States of Matter and Kinetic Theory
Gas-law and solid-state questions test the ideal model, its limits, and the link between microscopic motion and measurable properties. Expect the ideal gas equation, partial pressures, effusion, real-gas deviations and unit-cell densities.
Ideal gas behaviour
PV = nRT with T in kelvin (R = 8.314 J K-1 mol-1, or 0.08314 dm3 bar K-1 mol-1). Dalton's law gives P_total = sum of P_i, with P_i = x_i P_total. The kinetic model yields v_rms = sqrt(3RT/M): heavier molecules move more slowly at the same temperature, and mean kinetic energy depends only on T. Graham's law for effusion gives r1/r2 = sqrt(M2/M1). Raising T shifts the Maxwell-Boltzmann distribution right and flattens it, which is why rates rise steeply with temperature.
Real gases and phase behaviour
van der Waals: (P + an2/V2)(V - nb) = nRT. The a term corrects for attractions and lowers pressure; b corrects for finite molecular volume. Real gases approach ideal behaviour at low pressure and high temperature. On a phase diagram the triple point and critical point bound the liquid region; above the critical temperature a supercritical fluid cannot be liquefied by pressure alone.
Unit cells
Primitive cubic Z = 1, body-centred Z = 2, face-centred Z = 4, with packing efficiencies of 52%, 68% and 74%. For fcc, atoms touch along the face diagonal, so 4r = a·sqrt(2). Density rho = ZM/(N_A a3) with a in cm.
Worked example
Copper is fcc with a = 408 pm and M = 63.55 g mol-1. Find its density. a = 4.08 × 10-8 cm, so a3 = 6.79 × 10-23 cm3. Mass per unit cell = 4 × 63.55/6.022 × 1023 = 4.22 × 10-22 g. rho = 4.22 × 10-22/6.79 × 10-23 = 6.22 g cm-3, matching the handbook value.
Traps
- Using degrees Celsius in PV = nRT or in v_rms.
- Leaving a in picometres and then mixing it with g cm-3.
- Assuming both van der Waals corrections raise the pressure; attractions lower it.
- Inverting the molar masses in Graham's law, or omitting the square root.
- Claiming liquid and gas remain distinguishable at the critical point; they do not.