forces < dipole-dipole < hydrogen bonding. Stronger interactions lead to higher boiling points and lower vapour pressures.

Gas laws describe ideal gas behavior:

• Boyle's law: PV = constant (at constant T and n)
• Charles's law: V/T = constant (at constant P and n)
• Gay-Lussac's law: P/T = constant (at constant V and n)
• Avogadro's law: V ∝ n (at constant T and P)

These combine into the ideal gas equation: PV = nRT, where R = 0.0821 L·atm/(mol·K) = 8.314 J/(mol·K).

Dalton's law of partial pressures: $P_{total}$ = p₁ + p₂ + ...; partial pressure $p_{i}$ = $x_{i}$ × $P_{total}$ ($x_{i}$ = mole fraction).

Graham's law of diffusion: r₁/r₂ = √(M₂/M₁) — rate inversely proportional to square root of molar mass.

Solved Example 1: Volume of 2 mol ideal gas at 27°C and 2 atm. V = nRT/P = (2 × 0.0821 × 300)/2 Dimensional analysis: mol × L·atm/(mol·K) × K / atm = L ✓ V = 24.63 L

Solved Example 2: Rate of diffusion of gas A is twice that of gas B ($M_{B}$ = 64 g/mol). Find $M_{A}$. $\frac{r_{A}}{r_{B}}$ = √($\frac{M_{B}}{M_{A}}$) → 2 = √(64/$M_{A}$) → 4 = 64/$M_{A}$ → $M_{A}$ = 16 g/mol (likely CH₄)

Solved Example 3: Total pressure of 2 mol N₂ + 3 mol O₂ in 10 L at 300 K. $n_{total}$ = 5 mol; P = nRT/V = (5 × 0.0821 × 300)/10 = 12.315 atm $p_{N}$₂ = ($\frac{2}{5}$) × 12.315 = 4.926 atm; $p_{O}$₂ = ($\frac{3}{5}$) × 12.315 = 7.389 atm ✓

Kinetic molecular theory postulates random motion, elastic collisions, negligible intermolecular forces, and KE ∝ T. From PV = ($\frac{1}{3}$)mnc², combined with PV = nRT: Derivation of $v_{rms}$: PV = ($\frac{1}{3}$)Nmc² and PV = nRT, so ($\frac{1}{3}$)Nmc² = nRT.
Since N = nNₐ and m = M/Nₐ: ($\frac{1}{3}$)(nNₐ)(M/Nₐ)c² = nRT → c² = 3RT/M → $v_{rms}$ = √(3RT/M).

Molecular speeds: $v_{rms}$ = √(3RT/M), $v_{avg}$ = √(8RT/πM), $v_{mp}$ = √(2RT/M). Order: $v_{rms}$ > $v_{avg}$ > $v_{mp}$ (ratio 1.224 : 1.128 : 1 relative to $v_{mp}$).

Real gases deviate from ideal behavior at high pressure and low temperature. The van der Waals equation accounts for this: (P + a/V²)(V − b) = RT (per mole), where 'a' corrects for intermolecular attraction and 'b' for molecular volume.

The compressibility factor Z = PV/(nRT): Z = 1 (ideal), Z < 1 (attractive forces dominate, gas more compressible than ideal), Z > 1 (repulsive forces dominate, gas less compressible). Exception: H₂ and He show Z > 1 at all pressures because their 'a' is very small.

Boyle temperature $T_{B}$ = a/(Rb) — the temperature at which a real gas behaves ideally over a wide pressure range.

Critical constants: $T_{c}$ = 8a/(27Rb), $P_{c}$ = a/(27b²), $V_{c}$ = 3b; relationship: $\frac{P_{cV_c}}{T_{c}}$ = 3R/8.

Liquefaction requires high pressure and low temperature (below $T_{c}$). The Joule-Thomson effect cools gases on expansion below the inversion temperature. Andrews isotherms for CO₂ ($T_{c}$ = 31.1°C) illustrate gas-liquid-supercritical behavior.

Liquid state properties: Vapour pressure increases with temperature (more molecules escape); surface tension decreases with temperature (weaker cohesive forces); viscosity decreases with temperature for liquids (more kinetic energy overcomes intermolecular forces).

The key testable concept is ideal gas equation numericals, Graham's law calculations, and the relationship between molecular speeds ($v_{rms}$ > $v_{avg}$ > $v_{mp}$).