Dalton's Law of Partial Pressures

Why is it hard to breathe on a mountaintop?

As you saw in 1.7.1, about 21% of air is oxygen, at sea level and on top of Mount Everest alike. Yet most climbers above 8000 m carry bottled oxygen. If the fraction of oxygen doesn’t change, what does? To answer that, you need to know how much of the pressure of a mixture comes from each gas.

Dalton’s law of partial pressures

The partial pressure分压 of a gas in a mixture is the pressure it would exert if it filled the container alone at the same temperature. In 1801, John Dalton found that

P_\text{total} = P_\text{A} + P_\text{B} + P_\text{C} + \cdots

The total pressure of a mixture is the sum of the partial pressures. This is Dalton’s law of partial pressures道尔顿分压定律. Particles of an ideal gas don’t interact, so each gas hits the walls as if the others weren’t there.

From PV = nRT, each partial pressure depends only on that gas’s moles: P_A = n_ART/V. The fraction of the total moles that belongs to one gas is its mole fraction摩尔分数, X_A = n_A/n_total. So

P_\text{A} = X_\text{A} \cdot P_\text{total}

For gases, the volume fraction equals the mole fraction (Avogadro’s law). Air is about 21% oxygen by volume, so its mole fraction is 0.21, and in air at 1.00 atm the partial pressure of oxygen is about 0.21 atm.

Thin air at altitude

The higher you go, the lower the total air pressure, but the composition of air stays almost the same. The partial pressure of oxygen falls in proportion: on Everest’s summit the total pressure is about a third of sea-level pressure, and so is the partial pressure of oxygen. Your lungs take in oxygen according to its partial pressure, so there is far less to breathe.

Collecting gas over water

Gases such as oxygen and hydrogen are often collected by displacing water. The collected gas is mixed with water vapor, at a pressure equal to the vapor pressure of water水的饱和蒸气压 at that temperature.

What is vapor pressure? At the surface of a liquid, molecules keep breaking free and escaping as vapor, and vapor molecules keep hitting the surface and rejoining the liquid. In a closed space, escaping wins at first, so the vapor builds up. Once the two rates are equal, the amount of vapor stops changing, and the pressure of the vapor above the liquid is then the vapor pressure. For a given liquid it depends only on temperature: the hotter the liquid, the more molecules break free, and the higher the vapor pressure. A liquid boils when its vapor pressure equals the pressure around it. Water’s vapor pressure reaches 760 torr (101.3 kPa) at 100 °C, so at sea level water boils at 100 °C; high in the mountains, where the air pressure is lower, it boils below 100 °C.

When the water levels inside and outside the collecting tube are equal, the total pressure inside equals the air pressure:

P_\text{gas} = P_\text{atm} - P_{\text{H}_2\text{O}}

The vapor pressure of water depends only on temperature, so you look it up: 17.5 torr at 20 °C, 22.4 torr at 24 °C, and 23.8 torr at 25 °C. US lab reports almost always make this correction; Chinese textbooks usually skip it.

This lesson uses the atomic masses on the AP periodic table (H 1.008, He 4.00, N 14.01, O 16.00) and R = 62.36 L·torr/(mol·K).

There are three modes: a gas mixture, air at different altitudes, and collecting over water. Start with the mixture and check that the partial pressures add up to the total:

Try it: Dalton's law of partial pressures

A 10.0-L container at 25 °C. Change the amount of each gas and watch its partial pressure and the total pressure.

●He●N₂●O₂

Gas	n (mol)	Mole fraction X	Partial pressure (atm)

Example 1: a diving gas mixture

A diving tank holds 4.00 mol of He and 1.00 mol of O₂ at a total pressure of 2.00 × 10² atm. What is the partial pressure of each gas?

Step-by-step solution

The rule. Dalton’s law: each gas’s partial pressure is its mole fraction times the total pressure, P_A = X_AP_total.
Mole fractions.
$\begin{aligned} n_\text{total} &= 4.00 + 1.00 \\ &= 5.00\ \text{mol} \\ X(\ce{He}) &= 4.00 / 5.00 \\ &= 0.800 \\ X(\ce{O2}) &= 1.00 / 5.00 \\ &= 0.200 \end{aligned}$
Partial pressures. All in atm:
$\begin{aligned} P(\ce{He}) &= 0.800 \times 2.00 \times 10^{2} \\ &= \boxed{1.60 \times 10^{2}\ \text{atm}} \\ P(\ce{O2}) &= 0.200 \times 2.00 \times 10^{2} \\ &= \boxed{40.0\ \text{atm}} \end{aligned}$
Check. 160 atm + 40.0 atm = 200 atm, the total.
Think about it. The 4 : 1 ratio of partial pressures is the mole ratio. Which gas it is doesn’t matter; partial pressure depends only on the number of particles.

Example 2: hydrogen collected over water

Zinc reacts with hydrochloric acid, and 312 mL of hydrogen is collected over water at 24 °C and 748 torr, with the water levels equal. The vapor pressure of water at 24 °C is 22.4 torr. How many moles of hydrogen were collected?

Step-by-step solution

Why subtract water vapor? Gas collected over water is a mixture of H₂ and water vapor. Equal water levels mean the total pressure inside equals the air pressure, 748 torr, and 22.4 torr of that comes from water vapor.
Partial pressure of H₂. The subtraction keeps no decimal places:
$\begin{aligned} P(\ce{H2}) &= 748 - 22.4 \\ &= 726\ \text{torr} \end{aligned}$
Units. V = 0.312 L and T = 24 + 273 = 297 K. With P in torr, R = 62.36 L·torr/(mol·K).
Moles. Use the partial pressure of H₂, not the total:
$\begin{aligned} n &= \frac{P(\ce{H2})\, V}{RT} \\ &= \frac{726 \times 0.312}{62.36 \times 297} \\ &= \boxed{1.22 \times 10^{-2}\ \text{mol}} \end{aligned}$
Without the correction. Using 748 torr gives 1.26 × 10⁻² mol, about 3% too high.
A common mistake. Substituting 312 for V without converting mL to L, which makes n 1000 times too large.

Practice: find the partial or total pressure.

Total 1.00 atm, X(O₂) = 0.20: P(O₂)
Total 2.00 atm, X = 0.25: the gas's partial pressure
1 mol He and 3 mol Ne, total 1.00 atm: P(He)
Same mixture: P(Ne)
N₂ 0.60 atm, O₂ 0.15 atm, Ar 0.25 atm: total pressure
Equal moles of two gases, total 1.00 atm: one gas's partial pressure
Total 0.80 atm, X(CO₂) = 0.50: P(CO₂)
Only gases A and B, total 1.25 atm, P(A) = 1.00 atm: P(B)

Common mistakes

Mistake 1: “Heavier gases have bigger partial pressures.” At the same T and V, partial pressure depends only on moles, not on which gas it is. 0.1 mol of H₂ and 0.1 mol of CO₂ in the same container have the same partial pressure.

Mistake 2: “Gas collected over water is pure, so its pressure equals the air pressure.” The collected gas contains water vapor. The total pressure equals the air pressure, so subtract the vapor pressure of water to get the gas’s own partial pressure.

Mistake 3: “The partial pressure of oxygen is 0.21 atm everywhere.” 21% is a mole fraction; partial pressure = mole fraction × total pressure. At high altitude the total pressure is lower, and so is the partial pressure of oxygen.

Quick check

A container holds 0.20 mol H₂ and 0.30 mol N₂ at a total pressure of 1.00 atm. What is the partial pressure of H₂?

O₂ is collected over water at 25 °C. The air pressure is 760. torr and the vapor pressure of water is 23.8 torr. What is the partial pressure of O₂?

Summary

Dalton’s law: the total pressure of a gas mixture is the sum of the partial pressures, P_total = P_A + P_B + ⋯.
Partial pressure = mole fraction × total pressure, P_A = X_AP_total. For gases, volume fraction equals mole fraction; at altitude, lower total pressure means lower oxygen partial pressure.
For gas collected over water, P_gas = P_atm − P_H₂O, with the vapor pressure of water looked up by temperature.

On to AP (for later)

The AP equation sheet gives P_A = P_total × X_A and P_total = P_A + P_B + P_C + ⋯. AP questions often show a particle diagram with several colors of spheres and ask you to count mole fractions and find partial pressures. Collecting gas over water is a lab favorite, and subtracting the vapor pressure of water is the step that gets checked.

Data sources: vapor pressures of water from the IAPWS-95 formulation, as tabulated in the CRC Handbook.

Dalton's Law of Partial Pressures道尔顿分压定律

Learning goals

Before you start

Why is it hard to breathe on a mountaintop?