Article — Graham’s Law of Diffusion Calculator
How Graham's law of diffusion works
Graham's law of diffusion says the rate at which a gas diffuses or effuses is inversely proportional to the square root of its molar mass. For two gases at the same temperature and pressure, r1/r2 = √(M2/M1). Hydrogen diffuses about 4 times faster than oxygen because its molar mass is 16 times smaller and the square root of 16 is 4.
Thomas Graham published the relation in 1846 after measuring how quickly gases passed through plaster-of-paris stoppers. The law became the kinetic-theory cornerstone for understanding gas behavior and the practical basis for the uranium enrichment that powered the first nuclear weapons.
What is Graham's law of diffusion
Diffusion is the gradual spread of one gas through another (perfume across a room, ammonia along a corridor). Effusion is the escape of a gas through a small hole into a region of lower pressure (helium leaking from a balloon, gas escaping a vacuum chamber). Both rates depend on the average molecular speed, which in turn depends on temperature and molar mass.
The law applies at constant temperature. Both gases share the same average kinetic energy, so the lighter gas must move faster to carry the same energy — the same way a tennis ball must move faster than a bowling ball to have the same kinetic energy.
The Graham's law formula
The relation is:
r1 / r2 = √(M2 / M1)
Where r1 and r2 are diffusion or effusion rates and M1, M2 are molar masses (g/mol). Equivalent forms:
- Solving for r2 — r2 = r1 · √(M1/M2)
- Time form — if a volume effuses in t1 for gas 1 and t2 for gas 2, then t2/t1 = √(M2/M1)
- Unknown mass — if you measure r1/r2 for a known gas (e.g. O2) and an unknown, you can solve for the unknown's molar mass
Diffusion vs effusion in Graham's law
Strictly Graham's law was derived for effusion — gas escape through an opening smaller than the mean free path. Diffusion through another gas adds collision effects that complicate the picture. In practice the same square-root scaling works well for both, because collisions affect both gases similarly and the ratio remains close to √(M2/M1).
The classic ammonia-HCl demonstration places cotton wool soaked in NH3 at one end of a tube and cotton wool soaked in concentrated HCl at the other. A white ring of NH4Cl forms where the gases meet, displaced toward the HCl end. NH3 (17.03 g/mol) diffuses faster than HCl (36.46 g/mol) by a factor of √(36.46/17.03) = 1.46.
Graham's law from kinetic theory
The kinetic theory of gases gives the average molecular speed as:
vavg = √(8RT / πM)
(or, for the RMS speed, √(3RT/M)). Both have the 1/√M scaling that Graham's law captures. At a fixed temperature the ratio of speeds for two gases is exactly √(M2/M1), independent of which speed (mean, RMS, most probable) you choose. T cancels, which is why Graham's law has no temperature dependence.
The diffusion or effusion rate is proportional to the average speed because faster molecules hit the boundary or hole per unit time more often. The proportionality is the same for any gas, so the ratio of rates equals the ratio of speeds equals √(M2/M1).
Graham's law and uranium enrichment
Natural uranium contains 0.72% U-235 (the fissile isotope) and 99.27% U-238. To make weapons or fuel reactors that use enriched fuel, the U-235 fraction must be raised. The first method to scale up was gaseous diffusion: convert uranium to UF6 (gaseous above 56 °C), then push it through porous barriers.
The molar masses of 235UF6 and 238UF6 are 349.03 and 352.04 g/mol respectively. The single-stage enrichment factor is √(352.04/349.03) = 1.0043 — only 0.43% per stage. Reaching weapon-grade U-235 (90%+) required thousands of cascaded stages. The K-25 plant at Oak Ridge during the Manhattan Project consumed more electricity than a major city.
Modern enrichment uses gas centrifuges, which give per-stage factors of about 1.5 — far better than diffusion. But the underlying physics is still Graham's law: the lighter isotopomer outpaces the heavier one in any process that exploits molecular speed.
Worked examples for Graham's law
How much faster does H2 effuse than O2? r(H2)/r(O2) = √(32.00 / 2.016) = √15.87 = 3.98. H2 effuses about 4 times faster.
Unknown gas effuses 1.4 times slower than CO2. r(CO2) / r(X) = 1.4 = √(MX/MCO2). So MX = 1.42 × 44.01 = 86.3 g/mol. Likely candidate: SF2O or a heavy hydrocarbon.
A balloon filled with helium loses pressure 2.7 times faster than one filled with air. √(Mair/MHe) = √(28.97/4.003) = 2.69. The measurement confirms helium's escape rate predicted by Graham's law.
To convert rate ratio into time ratio, just invert. If H2 effuses 4 times faster than O2, the same volume of H2 escapes in 1/4 the time. The same square-root-of-molar-mass formula applies, just multiplied or divided depending on whether you want speed or time.
Graham's law limitations
The law assumes ideal gas behavior, no chemical interaction between the gases, free molecular flow (the opening or mean free path is small compared with the apparatus), and constant temperature. Deviations appear in three regimes:
At high pressure or low temperature gases stop behaving ideally — molecular interactions slow diffusion in ways that depend on the specific gases, not just the molar masses. The simple 1/√M scaling becomes only an approximation, and detailed transport calculations (Chapman-Enskog) are required for accurate diffusion coefficients.
- Polyatomic gases at high T — rotational and vibrational excitation give extra degrees of freedom that complicate the simple kinetic picture, but the speed scaling stays approximate
- Charged species — ions follow different rules dominated by electrostatic forces, not Graham's law
- Surface diffusion — molecules adsorbed on a solid surface diffuse by hopping, with rates governed by activation energy rather than molecular speed
- Cluster formation — in saturated vapor near condensation small clusters change the effective M
For most general chemistry homework, Graham's law works exactly as written. For research-grade diffusion calculations in real mixtures, the Chapman-Enskog theory and the Maxwell-Stefan equations replace the simple square-root form. They predict diffusion coefficients in cm2/s rather than relative rates, and they account for cross-collisions, temperature dependence, and non-ideality. The simple Graham's law remains the right teaching tool because it captures the essence in one square root.