What it does
Gas molecules are fast (hundreds of metres a second at room temperature), yet a smell takes seconds to cross a room. The reason is that each molecule keeps bumping into others and setting off in a new direction. The average distance it flies between two collisions is its mean free path, λ.
The top picture is a flat box of molecules bouncing off each other like billiard balls. One is marked and leaves a trail, so you can see its zig-zag. The page counts every collision and divides the total distance flown by the number of flights, so you watch λ being measured, and compares it with the formula. Crowd the box or make the molecules bigger and the flights get shorter.
The second card gives the real numbers for a gas: λ = kT / (√2 π d² p) filled in with your pressure, temperature and molecule size, plus the number of molecules in a cubic metre, their mean speed, and how many collisions each makes every second.
It serves Class 11 Physics, Kinetic Theory (mean free path, and the behaviour of gases), with links to Class 9 Science (particles of matter are moving and have spaces between them). Start with kinetic theory for molecules hitting the walls to make pressure, and go on to equipartition for how molecules share their energy.
How to use it
- How crowded sets the number of molecules in the box (like raising the pressure). Molecule size sets their diameter. Changing either starts the box again.
- Pause and Start again control the animation. The line under the box shows the collisions so far, the measured average flight and the formula value, in box widths.
- In The real numbers, pick a pressure with the buttons (Sea level, Top of Everest, Vacuum pump, Very high vacuum) or the slider, which moves in powers of ten.
- Set the temperature and choose the gas.
Worked example
With the page as it opens: nitrogen (d = 0.37 nm, about 28 g/mol) at 101.325 kPa and 300 K.
λ = 1.381×10⁻²³ × 300 / (1.414 × π × (0.37×10⁻⁹)² × 101 325) = 67.2 nm, about 180 times the molecule’s own size. There are n = p/kT = 2.45×10²⁵ molecules per m³, about 3.4 nm apart. They move at a mean speed of 476 m/s, so each has about 476 / 67.2×10⁻⁹ = 7.09×10⁹ collisions per second.
At a very high vacuum of 10⁻⁶ Pa the same molecules fly about 6.8 km between collisions, far longer than any container: they hit the walls, not each other.
In the box, 60 molecules each 2 % of the box wide give a formula value of 0.177 box widths; the count settles close to that after a few hundred collisions.
Check yourself
Five fresh questions: a mean free path, the number of molecules per cubic metre, what happens with more pressure, what happens with bigger molecules, and collisions per second. Answers are checked on the page and the working is shown. Nothing is saved.
Tips
- λ goes as 1/p and as 1/d²: double the pressure, half the path; double the diameter, a quarter of the path.
- At fixed pressure λ grows with T, because the same pressure at a higher temperature means fewer molecules per cubic metre.
- The √2 comes from the other molecules moving too: the average speed of one molecule relative to another is √2 times the mean speed.
FAQ
Why is the measured value in the box a bit shorter than the formula when it is crowded? The formula assumes the molecules take up a tiny share of the space. In a crowded box their own size matters, so flights are shorter. Real air at normal pressure is far emptier than the box.
Why does the flat box use a different formula? In a flat world a molecule sweeps out a strip of width 2d, not a tube of cross-section πd², so λ = 1 / (2√2 n d), with n counted per unit area.
Where does mean free path matter? Vacuum pumps, light bulbs, particle accelerators and the upper atmosphere: once λ is longer than the container, a gas stops behaving like a fluid.
Where the numbers come from
- Mean free path λ = kT / (√2 π d² p) = 1 / (√2 n π d²), number density n = p/kT, mean speed √(8kT/πm); flat (2-D) hard-disk version λ = 1 / (2√2 n d): standard kinetic theory.
- Boltzmann constant k = 1.380649×10⁻²³ J/K (exact, SI 2019); atomic mass unit 1.66053906660×10⁻²⁷ kg (CODATA 2018).
- Molecular diameters (nitrogen 0.37 nm, oxygen 0.36, helium 0.22, argon 0.36, carbon dioxide 0.46 nm) are rounded typical values for teaching.
- Checked against an independent Python calculation, and the 2-D formula against a separate hard-disk simulation.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.