Rule of thumb · Materials and ElementsNº 96 / 238

At one temperature lighter gas molecules are faster, as one over the square root of their mass

How fast do gas molecules move?

At a given temperature every gas molecule has the same average kinetic energy, so its speed goes as one over the square root of its mass: at 20 °C the rms speed is 1,905 m/s for hydrogen, 1,352 for helium, 511 for nitrogen and 408 for carbon dioxide, and a helium balloon leaks 2.6 times as fast as a nitrogen one.

Why it works

Temperature is the mean kinetic energy, (3/2)kT per molecule, so the root-mean-square speed is the square root of 3RT/M. Through a small hole the rate is set by how often the molecules arrive, which goes as the speed (Graham’s law), so helium (4.0 g/mol) leaks √(28.0/4.0) = 2.6 times as fast as nitrogen (28.0).

When it fails

The speeds are rms averages of a lopsided spread (the most probable is 0.82 of it and a few molecules go several times faster). A real balloon leaks mostly by dissolving in the skin and passing through it, which does not follow Graham’s law, so 2.6 is the limit for a pinhole.

Do it exactly

Estimate with the rule, then check it against the calculator that models it properly.

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