What this calculator does
Graham law says the rate of effusion is inversely proportional to the square root of molar mass. Hydrogen at 2.016 g/mol effuses 3.984 times faster than oxygen at 32.
The square root is what keeps the effect modest. Oxygen is nearly sixteen times heavier than hydrogen but effuses only about four times slower, because the ratio enters under a root. Separating gases by this route is therefore possible but slow, which is exactly why uranium enrichment by gaseous diffusion required thousands of stages.
The formula
The rate ratio is the square root of the second molar mass divided by the first. It follows from kinetic theory: at a given temperature all gases have the same average kinetic energy, so lighter molecules must move faster, by a factor of the inverse square root of their mass. The law applies to effusion through a small hole and approximately to diffusion.
| Term | Meaning |
|---|---|
| Effusion | Escape through a hole small compared with the mean free path. |
| Diffusion | Spreading through another gas. Graham law applies approximately. |
| Molar mass | The only property that matters here, since temperature is assumed equal. |
| Enrichment factor | The separation achieved per stage, which for similar masses is very close to 1. |
The inputs explained
| Field | What to enter |
|---|---|
| Molar mass of gas 1 (M₁) (g/mol) | Molar mass of the first gas. |
| Molar mass of gas 2 (M₂) (g/mol) | Molar mass of the second gas. |
| Known rate of gas 1 (units) | A known rate for gas 1, used to scale the answer for gas 2. The units are arbitrary. |
When to use it
Comparing gas escape rates
Which of two gases leaks from a container faster, and by how much.
Identifying an unknown gas
Measuring an effusion rate against a known gas gives the unknown molar mass.
Understanding isotope separation
The tiny mass difference between uranium isotopes gives an enrichment factor barely above 1, which explains the scale of the plants required.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How fast does each gas effuse against oxygen?
A range of gases against the same reference.
| Molar mass of gas 1 | Rate₁ / Rate₂ | Rate of gas 2 | √(M₂/M₁) used |
|---|---|---|---|
| 2.016 g/mol | 3.984 | 1.004 units | 3.984 |
| 4.003 g/mol | 2.827 | 1.415 units | 2.827 |
| 16.04 g/mol | 1.412 | 2.832 units | 1.412 |
| 44.01 g/mol | 0.853 | 4.691 units | 0.8527 |
Questions
What is the difference between effusion and diffusion?
Effusion is escape through a hole small enough that molecules pass one at a time without colliding. Diffusion is spreading through another gas, involving constant collisions. Graham law is exact for effusion and only approximate for diffusion, though it captures the right trend.
Why does the square root appear?
Because at a given temperature all gases have the same average kinetic energy, and kinetic energy is one half mv². Equal energy with different masses means velocity goes as the inverse square root of mass, and rate follows velocity.
Why was uranium enrichment so difficult?
Because the two isotopes of uranium hexafluoride differ in mass by less than 1%, giving a separation factor of about 1.0043 per stage. Reaching useful enrichment requires thousands of stages in cascade, which is why gaseous diffusion plants were enormous and energy-hungry.
Does temperature affect the ratio?
No, the ratio is temperature independent, because both gases speed up by the same factor when heated. Temperature affects the absolute rates but cancels entirely from the comparison, which is what makes the law so simple.
For finding a gas molar mass from measurements, see the molar mass of a gas calculator. For gas mixtures, see the partial pressure calculator.