Pressure in gases: Boyle’s law

1. Definitions (Must Know)

A. Pressure, p

• Pressure, p: force per unit area normal to a surface, p = F/A (unit: Pa).

B. Gas pressure (particle model)

Gas pressure is due to molecules colliding with the container walls and changing momentum.

At fixed temperature, gas pressure increases if:

• there are more molecules per unit volume (higher number density), or
• molecules move faster (higher average speed).

C. Boyle’s law

Boyle’s law states that the pressure of a fixed amount of gas is inversely proportional to the volume of the gas when the temperature is held constant.

pV = constant

p₁V₁ = p₂V₂

where p₁,p₂ are pressures and V₁,V₂ are volumes.

The pressures in a gas law must be absolute pressures, measured from a vacuum. If a question gives gauge pressure, first use: pₐbₛₒₗᵤₜₑ = pgₐᵤgₑ + pₐₜₘₒₛₚₕₑᵣᵢc

Scroll diagram horizontally to read all labels.

The diagram shows compression of gas in a cylinder while its temperature is kept constant. As the volume decreases, the absolute pressure increases so that pV remains constant.

If you plot p against 1/V, you get a straight line through the origin (gradient k).

Scroll diagram horizontally to read all labels.

An intuitive explanation comes from the particle model: pressure is due to molecules colliding with the container walls and changing momentum.

At constant temperature, the mean molecular kinetic energy is unchanged. If the volume is halved, the number density doubles, so the rate of momentum transfer to each unit area of wall doubles. Hence V → 1/2V Rightarrow p → 2p.

2. Key Ideas (What Earns Marks)

• State the conditions: fixed mass and constant temperature.
• Use Boyle’s law:
  • p₁V₁ = p₂V₂
• Fast factor check:
  • V decreases by a factor f Rightarrow p increases by a factor f.
• Graphs:
  • p vs V: inverse curve,
  • p vs 1/V: straight line through the origin.
• Use absolute pressure, not gauge pressure.

3. Detailed Explanations

A. What “fixed mass” and “constant temperature” mean

• Fixed mass: no gas enters or leaves, so the amount of gas (n or N) is constant.
• Constant temperature (isothermal): the gas stays at the same thermodynamic temperature T (in K).

B. Deriving Boyle’s law from the ideal gas equation

For an ideal gas: pV = nRT

If the mass is fixed (n constant) and temperature is constant (T constant), then nRT is constant, so: pV = constantRightarrow ppropto 1/V

C. Particle explanation (qualitative)

At constant temperature, the average kinetic energy of the molecules stays constant, so typical molecular speeds are similar.

When the gas is compressed into a smaller volume:

• number density increases (more molecules per m³),
• collision rate with the walls increases,
• so the total rate of momentum transfer to the walls increases,
• therefore the pressure increases.

D. Units you must be careful with

• SI units are Pa and m³.
• In p₁V₁ = p₂V₂, pressure units may cancel and volume units may cancel, but each type of unit must be consistent on both sides.
• Useful conversions:
  • 1 L = 10⁻³ m³
  • 1 cm³ = 10⁻⁶ m³

4. Common Mistakes

• Using Boyle’s law even though temperature changes (e.g. rapid compression without time to cool).
• Treating “fixed mass” as optional (Boyle’s law fails if gas leaks).
• Mixing pressure or volume units between the two states.
• Substituting gauge pressure directly into Boyle’s law instead of converting to absolute pressure.

5. Exam Tips

• Write “fixed mass, constant temperature” before you use p₁V₁ = p₂V₂.
• In factor questions, use ratios to save time:
  • p₂ = p₁V₁/V₂
• If asked to verify Boyle’s law from data, show that pV is (approximately) constant.
• Label graph axes precisely: p against V is a curve, while p against 1/V is linear.

6. Worked Examples

Example 1: Compressing air with a piston (factor change)Core

Air at a pressure of 1.0 × 10⁵ Pa is confined in a cylinder that is fitted with a movable piston. The air is then compressed by pushing the piston, so that the same mass of air now occupies one-fifth of the original volume without any change in temperature. Calculate the new pressure of the air.

Show Answer

Use Boyle’s law (fixed mass, constant temperature): p₁V₁ = p₂V₂

Here V₂ = 1/5V₁, so: p₂ = p₁V₁/V₂ = p₁V₁/1/5V₁ = 5p₁

p₂ = 5(1.0 × 10⁵) = 5.0 × 10⁵ Pa

Example 2: Find the new volume after a pressure changeCore

A fixed mass of gas occupies a volume of 40 cm³ at a pressure of 1.0 × 10⁵ Pa. Determine its volume when the pressure is 2.0 × 10⁵ Pa, assuming constant temperature.

Show Answer

Use Boyle’s law: p₁V₁ = p₂V₂ Rightarrow V₂ = p₁V₁/p₂

V₂ = (1.0 × 10⁵)(40)/2.0 × 10⁵ = 20 cm³

Example 3: Check whether data obeys Boyle’s lawCore

A fixed mass of gas at constant temperature has the following readings:

• V = 30 cm³, p = 1.6 × 10⁵ Pa
• V = 40 cm³, p = 1.2 × 10⁵ Pa

Show that Boyle’s law is consistent with the data.

Show Answer

Compute pV for each pair:

• (1.6 × 10⁵)(30) = 4.8 × 10⁶ Pa cm³
• (1.2 × 10⁵)(40) = 4.8 × 10⁶ Pa cm³

pV is the same (within rounding), so the data is consistent with pV = constant.

Example 4: Convert gauge pressure before using Boyle’s lawCore

Air occupies 60 cm³ at atmospheric pressure 100 kPa. It is compressed isothermally to 40 cm³. Find the final absolute pressure and final gauge pressure, taking atmospheric pressure as 100 kPa.

Show Answer

The initial absolute pressure is p₁ = 100 kPa.

p₁V₁ = p₂V₂ p₂ = (100)(60)/40 = 150 kPa

The final absolute pressure is 150 kPa. Therefore: pgₐᵤgₑ = pₐbₛₒₗᵤₜₑ-pₐₜₘₒₛₚₕₑᵣᵢc = 150-100 = 50 kPa

7. Mind Stretchers

Mind stretcher 1: Same temperature, different volumeExtension

Containers A and B contain the same type of gas, with the same number of molecules, at the same temperature. Container B has a greater volume than container A. Compare the gas pressures.

Show Answer

Using the ideal gas equation pV = NkT:

For both containers, N and T are the same, so pV is the same.

Therefore ppropto 1/V. Since VB>VA, we have pB<pA.

Mind stretcher 2: Why must Boyle’s law be isothermal?Extension

Explain why Boyle’s law requires constant temperature.

Show Answer

From the ideal gas equation pV = nRT, if the mass is fixed then pV is proportional to T.

So for pV to stay constant when V changes, T must stay constant. In a rapid compression, temperature can rise, so pressure increases more than Boyle’s law predicts.

8. Practice, Quiz and Next Step

Close your notes and use Pressure in gases: Boyle’s law in the supplied context below. This requires a constructed explanation or working, not recognition of an option.

Fresh context: An unfamiliar data set or physical system requires you to apply Pressure in gases: Boyle’s law while stating the model, regime and assumptions.

1. Retrieve: define pressure in gases: boyle’s law in your own words, including units, sign or conditions where relevant.
2. Represent: Choose and label an appropriate diagram, graph, table or symbolic model; derive or justify the relationship used.
3. Apply: Reach a conclusion, then evaluate it using units, uncertainty, a limiting case and one practical or modelling limitation.

Check the response before looking back

• The model, regime, coordinates and assumptions are explicit.
• The derivation or multi-step reasoning is visible rather than implied.
• The conclusion is tested against units, data quality and a limiting case.
• A practical control, uncertainty or model limitation is evaluated where applicable.

If one check fails, name that exact gap, revisit the matching explanation or worked example, and redo the task with different values or a different situation. Then use theA-Level Physics course hub orpractice browser for an independent re-test.