F. Gases and Atmospheric Chemistry · Gases and Atmospheric ChemistryDalton's Law of Partial Pressures
F3.5 F2.3 F3.4About 65 minutes
Learning goals- I can state Dalton's law of partial pressures and explain it with the KMT.
- I can calculate partial pressures from mole fractions or percentages.
- I can correct for water vapour when a gas is collected over water.
- I can explain why breathing is affected at high altitude and at depth.
Words to know
partial pressure
The pressure one gas in a mixture would exert if it were alone in the container at the same temperature.
Dalton's law of partial pressures
P(total) = P₁ + P₂ + P₃ + ... for a mixture of non-reacting gases.
mole fraction (x)
Moles of one component ÷ total moles in the mixture.
water displacement
Collecting a gas by letting it push water out of an inverted, water-filled container.
vapour pressure
The pressure of a vapour in equilibrium with its liquid; for water it depends only on temperature.
What does a mountaineer breathe?
Near the summit of Mount Logan, air is still about 21 % oxygen, just like the air in your kitchen. Yet climbers gasp for breath and many use bottled oxygen. If the percentage of oxygen is the same, what has changed? The answer involves the pressure of each gas separately in a mixture. John Dalton, the same scientist behind the early atomic theory, worked this out in 1801.
Dalton's law of partial pressures
In a mixture of gases that don't react, each gas exerts the pressure it would have if it were alone in the container at the same temperature. This is its partial pressure. The total pressure is the sum:
P(total) = P₁ + P₂ + P₃ + ...
KMT explanation: ideal gas particles don't attract or repel each other (postulate 4), so each gas's particles hit the walls just as they would on their own. The collisions simply add up.
Partial pressure and mole fraction
Because pressure is proportional to the number of particles (at constant V and T), each gas's share of the pressure equals its share of the moles. The mole fraction, x, of a gas is its moles divided by the total moles:
x(A) = n(A) ÷ n(total) and P(A) = x(A) × P(total)
For gases, percent by volume equals percent by moles (Avogadro). Dry air is about 78.08 % N₂, 20.95 % O₂ and 0.93 % Ar (plus small amounts of CO₂ and other gases). At sea level, P(O₂) ≈ 0.2095 × 101.3 kPa ≈ 21.2 kPa.
At about 6000 m, near the top of Mount Logan, the total pressure is only about 47 kPa. The air is still 20.95 % oxygen, but P(O₂) ≈ 9.9 kPa, less than half the sea-level value. What our lungs care about is the partial pressure of oxygen, because that is what pushes O₂ into the blood.
Worked exampleExample 1: Partial pressures from moles
A tank contains 0.40 mol of helium and 0.60 mol of argon at a total pressure of 250 kPa. Find the partial pressure of each gas.
- n(total) = 0.40 + 0.60 = 1.00 mol
- x(He) = 0.40 ÷ 1.00 = 0.40; x(Ar) = 0.60
- P(He) = 0.40 × 250 kPa = 100 kPa; P(Ar) = 0.60 × 250 kPa = 150 kPa
- Check: 100 + 150 = 250 kPa ✓
P(He) = 100 kPa, P(Ar) = 150 kPa
Collecting a gas over water
In the lab, gases such as hydrogen and oxygen are often collected by water displacement: the gas bubbles into an upside-down container full of water and pushes the water out. But water evaporates, so the collected gas is always mixed with water vapour. To find the pressure of the dry gas:
P(gas) = P(total) − P(H₂O)
The pressure of water vapour depends only on temperature. If the water level inside the container is the same as outside, P(total) equals the atmospheric pressure.
| Temperature (°C) | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 30 |
|---|
| P(H₂O) (kPa) | 2.34 | 2.49 | 2.64 | 2.81 | 2.99 | 3.17 | 3.36 | 4.25 |
Worked exampleExample 2: Hydrogen collected over water
A student collects 245 mL of hydrogen over water at 22 °C. The atmospheric pressure is 100.4 kPa and the water levels are equal. How many moles of H₂ were collected?
- P(H₂O) at 22 °C = 2.64 kPa (from the table)
- P(H₂) = 100.4 − 2.64 = 97.8 kPa
- V = 0.245 L, T = 295 K
- n = PV ÷ RT = (97.8)(0.245) ÷ [(8.314)(295)] = 0.00977 mol
9.77 × 10⁻³ mol of H₂. (Forgetting to subtract the water vapour would give an answer about 3 % too high.)
Hands-on activityHow much of the air is oxygen?
You need: a grown-up, a pad of plain steel wool (no soap), vinegar, a tall clear glass or narrow jar, a shallow dish of water, a ruler, a marker, dish gloves.
Safety: wear gloves when handling steel wool (it splinters). Do not taste anything; throw the steel wool in the garbage afterwards.
- Soak a small piece of steel wool in vinegar for one minute (this removes a coating so it rusts quickly), then squeeze it out.
- Push it to the bottom of the glass so it stays there when the glass is turned over.
- Turn the glass upside down and stand it in the dish of water. Mark the water level inside with the marker.
- Leave it for a day. Mark the new water level.
- Measure the height of the air space at the start and the height the water rose. Calculate the fraction.
What to look for: the steel wool rusts, using up the oxygen in the trapped air. With the oxygen gone, its partial pressure is gone, so the gas inside pushes less and the outside air pushes water up into the glass until the pressures balance. Water rises to fill roughly one-fifth of the air space (about 21 %). Results are often a little lower because the reaction may not use up all of the oxygen.
Just for funJohn Dalton was colour-blind, and he wrote one of the first scientific descriptions of colour blindness in 1794. He thought it was caused by blue-tinted fluid in his eyes. When his eyes were examined after his death, they weren't tinted at all. Even great scientists' hypotheses get tested!
Your turn
1State Dalton's law of partial pressures.
The total pressure of a mixture of non-reacting gases equals the sum of the partial pressures of the individual gases.
2A mixture contains O₂ at 30.0 kPa, N₂ at 65.0 kPa and CO₂ at 5.0 kPa. What is the total pressure?
30.0 + 65.0 + 5.0 = 100.0 kPa
3A tank holds 2.0 mol He, 3.0 mol Ne and 5.0 mol Ar at a total pressure of 300 kPa. Find each partial pressure.
Total 10.0 mol. He: 0.20 × 300 = 60 kPa; Ne: 0.30 × 300 = 90 kPa; Ar: 0.50 × 300 = 150 kPa.
4Oxygen is collected over water at 25 °C. The total pressure is 101.3 kPa. What is the pressure of the dry oxygen?
5Air is about 21 % oxygen. What is the partial pressure of oxygen when the total pressure is 101.3 kPa?
6Use the KMT to explain why partial pressures simply add up.
Ideal gas particles don't attract or repel each other, so each gas's particles hit the walls as if the other gases weren't there. The total number of collisions is the sum of each gas's collisions.
More practice: warm-up
1Write Dalton's law as an equation.
P(total) = P₁ + P₂ + P₃ + ...
2Gas A exerts 40 kPa and gas B exerts 55 kPa in the same container. Find the total pressure.
The moles of one component divided by the total moles of the mixture.
4What is the vapour pressure of water at 20 °C (from the table)?
5Why must you subtract water's vapour pressure when a gas is collected over water?
The collected gas is mixed with water vapour, which contributes part of the total pressure.
More practice: on your own
150.0 mL of oxygen is collected over water at 23 °C when the atmospheric pressure is 99.8 kPa. How many moles of O₂ were collected?
P(O₂) = 99.8 − 2.81 = 97.0 kPa. n = (97.0)(0.0500) ÷ [(8.314)(296)] = 1.97 × 10⁻³ mol
2120 mL of hydrogen is collected over water at 20 °C when the atmospheric pressure is 101.0 kPa. How many moles of H₂ is this?
P(H₂) = 101.0 − 2.34 = 98.66 kPa. n = (98.66)(0.120) ÷ [(8.314)(293)] = 4.86 × 10⁻³ mol
3Air is 0.93 % argon. What is the partial pressure of argon at 101.3 kPa?
0.0093 × 101.3 = 0.94 kPa
4A scuba diver at about 40 m depth breathes air at a total pressure of about 500 kPa. What is the partial pressure of oxygen in that air (21 % O₂)? Compare it with sea level.
0.21 × 500 = 105 kPa, about five times the sea-level value of about 21 kPa. Very high oxygen partial pressures can be harmful, which is one reason deep dives use special gas mixtures.
5Explain why climbers at 6000 m feel short of breath even though air there is still about 21 % oxygen.
The total pressure is less than half that at sea level, so the partial pressure of oxygen is less than half as well (about 10 kPa instead of 21 kPa). Lower P(O₂) pushes less oxygen into the blood in the lungs.
Challenge
1Suppose a dive plan requires the partial pressure of oxygen to stay at or below 140 kPa. What is the highest total pressure at which a diver could breathe ordinary air (21 % O₂)?
P(total) = 140 ÷ 0.21 = 667 kPa (about 6.6 atm).
2350 mL of hydrogen is collected over water at 21 °C and 100.2 kPa. What mass of hydrogen was collected?
P(H₂) = 100.2 − 2.49 = 97.71 kPa. n = (97.71)(0.350) ÷ [(8.314)(294)] = 0.01399 mol. m = 0.01399 × 2.016 = 0.0282 g
3A container holds equal masses of helium and methane (CH₄) at a total pressure of 100 kPa. Find the partial pressure of each gas.
Take 1 g of each: n(He) = 1 ÷ 4.00 = 0.250 mol; n(CH₄) = 1 ÷ 16.05 = 0.0623 mol. x(He) = 0.250 ÷ 0.312 = 0.800. P(He) = 80.0 kPa, P(CH₄) = 20.0 kPa. Equal masses do not mean equal partial pressures: the lighter gas has more moles.
Quiz
1. A mixture has P(N₂) = 70 kPa, P(O₂) = 25 kPa and P(Ar) = 5 kPa. The total pressure is:
2. A gas is 30 % of the moles in a mixture at 200 kPa. Its partial pressure is:
3. Hydrogen is collected over water at 25 °C with P(total) = 101.0 kPa. P(H₂) is:
4. Why is it harder to breathe at high altitude?
5. Dalton's law works because, in an ideal gas mixture:
6. The vapour pressure of water depends on:
Big idea (F3.5, F2.3): Dalton's law, P(total) = P₁ + P₂ + ..., with partial pressure = mole fraction × total pressure. Used for air, breathing at altitude and depth, and collecting gases over water.
How to help: the rusting steel wool activity shows that about one-fifth of air is oxygen. Ask your teen to explain why the water rises using the words "partial pressure".
Common difficulties:
- Forgetting to subtract water vapour pressure when a gas is collected over water.
- Confusing percentage and partial pressure at altitude: the percentage of O₂ stays the same; its partial pressure drops.
- Assuming equal masses mean equal partial pressures. Partial pressure depends on moles.
Safety: wear gloves with steel wool.