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A helium balloon goes flat faster than an air-filled one, and a gas leak spreads through a room even when no air is moving. Both come down to one rule found by the chemist Thomas Graham in the early 1830s. This post covers what he measured, how the law follows from kinetic theory, how to use it in problems, and the second field of chemistry he started thirty years later.
Who was Thomas Graham?
Thomas Graham lived from 1805 to 1869. A two-part review of his science by Jaime Wisniak in *Educación Química* describes him as known for founding colloid chemistry and for fundamental research on the diffusion of gases, liquids and solutions, dialysis and osmosis72481-9). His key papers appeared in the *Transactions of the Royal Society of Edinburgh* and the *Philosophical Transactions* of the Royal Society of London, and his own words are the clearest guide to what he found.
What did Graham's 1833 paper say?
His paper On the Law of the Diffusion of Gases, printed in volume 12 of the *Transactions of the Royal Society of Edinburgh* (a version also appeared in the Philosophical Magazine in April 1833), states its aim at the start: to establish "with numerical exactness" a law. In his words, the intermixture of two gases happens by an exchange of small volumes which are, "in the case of each gas, inversely proportional to the square root of the Density of that gas."
He then listed his "equivalent volumes of diffusion", with air as 1: hydrogen 3.7947, water vapour 1.2649, nitrogen 1.0140, oxygen 0.9487 and carbonic acid (carbon dioxide) 0.8091. He noted that these are the reciprocals of the square roots of the densities, taking air as 1.
You can check his arithmetic. For oxygen, 1/0.9487² = 1.111, so Graham was taking oxygen to be about 1.11 times as dense as air. For hydrogen, 1/3.7947² = 0.0694, about one-fourteenth as dense as air.
What about effusion through a hole?
Graham extended the work in two papers called "On the motion of gases". The second, published in the Philosophical Transactions in 1849, states that experiments on gases rushing into a vacuum or through a hole in a thin plate agree with the law "that the times of passage for equal volumes are proportional to the square roots of the densities". He adds that this law of "Effusion" is also the law of diffusion of one gas into another.
Why does the square root appear?
Kinetic theory explains it. At a given temperature, the average kinetic energy of the molecules, ½m⟨v²⟩, is the same for every gas. So a lighter molecule must move faster, with speed proportional to 1/√m. The NCERT Class 11 Physics chapter on kinetic theory makes this point in Example 12.6 and adds that the rate of diffusion of gases is inversely proportional to the square root of their masses. Our guide to the kinetic theory of gases covers the derivation.
Worked examples
1. Hydrogen against oxygen. Using M = 2 for H₂ and 32 for O₂: rate(H₂)/rate(O₂) = √(32/2) = √16 = 4. Hydrogen diffuses four times as fast.
2. Effusion times. As sample numbers, suppose 100 mL of oxygen takes 60 s to pass through a pinhole. The same volume of hydrogen takes 60/4 = 15 s.
3. Finding a molar mass. An unknown gas takes 1.5 times as long as oxygen to effuse. Then √(M/32) = 1.5, so M = 32 × 2.25 = 72 g/mol.
4. Uranium enrichment. NCERT's Example 12.6 compares the two forms of uranium hexafluoride, with molecular masses 349 (uranium-235) and 352 (uranium-238). The speed ratio is √(352/349) = √1.0086 ≈ 1.0043, a difference of only about 0.43%. NCERT explains that this is why separation through porous barriers must be repeated many times.
Common mistakes and how to avoid them
Mixing rate and time. A faster gas takes less time. If your lighter gas comes out slower, you have inverted the ratio.
Forgetting the square root. Hydrogen is 16 times lighter than oxygen, but only 4 times faster.
Using atomic instead of molecular mass. Oxygen gas is O₂, so M = 32, not 16.
Comparing at different temperatures. The law assumes the same temperature and pressure.
What did Graham discover about colloids?
In 1861 Graham turned to diffusion in liquids. His paper Liquid diffusion applied to analysis divided substances into fast-diffusing ones that crystallise and very slow ones such as gelatine, starch and gums. "As gelatine appears to be its type," he wrote, "it is proposed to designate substances of the class as colloids". The fast class he named crystalloids. Our post on John Tyndall and the scattering of light by colloids picks up the story.
If molecular speeds are hard to imagine, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read.
What can a student learn from how Graham worked?
Graham did not stop at saying light gases spread faster. He measured enough gases, carefully enough, to find the exact form of the rule, and then tested the same rule in a new setting, effusion. Measure several cases, look for the exact relationship, then test it somewhere new.
Practice routine:
Compute the rate ratio for helium (M = 4) against nitrogen (M = 28).
A gas effuses twice as fast as sulphur dioxide (M = 64). Find its molar mass.
Explain in two sentences why a helium balloon deflates faster than an air balloon.
Next step: solve the three practice problems and check each by asking whether the lighter gas came out faster.
