Mean Free Path

How It Works — Step by Step

1. 1

   One Molecule Alone

   Start with a single red molecule in an empty box. It flies in a straight line and only changes direction when it bounces off a wall, so its free path, the distance it travels between two collisions with other molecules, never ends. Move the "Molecules" slider up to add a crowd and watch gold dots appear on the red trail where collisions happen.

2. 2

   A Crowd of Frozen Targets

   Here the other molecules are frozen and only the red tracer moves. Two molecules of diameter d touch when their centres come within d of each other, so the tracer hits any centre inside the yellow collision tube, a band 2d wide. That width is the collision cross-section, σ = 2d. Flying a distance L sweeps an area 2d · L holding about n · σ · L centres, so one hit takes λ = 1 / (n · σ).

3. 3

   Everyone Moves: the √2

   Now "Freeze others" is off, so every molecule moves and partners also rush toward the tracer, and the average closing speed between two molecules is √2 times their average speed. Collisions come √2 times more often, so the free path shrinks to λ = 1 / (√2 · n · σ). Compare the "Predicted mean free path" and "Measured mean free path" readouts as the measurement builds up.

4. 4

   No Two Paths Are Alike

   The "Free paths measured" chart collects every measured free path. Most are short, a few are long, and the bars fall away like the green curve, an exponential. The mean free path is the average, marked by the dashed green line; the red line is the measured average. About 37% of free paths are longer than λ, which you can check in the "Paths longer than λ" readout.

5. 5

   Crowd and Size

   Move the "Molecules" slider from 40 to 80: the number density n doubles and the predicted path halves, from 159 to 80 units. Now move the "Molecule size" slider from 8 to 4: σ halves and the path doubles. In this flat box σ = 2d, but real molecules are spheres with σ = πd², so doubling a real molecule’s diameter cuts its free path to a quarter.

6. 6

   Worked Example: Air in This Room

   Air at room temperature and normal pressure holds about n = 2.5 × 10²⁵ molecules per cubic metre, each about d = 0.37 nm across. In 3D, σ = πd² = 4.3 × 10⁻¹⁹ m². So λ = 1 / (1.41 × 2.5 × 10²⁵ × 4.3 × 10⁻¹⁹) ≈ 66 nm, about 180 molecule widths. At an average speed of 470 m/s, each molecule collides about 7 billion times every second.

7. 7

   Thin Gas, Long Paths

   Pumping gas out of a chamber lowers n, so λ grows in proportion: at one millionth of normal air pressure it is about 66 mm. Here only 8 molecules share the box and the predicted path, 795 units, is longer than the box itself, so the tracer meets walls far more often than partners. Vacuum tubes and particle beams rely on exactly this.