Thermal expansion

Expansion from heating

Most materials expand when heated and contract when cooled. This behavior is called thermal expansion. It occurs because atoms and molecules in a solid or liquid tend to occupy slightly larger average separations at higher temperatures.

Thermal expansion matters in bridges, railroads, buildings, engines, thermometers, cookware, and precision instruments. Even small fractional changes can be important over large lengths or tight tolerances.

Linear expansion

For a solid object whose length changes with temperature, the approximate relationship is

$Delta L=alpha L_0Delta T$

Here $L_0$ is the original length, $Delta L$ is the change in length, $Delta T$ is the temperature change, and $alpha$ is the coefficient of linear expansion.

The final length is

$L=L_0(1+alphaDelta T)$

This model assumes the temperature change is not too large and $alpha$ is approximately constant.

Area and volume expansion

If a solid expands uniformly in all directions, area and volume also change. For small temperature changes, the area expansion coefficient is approximately

$2alpha$

so

$Delta Aapprox 2alpha A_0Delta T$

The volume expansion coefficient for an isotropic solid is approximately

etaapprox 3alpha

so

Delta V=eta V_0Delta T

Liquids are usually described using volume expansion because they take the shape of their containers.

Microscopic origin

If atoms in a solid were connected by perfectly symmetric springs, heating would increase vibration amplitude but not average separation. Real interatomic potentials are asymmetric. At higher vibrational energies, atoms spend more time at larger separations, increasing the average spacing.

This microscopic asymmetry explains why expansion is common. It also explains why expansion coefficients differ among materials.

Holes expand too

A common puzzle asks what happens to a hole in a metal plate when the plate is heated. The hole expands as if it were made of the same material. The entire geometry scales up. This is important for fitting metal rings, bearings, and machine parts.

Thermal stress

If a material is prevented from expanding or contracting, temperature changes can create large stresses. A bridge fixed rigidly at both ends would experience stress during seasonal temperature changes. To prevent damage, engineers use expansion joints.

Railroad tracks, concrete roads, pipelines, and building materials must account for thermal expansion. If constrained, they may buckle, crack, or warp.

Bimetallic strips

A bimetallic strip consists of two bonded metals with different expansion coefficients. When heated, one metal expands more than the other, causing the strip to bend. This principle is used in some thermostats and temperature switches.

The bending turns temperature change into mechanical motion.

Water's unusual expansion

Water has an important anomaly: it is densest near $4^circ C$. As water cools below this temperature, it expands until it freezes. Ice is less dense than liquid water, so it floats.

This unusual behavior matters greatly for life in lakes and ponds. Ice forms at the surface, insulating the water below.

The big idea

Thermal expansion is the change in size caused by temperature change. Linear, area, and volume expansion are described by simple proportional models for modest temperature ranges. The effect comes from microscopic interatomic behavior and has major engineering and natural consequences.