Graham's Law Calculator – Effusion Rates, Unknown Gases and the Ammonia Ring
Graham's law of effusion says that at the same temperature and pressure, a gas escapes through a pinhole at a rate inversely proportional to the square root of its molar mass. This Graham's law calculator compares two gases, identifies an unknown gas from its effusion rate or time, predicts where the white ring forms in the ammonia and hydrogen chloride tube, and counts the stages needed to enrich uranium by gaseous diffusion.
r₁ / r₂ = √(M₂ / M₁) t₁ / t₂ = √(M₁ / M₂) (same amount effused)Effusion versus diffusion
The Scottish chemist Thomas Graham studied how gases spread out and escape in the 19th century. Two processes are easy to confuse. Effusion is gas leaking through a hole smaller than the distance a molecule travels between collisions, so molecules pass through one at a time. Diffusion is one gas spreading through another, with endless collisions along the way. Graham's law is exact for effusion and only approximate for diffusion, which is why the tube mode carries a caution.
Why the square root of the mass?
At the same temperature every gas has the same average kinetic energy, ½mv² = (3/2)kT. A heavier molecule must therefore move more slowly, and its speed scales as 1/√m. Faster molecules hit the wall, and so find the pinhole, more often. That is why oxygen, almost 16 times heavier than hydrogen, effuses only about 4 times more slowly. The speed-distribution chart in the calculator shows this directly, and the Ideal Gas Law Calculator works out the speeds of a single gas in its kinetic-theory mode.
Identifying an unknown gas
Squaring the law gives M₂ = M₁ × (r₁ / r₂)², or with times M₂ = M₁ × (t₂ / t₁)². If nitrogen takes 38.0 s to effuse and an unknown gas takes 64.5 s for the same amount, the unknown has a molar mass of about 80.7 g/mol, very close to hydrogen bromide. The calculator lists the nearest gases within 5 % so you can name the unknown.
The ammonia and hydrogen chloride tube
Cotton plugs soaked in ammonia and hydrochloric acid are pushed into opposite ends of a glass tube. Where the gases meet they react to form solid ammonium chloride, a white ring. Ammonia is the lighter gas, so it covers more of the tube and the ring forms nearer the HCl end, about 59 % of the way from the ammonia. In a real tube the gases diffuse through air, so the ring forms near the predicted point rather than exactly on it.
When the temperatures differ
The often-quoted rate ∝ √(T/M) quietly assumes both gases have the same number of molecules per litre. At equal pressure a hotter gas is less dense, and the molar effusion rate goes as 1/√(MT), so the hotter gas effuses more slowly in moles per second. The advanced panel makes you choose which quantity is held equal.
Uranium enrichment by gaseous diffusion
The Manhattan Project separated uranium isotopes by pushing uranium hexafluoride through porous barriers. Molecules of ²³⁵UF₆ and ²³⁸UF₆ differ in mass by less than 1 %, so an ideal stage enriches by a factor of only α ≈ 1.0043. Reaching reactor-grade uranium took hundreds to thousands of stages in series. Gas centrifuges, which need far fewer stages, later replaced diffusion plants.
Everyday effusion
A helium party balloon goes flat overnight while an air-filled one lasts for days: helium atoms are light and fast and leak through the rubber more easily. Sound also travels faster in light gases, which is why helium raises the pitch of your voice. The Speed of Sound Calculator works that out.