F. Gases and Atmospheric Chemistry · Gases and Atmospheric Chemistry

Boyle's Law: Pressure and Volume

F3.5 F3.4 F2.3About 65 minutes

Learning goals

I can state Boyle's law and write it as P₁V₁ = P₂V₂.
I can explain Boyle's law using the kinetic molecular theory.
I can recognize graphs of an inverse relationship.
I can solve Boyle's law problems with consistent units.

Words to know

Boyle's law
At constant temperature and amount of gas, volume is inversely proportional to pressure: P₁V₁ = P₂V₂.

inverse proportion
A relationship in which one quantity multiplied by the other is constant; when one doubles, the other halves.

compress
To squeeze into a smaller volume.

hyperbola
The curved graph shape of an inverse proportion such as P versus V.

Squeeze a syringe

Put your finger over the tip of a plastic syringe (no needle) and push the plunger. It gets harder and harder to push. Let go, and the plunger springs back. The trapped air has a pressure that rises as its volume falls.

In 1662 the Irish-born scientist Robert Boyle published careful measurements of this relationship using a J-shaped glass tube and mercury. His result is the first of the gas laws.

Remember from Grade 8

In the Grade 8 Fluids unit you saw that squeezing a gas into a smaller space raises its pressure, because the particles hit the walls more often. Now you will make that relationship quantitative: you'll calculate exactly how much the pressure changes.

Boyle's law

For a fixed amount of gas at constant temperature, the volume is inversely proportional to the pressure.

Double the pressure → the volume is halved.
Triple the pressure → the volume becomes one-third.

Mathematically, P × V is constant, so for a change from state 1 to state 2:

P₁V₁ = P₂V₂ (constant T and n)

The units of P must match on both sides (both kPa, or both atm), and the units of V must match (both L, or both mL). Temperature isn't in the equation, but it must stay constant.

Pressure (kPa)	Volume (L)	P × V (kPa·L)
50	6.0	300
100	3.0	300
150	2.0	300
200	1.5	300
250	1.2	300
300	1.0	300

Data for a gas sample at constant temperature. Every product P × V is the same: 300 kPa·L.

Left: plotting pressure against volume gives a curve (a hyperbola), the shape of an inverse relationship. Right: plotting pressure against 1/V gives a straight line through the origin, which proves P ∝ 1/V.

Why? The KMT explanation

At constant temperature, the particles keep the same average kinetic energy, so each collision hits the wall just as hard as before. But if the volume is halved, the same number of particles are packed into half the space. Each particle reaches a wall twice as often, so there are twice as many collisions per second on each square centimetre of wall. Pressure doubles.

Halving the volume of a gas at constant temperature doubles the number of wall collisions per second, so the pressure doubles.

Worked example

Example 1: A balloon goes up

A balloon holds 2.50 L of helium at 101 kPa. It rises to a height where the pressure is 85.0 kPa. Assuming the temperature does not change, what is its new volume?

List the variables: P₁ = 101 kPa, V₁ = 2.50 L, P₂ = 85.0 kPa, V₂ = ?
Predict first: the pressure drops, so the volume should increase.
Rearrange P₁V₁ = P₂V₂: V₂ = P₁V₁ ÷ P₂
V₂ = (101 kPa × 2.50 L) ÷ 85.0 kPa = 2.97 L
Check: 2.97 L > 2.50 L, matching the prediction.

V₂ = 2.97 L

Worked example

Example 2: Different units

A 450 mL sample of air at 760 mmHg is compressed to 150 mL at constant temperature. Find the new pressure in mmHg and in atm.

P₁ = 760 mmHg, V₁ = 450 mL, V₂ = 150 mL, P₂ = ?
The volumes are both in mL, so no conversion is needed (the units cancel).
P₂ = P₁V₁ ÷ V₂ = (760 mmHg × 450 mL) ÷ 150 mL = 2280 mmHg
2280 mmHg × (1 atm ÷ 760 mmHg) = 3.00 atm
Check: the volume became one-third, so the pressure tripled. ✓

P₂ = 2280 mmHg = 3.00 atm

Worked example

Example 3: Why scuba divers must never hold their breath

Every 10 m of seawater adds roughly 100 kPa of pressure. At a depth of 20 m, the pressure on a diver is about 300 kPa. A diver's lungs hold 4.0 L of air at that depth. If the diver held their breath and rose to the surface (100 kPa), what volume would the air try to expand to?

P₁ = 300 kPa, V₁ = 4.0 L, P₂ = 100 kPa
V₂ = P₁V₁ ÷ P₂ = (300 × 4.0) ÷ 100 = 12 L
Human lungs cannot hold 12 L. The expanding air could badly injure the lungs.

About 12 L, three times the original volume. That is why scuba training teaches divers to breathe out continuously as they ascend.

Hands-on activity

Build a Cartesian diver

You need: a clear 1 L or 2 L plastic pop bottle with its cap, water, and a "diver": a plastic ketchup or soy-sauce packet that floats, or a plastic pen cap (with no hole at the top) weighed down with a small lump of modelling clay so it just barely floats.

Fill the bottle to the very top with water.
Put in the diver. If you're using a pen cap, open end down, adjust the clay until the cap barely floats with a small air bubble trapped inside.
Screw the cap on tightly.
Squeeze the bottle hard. Then let go. Watch the air bubble inside the diver closely.
Try to make the diver hover in the middle.

What to look for: when you squeeze, the pressure in the water rises. The trapped air in the diver is compressed (Boyle's law), water moves into the space, the diver gets denser and sinks. Release, the air expands, and it floats back up. In the pen cap you can actually see the bubble shrink and grow.

Explain: write one sentence using Boyle's law and one using the KMT.

Just for fun

Robert Boyle's assistant on his air-pump experiments was Robert Hooke, who later became famous for Hooke's law of springs. So in one lab you had two Roberts, an air pump, and a lot of laws. History does not record whether anyone ever said, "Just call me Bob."

Your turn

State Boyle's law in words.

A gas occupies 500 mL at 100 kPa. What is its volume at 250 kPa (constant T)?

A 1.20 L sample of gas at 98.0 kPa is compressed to 0.800 L. Find the new pressure.

A 15 L tank of gas at 101.3 kPa is squeezed into 5.0 L. What is the new pressure?

What shape is the graph of P versus V for Boyle's law? What graph would give a straight line?

Use the KMT to explain why pressure rises when a gas is compressed at constant temperature.

More practice: warm-up

If the pressure on a gas doubles (constant T), what happens to its volume?

If the volume of a gas is tripled (constant T), what happens to its pressure?

Write the equation for Boyle's law.

Which two quantities must stay constant for Boyle's law?

P₁ = 200 kPa, V₁ = 3.0 L, P₂ = 100 kPa. Find V₂.

More practice: on your own

A weather balloon has a volume of 10 L at 101.3 kPa. At high altitude its volume is 50 L. Assuming constant temperature, what is the pressure there?

A 2.4 L sample of gas at 250 kPa is compressed to 1.6 L at constant temperature. Predict whether the pressure rises or falls, then calculate it.

A syringe contains 25.0 mL of air at 100.0 kPa. The plunger is pushed until the pressure is 333 kPa. What is the new volume?

Why does a bag of chips sealed at sea level puff up when it is taken up a mountain?

A student plots P versus V for her data and gets a curve. Explain how she could use her data to show that P and V are inversely proportional.

Challenge

A diver at 30 m depth (about 400 kPa) releases a 0.50 L air bubble. What will its volume be just below the surface (about 100 kPa), assuming constant temperature? Give one reason the real bubble might be a bit different.

A gas sample's volume is reduced by 25 % at constant temperature. By what percentage does its pressure change?

A 3.0 L bottle of air at 99 kPa is connected by a valve to an empty (evacuated) 1.0 L bottle. The valve is opened. What is the final pressure throughout both bottles at the same temperature?

Check your understanding

Quiz

Boyle's law states that, at constant temperature, the volume of a gas is:

A gas occupies 6.0 L at 100 kPa. At constant temperature, what is its volume at 300 kPa?

Which graph is a straight line through the origin for a gas obeying Boyle's law?

A gas is compressed from 4.0 L to 1.0 L at constant temperature. Its pressure:

According to the KMT, compressing a gas at constant temperature raises its pressure because:

Why must a scuba diver breathe out while rising to the surface?

For parents and teachers

Big idea (F3.5, F3.4, F2.3): Boyle's law, P₁V₁ = P₂V₂ at constant temperature and amount. Students should be able to state it, explain it with the kinetic molecular theory, recognize its graphs and solve problems.

How to help: have your teen predict whether the answer should be bigger or smaller before calculating. If the prediction and the answer disagree, the equation was rearranged wrongly.

Common difficulties:

Mixing units (kPa on one side and atm on the other). Units of P must match each other; units of V must match each other.
Rearranging incorrectly. V₂ = P₁V₁ ÷ P₂, not P₂V₁ ÷ P₁.
Forgetting the conditions: Boyle's law needs constant temperature and amount of gas.

Activity note: the Cartesian diver uses only water and a bottle. Make sure the pen cap has no hole in its top, or the air bubble will escape.