Kinetic theory of gases

The kinetic theory explains gas behaviour from the motion of particles. Gas pressure comes from countless molecules bouncing off the walls, and temperature is a measure of their average kinetic energy. From a few simple assumptions we can work out how fast air molecules move: hundreds of metres per second.

Key facts

  • Assumptions: many molecules in random motion; elastic collisions; negligible molecular volume; no forces except during collisions.
  • Pressure is caused by molecules colliding with the walls and changing momentum.
  • pV = ⅓Nm⟨c²⟩, so p = ⅓ρ⟨c²⟩ and c_rms = √(3p/ρ).
  • Mean kinetic energy of a molecule = (3/2)kT, proportional to absolute temperature.
  • c_rms = √(3RT/M) = √(3kT/m).
  • Internal energy of a monatomic ideal gas: U = (3/2)nRT.

Worked examples

Example 1

Find the mean kinetic energy of an air molecule at 300 K.

  1. The mean kinetic energy depends only on the kelvin temperature. E = 1.5 × 1.38 × 10^−23 × 300 = 6.21 × 10^−21 J

Answer: 6.21 × 10^−21 J

Example 2

Find the r.m.s. speed of nitrogen molecules (M = 0.028 kg/mol) at 300 K.

  1. Molar mass in kg/mol. c = √(3 × 8.31 × 300 ÷ 0.028) = 517 m/s

Answer: 517 m/s

Example 3

A gas is heated from 27°C to 927°C. By what factor does the r.m.s. speed of its molecules increase?

  1. Convert to kelvin: 300 K → 1200 K, so T is 4 times bigger. 1200 ÷ 300 = 4
  2. c_rms ∝ √T, so each arrow is twice as long. √4 = 2

Answer: 2