Thermal Expansion of Solids and Liquids – Quantity of Heat and Heat Transfer (Senior 3 Physics) | Ganzaa

Quantity of Heat and Heat Transfer

Lessons

Senior 3 · Physics

Quantity of Heat and Heat Transfer

5 lessons

Lesson 5 of 5

Thermal Expansion of Solids and Liquids

25 min

The ticking roof

On a sunny morning, a house with an iron-sheet roof often ticks and cracks for an hour or so. Nobody is walking on it. As the Sun warms the metal sheets they get slightly longer and wider, the nails and wooden battens hold them back, and every so often a sheet slips a little with a sharp click. In the evening, as the roof cools and shrinks, the ticking can start again.

Almost every substance expands when it is heated and contracts when it cools. In S1 you saw this as a property of matter; in this lesson you will calculate it, and see how engineers build around it or use it.

Why things expand

When a solid is heated, its particles vibrate with larger amplitude and push one another slightly further apart, so the whole body gets bigger. In a liquid the particles move about more and take up more room. The change is small for solids, larger for liquids and very large for gases (S2's gas laws).

Linear expansion

Experiments show that the increase in length $\Delta L$ of a rod is proportional to its original length $L_{0}$ and to its rise in temperature $\Delta T$ :

\Delta L = \alpha\,L_0\,\Delta T \qquad\text{so}\qquad L = L_0\,(1 + \alpha\,\Delta T)

The constant $\alpha$ depends on the material.

Definition

Linear expansivity (coefficient of linear expansion, $\alpha$ ): the increase in length per unit original length per kelvin rise in temperature, $\alpha = \frac{\Delta L}{L_0\,\Delta T}$ . SI unit: per kelvin ( $\mathrm{K^{-1}}$ ). Example: for steel $\alpha = 1.2 \times 10^{-5}\ \mathrm{K^{-1}}$ , so each metre of steel grows by 0.012 mm for each kelvin.

Area (superficial) expansivity $\beta = \frac{\Delta A}{A_0\,\Delta T} \approx 2\alpha$ . Volume (cubic) expansivity $\gamma = \frac{\Delta V}{V_0\,\Delta T} \approx 3\alpha$ for a solid.

Expansivities of some materials (approximate, near room temperature)

Solid	$\alpha$ ( $\mathrm{K^{-1}}$ )	Liquid	$\gamma$ ( $\mathrm{K^{-1}}$ )
Aluminium	$2.4 \times 10^{-5}$	Petrol	$9.5 \times 10^{-4}$
Brass	$1.9 \times 10^{-5}$	Ethanol	$1.1 \times 10^{-3}$
Copper	$1.7 \times 10^{-5}$	Water (at 20 °C)	$2.1 \times 10^{-4}$
Steel, iron, concrete	$1.2 \times 10^{-5}$	Mercury	$1.8 \times 10^{-4}$
Ordinary glass	$0.9 \times 10^{-5}$
Heat-resistant (borosilicate) glass	$0.3 \times 10^{-5}$

Liquids expand far more than solids: petrol's $\gamma$ is about 26 times the volume expansivity of steel ( $3.6 \times 10^{-5}$ ). Steel and concrete have almost the same $\alpha$ , which is why steel rods can reinforce concrete without cracking it as the temperature changes.

Worked example 1: a steel footbridge beam

A steel beam in a footbridge is 12 m long at 15 °C on a cool night. In the afternoon sun it reaches 45 °C. By how much does it lengthen?

\Delta L = \alpha\,L_0\,\Delta T = 1.2 \times 10^{-5} \times 12 \times (45 - 15) = 4.3 \times 10^{-3}\ \text{m} \approx 4.3\ \text{mm}

The beam must rest on a roller or sliding bearing at one end, with a gap of at least 5 mm, or it will push against the supports and bend them.

Area and volume expansion

Heat a square plate of side $L_{0}$ . Each side becomes $L_0(1 + \alpha\,\Delta T)$ , so the area becomes

A = L_0^{2}(1 + \alpha\,\Delta T)^{2} = A_0\left(1 + 2\alpha\,\Delta T + \alpha^{2}\Delta T^{2}\right) \approx A_0\,(1 + 2\alpha\,\Delta T)

because $\alpha\,\Delta T$ is tiny and its square is tinier still. So $\beta \approx 2\alpha$ and, by the same reasoning for a cube, $\gamma \approx 3\alpha$ :

\Delta A = \beta\,A_0\,\Delta T \qquad \Delta V = \gamma\,V_0\,\Delta T

A hole in a plate expands exactly as if it were made of the plate's material: heating a metal ring makes the hole bigger, not smaller. This is how the classic ball-and-ring demonstration works: a ball that just passes through a cold ring will not pass once the ball is heated, but will pass again if the ring is heated too.

Worked example 2: an aluminium sheet

An aluminium sheet measures 2.0 m by 1.0 m at 20 °C. Find the increase in its area when it is heated to 60 °C.

\beta = 2\alpha = 2 \times 2.4 \times 10^{-5} = 4.8 \times 10^{-5}\ \mathrm{K^{-1}}

\Delta A = \beta\,A_0\,\Delta T = 4.8 \times 10^{-5} \times 2.0 \times 40 = 3.84 \times 10^{-3}\ \mathrm{m^2} \approx 38\ \mathrm{cm^2}

Worked example 3: a full fuel tank in the sun

A steel fuel tank is filled to the brim with 50 L of petrol at 15 °C, and the vehicle is then parked in the sun until the tank and petrol reach 35 °C. How much petrol overflows?

\Delta V_{\text{petrol}} = 9.5 \times 10^{-4} \times 50 \times 20 = 0.95\ \text{L}

\Delta V_{\text{tank}} = 3 \times 1.2 \times 10^{-5} \times 50 \times 20 = 0.036\ \text{L}

\text{overflow} = 0.95 - 0.036 \approx 0.91\ \text{L}

The container expands too, but far less than the liquid. This is why fuel tanks, jerrycans and bottles are never filled right to the top.

Worked example 4: measuring $\alpha$

A metal rod 0.800 m long at 20 °C is heated in a steam jacket to 95 °C (the boiling point in Kigali). A micrometer screw gauge shows that it grows by 1.02 mm. Find $\alpha$ and identify the metal.

\alpha = \frac{\Delta L}{L_0\,\Delta T} = \frac{1.02 \times 10^{-3}}{0.800 \times 75} = 1.7 \times 10^{-5}\ \mathrm{K^{-1}}

This matches copper.

Bimetallic strips: expansion put to work

A bimetallic strip is two strips of different metals, such as brass and iron, fixed firmly together along their length. When it is heated, the brass expands more than the iron, so the strip must bend, with the brass on the outside of the curve. Cooled below room temperature, it bends the other way.

This makes a simple automatic switch, a thermostat: in an electric iron, an oven or a water heater, the strip bends away from the contact when the set temperature is reached and switches the heater off; as it cools it straightens and switches the heater on again. Bimetallic strips are also used in fire alarms, in dial thermometers, and in the thermal trips of some circuit breakers (unit 7).

Allowing for expansion

Bridges and long roads have expansion joints: gaps or interlocking steel "fingers" that let each section lengthen and shorten.
Overhead power lines are strung with a sag, so they can contract on cool nights without snapping or pulling down their pylons (unit 6).
Concrete paths and floors are laid in slabs with narrow joints between them.
Glassware: hot water poured into a thick glass can crack it, because the inside expands before the outside warms. Laboratory glass is made of borosilicate glass, whose tiny $\alpha$ keeps the stresses small.
Tight lids: running hot water over a metal lid stuck on a glass jar loosens it, because the metal expands more than the glass.

The strange behaviour of water

As S1 mentioned, water does not follow the rule between 0 °C and 4 °C: when it is heated in this range it contracts, so water is densest at about 4 °C. Above 4 °C it expands like other liquids. In cold countries this is what keeps fish alive in winter: the densest water (4 °C) sinks to the bottom of a lake, and the ice forms on top and floats, insulating the water below.

Warning

Watch the units and the power of ten. Expansivities are tiny numbers such as $1.2 \times 10^{-5}\ \mathrm{K^{-1}}$ ; a calculator slip of one power of ten gives an answer ten times too big. Keep $L_{0}$ in metres and convert $\Delta L$ to millimetres only at the end. And for area or volume, use $2\alpha$ or $3\alpha$ , not $\alpha$ .

Tip

Same shape as $Q = mc\,\Delta T$ . Every expansion formula reads "change = constant × original amount × temperature change": $\Delta L = \alpha L_0 \Delta T$ , $\Delta A = \beta A_0 \Delta T$ , $\Delta V = \gamma V_0 \Delta T$ , with $\beta \approx 2\alpha$ and $\gamma \approx 3\alpha$ for solids.

Thermal expansion explained using the ring and ball experiment (Mwalimu Ndolo). Watch what happens when the ball is heated, and then when it cools again.

✦

Practice

Define linear expansivity and give its SI unit.
A copper wire is 50 m long at 20 °C. How much longer is it at 40 °C?
Concrete road slabs are 10 m long. The temperature of the road ranges over 30 K during the year. What is the smallest gap to leave between slabs?
A steel plate has an area of 0.50 m² at 20 °C. Find the increase in its area at 220 °C.
A 20 L jerrycan is filled with petrol at 15 °C and left in the sun until the petrol reaches 40 °C. Ignoring the jerrycan's own expansion, how much petrol would overflow?
A brass rod 0.500 m long grows by 0.71 mm when heated from 20 °C to 95 °C. Find its linear expansivity.
Explain, with a sketch, how a bimetallic strip switches off an electric iron when it becomes too hot. Which metal must be on the side nearer the contact?
A metal ring is stuck tightly on a glass bottle. A pupil suggests heating the whole bottle and ring together in hot water. Will this help? Explain.

Show answersHide answers

The increase in length per unit original length per kelvin rise in temperature, $\alpha = \frac{\Delta L}{L_0\,\Delta T}$ ; $\mathrm{K^{-1}}$ .
$\Delta L = 1.7 \times 10^{-5} \times 50 \times 20 = 0.017\ \text{m} = 17\ \text{mm}$ .
$\Delta L = 1.2 \times 10^{-5} \times 10 \times 30 = 3.6 \times 10^{-3}\ \text{m}$ , so at least 3.6 mm (in practice a little more).
$\Delta A = 2 \times 1.2 \times 10^{-5} \times 0.50 \times 200 = 2.4 \times 10^{-3}\ \mathrm{m^2} = 24\ \mathrm{cm^2}$ .
$\Delta V = 9.5 \times 10^{-4} \times 20 \times 25 \approx 0.48\ \text{L}$ .
$\alpha = \frac{0.71 \times 10^{-3}}{0.500 \times 75} \approx 1.9 \times 10^{-5}\ \mathrm{K^{-1}}$ .
When the iron is cold, the straight strip touches the contact and current flows through the heating element. As it gets hotter, one metal expands more than the other, so the strip bends away from the contact, breaking the circuit. The metal that expands more (e.g. brass) ends up on the outside of the curve, so the strip bends away from it. To bend away from the contact, the brass must therefore be on the side nearer the contact. As the iron cools, the strip straightens and the heater switches on again.
Yes, a little: both expand, but the metal ring expands more than the glass (its $\alpha$ is about twice as large), so the ring loosens. It works better to heat only the ring, for example by pouring hot water over the ring alone, so that the glass stays cool and does not expand at all.

Key points

Linear: $\Delta L = \alpha L_0 \Delta T$ ; area: $\Delta A = \beta A_0 \Delta T$ with $\beta \approx 2\alpha$ ; volume: $\Delta V = \gamma V_0 \Delta T$ with $\gamma \approx 3\alpha$ for solids.
Liquids expand far more than solids, so containers are never filled to the brim.
A bimetallic strip bends with the more-expanding metal on the outside; it is the heart of many thermostats.
Structures need expansion gaps; water is densest at 4 °C.

1The ticking roof

2Why things expand

3Linear expansion

4Expansivities of some materials (approximate, near room temperature)

5Area and volume expansion

6Bimetallic strips: expansion put to work

7Allowing for expansion

8The strange behaviour of water