Thermal expansion

Almost all bodies expand as temperature increases. The reason is microscopic: the oscillations of atoms about their equilibrium position become wider, and — because the interatomic potential is asymmetric — the mean equilibrium position shifts towards greater distances. The body, overall, lengthens.

Principle — Linear expansion of solids

A rod of initial length $\ell_0$ at temperature $T_0$, heated to $T_0 + \Delta T$, takes on the length $\ev{\ell = \ell_0\,(1 + \lambda\,\Delta T)}$ where $\lambda$ is the material’s linear expansion coefficient (unit $\text{K}^{-1}$).

The actual elongation is thus $\Delta\ell = \ell_0\,\lambda\,\Delta T$: proportional to the starting length and to the temperature change. An analogous law holds for volumetric quantities:

$\ev{V = V_0\,(1 + \alpha\,\Delta T)}$

with volumetric expansion coefficient $\alpha = 3\lambda$ (for small changes). The factor of $3$ arises because a cube expands simultaneously along the three dimensions.

The coefficients $\lambda$ vary a great deal from material to material:

Material	$\lambda$ ($10^{-6}\,\text{K}^{-1}$)
Aluminium	24
Copper	17
Steel	12
Pyrex glass	3
Diamond	1

Why Pyrex withstands thermal shock

Pyrex glass has a very low $\lambda$: when heated, it expands very little, and internal stresses between parts at different temperatures stay low. This is why it withstands thermal shock far better than ordinary glass.

Expansion is far from negligible in structures: a railway track tens of metres long can lengthen by more than a centimetre between winter and summer. This is why expansion joints, small gaps between one rail and the next, are left in place — without them the track would buckle (see the railway track example).

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Topics: Thermology Concepts: Thermal expansion

Related exercises: Worked exercise — The railway track in winter · Problem — Three bars at the same temperature · Problem — Aluminium with doubled lambda