Thermal expansion of solids: linear, superficial and cubical
Solids expand when heated. The increase in length, area or volume depends on the original size, the material and the rise in temperature.
Why solids expand
In a solid, the particles are held close together and keep vibrating. When the solid is heated, they vibrate more strongly and push one another a little farther apart. So the solid expands: its length, area and volume increase. This is thermal expansion. On cooling, the solid contracts (becomes smaller). The coefficient of expansion is the expansion of each unit length, area or volume for a 1 K rise in temperature. It is different for different materials.
Three kinds of expansion in solids
| Kind | What increases | Formula | Coefficient |
|---|---|---|---|
| Linear expansion | Length | ΔL/L₀ = αL ΔT | αL: coefficient of linear expansion |
| Superficial (areal) expansion | Area of the surface | ΔA/A₀ = αA ΔT | αA: coefficient of superficial expansion |
| Cubical expansion | Volume | ΔV/V₀ = αV ΔT | αV: coefficient of cubical expansion |
Symbols, units and examples
- L₀, A₀ and V₀ are the original length, area and volume. ΔL, ΔA and ΔV are the increases. ΔT is the rise in temperature.
- Each coefficient has the unit K⁻¹. A rise of 1 °C is the same as a rise of 1 K.
- Railway tracks: small gaps are left between rails so they can expand in summer without bending.
- Bridges and flyovers have expansion joints that let the concrete and steel expand and contract.
- Power lines are left slightly loose. They sag more in summer, when they expand.
- A tight metal lid on a glass bottle opens easily after it is dipped in hot water. The metal expands more than the glass.
Linear expansion of a rail
A steel rail is 10 m long at 25 °C. How much longer is it at 45 °C? (αL for steel = 1.2 × 10⁻⁵ K⁻¹)
- Given: L₀ = 10 m, ΔT = 45 − 25 = 20 °C = 20 K, αL = 1.2 × 10⁻⁵ K⁻¹.
- Formula: ΔL/L₀ = αL ΔT, so ΔL = αL × L₀ × ΔT.
- ΔL = 1.2 × 10⁻⁵ K⁻¹ × 10 m × 20 K = 240 × 10⁻⁵ m.
- ΔL = 2.4 × 10⁻³ m.
The rail becomes 2.4 × 10⁻³ m (2.4 mm) longer. This is why small gaps are left between rails.
Activity: the ball and ring
- 1Take a metal ball that just passes through a metal ring.
- 2Heat the ball over a flame for a few minutes, holding it with tongs.
- 3★Try to pass it through the ring. Observe: it no longer passes, because its volume has increased (cubical expansion).
- 4Let it cool and try again. Observe: it passes, because it has contracted.
★ key step
The gaps between rails, the expansion joints on flyovers and sagging power lines in May are all there because solids expand in the heat. A tight lid on a pickle jar loosens when you pour hot water over it.
2 marks: “Define the coefficient of linear expansion. Give its unit.” The increase in length per unit original length for a 1 K rise in temperature; αL = ΔL/(L₀ΔT); unit K⁻¹.
2 marks: “Why are small gaps left between railway rails?” Rails expand in summer heat. The gaps give them room to expand, so they do not bend or buckle.
4 marks: “Name the three types of thermal expansion in solids and write their formulas.” Linear: ΔL/L₀ = αL ΔT; superficial: ΔA/A₀ = αA ΔT; cubical: ΔV/V₀ = αV ΔT; explain each symbol; unit K⁻¹; one example each.
Line, Surface, Cube: Linear (length), Superficial (area), Cubical (volume), for one, two and three dimensions.