Effusion is the escape of gas molecules through a hole so small that they pass one at a time, without colliding on the way – helium leaking out of a balloon, or gas seeping through a pinhole into a vacuum. Thomas Graham found in 1846 that, at the same temperature and pressure, the rate of effusion is inversely proportional to the square root of the gas’s density, and therefore of its molar mass.
The kinetic theory of gases explains why: at a given temperature all gases have the same average kinetic energy ½mv², so lighter molecules must move faster, with mean speed proportional to 1/√M. Hydrogen (2 g/mol) effuses four times faster than oxygen (32 g/mol), and helium about 2.65 times faster than air – which is why helium balloons go flat in a day while air-filled ones last for weeks.
The same square-root rule holds approximately for the diffusion of one gas into another, and it has been used to measure molar masses (time how long equal volumes take to effuse) and, on an industrial scale, to enrich uranium: ²³⁵UF₆ effuses only 0.43% faster than ²³⁸UF₆, so the gas has to pass through more than a thousand barrier stages.
How to use the Rate of Effusion Calculator
- Calculate
- Gas 1
- Molar mass of gas 1
- Gas 2
- Molar mass of gas 2
- Rate of effusion of gas 1 — Any unit – the same as gas 2
- Rate of effusion of gas 2 — Any unit – the same as gas 1
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose what you want to find, pick the two gases from the list (or enter custom molar masses), and type the known rate. Rates can be in any unit (mL/min, mol/s, relative) as long as both use the same one. The calculator returns the missing value, the rate ratio and the matching ratio of effusion times.
Worked example
Hydrogen (2.016 g/mol) against oxygen (31.998 g/mol), with oxygen effusing at 1 mL/min: rate(H₂) = 1 × √(31.998 / 2.016) = 3.984 mL/min. Equal volumes take 0.251 times as long for hydrogen.
Identifying an unknown gas: if it effuses 4 times as fast as O₂ (rate 4 against 1), M = 31.998 × (1/4)² = 2.0 g/mol – it must be hydrogen.
Relative effusion rates (air = 1)
| Gas | M (g/mol) | Rate relative to air |
|---|---|---|
| Hydrogen H₂ | 2.016 | 3.790 |
| Helium He | 4.0026 | 2.690 |
| Methane CH₄ | 16.043 | 1.344 |
| Ammonia NH₃ | 17.031 | 1.304 |
| Water vapour H₂O | 18.015 | 1.268 |
| Neon Ne | 20.18 | 1.198 |
| Carbon monoxide CO | 28.01 | 1.017 |
| Nitrogen N₂ | 28.014 | 1.017 |
| Air (mean) | 28.96 | 1.000 |
| Oxygen O₂ | 31.998 | 0.951 |
| Hydrogen chloride HCl | 36.461 | 0.891 |
| Argon Ar | 39.948 | 0.851 |
| Carbon dioxide CO₂ | 44.009 | 0.811 |
| Chlorine Cl₂ | 70.906 | 0.639 |
| Krypton Kr | 83.798 | 0.588 |
| Xenon Xe | 131.293 | 0.470 |
| Sulfur hexafluoride SF₆ | 146.06 | 0.445 |
| Uranium hexafluoride ²³⁵UF₆ | 349.034 | 0.288 |
| Uranium hexafluoride ²³⁸UF₆ | 352.041 | 0.287 |
Tips
- Keep temperature in kelvin and pressure in consistent units.
- For gases, ideal-gas behaviour is a good approximation near room conditions.
- Half-life and rate constant are linked by t½ = ln 2 / k for first-order processes.
- Everything is calculated in your browser.
Common mistakes to avoid
- Using the mass number instead of the molar mass in Graham’s law.
- Mixing partial pressures and mole fractions.
- Forgetting that osmotic pressure uses the total particle concentration (van ‘t Hoff factor).
Inputs explained
| Field | What to enter |
|---|---|
| Calculate | Choose one: Rate of gas 1, Rate of gas 2, Molar mass of gas 1 (from both rates), Molar mass of gas 2 (from both rates) … |
| Gas 1 | Choose one: Hydrogen H₂ – 2.016 g/mol, Helium He – 4.0026 g/mol, Methane CH₄ – 16.043 g/mol, Ammonia NH₃ – 17.031 g/mol … |
| Molar mass of gas 1 | Units: g/mol, kg/mol |
| Gas 2 | Choose one: Hydrogen H₂ – 2.016 g/mol, Helium He – 4.0026 g/mol, Methane CH₄ – 16.043 g/mol, Ammonia NH₃ – 17.031 g/mol … |
| Molar mass of gas 2 | Units: g/mol, kg/mol |
| Rate of effusion of gas 1 | A number. Any unit – the same as gas 2 |
| Rate of effusion of gas 2 | A number. Any unit – the same as gas 1 |
What each result means
- Rate ratio rate₁ / rate₂ = sqrt([Molar mass of gas 2]÷[Molar mass of gas 1])
- Effusion time ratio t₁ / t₂ = 1÷[Rate ratio rate₁ ÷ rate₂]
Formula
rate₁ / rate₂ = √(M₂ / M₁) · time₁ / time₂ = √(M₁ / M₂) · M₁ = M₂ · (rate₂ / rate₁)²Worked example
Inputs — Calculate: Rate of gas 1; Gas 1: Hydrogen H₂ – 2.016 g/mol; Gas 2: Oxygen O₂ – 31.998 g/mol; Rate of effusion of gas 2: 1.
Results — Molar mass of gas 1 = 2.016 g/mol; Molar mass of gas 2 = 31.998 g/mol; Rate ratio rate₁ / rate₂ = 3.98397; Rate of gas 1 = 3.98397; Rate of gas 2 = 1; Result = 3.98397; Effusion time ratio t₁ / t₂ = 0.25101; Faster gas = Gas 1 effuses faster.
Key terms
- Effusion
- Escape of gas molecules through an opening smaller than their mean free path.
- Molar mass M
- Mass of one mole of a substance (g/mol).
- Rate of effusion
- Amount of gas escaping per unit time (volume, moles or molecules per second).
- Mean free path
- Average distance a molecule travels between collisions.
Frequently asked questions
What is Graham's law?
At the same temperature and pressure, rate₁/rate₂ = √(M₂/M₁): the rate of effusion is inversely proportional to the square root of molar mass.
How much faster does hydrogen effuse than oxygen?
√(31.998/2.016) = 3.98 – almost exactly four times faster.
What is the difference between effusion and diffusion?
Effusion is escape through a tiny hole into a vacuum without collisions; diffusion is gradual mixing through another gas with many collisions. Graham's law is exact for effusion and approximate for diffusion.
How do you find a molar mass with Graham's law?
Measure the rates (or times) for the unknown and a known gas: M_unknown = M_known × (rate_known / rate_unknown)². If an unknown gas effuses at 0.5 times the rate of O₂, M = 32 × 4 = 128 g/mol.
Why does a helium balloon deflate faster than an air balloon?
Helium (4.0 g/mol) effuses √(28.96/4.0026) = 2.69 times faster than air through the rubber.
How was Graham's law used to enrich uranium?
²³⁵UF₆ (349.03 g/mol) effuses √(352.04/349.03) = 1.0043 times faster than ²³⁸UF₆, so gaseous-diffusion plants used over 1000 stages in series.
Does temperature matter?
Both gases must be at the same temperature. If they are not, the rate also scales with √T.
What formula does the Rate of Effusion Calculator use?
rate₁ / rate₂ = √(M₂ / M₁) · time₁ / time₂ = √(M₁ / M₂) · M₁ = M₂ · (rate₂ / rate₁)².