Dalton’s Law of Partial Pressures and Mole Fraction
Dalton’s law of partial pressures states that the total pressure of a mixture of non-reacting gases equals the sum of the pressures that each gas would exert alone, and each partial pressure is the mole fraction of the gas multiplied by the total pressure. In brief: Ptotal = P1 + P2 + P3 + …, and Pi = χi × Ptotal; oxygen makes up 20.95 % of dry air, so at 760 mmHg its partial pressure is about 159 mmHg.
Dalton’s law is important in respiration, diving, meteorology and every laboratory that collects gases over water. It works because the particles of an ideal gas do not interact, so each gas behaves as if it alone filled the container. This article states the law, explains mole fraction, gives a table for air, and solves examples including the collection of a gas over water.
What does Dalton’s law say?
The partial pressure of a gas in a mixture is the pressure it would exert if it occupied the whole container by itself at the same temperature. For a mixture of non-reacting gases:
Ptotal = PA + PB + PC + ...
For each component, Pi = niRT / V, so the partial pressure is proportional to the number of moles of that gas. Because the molecules of an ideal gas do not attract each other, adding another gas does not change the pressure of the first.
What is mole fraction?
The mole fraction χi (Greek chi) is the number of moles of a component divided by the total number of moles:
χi = ni / ntotal
Mole fractions of all components add up to 1. Dividing Pi = niRT/V by Ptotal = ntotalRT/V gives Pi = χi × Ptotal. For gases, the mole fraction equals the volume fraction, so a gas that makes up 21 % of the volume of a mixture has χ = 0.21.
Partial pressures in air
| Gas | Mole fraction | Partial pressure (mmHg) |
|---|---|---|
| Nitrogen, N2 | 0.7808 | 593.4 |
| Oxygen, O2 | 0.2095 | 159.2 |
| Argon, Ar | 0.0093 | 7.1 |
| Carbon dioxide, CO2 | 0.0004 | 0.3 |
The values are rounded; the carbon dioxide content is currently a little over 0.04 % and rising. At the top of Mount Everest, where the total pressure is about 250 mmHg, the oxygen fraction is still 0.2095, but its partial pressure is only about 52 mmHg, which explains why climbers struggle for breath.
How do you calculate partial pressures?
Worked example 1: three gases in a tank
A 20.0 L tank at 300 K contains 2.00 mol of helium, 3.00 mol of neon and 5.00 mol of argon. Find the total pressure and each partial pressure.
- ntotal = 2.00 + 3.00 + 5.00 = 10.00 mol.
- Ptotal = nRT / V = 10.00 × 0.08206 × 300 / 20.0 = 12.3 atm.
- χ(He) = 0.200, χ(Ne) = 0.300 and χ(Ar) = 0.500.
- P(He) = 0.200 × 12.3 = 2.46 atm, P(Ne) = 3.69 atm and P(Ar) = 6.16 atm.
The three partial pressures add up to 12.31 atm, which confirms the result.
Worked example 2: from a given total pressure
A diver breathes a mixture with 10 % oxygen by volume at a depth where the total pressure is 6.0 atm. The partial pressure of oxygen is 0.10 × 6.0 = 0.60 atm, which is a safe value, since oxygen partial pressures above about 1.4 atm risk oxygen toxicity.
How do you correct for water vapor?
When a gas is collected by bubbling it through water, the gas is saturated with water vapor. The pressure measured in the collection vessel is the sum of the gas pressure and the vapor pressure of water at that temperature.
Worked example 3: hydrogen over water
Hydrogen produced by a reaction is collected over water at 25 °C. The total pressure is 745 mmHg. What is the pressure of the dry hydrogen?
P(H2) = Ptotal − P(H2O) = 745 − 23.8 = 721.2 mmHg, which is 0.949 atm.
Vapor pressure of water
| Temperature (°C) | Vapor pressure (mmHg) |
|---|---|
| 0 | 4.6 |
| 10 | 9.2 |
| 20 | 17.5 |
| 25 | 23.8 |
| 30 | 31.8 |
| 40 | 55.3 |
| 50 | 92.5 |
| 100 | 760 |
Who was John Dalton?
John Dalton (1766–1844), an English chemist and teacher, published the law of partial pressures in 1801 while studying the composition of the atmosphere. He wanted to know why the gases in air do not separate by weight. His answer was that each gas acts independently of the others. Dalton is better known for his atomic theory, which he introduced a few years later, and for the first scientific description of color blindness, which he himself had.
Where is the law used?
- Physiology: gas exchange in the lungs is driven by differences in partial pressure.
- Diving: mixtures such as nitrox and trimix are chosen to keep the partial pressure of each gas within safe limits.
- Laboratory work: gases collected over water must be corrected for vapor pressure.
- Industry: gas separation and chemical reactor design involve partial pressures in equilibrium expressions.
Which mistakes are common?
- Forgetting water vapor: a gas collected over water is not dry.
- Adding mole fractions to pressures: mole fractions are dimensionless and add up to 1.
- Using mass fractions: the law needs moles (or volume fractions), not masses.
- Applying the law to reacting gases: if the gases react, the moles change.