Effusion Rate Comparator

Physics Thermodynamics • Kinetic Theory of Ideal Gases

Written by STEM Calculators Team Published January 3, 2026 Updated February 24, 2026

8. Effusion Rate Comparator

Graham’s law (same temperature): \(r \propto 1/\sqrt{M}\). So \(\dfrac{r_A}{r_B}=\sqrt{\dfrac{M_B}{M_A}}\) and \(\dfrac{t_A}{t_B}=\sqrt{\dfrac{M_A}{M_B}}\).

Inputs support: pi, e, sqrt(), sin, cos, exp, log. Use * for multiplication.

Inputs

Temperature \(T\) (K) Compare Speed (display only) Molar mass unit (inputs) Normalize graph rates to In-plot labels Shade curve area

Graham’s law ratios don’t depend on hole size (for the ratio at the same \(T\)). Use the optional Knudsen check if you want a quick “effusion regime” sanity check.

Gas mixer (pick gases + molar masses)

The main ratio uses Gas 1 vs Gas 2 (first two rows).

Gas

Molar mass \(M\)

Unit

2 gas(es)

Optional: Knudsen (effusion-regime) check

Pressure \(P\) (kPa) Hole diameter \(d_h\) (mm) Molecular diameter \(d\) (m)

Uses \(\lambda=\frac{1}{\sqrt{2}\pi d^2 n}\) with \(n=\frac{P}{kT}\) and \(Kn=\lambda/d_h\). Rough: \(Kn \gtrsim 10\) → effusion/Knudsen regime likely.

Graph + animation controls

Progress \(\tau\in[0,1]\) Play speed

Plot supports drag-to-pan, wheel-to-zoom, and double-click (or “Reset view”).

Ready

Steps

Enter gases and click Solve.

Molecule “speed race” (illustrative)

Faster gas advances more per \(\tau\) (visual intuition)

Progress: \(\tau=0.650\)

Graph: relative effusion rate vs molar mass

Curve \(r \propto 1/\sqrt{M}\) + your gases marked

Hover: …

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Theory

Graham’s law (why lighter gases effuse faster)

Effusion is the escape of gas molecules through a tiny hole into vacuum (or a region of much lower pressure), ideally when the hole is smaller than the mean free path so molecules pass without colliding in the opening.

Key proportionality

The effusion rate is proportional to a characteristic molecular speed, which for an ideal gas scales like \(v \propto 1/\sqrt{M}\). Therefore,

\[ r \propto \frac{1}{\sqrt{M}} \quad \Rightarrow \quad \frac{r_A}{r_B} = \sqrt{\frac{M_B}{M_A}}. \]

Time to effuse the same amount

If “rate” is inversely proportional to time, then \[ t \propto \sqrt{M} \quad \Rightarrow \quad \frac{t_A}{t_B} = \sqrt{\frac{M_A}{M_B}}. \]

The ratio does not depend on hole size or pressure (when comparing two gases at the same temperature). Those factors cancel.

Speeds used (mean, rms, most probable)

The calculator can display one of these ideal-gas speeds (all scale as \(\sqrt{T/M}\)):

\[ v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}},\qquad \langle v\rangle=\sqrt{\frac{8RT}{\pi M}},\qquad v_{mp}=\sqrt{\frac{2RT}{M}}. \]

The ratio of effusion rates only needs the \(\,1/\sqrt{M}\,\) scaling, so it’s independent of which exact speed you prefer.

Knudsen criterion (when “effusion” is a good model)

A rough check uses the Knudsen number \(Kn=\lambda/d_h\) where \(d_h\) is the hole diameter and \(\lambda\) is the mean free path.

\[ n=\frac{P}{kT},\qquad \lambda=\frac{1}{\sqrt{2}\,\pi d^2 n},\qquad Kn=\frac{\lambda}{d_h}. \]

Rule-of-thumb: \(Kn\gtrsim 10\) suggests molecules traverse the opening without many collisions, so Graham’s law behavior is more reliable.

Application note: separation factor (e.g., uranium enrichment)

In gas diffusion, a “single-stage” separation factor is often approximated by the same square-root mass ratio: \[ \alpha \approx \sqrt{\frac{M_{\text{heavy}}}{M_{\text{light}}}}. \] Because the difference between \(^{235}\mathrm{UF_6}\) and \(^{238}\mathrm{UF_6}\) is small, many stages are required for significant enrichment.