Kinetic Theory - Quick Revision
Model
- A gas is a large number of tiny molecules in random motion, with negligible size and only elastic collisions (no force except during collisions).
- Ideal gas: PV = nRT = N kB T.
Pressure and temperature
- P = (1/3) rho vbar^2 = (1/3)(N/V) m vbar^2 (kinetic origin of pressure).
- Kinetic interpretation: (1/2) m vbar^2 = (3/2) kB T - mean molecular KE depends only on T.
- rms speed vrms = sqrt(3RT/M) = sqrt(3kBT/m); lighter gases move faster.
Equipartition and specific heats
- Equipartition: each quadratic degree of freedom carries (1/2)kB T per molecule.
- Degrees of freedom f: monatomic 3, diatomic 5 (+2 vibrational at high T).
- Cv = (f/2)R, Cp = (f/2+1)R, gamma = 1 + 2/f (monatomic 1.67, diatomic 1.4).
- Internal energy U = (f/2) nRT.
Mean free path
- lambda = 1/(sqrt2 n pi d^2) = kB T/(sqrt2 pi d^2 P) - average distance between collisions; smaller at higher pressure/density.