Graham's Law Effusion Calculator

Calculate relative effusion rates and separation factors using Graham's law of effusion.

U-235F6: 349, UF6: 352
Multiple stages for isotope separation
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Why Lighter Gases Effuse Faster

At a given temperature, all gas molecules have the same average kinetic energy. Since KE = ½mv², a lighter molecule must move faster to carry the same energy — and speed determines how often it finds the hole.

rate1/rate2 = √(M2/M1)

Note the inverse square root. Quadrupling the molar mass halves the rate, so the dependence is weaker than mass alone would suggest. Hydrogen effuses four times faster than oxygen despite being sixteen times lighter.

Gas pairMass ratioRate ratio
H2 / O21 : 164.00
He / N21 : 72.65
235UF6 / 238UF6349 : 3521.0043

Uranium Enrichment: Why It Is So Hard

That last row is one of the most consequential numbers in twentieth-century engineering. A single effusion stage separates U-235 from U-238 by a factor of just 1.0043 — four parts in a thousand.

Reaching reactor-grade enrichment therefore requires the stages to be cascaded, with each one's output feeding the next. Because enrichment multiplies, n stages give a factor of 1.0043n, and roughly 1,200 stages are needed to go from natural 0.7% to 3–5% reactor grade. The Manhattan Project's gaseous diffusion plant at Oak Ridge covered 44 acres for this reason.

Uranium hexafluoride is used because fluorine has only one stable isotope. Any other element in the compound would contribute its own isotopic spread and blur the tiny mass difference that the separation depends on.

Worked Examples

Example 1: U-235F6 vs U-238F6 separation
M1=349, M2=352 (UF6 masses)
Result: Rate ratio=1.00429 per stage
Need ~1400 stages for 90% enrichment
Example 2: H2 vs O2 effusion rate
M1=2, M2=32
Result: Rate ratio=sqrt(32/2)=4
H2 effuses 4x faster than O2
Example 3: Cascade multiplication
1.0043 per stage, 1,200 stages
Result: Factor of about 170
Because stages multiply, 1.00431200 gives the enrichment needed to reach reactor grade from natural uranium.
Example 4: Why UF6 specifically
Fluorine has one stable isotope; uranium is the only varying element
Result: Mass difference is clean
Using an element with multiple isotopes would blur the 3 u difference the whole separation depends on.

Common Mistakes

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Confusing effusion with diffusion

Effusion is escape through a small hole into vacuum; diffusion is mixing through another gas. Graham's law strictly describes effusion, though it approximates diffusion.

⚠️
Using mass ratio instead of its square root

The relationship is inverse square root. A sixteen-fold mass difference gives a fourfold rate difference, not sixteenfold.

⚠️
Inverting the ratio

The lighter gas is faster, so the heavier mass goes in the numerator under the square root. Getting this backwards inverts the answer.

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Applying it at high pressure

Effusion assumes molecules pass through individually without colliding near the aperture. At high pressure the flow becomes hydrodynamic and the law fails.

Frequently Asked Questions

Why uranium hexafluoride for enrichment?
UF6 is the only uranium compound that is a gas at near room temperature (sublimes at 56C). This allows gaseous diffusion or centrifuge separation based on mass difference.
Graham's law derivation?
All gases at same T have same average kinetic energy: 1/2 m*v^2 = 3/2 kT. So v = sqrt(3kT/m). Heavier molecules move slower. Effusion rate proportional to mean speed, so rate proportional to 1/sqrt(M).
What is the difference between effusion and diffusion?
Effusion is escape through a small orifice into vacuum. Diffusion is spreading through another gas. Graham's law describes effusion exactly and diffusion approximately.
Why is uranium enrichment so difficult?
Because the mass difference between 235UF6 and 238UF6 gives a separation factor of only 1.0043 per stage. Around 1,200 cascaded stages are needed.
Why use uranium hexafluoride?
Fluorine has a single stable isotope, so uranium is the only source of mass variation. It is also volatile enough to handle as a gas at moderate temperature.
Why is the relationship a square root?
Because kinetic energy depends on mass times velocity squared. Equal kinetic energy means velocity scales with the inverse square root of mass.

Formula Explorer connections

Interpretation: This relationship connects pressure, volume, temperature, amount or phase composition for gases and volatile mixtures. Assumption: Use absolute temperature and compatible pressure-volume units. Ideal behavior weakens at high pressure, low temperature, strong intermolecular attraction or near phase change.

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