Thermal expansion

Most things expand when heated, because their particles jiggle harder and push one another a little further apart. The fractional change grows with the temperature rise:

• Length: ΔL = α_l L ΔT (α_l is the coefficient of linear expansion).
• Area: ΔA ≈ 2α_l A ΔT.
• Volume: ΔV = α_V V ΔT, with α_V = 3α_l.

Where do 2 and 3 come from? Heat a unit square and each side becomes (1 + α); its area becomes (1 + α)² ≈ 1 + 2α, since α² is tiny. A unit cube becomes (1 + α)³ ≈ 1 + 3α. So the three coefficients are in the ratio 1 : 2 : 3.

Example: a steel bridge 100 m long (α = 1.2 × 10⁻⁵ K⁻¹) faces a 40 °C swing between winter and summer. It changes length by 100 × 1.2 × 10⁻⁵ × 40 = 4.8 cm, so engineers leave expansion joints. Railway lines have small gaps for the same reason.

Every length of a body grows by the same fraction, holes included: a hole in a heated metal plate gets bigger, as if it were made of the metal itself. That is why a tight lid comes off a jar after you run hot water over it.

Liquids expand more than their containers. A steel tank brim-full of 75 L of petrol at 10 °C, warmed to 30 °C, grows by 75 × 3.6 × 10⁻⁵ × 20 ≈ 0.05 L, but the petrol (α_V ≈ 9.5 × 10⁻⁴ K⁻¹) grows by about 1.4 L, so about 1.37 L spills over.

Since the mass stays the same while the volume grows, the density falls: Δρ/ρ ≈ –α_V ΔT.

Water is unusual. From 0 °C to 4 °C it contracts when warmed, so it is densest at 4 °C. In winter, water at 4 °C sinks to the bottom of a lake while colder water floats on top and freezes. The lake freezes from the top down, and fish survive in the water below.

If a rod is clamped at both ends so it cannot expand, a thermal stress builds up instead: stress = YαΔT, and the force is YAαΔT. A clamped steel rod (Y = 2 × 10¹¹ Pa) heated by 50 °C feels 2 × 10¹¹ × 1.2 × 10⁻⁵ × 50 = 1.2 × 10⁸ Pa, enough to buckle rails if no gaps were left.