Mean free path: follow one molecule.

One tracked molecule zigzags among others; its measured average distance between hits matches l = 1/(√2 nπd²).

Double the diameter and predict the new mean free path (¼), then confirm with the tracked molecule.

Molecules are thousands of diameters apart (confusing spacing with 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.

Teaching this idea? Put the simulation on your own page or LMS with this code.