Lesson 3 · 20 min

Mixtures, partial pressures, and effusion

Each gas in air contributes its own share of pressure, and light molecules move faster, so they leak through a pinhole faster — the principle behind uranium enrichment.

In this lesson

Use Dalton’s law and mole fractions to find partial pressures in a mixture
Compare effusion rates of different gases with Graham’s law
Distinguish effusion from diffusion and see why uranium enrichment needs thousands of stages

At sea level air presses at about 101 kPa, of which about 21 kPa comes from oxygen. At 5000 m altitude the total falls to about 54 kPa and the oxygen share to about 11 kPa — the fraction of oxygen in the air has not changed at all, yet people gasp for breath. The key: each gas contributes its own share of the pressure.

Predict first

A box holds heavy particles (mass 1) and light ones (mass ¼) at the same temperature. Roughly how many times faster, on average, are the light particles?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

The same 2 times 4 times 16 times

Dalton’s law of partial pressures

In an ideal gas mixture each gas behaves as though the others were not there. The pressure it would exert alone in the whole container is its partial pressure, and the total is the sum:

P_\text{total} = \sum_i p_i,\qquad p_i = x_i P_\text{total},\qquad x_i = \frac{n_i}{n_\text{total}}

$p_i$: partial pressure of component $i$
$x_i$: mole fraction of component $i$

In the mixture box there are equal numbers of heavy and light particles, so the two measured partial pressures come out nearly equal — even though the light particles move twice as fast. No contradiction: they hit the walls more often but deliver less momentum per hit, and the two effects cancel exactly. Partial pressure depends on the number of molecules, not their mass.

Worked example

Oxygen at altitude

Dry air is 0.2095 $\ce{O2}$ by mole fraction. Find the oxygen partial pressure at sea level (101.3 kPa) and in Lhasa (about 65 kPa).

Worked example

Collected over water: hydrogen carries water vapour

0.0400 g of magnesium reacts with excess hydrochloric acid and the hydrogen is collected over water in a gas burette. With the levels inside and outside matched it reads 40.94 mL; the water is at 22.0 °C and the barometer reads 101.30 kPa. Find the partial pressure of the hydrogen and the gas constant R.

Graham’s law: light molecules leak faster

Molecules escaping through a tiny hole in the wall is effusion. Its rate is proportional to how often molecules hit the hole, hence to their mean speed. At the same temperature and pressure:

\frac{\text{rate}_1}{\text{rate}_2} = \sqrt{\frac{M_2}{M_1}}

Graham’s law (1848).

$M$: molar mass

Concept check

Under the same conditions, how many times faster does helium (4.0 g/mol) effuse than oxygen (32.0 g/mol)?

8 times 2.83 times 0.35 times The same

Try the calculation

An unknown gas effuses at 0.707 times the rate of oxygen. What is its molar mass in g/mol?

Key takeaways

In an ideal mixture each gas contributes independently: $p_i = x_i P$ , and the total is their sum.
Partial pressure depends on the number of molecules, not their mass.
Effusion rate is inversely proportional to $\sqrt{M}$ (Graham); isotope separation exploits the tiny difference.

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Data and references

P. Atkins, J. de Paula, J. Keeler, Atkins’ Physical Chemistry, 11th ed. (2018), Topics 1A and 16A.
U.S. Standard Atmosphere, 1976 (NOAA/NASA/USAF) — pressure at altitude; dry-air O₂ mole fraction 0.2095.
M. Benedict, T. H. Pigford, H. W. Levi, Nuclear Chemical Engineering, 2nd ed., McGraw-Hill (1981), Ch. 12 (isotope separation).