Thermal expansion

Warmed things grow

Warm a solid, and its atoms jiggle more, pushing each other a little farther apart. Every length grows by the same fraction:

$\alpha$ is the coefficient of linear expansion: steel $1.2 \times 10^{-5}\ \text{K}^{-1}$, copper $1.7 \times 10^{-5}\ \text{K}^{-1}$, aluminium $2.4 \times 10^{-5}\ \text{K}^{-1}$, glass $0.9 \times 10^{-5}\ \text{K}^{-1}$.

Areas and volumes grow too:

Liquids have only a volume coefficient: mercury $1.8 \times 10^{-4}\ \text{K}^{-1}$.

You should find that aluminium grows twice as much as steel, so its pipes need bigger loops to take up the change.

Watch out. A hole grows with the material around it, as if it were made of that material.

Where it matters

• Gaps in rails and joints in bridges let them grow on hot days.
• A bimetallic strip of two metals bends when heated, the one that grows more on the outside: it switches thermostats.
• A tight metal lid on a glass jar loosens under hot water, because the metal grows more than the glass.

Water is odd

From $0\ ^\circ\text{C}$ to $4\ ^\circ\text{C}$, water shrinks as it warms; above $4\ ^\circ\text{C}$ it grows like other liquids. So water is densest at $4\ ^\circ\text{C}$. In winter, a pond's water at $4\ ^\circ\text{C}$ sinks, colder water stays on top and freezes, and the ice shelters the life below.

A worked example

An aluminium window frame is $1.6\ \text{m}$ wide. How much wider does it get on a day $35\ \text{K}$ warmer, and how much does its glass pane grow?

1. Frame. $2.4 \times 10^{-5}\ \text{K}^{-1} \times 1.6\ \text{m} \times 35\ \text{K} = 1.34 \times 10^{-3}\ \text{m}$, about $1.3\ \text{mm}$.
2. Glass. $0.9 \times 10^{-5}\ \text{K}^{-1} \times 1.6\ \text{m} \times 35\ \text{K} = 0.50\ \text{mm}$.
3. So the pane is fitted with a little room to spare, in a soft seal.

Practise with new numbers

Try it in Python

The program compares the frame and its glass.

Remember these

• $\Delta L = \alpha L\,\Delta T$; area $2\alpha$; volume $\gamma = 3\alpha$.
• Leave gaps; bimetallic strips bend toward the metal that grows less.
• Water is densest at $4\ ^\circ\text{C}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A copper pipe $8.0\ \text{m}$ long warms by $60\ \text{K}$. How much does it grow?

2. A metal has $\alpha = 2.0 \times 10^{-5}\ \text{K}^{-1}$. What is its volume coefficient?

3. Why does a glass tumbler sometimes crack when hot water is poured into it?

4. Why does ice float on water?