Thermal expansion

Linear expansion

Narrated video. Transcript:

Railway tracks are laid with small gaps between the rails, and long bridges rest on rollers at one end. Why? Because things get longer when they get hotter.

Heating a substance usually makes it expand. A change in length is linear expansion, a change in area is area expansion, and a change in volume is volume expansion.

Watch a metal rod of length L being heated. It gets longer by a small amount ΔL. Experiments show that ΔL over L is proportional to the rise in temperature, ΔT.

So ΔL over L equals α ΔT. The constant α is the coefficient of linear expansion, or linear expansivity. It is a property of the material.

Its unit is per kelvin, and the values are tiny: about 2.5 × 10⁻⁵ per kelvin for aluminium, 1.2 × 10⁻⁵ for iron, and only 0.32 × 10⁻⁵ for pyrex glass.

So a one-metre aluminium rod heated by 100 kelvin grows by only 2.5 millimetres. In our animations the growth is exaggerated, so that you can see it.

Pyrex expands far less than metals. That is why a pyrex dish can go into a hot oven without cracking.

Now a worked example. A blacksmith must fit an iron ring onto the wooden wheel of a cart. At 27 degrees Celsius, the wheel rim is 5.243 metres across and the ring only 5.231 metres.

A diameter expands just like a length, so the ring must grow by 0.012 metres. ΔT equals ΔL over α L: 0.012 divided by 1.20 × 10⁻⁵ times 5.231, which is about 191 kelvin.

So the ring is heated to about 27 plus 191, that is 218 degrees Celsius. It slips on, then cools and shrinks to grip the wheel tightly.

Area and volume expansion

Narrated video. Transcript:

Now heat a square metal plate. Both its sides grow, so its area grows too. ΔA over A is about 2 α ΔT.

For volume, we write ΔV over V equals α V times ΔT. The coefficient α V, which we will call β, is the coefficient of volume expansion.

For a solid there is a neat link between β and α. Take a cube of side l. When heated, each side becomes l plus Δl, where Δl equals α l ΔT.

The new volume is l plus Δl, all cubed. Expand it: l cubed, plus 3 l squared Δl, plus terms with Δl squared and Δl cubed.

Δl is so tiny compared with l that those last terms can be ignored. So ΔV equals 3 l squared Δl, which is 3 α V ΔT.

That gives β equals 3 α. The volume expansivity of a solid is three times its linear expansivity, just as its area expansivity is twice it.

Liquids have no fixed shape, so only their volume expansion makes sense. Their β is usually much larger than a solid’s. That is why a liquid thermometer works: the liquid expands far more than the glass tube around it.

For an ideal gas at constant pressure, P V equals μ R T gives ΔV over V equals ΔT over T. So β for a gas is 1 over T.

At 0 degrees Celsius, that is 1 over 273, about 3.7 × 10⁻³ per kelvin, much larger than for solids and liquids. And unlike theirs, it gets smaller as the gas gets hotter.

Water’s odd behaviour and thermal stress

Narrated video. Transcript:

Water is strange. Heat water from 0 degrees Celsius to 4 degrees, and it contracts. Only above 4 degrees does it expand like other liquids.

So water has its smallest volume, and its greatest density, at 4 degrees Celsius. This is called the anomalous expansion of water.

Watch a lake in winter. The surface water cools, becomes denser and sinks, until the whole lake is at 4 degrees.

Now the surface cools further. Water below 4 degrees is less dense, so it stays on top, and finally freezes. The ice floats and insulates the water underneath.

The bottom of the lake stays at about 4 degrees all winter. If water behaved like other liquids, lakes would freeze from the bottom up, and fish could not survive.

Next, what if something is not allowed to expand? Clamp a steel rod between two rigid walls, and heat it.

The rod tries to grow by α L ΔT, but the walls push it back to its original length. That is the same as compressing it by a strain of α ΔT.

Stress equals Young’s modulus times strain, so the thermal stress is Y α ΔT. The force on the walls is Y α ΔT times the area of the rod.

For steel, Y is about 2 × 10¹¹ pascals and α is 1.2 × 10⁻⁵ per kelvin. A rise of just 20 kelvin gives a stress of 4.8 × 10⁷ pascals, nearly five hundred times atmospheric pressure.

That is enough to buckle a railway track. This is why rails have expansion gaps, and why concrete roads and bridges have expansion joints.

Try it: heat a rod or a cube

Interactive simulation. Things to try:

• The expansion is drawn 100 times larger than life.
• Compare aluminium, steel and pyrex for the same temperature rise.
• Tick “Clamp” to stop the rod expanding and see the thermal stress.
• Switch to Cube to check that ΔV/V is three times ΔL/L.

Try it: water near 4 °C

Interactive simulation. Things to try:

• Cool the water slowly from 12 °C towards 0 °C and watch the volume.
• Below 4 °C the volume grows again as the water gets colder.
• At 0 °C the lake freezes at the top while the bottom stays at 4 °C.

A hole in a hot plate

A steel plate has a circular hole cut in it. When the plate is heated, what happens to the hole?

1. It gets smaller, because the metal expands into it
2. It stays the same size
3. It gets bigger
4. It depends on the size of the plate

Gap between rails

Steel rails 12.5 m long are laid at 20 °C. On the hottest day the rails reach 45 °C. Taking α = 1.2 × 10⁻⁵ K⁻¹, what is the smallest gap to leave between neighbouring rails?

1. 0.375 mm
2. 3.75 mm
3. 6.75 mm
4. 37.5 mm

Summary

• Linear expansion: ΔL/L = α ΔT
• Area expansion: ΔA/A ≈ 2α ΔT
• Volume expansion: ΔV/V = β ΔT, β = 3α (solids)
• Ideal gas at constant P: β = 1/T
• Thermal stress (clamped): σ = Y α ΔT

• α is tiny (about 10⁻⁵ K⁻¹ for metals), so the squares and cubes of αΔT can be ignored. That is why β = 3α.
• Liquids expand more than solids, and gases much more, with β = 1/T.
• Water contracts from 0 °C to 4 °C and is densest at 4 °C, so lakes freeze from the top down.
• A body that is not allowed to expand develops a thermal stress YαΔT. Rails and bridges need expansion gaps.