Mean free path

Gas molecules move at hundreds of metres a second, yet a smell takes a long time to cross a room. The reason is collisions: each molecule keeps hitting others and changing direction, so it follows a zigzag made of short straight pieces.

The mean free path l is the average length of those straight pieces: the distance a molecule goes between one collision and the next.

Picture a molecule of diameter d moving along. It hits any molecule whose centre comes within d of its path, so in each second it sweeps out a cylinder of cross-section πd². The more molecules per volume (n) and the bigger they are, the sooner it hits one:

• l = 1/(√2 nπd²). The √2 appears because the other molecules are moving too.
• Since n = P/(k_BT), this is also l = k_BT/(√2 πd²P).
• So l is longer for a thinner gas (lower P, or higher T at fixed P) and for smaller molecules.

Example: air at 300 K and 1 atm has n = P/(k_BT) ≈ 1.013 × 10⁵/(1.38 × 10⁻²³ × 300) ≈ 2.4 × 10²⁵ molecules per m³. With d = 3 Å, l = 1/(1.414 × 3.14 × 2.4 × 10²⁵ × 9 × 10⁻²⁰) ≈ 1 × 10⁻⁷ m, several hundred molecular diameters.

In a good vacuum of 10⁻⁴ Pa, about a billion times lower than the air's pressure, the same formula gives l of about 100 m. Then a molecule crosses a whole vacuum tube without meeting another one.

The collision frequency is v/l, and the time between collisions is l/v. A collision itself lasts only about d/v, so a molecule spends nearly all its time moving freely, and only a tiny fraction actually colliding.