Effusion Rate Calculator | Graham's Law of Effusion

What is an effusion rate calculator?

An effusion rate calculator finds how fast one gas escapes through a tiny hole compared to another gas. It uses Graham's law of effusion, which links the escape speed of a gas to its molar mass and temperature. Effusion happens when gas molecules pass one at a time through an opening so small that the molecules do not collide with each other as they go through.

Graham's law of effusion formula

Graham's law compares the effusion rates of two gases, gas 1 and gas 2:

$\frac{\text{Rate}_1}{\text{Rate}_2} = \sqrt{\frac{M_2}{M_1}} \times \sqrt{\frac{T_1}{T_2}}$

• Rate₁ and Rate₂ are the effusion rates of gas 1 and gas 2.
• M₁ and M₂ are the molar masses of gas 1 and gas 2, in grams per mole (g/mol).
• T₁ and T₂ are the absolute temperatures of gas 1 and gas 2, in kelvin (K).

A result above 1 means gas 1 effuses faster than gas 2. A result below 1 means gas 2 is faster. A result of exactly 1 means both gases effuse at the same rate.

The rate of effusion depends on how often molecules hit the hole, which depends on their density and speed. Kinetic theory gives the number of molecules striking a unit area of wall per second as one quarter of the number density times the mean molecular speed, and the mean speed is √(8RT/πM). The temperature term in this formula changes only the speed. It therefore assumes both gases have the same number of molecules per unit volume. Two gases at the same pressure but different temperatures do not meet that assumption, because the hotter gas is less dense.

Why the formula works

The formula comes from the kinetic theory of gases. All gases at the same temperature have the same average kinetic energy, ½mv² = (3/2)kT, where m is the mass of one molecule, v is its average speed, k is the Boltzmann constant, and T is the absolute temperature. Solving for speed gives v = √(3kT/m). A lighter molecule must move faster than a heavier one to hold the same kinetic energy. Since the rate of effusion rises with molecular speed, and molar mass scales with molecular mass, dividing the speed of gas 1 by the speed of gas 2 gives the formula above.

Two simpler cases

• Same temperature. If T₁ equals T₂, the temperature terms cancel and the formula becomes Rate₁/Rate₂ = √(M₂/M₁). Only the molar masses matter. This is Graham's law in its usual form, for two gases at the same temperature and pressure.
• Same molar mass. If M₁ equals M₂, the formula becomes Rate₁/Rate₂ = √(T₁/T₂). Only the temperatures matter.

How to use the effusion rate calculator

1. Enter the molar mass and temperature of gas 1.
2. Enter the molar mass and temperature of gas 2.
3. Read the relative effusion rate. It shows how many times faster gas 1 effuses than gas 2.

Both molar mass and temperature must be positive numbers. Temperature must be entered in kelvin, not Celsius or Fahrenheit, because the formula depends on absolute temperature.

Common gas molar masses

Worked examples

Example 1: helium versus methane, same temperature

Helium (He) has a molar mass of 4.00 g/mol. Methane (CH₄) has a molar mass of 16.04 g/mol. Both are at 298 K.

$\frac{\text{Rate}_{He}}{\text{Rate}_{CH_4}} = \sqrt{\frac{16.04}{4.00}} \times \sqrt{\frac{298}{298}} = \sqrt{4.01} \times 1 \approx 2.0025$

Helium effuses about 2.00 times faster than methane.

Example 2: hydrogen versus oxygen, different temperatures

Hydrogen (H₂) has a molar mass of 2.02 g/mol and a temperature of 400 K. Oxygen (O₂) has a molar mass of 32.00 g/mol and a temperature of 300 K.

$\sqrt{\frac{32.00}{2.02}} \approx 3.9801 \qquad \sqrt{\frac{400}{300}} \approx 1.1547$

$\frac{\text{Rate}_{H_2}}{\text{Rate}_{O_2}} = 3.9801 \times 1.1547 \approx 4.5959$

Hydrogen at 400 K effuses about 4.60 times faster than oxygen at 300 K. Rounding each square root to two decimal places before multiplying, as some worked examples online do, gives 3.98 × 1.15 = 4.58, which is slightly off. Keeping more decimal places until the final step avoids that error.

Applications of Graham's law

• Isotope separation. The Manhattan Project used gaseous diffusion of uranium hexafluoride to separate uranium-235 from uranium-238. The two isotopes form uranium hexafluoride gas with slightly different molar masses, so they effuse at slightly different rates.
• Leak detection. Helium is a common tracer gas for finding leaks in vacuum systems and pressure vessels because its low molar mass lets it effuse through small leaks quickly.
• Membrane gas separation. Industrial membranes that purify or separate gas mixtures rely partly on the different effusion rates of the component gases.

History

The Scottish chemist Thomas Graham measured how fast different gases escaped through small openings. He found that the rate of effusion is inversely proportional to the square root of the gas's molar mass. His results later helped support the kinetic theory of gases, developed further by James Clerk Maxwell and Ludwig Boltzmann.

Frequently asked questions

What is the difference between effusion and diffusion? Effusion is the escape of gas molecules through a small hole into a vacuum or region of lower pressure, where the hole is smaller than the average distance a molecule travels between collisions. Diffusion is the spreading of gas molecules through another gas or medium, driven by differences in concentration, and it involves molecules colliding with each other.

Why must temperature be in kelvin? The kinetic energy of gas molecules is proportional to absolute temperature. Celsius and Fahrenheit scales do not start at absolute zero, so using them would give the wrong ratio.

Does pressure affect the relative effusion rate? If both gases are at the same temperature and pressure, pressure cancels out of the ratio. When the temperatures differ, the calculator's temperature term assumes both gases have the same molecular density, not the same pressure.

Can Graham's law find the molar mass of an unknown gas? Yes. If the effusion rate of an unknown gas is measured against a gas of known molar mass, the formula can be rearranged:

$M_{unknown} = M_{known} \times \left(\frac{\text{Rate}_{known}}{\text{Rate}_{unknown}}\right)^2 \times \frac{T_{unknown}}{T_{known}}$

Does Graham's law apply to liquids? No. It applies only to gases. Liquid molecules are packed much closer together and interact far more strongly, so the same formula does not describe how liquids move through small openings.

How much faster does a gas effuse if its temperature doubles? The effusion rate is proportional to the square root of the absolute temperature. Doubling the temperature multiplies the rate by √2, or about 1.41, if the molar mass and the molecular density stay the same.

References

1. OpenStax. Chemistry 2e, section 9.4. Effusion and Diffusion of Gases — Graham's law, rate ∝ 1/√ℳ for gases at the same temperature and pressure; gaseous diffusion of UF₆ in the Manhattan Project.
2. Eames, E. V. (City College of San Francisco), via LibreTexts. General Chemistry Supplement: Diffusion and Effusion. Diffusion and Effusion — effusion rate depends on how often particles hit the hole, which depends on their density and speed.
3. National Institute of Standards and Technology. NIST Chemistry WebBook: Helium. Helium — molecular weight 4.002602.
4. National Institute of Standards and Technology. NIST Chemistry WebBook: Methane. Methane — molecular weight 16.0425.
5. LibreTexts. Physical Chemistry, section 27.4. The Frequency of Collisions with a Wall — wall-collision flux z_w = ¼ (N/V)⟨v⟩ with ⟨v⟩ = (8RT/πM)^½; basis for the equal-number-density assumption behind the temperature term.