Hong Life CalculatorsUnderstanding the Graham's law of effusion calculator — concept and principle
Graham's law compares how fast two gases escape through a tiny hole, a process called effusion. Enter the molar masses of two gases and the calculator returns the ratio of their effusion rates. It is used in chemistry classes, for estimating molar masses from timing experiments, and to understand why light gases such as helium leak out of balloons so quickly.
How it is calculated
r₁ / r₂ = √(M₂ / M₁)
- M₁, M₂: molar masses of gas 1 and gas 2 in g/mol (any unit works, since only their ratio enters).
- r₁ / r₂: how many times faster gas 1 effuses than gas 2 under the same temperature and pressure.
Because time is inversely proportional to rate, the time needed for equal amounts to escape follows the reverse ratio: t₁ / t₂ = √(M₁ / M₂).
The principle
At the same temperature, all gas molecules have the same average kinetic energy, ½mv². A lighter molecule must therefore move faster: the typical speed is proportional to 1/√m. Effusion through a hole much smaller than the distance molecules travel between collisions is simply molecules hitting the hole one by one, so the escape rate is proportional to their average speed. Taking the ratio for two gases gives √(M₂/M₁). Thomas Graham reported this square-root relationship experimentally in the 1840s, before kinetic theory explained it.
Worked example
- Inputs: M₁ = 2.016 g/mol (hydrogen, H₂), M₂ = 32.00 g/mol (oxygen, O₂).
- Mass ratio: 32.00 / 2.016 = 15.873.
- Square root: √15.873 = 3.984095364.
- Hydrogen effuses about 3.98 times faster than oxygen, so the same amount of oxygen takes about 3.98 times longer to escape.
| Molar mass gas 1 (g/mol) | Rate₁ / Rate₂ |
|---|---|
| 0.504 | 7.968190729 |
| 1.01 | 5.634361698 |
| 1.51 | 4.600437062 |
| 2.02 | 3.984095364 |
| 2.52 | 3.563483225 |
| 3.02 | 3.253000243 |
| 4.03 | 2.817180849 |
How to read the result
A ratio above 1 means gas 1 is faster; below 1 means it is slower. Because of the square root, large differences in mass give only moderate differences in rate: a sixteen-fold heavier gas is only four times slower. Very small differences are what made gaseous-diffusion uranium enrichment so demanding: ²³⁵UF₆ and ²³⁸UF₆ differ in rate by only about 0.4%, so the gas had to pass through many stages in series.
Common mistakes
- Inverting the ratio and writing √(M₁/M₂) for the rate; that formula gives the ratio of times.
- Using atomic mass instead of molecular mass, for example 16 for oxygen gas instead of 32.
- Comparing gases at different temperatures; the law assumes both are at the same temperature.
Frequently asked questions
What is the difference between effusion and diffusion?
Effusion is escape through a tiny hole into a vacuum or low pressure; diffusion is gradual mixing of gases. Graham's law is exact for effusion and only approximate for diffusion.
How can I find an unknown molar mass with this law?
Measure the effusion times of the unknown and a known gas under identical conditions; then M_unknown = M_known × (t_unknown / t_known)².
Sources: P. Atkins and J. de Paula, Atkins' Physical Chemistry, Oxford University Press · OpenStax, Chemistry 2e, Section 9.4 "Effusion and Diffusion of Gases"