Last updated: July 3, 2026
Rate of Effusion Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The rate of effusion calculator applies Graham's law, r₁/r₂ = √(M₂/M₁), to compare two gases from their molar masses. Enter M₁ and M₂ in g/mol to get the dimensionless rate ratio and its reciprocal.
The relative rate of effusion of gas one to gas two equals the square root of the molar mass of gas two divided by the molar mass of gas one.
Key Takeaways
- Graham's law compares gas effusion rates with r₁/r₂ = √(M₂/M₁).
- A lighter gas effuses faster because its molecules move faster at the same temperature.
- The rate ratio is dimensionless and can multiply a known rate of gas 2.
- Keep gas order consistent: swapping gas 1 and gas 2 gives the reciprocal ratio.
- The law is best for ideal gases escaping through a small opening under matched conditions.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
rate1/rate2 = sqrt(M2/M1)
Where:
- r_1=Effusion rate of gas 1(same rate unit as gas 2)
- r_2=Effusion rate of gas 2(same rate unit as gas 1)
- M_1=Molar mass of gas 1(g/mol)
- M_2=Molar mass of gas 2(g/mol)
Worked Examples
Hydrogen gas compared with oxygen gas
Hydrogen is much lighter than oxygen, so it effuses faster through the same small opening.
- 1Identify M₁ = 2.016 g/mol for H₂ and M₂ = 32.00 g/mol for O₂.
- 2Apply Graham's law: r₁/r₂ = √(M₂/M₁).
- 3Compute √(32.00/2.016) = √15.873 ≈ 3.984.
Helium compared with methane
Helium effuses about twice as fast as methane under identical conditions.
- 1Use M₁ = 4.003 g/mol for He and M₂ = 16.04 g/mol for CH₄.
- 2Substitute into r₁/r₂ = √(M₂/M₁).
- 3Compute √(16.04/4.003) = √4.007 ≈ 2.002.
Carbon dioxide compared with nitrogen
Carbon dioxide is heavier than nitrogen, so gas 1 effuses more slowly than gas 2.
- 1Use M₁ = 44.01 g/mol for CO₂ and M₂ = 28.01 g/mol for N₂.
- 2Apply r₁/r₂ = √(28.01/44.01).
- 3Compute √0.6365 ≈ 0.7978, so CO₂ effuses at about 79.8% of the N₂ rate.
Introduction
The rate of effusion calculator compares how quickly two gases escape through a tiny opening under the same temperature and pressure. It uses Graham's law, r₁/r₂ = √(M₂/M₁), which says lighter gases effuse faster because their average molecular speeds are higher. Use it with the molar mass calculator to find gas molar masses or the partial pressure calculator when gas mixtures are involved. The relationship follows kinetic molecular theory as summarized by LibreTexts gas effusion notes and standard IUPAC gas terminology.
What Graham's law says
Graham's law states that, at the same temperature, the rate of effusion of a gas is inversely proportional to the square root of its molar mass. For two gases, that becomes r₁/r₂ = √(M₂/M₁). The heavier molar mass goes in the numerator only because the formula reports the rate of gas 1 relative to gas 2.
If r₁/r₂ is greater than 1, gas 1 effuses faster than gas 2.
If r₁/r₂ is less than 1, gas 1 effuses more slowly than gas 2.
If both molar masses match, the ratio is exactly 1.
The result is dimensionless because it compares two rates.
How to calculate relative effusion rate
Start by assigning gas 1 and gas 2 consistently. Enter their molar masses in g/mol, divide M₂ by M₁, and take the square root. The ratio can multiply any known rate of gas 2 to estimate gas 1's rate under identical conditions.
Find M₁ for gas 1 and M₂ for gas 2.
Compute the mass ratio M₂/M₁.
Take the square root to obtain r₁/r₂.
Interpret the ratio as faster than, slower than, or equal to gas 2.
Do not reverse M₁ and M₂ unless you intentionally want the reciprocal ratio r₂/r₁.
Why lighter gases effuse faster
At a fixed temperature, gases have the same average kinetic energy. Since kinetic energy depends on mass and speed, lighter molecules must move faster on average than heavier molecules. Faster molecules collide with a pinhole more often and pass through it more rapidly. The square-root dependence comes from root-mean-square speed in kinetic molecular theory.
For gas speed comparisons, this same mass dependence connects to diffusion and transport calculations such as the diffusion coefficient calculator.
Common effusion comparisons
The table below gives typical Graham's-law ratios for gases often used in teaching examples. Values assume identical temperature, pressure, and orifice geometry.
| Gas 1 | Gas 2 | M₁ (g/mol) | M₂ (g/mol) | r₁/r₂ |
|---|---|---|---|---|
| H₂ | O₂ | 2.016 | 32.00 | 3.984 |
| He | CH₄ | 4.003 | 16.04 | 2.002 |
| CO₂ | N₂ | 44.01 | 28.01 | 0.7978 |
| NH₃ | HCl | 17.03 | 36.46 | 1.463 |
Using a known rate
If you know one effusion rate, multiply or divide by the ratio. For example, if gas 2 effuses at 5.0 mL/s and r₁/r₂ = 2.0, then gas 1 effuses at 10.0 mL/s. If you know gas 1's rate instead, divide by r₁/r₂ to get gas 2's rate. Keep the same physical setup and rate unit throughout.
Assumptions and limitations
Graham's law works best for ideal gases effusing through a small hole from a container into a lower-pressure region. It assumes the gases are at the same temperature and that the opening is small enough for molecular effusion rather than bulk flow. Real-gas effects, leaks with large openings, adsorption on surfaces, or temperature gradients can change measured rates. For amount conversions before comparing gases, use the mole calculator or average atomic mass calculator as needed.
Use molar masses in the same unit, normally g/mol.
Compare gases at the same temperature.
Use only for similar orifice geometry and pressure conditions.
Treat very high-pressure or strongly interacting gases cautiously.
Quick Reference Card
Rate of Effusion — Quick Reference
Quick reference • Rate of Effusion Calculator
r₁/r₂ = √(M₂/M₁)Valid range: Use positive molar masses; best for ideal gases at the same temperature effusing through the same small opening.
Common Values
⚠ Watch Out
- •Molar masses must be positive numbers.
- •Do not compare gases at different temperatures using this simple ratio.
- •Large leaks may involve bulk flow rather than true effusion.
- •Swapping gas order changes the answer to the reciprocal.
- •Real gases at high pressure can deviate from ideal Graham's-law behavior.
Pro Tips
- →Use g/mol for both gases so the units cancel cleanly.
- →Keep extra digits through the square-root step, then round the final ratio.
- →If the result is less than 1, gas 1 is slower; take the reciprocal to see how much faster gas 2 is.
- →Pair with a molar mass calculator when comparing molecular formulas.
- →State the gas order in reports to avoid ambiguity.
FAQs
What is effusion?
Effusion is the escape of gas molecules through a tiny opening into a lower-pressure region, ideally without many molecule-molecule collisions in the opening.
What formula does this calculator use?
It uses Graham's law: r₁/r₂ = √(M₂/M₁), where M₁ and M₂ are the molar masses of gas 1 and gas 2.
Why is the heavier gas in the numerator?
The expression reports gas 1's rate relative to gas 2. Because rate is inversely proportional to √M, the ratio becomes √(M₂/M₁).
What does a ratio greater than 1 mean?
A ratio greater than 1 means gas 1 effuses faster than gas 2. For H₂ versus O₂, r₁/r₂ is about 3.984, so H₂ effuses nearly four times as fast.
Can I use molecular mass instead of molar mass?
Yes, if both masses use consistent relative units. In chemistry, molar mass in g/mol is the standard input and gives the same ratio as relative molecular mass.
Is effusion the same as diffusion?
No. Effusion is escape through a small opening, while diffusion is spreading through space or another gas. Both are faster for lighter gases, but their experimental conditions differ.