Rate of Effusion Calculator

Effusion is the escape of gas molecules through an opening small enough that they leave one at a time without colliding in the hole. Kinetic theory gives the number of molecules striking unit area of wall per second as one quarter of the number density times the mean molecular speed, and that mean speed is sqrt(8*R*T/(pi*M)). Every other factor is shared by two gases held at the same temperature and pressure, so only the square root of molar mass survives and the ratio collapses to rate1/rate2 = sqrt(M2/M1). Lighter molecules simply hit the aperture more often. Hydrogen at 2.016 g/mol therefore escapes about 3.98 times faster than oxygen at 32.00 g/mol, and the related root-mean-square speed sqrt(3*R*T/M) is roughly 515 m/s for nitrogen and 1360 m/s for helium at 298 K. The critical limitation is the Knudsen condition: the hole must be small compared with the mean free path, which is only about 68 nm for air at 1 atm and 25 degrees C. Through a larger opening the gas moves as a bulk fluid and the flow becomes pressure-driven rather than molecular, so the square-root law no longer holds and the two gases pass at nearly equal rates. Separation factors are also unforgiving. Uranium hexafluoride made from U-235 has molar mass 349.0 against 352.0 for the U-238 form, giving a single-stage factor of sqrt(352.0/349.0) = 1.0043, which is why thousands of cascaded stages were required for enrichment.

Thomas Graham, working in Glasgow at Anderson's Institution and later at University College London, established the relationship experimentally. His studies on the diffusion of gases through porous plaster plugs, published in the Philosophical Magazine in the early 1830s, produced the law that rates vary inversely with the square root of density, and he extended the result to effusion through fine apertures in work published in the 1840s. Graham had no molecular theory to lean on; his conclusion rested on painstaking measurement, and it later became one of the observations that kinetic theory had to reproduce when Clausius and Maxwell developed it in the 1850s and 1860s. Graham also coined the terms colloid and dialysis, and he ended his career as Master of the Mint. The molecular-flow regime was put on a formal footing by the Danish physicist Martin Knudsen around 1909, who characterised flow through apertures and tubes when the mean free path exceeds the geometry, and whose effusion cell is still used to measure very low vapour pressures. The most consequential application came during the Second World War, when the Manhattan Project built the K-25 gaseous diffusion plant at Oak Ridge, Tennessee, using uranium hexafluoride and cascades of porous barriers. Gaseous diffusion remained in commercial service for decades before gas centrifuges displaced it on energy grounds.