Physics · Lesson 13

Thermal expansion & gas laws

JAMBWAECNECO

What the syllabus asks

JAMB: Thermal expansion: solids, liquids, linear, area and cubic expansivities and their relationship, anomalous expansion of water; gas laws: Boyle’s, Charles’, pressure law, general gas equation, ideal gas, kinetic theory explanation

WAEC: Thermal expansion of solids (linear, area, volume expansivity), applications and problems of expansion, bimetallic strip, expansion of liquids (real and apparent), anomalous expansion of water; gas laws (Boyle’s, Charles’, pressure law, general gas law), absolute zero and the Kelvin scale, kinetic explanation

NECO: Thermal expansion of solids (linear, area, volume expansivity), applications and problems of expansion, bimetallic strip, expansion of liquids (real and apparent), anomalous expansion of water; gas laws (Boyle’s, Charles’, pressure law, general gas law), absolute zero and the Kelvin scale, kinetic explanation

What you will learn

Explain thermal expansion using the particle model.
Define and use linear, area and cubic expansivity, and relate β = 2α and γ = 3α.
Describe applications and problems of expansion and how they are handled.
Distinguish real and apparent expansion of a liquid and describe the anomalous expansion of water.
State Boyle’s law, Charles’ law and the pressure law, and solve problems with them.
Use the general gas equation with temperatures in kelvin.
Explain the gas laws using the kinetic theory of gases.

Key points

Heating makes particles vibrate or move more and spread out; density falls.
α = ΔL/(L₀Δθ) in K⁻¹; ΔL = αL₀Δθ.
β = 2α, γ = 3α.
Allow for expansion in rails, bridges, cables; use it in bimetallic strips, shrink fits and rivets.
γ_real = γ_apparent + γ_container.
Water is densest at 4 °C; ice floats; ponds freeze from the top.
T(K) = θ(°C) + 273; absolute zero = 0 K = −273 °C.
Boyle: P₁V₁ = P₂V₂ at constant T.
Charles: V₁/T₁ = V₂/T₂ at constant P; pressure law: P₁/T₁ = P₂/T₂ at constant V.
General: P₁V₁/T₁ = P₂V₂/T₂; gas pressure comes from particle collisions with the walls.

Start here: Why things expand when heated

On a hot afternoon in Kano, the steel gate of a compound can be hard to close, yet at dawn it swings shut easily. Electric cables between electricity poles sag more in the dry-season heat than in the cool harmattan morning. These are everyday signs of thermal expansion.

Thermal expansion: the increase in the size (length, area or volume) of a substance when its temperature rises.

The particle model explains it. In a solid, particles vibrate about fixed positions, held by strong forces. When heated, they gain energy and vibrate with larger amplitude, so on average they push one another slightly further apart. The whole solid gets a little bigger. In liquids and gases the particles move about; heating makes them move faster and spread out more, so liquids expand more than solids and gases expand far more than liquids.

Note that the mass does not change during expansion; only the volume increases. So density (mass ÷ volume) decreases when a substance is heated. This is why hot air rises and why a hot-air balloon floats.

⚠️ Common mistake: saying the particles themselves get bigger. The particles stay the same size; the spaces between them increase.

Try it yourself. A metal ball just passes through a ring when both are cold. Heat only the ball. What happens, and why?

Show solution

The ball no longer passes through, because its particles vibrate more and move apart, so its diameter increases.

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Practice questions

1. A copper wire 5.0 m long is heated through 40 K. Find its expansion (α = 1.7 × 10⁻⁵ K⁻¹).

Show worked answer

1.7 × 10⁻⁵ × 5 × 40 = 3.4 × 10⁻³ m = 3.4 mm.

2. A metal has α = 2.0 × 10⁻⁵ K⁻¹. Find β and γ.

Show worked answer

β = 4.0 × 10⁻⁵ K⁻¹, γ = 6.0 × 10⁻⁵ K⁻¹.

3. State two applications of a bimetallic strip.

Show worked answer

Any two: thermostat in an electric iron, fire alarm, bimetallic thermometer, flashing indicator.

4. Explain why fish survive in a frozen pond.

Show worked answer

Water at 4 °C (densest) stays at the bottom; ice floats and insulates the water below.

5. A gas occupies 120 cm³ at 2.0 × 10⁵ Pa. Find its volume at 8.0 × 10⁴ Pa at constant temperature.

Show worked answer

2.0 × 10⁵ × 120/8.0 × 10⁴ = 300 cm³.

Key words

Thermal expansion: The increase in size of a substance when its temperature rises.
Linear expansivity (α): The fractional increase in length per kelvin rise in temperature.
Area expansivity (β): The fractional increase in area per kelvin rise; β = 2α.
Cubic expansivity (γ): The fractional increase in volume per kelvin rise; γ = 3α.
Bimetallic strip: Two metals of different expansivity riveted together; it bends when heated.
Real expansion: The true expansion of a liquid, allowing for the expansion of its container.
Apparent expansion: The expansion of a liquid as observed, ignoring the container’s expansion.
Anomalous expansion: The contraction of water as it is heated from 0 °C to 4 °C.
Absolute zero: The lowest possible temperature, 0 K or −273 °C (more precisely −273.15 °C).
Kelvin (K): The SI unit of temperature; T = θ + 273.
Boyle’s law: At constant temperature, the pressure of a fixed mass of gas is inversely proportional to its volume.
Charles’ law: At constant pressure, the volume of a fixed mass of gas is directly proportional to its absolute temperature.

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