Thermal Expansion
Most things grow a little when warmed; water near freezing does the opposite.
Lesson at a glanceShow the key points
- Coefficient of linear expansion α = (1/L) dL/dT; for small constant α, L_θ = L₀(1 + αθ) (23.11).
- Coefficient of volume (cubical) expansion γ = (1/V) dV/dT; V_θ = V₀(1 + γθ) (23.12); γ = 3α.
- Water contracts from 0°C to 4°C (γ negative) and expands above: its density is greatest at 4°C.
- So lake bottoms stay near 4°C under surface ice, and aquatic life survives.
- A liquid in a flask: apparent expansion = real expansion − expansion of the container (the level may first dip).
Railway lines have small gaps between rails, bridges sit on rollers, and a tight jar lid loosens under hot water. Almost everything grows a little as it warms.
Linear expansion
A rod of length L at temperature T grows by ΔL when warmed by ΔT. The coefficient of linear expansion is
If α is small and nearly constant, with L₀ the length at 0°C and θ the Celsius temperature:
A steel rail (α = 1.2 × 10⁻⁵ K⁻¹) is 12.0 m long at 0°C. How much longer is it at 40°C?
Volume expansion
A brass ball (α = 1.9 × 10⁻⁵ K⁻¹) has volume 50.0 cm³ at 20°C. By how much does its volume grow at 120°C?
The odd behaviour of water
Heat 1 g of water from 0°C: its volume falls, from about 1.00013 cm³ to 1.00000 cm³ at 4°C, and only then rises. Between 0 and 4°C its expansion coefficient is negative, so water is densest at 4°C.
In winter, lake water at 4°C sinks to the bottom and stays there, while colder water and then ice float on top. The ice insulates the water below, and fish survive near the bottom.
Real and apparent expansion of a liquid
Heat a liquid in a flask with a graduated stem. At first the flask (heated from outside) expands before the liquid does, and the level drops, as though the liquid had shrunk. Then the liquid catches up and rises. What you read on the stem is the apparent expansion:
- 1α = (1/L) dL/dT; L_θ = L₀(1 + αθ).
- 2γ = (1/V) dV/dT; V_θ = V₀(1 + γθ); γ = 3α.
- 3Water contracts from 0°C to 4°C and is densest at 4°C, so lakes freeze from the top.
- 4A liquid's apparent expansion is its real expansion minus the container's.
Check yourself
Keep exploring
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