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: p_absolute = p_gauge + p_atmospheric

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).

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 → (1/2)V ⇒ 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 ⇒ 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 = constant ⇒ p ∝ 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

7. Mind Stretchers