Mean free path

Gas molecules move about as fast as sound, yet gas leaking from a kitchen cylinder takes a long time to reach the far corners, and the top of a smoke cloud can hold together for hours. The reason is collisions: molecules have a small but finite size, so their paths keep being deflected.

Model molecules as spheres of diameter d. A chosen molecule with average speed ⟨v⟩ hits any other whose centre comes within d of its own. In time Δt it sweeps a cylinder of volume πd²⟨v⟩Δt (Fig. 12.7).

With n molecules per unit volume it makes nπd²⟨v⟩Δt collisions in Δt, so the average time between collisions is τ = 1/(nπ⟨v⟩d²).

Mean free path, the average distance between successive collisions: l = ⟨v⟩τ = 1/(nπd²).

That picture freezes the other molecules. They move too, so the relative speed should be used; a more exact treatment gives l = 1/(√2 nπd²).

Air at STP: ⟨v⟩ = 485 m s⁻¹, n = (0.02 × 10²³)/(22.4 × 10⁻³) = 2.7 × 10²⁵ m⁻³ and d = 2 × 10⁻¹⁰ m give τ = 6.1 × 10⁻¹⁰ s and l = 2.9 × 10⁻⁷ m ≈ 1500d.

l falls as the number density or the molecular size grows. In a well-evacuated tube n is so small that l can reach the length of the tube.

Example 12.9: water vapour at 373 K, with the same d as air. n scales as 1/T: n = 2.7 × 10²⁵ × 273/373 ≈ 2 × 10²⁵ m⁻³, so l ≈ 4 × 10⁻⁷ m.

That is about 100 times the 40 Å (4 × 10⁻⁹ m) spacing found in Example 12.3. A mean free path this large is what makes a gas behave like a gas: it cannot be kept without a container.

Relations between bulk properties (viscosity, heat conduction, diffusion) and molecular size, worked out from kinetic theory, gave the first estimates of the size of molecules.