Graham's Law of Effusion

Temperature is a measure of average kinetic energy, and at a given temperature every gas in the room has the same average $\tfrac{1}{2}mv^2$ per molecule. If the energies match and the masses do not, the speeds must compensate: $v \propto 1/\sqrt{m}$. A molecule that finds a pinhole does so at a rate set by how fast it is travelling, so the escape rates of two gases stand in the inverse ratio of the square roots of their molar masses. The square root is the whole content of the law, and it comes from the square in the kinetic energy.

Helium against nitrogen: $\sqrt{28.01/4.003} = \sqrt{6.997} = 2.65$. Helium leaves 2.65 times faster, which is why a helium balloon is limp by morning while an air-filled one holds for weeks. Note that a factor of seven in mass buys only a factor of 2.65 in rate — the square root flattens everything, and that flattening has consequences.

Thomas Graham established the relation in 1848 by timing gases through a plaster plug. Its most consequential application came a century later. Uranium enrichment needs to separate ²³⁵U from ²³⁸U, and the only usable gaseous compound is uranium hexafluoride: 349.03 g/mol against 352.04. The separation factor is $\sqrt{352.04/349.03} = 1.0043$ — four-tenths of one percent per stage. To go from natural uranium to reactor grade takes over a thousand stages in cascade, and the K-25 plant built at Oak Ridge to do it enclosed some seventeen hectares under one roof and was for a time the largest building in the world. The square root is the reason it had to be that big.

Effusion is not diffusion, and Graham's law is a law about effusion. Effusion is escape through an opening small compared with the mean free path, so molecules leave one at a time without colliding on the way out — a pure speed contest. Diffusion is one gas spreading through another, and it is dominated by collisions, not by free flight. Graham's law describes diffusion only roughly, and the approximation degrades as pressure rises and collisions multiply. A leak through a crack in a fitting is usually closer to fluid flow than to either.

Two arithmetic traps. The subscripts are deliberately crossed, $r_1/r_2 = \sqrt{M_2/M_1}$, with 2 over 1 on the right — and inverting them is the commonest slip on this page. The sanity check is free: the lighter gas must always come out faster, so if your answer says otherwise, flip it. The second is confusing rate with time. If helium effuses 2.65 times faster, it takes $1/2.65 = 0.377$ times as long to release the same quantity. Rates and times are reciprocals, and a question worded "how long" wants the reciprocal of a question worded "how fast". Both gases must also be at the same temperature and pressure for the comparison to mean anything.