Thermal expansion

Most substances expand when heated and contract when cooled; the rise in size is thermal expansion. A tight metal lid loosens under hot water, mercury climbs a thermometer, and a partly blown balloon in warm water swells.

Linear expansion: a rod of length l grows by Δl when its temperature rises by ΔT, with Δl/l = α_l ΔT. α_l is the coefficient of linear expansion, a property of the material.

α_l in units of 10⁻⁵ K⁻¹, smallest first: lead 0.29, pyrex glass 0.32, iron 1.2, gold 1.4, copper 1.7, brass 1.8, silver 1.9, aluminium 2.5. Metals expand more than glass: copper about five times pyrex.

Area expansion: a flat sheet's area grows by Δa/a = 2α_l ΔT, so the coefficient of area expansion is 2α_l.

Volume expansion: ΔV/V = α_V ΔT, with α_V the coefficient of volume expansion. It is not strictly a constant: it depends on temperature and levels off only at high temperature, as the curve for copper shows.

α_V in K⁻¹: aluminium 7 × 10⁻⁵, brass 6 × 10⁻⁵, iron 3.55 × 10⁻⁵, glass (ordinary) 2.5 × 10⁻⁵, pyrex 1 × 10⁻⁵, hard rubber 2.4 × 10⁻⁴, invar 2 × 10⁻⁶, paraffin 58.8 × 10⁻⁵, mercury 18.2 × 10⁻⁵, water 20.7 × 10⁻⁵, ethanol 110 × 10⁻⁵.

For an isotropic solid α_V = 3α_l: a cube of side l heated by ΔT becomes (l + Δl)³ ≈ l³ + 3l²Δl, so ΔV/V = 3Δl/l = 3α_l ΔT.

Gases expand far more than solids or liquids. For an ideal gas at constant pressure PΔV = μRΔT, which gives α_V = 1/T: about 3.7 × 10⁻³ K⁻¹ at 0 °C, falling as T rises, and about 3300 × 10⁻⁶ K⁻¹ near room temperature, much larger than for any solid or liquid.

A thin plate expands like a magnified photograph: a hole in it grows too, because every length in the plate grows in the same ratio.