Graham’s Law of Effusion — Theory & Guide
Effusion is the escape of gas molecules through a tiny orifice into a vacuum.
At the same temperature and pressure, lighter gases effuse faster than heavier ones.
Graham’s law quantifies this trend and follows from kinetic–molecular theory.
\[
\frac{Q_A}{Q_B} \;=\; \sqrt{\frac{M_B}{M_A}}
\]
Here \(Q\) can be a rate of effusion, a molecular speed, the
distance traveled in a fixed time, or the amount effused in a fixed time.
\[
\frac{t_A}{t_B} \;=\; \sqrt{\frac{M_A}{M_B}}
\]
Times to effuse the same amount are inversely related to rates.
Connection to kinetic–molecular theory
From KMT, the rms speed of molecules in an ideal gas is
\[
u_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}
\]
At fixed \(T\) and with a sufficiently small orifice (molecules pass one by one without
collisions), the effusion rate is proportional to molecular speed, so \(Q\propto u_{\mathrm{rms}}
\propto 1/\sqrt{M}\). This yields the square-root dependence in Graham’s law.
Equivalent forms you will use
- Rates, speeds, distances, amounts (A divided by B):
\[
\frac{Q_A}{Q_B}=\sqrt{\frac{M_B}{M_A}}
\]
- Times (A divided by B) for the same amount:
\[
\frac{t_A}{t_B}=\sqrt{\frac{M_A}{M_B}}
\]
- Solve for an unknown molar mass:
\[
M_A=\frac{M_B}{\left(Q_A/Q_B\right)^2}
\]
\[
M_A=\left(\frac{t_A}{t_B}\right)^2 M_B
\]
How to reason quickly
- If \(M_A \lt M_B\), then \(Q_A/Q_B \gt 1\) and \(t_A/t_B \lt 1\) (lighter gas effuses faster and in less time).
- Mass units must match (both \(g\cdot\mathrm{mol^{-1}}\) or both \(kg\cdot\mathrm{mol^{-1}}\)); units cancel in ratios.
- Check magnitudes: a factor of \(4\) in molar mass gives a factor of \(2\) in rate or time ratios.
Worked micro-example
Compare \(\mathrm{H_2}\) and \(\mathrm{N_2}\) at the same \(T\):
\[
\frac{r_{\mathrm{H_2}}}{r_{\mathrm{N_2}}}
= \sqrt{\frac{M_{\mathrm{N_2}}}{M_{\mathrm{H_2}}}}
= \sqrt{\frac{28.014}{2.016}}
= 3.728
\]
Hydrogen effuses about \(3.73\times\) as fast as nitrogen; conversely,
\(t_{\mathrm{H_2}}/t_{\mathrm{N_2}} = 1/3.728\).
Assumptions & limitations
- Applies to effusion through a tiny orifice into vacuum; not reliable for bulk diffusion where collisions dominate.
- Valid when gases A and B are at the same temperature and pressure and behave ideally.
- Large holes, high pressures, or strong intermolecular forces can cause deviations.
