Dalton's law of partial pressures

Each gas pushes on its own: Dalton's law

Air looks like one gas, but it is a mixture: roughly 78 percent nitrogen, 21 percent oxygen, 1 percent argon, plus traces. Dalton's law says each component pushes independently, as if the others were absent:

$$P_{total} = P_1 + P_2 + P_3 + \cdots$$

The partial pressure $P_i$ of each gas is the pressure it alone would exert in the same container at the same temperature. Add them all and you recover the pressure gauge reading.

A worked example: a 10.0 L tank at $25^\circ\text{C}$ holds 0.300 mol N2 and 0.100 mol O2. Each gas obeys $PV = nRT$ alone: $P_{N_2} = (0.300 \times 0.08206 \times 298)/10.0 = 0.733$ atm, $P_{O_2} = (0.100 \times 0.08206 \times 298)/10.0 = 0.244$ atm. Total: $0.733 + 0.244 = 0.977$ atm. Measuring the tank would show 0.977 atm, with no hint that two gases share the space.

Mole fractions: shares of the pressure

Counting moles for every component is tedious, so chemists use the mole fraction, the fraction of total moles belonging to one gas:

$$X_i = \frac{n_i}{n_{total}}, \qquad P_i = X_i P_{total}$$

Mole fractions always sum to 1, and they convert a total pressure into partial pressures in one multiplication. For air at sea level, $X_{O_2} \approx 0.21$, so oxygen's partial pressure is $0.21 \times 1.00 = 0.21$ atm.

A worked example: a heliox diving mixture is 80 percent helium and 20 percent oxygen by moles, breathed at a total pressure of 5.0 atm (about 40 meters depth). $P_{O_2} = 0.20 \times 5.0 = 1.0$ atm and $P_{He} = 0.80 \times 5.0 = 4.0$ atm. The diver's body responds to those partial pressures, not the percentages: 1.0 atm of oxygen is safe, while the 4.0 atm of helium is inert and harmless. Gas blending for diving is mole-fraction arithmetic with life-or-death stakes.

Scuba diving: depth multiplies every partial pressure

Scuba diving is applied partial-pressure physics. Descend 10 meters and the total pressure rises by about 1 atm; at 30 meters a diver breathes air at 4.0 atm total. Percentages do not change, but every partial pressure quadruples.

Two limits follow. Oxygen toxicity strikes when $P_{O_2}$ exceeds about 1.4 atm, so air divers stay shallow enough that $0.21 \times P_{total}$ stays below it, which caps the total pressure near 6.7 atm, roughly 57 meters. Nitrogen narcosis, the 'rapture of the deep,' sets in as $P_{N_2}$ climbs past about 3 atm, impairing judgment like alcohol.

A worked example: can a diver safely breathe air at 40 meters (5.0 atm total)? $P_{O_2} = 0.21 \times 5.0 = 1.05$ atm, under the 1.4 atm toxicity limit, so oxygen is acceptable. But $P_{N_2} = 0.78 \times 5.0 = 3.9$ atm, well into narcosis territory. The verdict is no: technical divers switch to helium blends that cut both fractions, trading cost for consciousness.

Collecting gases over water: the vapor correction

The classic school experiment collects hydrogen over water by bubbling it into an inverted bottle. The trapped gas is not pure hydrogen; it is hydrogen plus water vapor, because water evaporates into any gas space above it. Dalton's law corrects the measurement:

$$P_{H_2} = P_{total} - P_{water}$$

Water's vapor pressure depends only on temperature: 23.8 mmHg at $25^\circ\text{C}$, 17.5 mmHg at $20^\circ\text{C}$.

A worked example: hydrogen collected over water at $25^\circ\text{C}$ shows a total pressure of 758 mmHg. $P_{H_2} = 758 - 23.8 = 734.2$ mmHg. If the bottle holds 0.450 L, the moles of dry hydrogen are $n = PV/RT = ((734.2/760) \times 0.450)/(0.08206 \times 298) = 0.0178$ mol. Skipping the correction would overcount by about 3 percent, enough to fail a careful lab. Always ask whether collected gas is 'wet' before doing stoichiometry.

Limits: reacting gases and moving totals

Dalton's law assumes the gases ignore each other, which fails if they react: ammonia and hydrogen chloride in one container form solid ammonium chloride, and partial pressures become meaningless. It also assumes ideal behavior, so high-pressure mixtures need fugacity corrections from the van der Waals world.

A subtler trap is confusing partial pressure with concentration. Doubling the container volume at fixed moles halves every partial pressure, just like diluting a solution. Students sometimes treat the mole fractions as fixed and forget the total changed.

A worked example: a mixture at 2.00 atm total has $X_A = 0.25$, so $P_A = 0.500$ atm. The container volume doubles at constant temperature. Total pressure halves to 1.00 atm, and $P_A = 0.25 \times 1.00 = 0.250$ atm. The mole fraction never moved; the pressure did. Track which quantity the problem actually changed before reaching for $P_i = X_iP_{total}$.

Common mistakes

Real-world applications

• Blending breathing gases for deep scuba dives

• Collecting gases over water in the laboratory

• Designing controlled atmospheres for food storage

• Monitoring oxygen levels in medical ventilators