Physics · 2 Thermal physics

2.2.1 Thermal expansion of solids, liquids and gases

Before you start

The particle model: in a solid the particles are close together in a fixed, regular arrangement; in a liquid they are close together but irregular and move around each other; in a gas they are far apart and move randomly.
A higher temperature means the particles have more kinetic energy on average, so they move faster.
Gas pressure is caused by gas particles colliding with the walls of their container.
Density is mass per unit volume: $ρ = m / V$ .
Working with temperatures below zero: a rise from $- 10$ °C to $40$ °C is a rise of $50$ °C.

Syllabus objectives covered (3)

Describe, qualitatively, the thermal expansion of solids, liquids and gases at constant pressure
Describe some of the everyday applications and consequences of thermal expansion
Explain, in terms of the motion and arrangement of particles, the relative order of magnitudes of the expansion of solids, liquids and gases as their temperatures rise

Part 1 of 6

What thermal expansion means

Thermal expansion is the increase in size of a material when its temperature rises. Heat a metal rod and it gets slightly longer. Heat a liquid and its volume increases. Heat a gas, and if it is free to, it spreads into a larger volume.
When a material cools, the reverse happens: it contracts, which means it gets smaller. Every 'expands when heated' in this lesson has a matching 'contracts when cooled', and a structure has to cope with both.
A solid expands in every direction at once. A metal rod gets longer, wider and thicker. For a long, thin object such as a rail or a cable, the change you notice is the change in length, because the length is by far its largest dimension.
The mass does not change. Nothing is added to the material: the same particles take up more space. Density is mass divided by volume, so the same mass in a larger volume means the density decreases.
$ρ = \frac{m}{V}$
Nearly every material behaves this way. Water between $0$ °C and $4$ °C is an unusual case: over that small range it contracts as it warms. The syllabus does not ask you about it, but it is why this lesson says 'nearly every' and not 'every'.
'Qualitatively' in the objective means you describe in words: expands, contracts, more, less, most. There is no equation for thermal expansion for you to recall. If a question gives you data, such as how much one metre of steel grows per degree, you work from that data directly.

Part 2 of 6

Solids, liquids and gases compared

Compare equal volumes heated through the same temperature rise. In general, a solid expands least, a liquid expands more, and a gas expands most of all.
The differences are large. Take $1000$ cm³ of each, heated from $20$ °C to $70$ °C, with the gas at constant pressure. Steel grows by about $2$ cm³. Water grows by about $21$ cm³. Air grows by about $170$ cm³. That is the 'relative order of magnitudes' the Extended objective asks about: here the gas grows about $8$ times as much as the liquid, and the liquid about $10$ times as much as the solid.
Different solids expand by different amounts for the same temperature rise. Brass, for example, expands more than steel. Different liquids also expand by different amounts. Keep this in mind for the bimetallic strip later in the lesson.
A solid: the ball-and-ring demonstration. A metal ball just passes through a metal ring when both are at room temperature. Heat the ball and it no longer fits through, because it has expanded. Let it cool and it passes through again.
Here is the awkward case. Heat the ring instead of the ball, and the hole gets bigger, not smaller. The whole ring expands outwards, hole included, like a photograph being enlarged. So the cold ball passes through the hot ring easily.
A liquid: fill a glass flask to the top with coloured water and fit a stopper with a narrow glass tube through it. Warm the flask and the water level in the tube rises. The glass expands too, but the water expands more, so the extra volume of water is pushed up the tube. The narrow tube turns a small change in volume into a large, easy-to-see change in height.
A gas: take a flask of air, with a stopper and a tube whose open end dips under water in a beaker. Hold the flask in your hands. The warmth of your hands alone is enough for the air to expand, and bubbles come out of the end of the tube. A small temperature rise gives a gas a change in volume you can see.
apparatus diagram — not drawn yet
A round-bottomed flask of air held upside-down in two hands, closed with a stopper through which a glass tube passes downwards into a beaker of water, with bubbles rising from the open end of the tube.
Just the warmth of a hand makes the air expand enough to push bubbles out of the tube.
Let the flask cool and the air contracts. Water is then pushed a little way up into the tube by the atmosphere, to fill the space the air no longer takes up.

Part 3 of 6

Why 'at constant pressure' matters for a gas

The objective says 'at constant pressure'. That condition matters most for a gas, because a gas only expands if whatever holds it lets it.
Picture a gas trapped in a cylinder under a piston that can slide freely. As the gas warms, it pushes the piston outwards and its volume increases. The pressure stays the same throughout. That is expansion at constant pressure.
Now seal the same gas inside a strong, rigid metal container. Its volume cannot change, so it cannot expand. Heating it raises its pressure instead. That is the constant-volume case from the gas section of the syllabus, and it is not thermal expansion.
So when you describe a gas, say that its volume increases when it is heated at constant pressure. 'A gas expands when heated' with no condition is only true if the gas is free to expand.

Part 4 of 6

The particle explanation: why solids expand least and gases most

Start with what heating does to every material. A rise in temperature gives the particles more kinetic energy on average, so they move faster.
The particles themselves do not get bigger. What changes is the average distance between them. Expansion is the particles moving further apart, not the particles swelling. Writing 'the particles expand' contradicts a correct point, so it can cancel a mark you would otherwise have scored.
How far apart they move depends on how strongly they are held together. That is the forces between the particles, and it is different in each state.
In a solid, the particles are packed close together in a regular arrangement. Strong forces hold each particle in a fixed position, and it can only vibrate about that position.
Heat the solid and its particles vibrate faster and with a larger amplitude. Each particle pushes its neighbours slightly further away, so the average separation increases a little. The strong forces pull the particles back towards their positions, so the increase is small. A solid expands least.
In a liquid, the particles are still close together, but in an irregular arrangement, and they move around each other. The forces between them are weaker than in a solid.
Heat the liquid and its particles move faster. The weaker forces hold them less tightly, so the same temperature rise moves them further apart than in a solid. A liquid expands more than a solid.
In a gas, the particles are far apart and move randomly at high speed. The forces between them are negligible, except when they collide.
Heat the gas and its particles move faster, so they hit the walls of the container harder and more often. At constant pressure, they push the piston or the walls outwards until the pressure is back to its original value. Nothing holds the particles together, so the gas spreads into a much larger volume. A gas expands most.
Put the two orders side by side. The forces between particles are strongest in a solid, weaker in a liquid, negligible in a gas. The expansion is the reverse: least in a solid, more in a liquid, most in a gas. The weaker the forces, the greater the expansion.
$forces: solid > liquid > gas expansion: solid < liquid < gas$
A full answer needs both halves. 'The particles move faster' is true of all three states, so on its own it cannot explain why the amounts differ. The forces between the particles are what explain the order.

Part 5 of 6

When expansion is a problem: leave room for it

If a material is stopped from expanding, it pushes on whatever is stopping it with a very large force. That force can bend, buckle or crack a structure. So engineers do not try to stop expansion. They leave room for it.
Bridges. A long steel or concrete bridge gets longer on a hot day and shorter on a cold night. One end is fixed, and the other rests on rollers so that it can move. There is a gap, called an expansion joint, between that end and the road.
apparatus diagram — not drawn yet
Side view of a bridge deck spanning two supports: the left end fixed to its support, the right end resting on a row of rollers on its support, with a small labelled gap (expansion joint) between the right end of the deck and the road.
One end is fixed and the other can roll, so the deck can lengthen into the gap instead of pushing on its supports.
Railways. Rails laid in separate lengths have small gaps between the lengths. On a hot day the rails expand into the gaps instead of pushing against each other and buckling.
Concrete roads and pavements are laid in slabs with gaps between them. The gaps are filled with a soft material that can be squashed as the slabs expand.
Overhead power cables are hung with some slack, and they sag more in summer than in winter. In cold weather they contract. A cable put up tight on a warm day would become tighter still as it cooled, and could snap or pull on its pylons.
Long pipes that carry hot water or steam include bends or loops. The loops flex as the pipe expands and contracts, so the pipe does not push its joints apart.
Bottles and other containers of liquid are not filled right to the top. The space above the liquid leaves room for it to expand on a warm day. Without it, the expanding liquid could push the lid off or crack the container.
Thick glass can crack when very hot water is poured into it. The inside surface heats up and expands before the outside does, and the uneven expansion strains the glass until it cracks.

Part 6 of 6

Expansion put to work

A liquid-in-glass thermometer uses the expansion of a liquid such as alcohol or mercury. When the temperature rises, the liquid in the bulb expands more than the glass does, so it is pushed up the narrow tube. The narrower the tube, the further the liquid moves for the same expansion.
A bimetallic strip is two strips of different metals, such as brass and steel, fixed firmly together along their whole length. When it is heated, the brass expands more than the steel. The two layers cannot slide past each other, so the strip bends.
apparatus diagram — not drawn yet
Two side views of a brass-and-steel bimetallic strip clamped at its left end: on the left, straight at room temperature with the brass layer underneath; on the right, after heating, curved upwards with the brass layer on the outside (lower, longer side) of the curve.
The metal that expands more ends up on the outside of the curve, because the outside of a curve is the longer side.
Which way it bends follows from one idea: the outside of a curve is longer than the inside. The brass has grown more, so it is the longer layer, and it ends up on the outside of the curve. The strip bends towards the steel side.
Cool the strip below the temperature at which it was straight, and it bends the other way. The brass contracts more, so it becomes the shorter layer and ends up on the inside of the curve.
In a thermostat, the free end of a bimetallic strip touches an electrical contact and completes the circuit of a heater. As the room warms, the strip bends away from the contact and breaks the circuit, so the heater switches off. As the room cools, the strip straightens, touches the contact again, and the heater switches back on.
Expansion can make a tight fit. To fix a metal ring tightly onto a wheel or a shaft, heat the ring so that it expands and slides on easily. As it cools, it contracts and grips tightly.
Rivets work the same way. A rivet is put in red hot and hammered flat. As it cools, it contracts and pulls the metal plates it joins tightly together.
A tight metal lid on a glass jar can be loosened by running hot water over it. For the same temperature rise the metal lid expands more than the glass, so the lid loosens on the rim.
Warm air is less dense than the cooler air around it, because it has expanded. This is why warm air rises. It is the idea behind a hot-air balloon, and you will meet it again in convection.

Worked examples

Example 1. Explain, in terms of the motion and arrangement of particles, why a liquid expands more than a solid but less than a gas, for the same rise in temperature.

Show the working

Plan the answer in two halves before you write. Half one is what heating does to the particles in every state. Half two is what is different between the states. The order of expansion comes from half two.
Half one: heating gives the particles more kinetic energy, so they move faster. In the solid they vibrate faster and with a larger amplitude. In all three states this pushes the particles further apart.
Half two, the solid: the particles are close together in fixed positions, held by strong forces. The forces pull them back, so their average separation increases only a little.
The liquid: the particles are close together but move around each other, and the forces between them are weaker than in the solid. The same temperature rise moves them further apart than in the solid.
The gas: the particles are far apart and the forces between them are negligible. At constant pressure nothing holds them together, so they spread out into a much larger volume.
Finish by linking the two orders in one sentence: the weaker the forces between the particles, the more the material expands. Never write that the particles get bigger.

Answer: When heated, the particles in all three states move faster (in the solid they vibrate with a larger amplitude), so they move further apart. In a solid, strong forces hold the particles in fixed positions, so they separate only slightly. In a liquid the forces are weaker, so the particles separate more. In a gas the forces are negligible, so at constant pressure the particles spread much further apart. The weaker the forces, the greater the expansion.

Example 2. A section of a steel bridge is $30$ m long. Each metre of the steel becomes $0.012$ mm longer for every $1$ °C rise in temperature. The temperature of the steel ranges from $- 10$ °C on a winter night to $40$ °C on a summer afternoon. (a) Calculate the increase in length of the section between its coldest and its hottest. (b) Explain why a gap is left at one end of the section.

Show the working

Find the temperature rise first. Subtracting a negative number adds it, so the rise is $50$ °C, not $30$ °C.
$Δ θ = 40 - (- 10) = 50^{\circ} C$
The data give the growth of one metre for one degree. For $50$ degrees, one metre grows $50$ times as much.
$0.012 \times 50 = 0.60 mm for each metre$
The section is $30$ m long, so it grows $30$ times as much as one metre does.
$30 \times 0.60 = 18 mm$
Check the unit. The data are per metre and give millimetres, so multiply by the length in metres and the answer comes out in mm. If your answer is $1000$ times too big, you used the length in mm instead of in m.
For (b), name the expansion and then the damage it would do. On hot days the steel expands and the section gets longer. The gap gives it room to do that. Without the gap, the expanding steel would push on the road and the supports with a very large force, and the bridge or its supports could buckle or crack.

Answer: (a) $18$ mm. (b) The steel expands on hot days; the gap gives the section room to get longer, so it does not push on its supports with very large forces and buckle or crack.

Example 3. A straight bimetallic strip is made of brass and steel fixed firmly together, with the brass underneath. One end is clamped and the strip is horizontal at $20$ °C. For the same temperature rise, brass expands more than steel. (a) Describe what happens to the strip when it is heated to $80$ °C. (b) Predict what happens to the strip when it is cooled to $0$ °C.

Show the working

Start from the rule: the metal that expands more ends up on the outside of the curve. The outside of a curve is longer than the inside, and the metal that has grown more is the longer layer.
On heating, the brass becomes longer than the steel. The brass is underneath, so the underneath must become the outside of the curve. The strip curves upwards, and its free end rises.
On cooling below $20$ °C, the brass also contracts more than the steel. Now the brass is the shorter layer, so it ends up on the inside of the curve. The strip curves downwards, and its free end falls.
The trap is to say that the strip bends towards the metal that expands more. On heating it bends away from the brass, towards the steel side, because the brass becomes the outer, longer side.

Answer: (a) The strip bends upwards into a curve, with the brass on the outside (underneath) of the curve. (b) The strip bends downwards, with the brass on the inside of the curve.

Where people go wrong

Mistake: Writing that the particles get bigger when a material is heated.

Instead: The particles stay the same size. They move faster or vibrate more, and the average distance between them increases.

Expansion is about the spaces between particles. A statement that the particles themselves expand contradicts the correct particle explanation, so it can cancel a mark you would otherwise score.

Mistake: Thinking that a heated object gains mass, or that its density stays the same.

Instead: The mass stays the same. The volume increases, so the density decreases.

Nothing is added to the material when it is heated. The same mass spread over a larger volume gives a smaller mass per unit volume.

Mistake: Saying that a bimetallic strip bends towards the metal that expands more.

Instead: It bends so that the metal that expands more is on the outside of the curve, which means it bends towards the other metal.

The outside of a curve is longer than the inside. The metal that has grown more is longer, so it must be on the outside.

Mistake: Saying that any gas expands when it is heated, even one sealed in a rigid container.

Instead: A gas expands when heated at constant pressure. In a sealed rigid container its volume cannot change, so its pressure increases instead.

Expansion needs room. The objective's condition, 'at constant pressure', is there because a gas that cannot push its container outwards cannot expand.

Mistake: Explaining the order of expansion only by saying that the particles move faster when heated.

Instead: Add the forces: strong in a solid, weaker in a liquid, negligible in a gas. The weaker the forces, the more the particles move apart.

The particles move faster in all three states. Something that is the same for all three cannot explain why the amounts are different.

Check yourself

A metal block is heated. State what happens to (a) its mass, (b) its volume, (c) its density.
Answer
(a) The mass stays the same. (b) The volume increases. (c) The density decreases, because the same mass now takes up a larger volume.
Equal volumes of a solid, a liquid and a gas at constant pressure are heated through the same temperature rise. Put them in order, starting with the one that expands least.
Answer
Solid (least), then liquid, then gas (most).
Some air is sealed inside a strong, rigid steel container and heated. Does the air expand? What happens instead?
Answer
No. The volume of the container cannot change, so the air cannot expand. Its pressure increases instead. A gas can expand only if it is free to push its container outwards, as it is at constant pressure.
Each metre of an aluminium rod becomes $0.023$ mm longer for every $1$ °C rise in temperature. The rod is $2.0$ m long. Calculate its increase in length when it is heated from $20$ °C to $120$ °C.
Answer
Temperature rise $= 120 - 20 = 100$ °C. Increase $= 2.0 \times 0.023 \times 100 = 4.6$ mm.
A student writes: 'A solid expands when it is heated because its particles get bigger.' Correct the statement.
Answer
The particles do not get bigger. When the solid is heated, its particles vibrate faster and with a larger amplitude, so the average distance between them increases.
Explain, in terms of particles, why a gas at constant pressure expands much more than a solid for the same temperature rise.
Answer
In both, heating makes the particles move faster. In the solid, strong forces hold the particles in fixed positions, so they move only slightly further apart. In the gas, the forces between the particles are negligible, so nothing holds them together and they spread much further apart.
Explain why overhead power cables are hung with some slack instead of being pulled tight.
Answer
In cold weather the cables contract and get shorter. A cable that was already tight would become even tighter and could snap or pull on its pylons. The slack leaves room for the contraction.
Explain why running hot water over a tight metal lid helps to open a glass jar.
Answer
For the same temperature rise, the metal lid expands more than the glass. The lid gets slightly bigger than the rim of the jar, so it loosens.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. Three samples, X, Y and Z, are a metal, a liquid and a gas, in some order. Each sample has a volume of $500$ cm³ at $20$ °C. Each is heated to $60$ °C, and the gas is kept at constant pressure. The increase in volume of each sample is: X, $68$ cm³; Y, $0.7$ cm³; Z, $7.6$ cm³.

(a) Identify which sample is the gas and which sample is the metal. 2 marks No calculator

Mark scheme

B1 the gas is X (largest increase in volume)
B1 the metal is Y (smallest increase in volume)

(b) State what happens to (i) the mass and (ii) the density of sample Z as it is heated. 2 marks No calculator

Mark scheme

B1 (i) the mass stays the same / does not change
B1 (ii) the density decreases

(c) Explain, in terms of the motion and arrangement of the particles, why sample X expands much more than sample Y. 3 marks No calculator

Mark scheme

B1 when heated, the particles move faster / gain kinetic energy / (in Y) vibrate with a larger amplitude
B1 in Y (the metal) the particles are close together and held in fixed positions by strong forces
B1 in X (the gas) the particles are far apart / the forces between the particles are (very) weak / negligible / the particles are free to move apart, so their separation increases much more; ORA; 'the particles get bigger' is CON

(d) Sample X is now sealed inside a strong, rigid metal container and heated again from $20$ °C to $60$ °C. Predict what happens to (i) the volume and (ii) the pressure of the gas. 2 marks No calculator

Mark scheme

B1 (i) the volume stays the same / does not change
B1 (ii) the pressure increases

Question 2. A thermostat in an electric heater contains a bimetallic strip made of brass and steel fixed firmly together. One end of the strip is fixed. At room temperature the strip is straight and its free end presses on a contact, which completes the heater circuit. The brass is on the side of the strip nearest the contact. For the same temperature rise, brass expands more than steel.

(a) Describe what happens to the shape of the strip as the room warms up. 2 marks No calculator

Mark scheme

B1 the strip bends / curves
B1 with the brass on the outside of the curve / bends towards the steel side / bends away from the contact

(b) Explain how the strip switches the heater off when the room becomes too warm. 2 marks No calculator

Mark scheme

B1 the free end of the strip moves away from the contact
B1 so the circuit is broken / there is no current in the heater

(c) State why the strip must be made of two different metals. 1 mark No calculator

Mark scheme

B1 the two metals expand by different amounts for the same temperature rise (so the strip bends); a strip of one metal would only get longer, owtte

Question 3. A steel road bridge is built in sections, each $40$ m long. Each metre of the steel becomes $0.012$ mm longer for every $1$ °C rise in temperature. Over a year, the temperature of the steel ranges from $- 8$ °C to $37$ °C.

(a) Calculate the increase in length of one section between its lowest and its highest temperature. 3 marks

Mark scheme

C1 temperature rise = 37 − (−8) = 45 °C
C1 40 × 0.012 × 45 seen, in any order (e.g. 0.54 × 40 or 0.48 × 45), or 40 × 0.012 × their temperature rise
A1 21.6 mm (accept 22 mm, 2.16 cm or 0.0216 m); unit required

(b) One end of each section is fixed, and the other end rests on rollers. Explain why the second end is not fixed as well. 2 marks No calculator

Mark scheme

B1 the section expands on hot days / contracts on cold days / changes length as the temperature changes
B1 the rollers let that end move, so there are no very large forces on the supports / the bridge does not buckle or crack

(c) Give one other example of something that is built to allow for thermal expansion or contraction. 1 mark No calculator

Mark scheme

B1 any one of: gaps between lengths of railway line / gaps between concrete road slabs / overhead cables hung with slack / pipes with expansion loops / containers of liquid not filled to the top / any other correct example, owtte; ignore other bridges

How this is examined

Why a gas expands far more than a solid, explained with particles: a two- or three-mark Explain that names two states and asks for the answer in terms of particles Three separate points: heating makes the particles move faster (in a solid, vibrate with a larger amplitude); the forces between particles are strong in a solid (weaker in a liquid) and negligible in a gas; so the gas particles move much further apart. Writing that the particles get bigger contradicts a correct point and cancels it.
How a bimetallic strip bends, and how that switches a circuit: a two- or three-mark Describe or Explain, with a diagram of a strip and a contact in a thermostat The two metals expand by different amounts for the same temperature rise; the strip bends with the metal that expands more on the outside of the curve; the free end moves away from the contact and breaks the circuit (or towards it and completes the circuit).
Why bridges, rails, roads, cables and pipes are built with gaps or slack: a two-mark Explain or Suggest about one named structure Two links: name the expansion on hot days (or the contraction on cold days); then name what the gap or slack prevents, such as buckling, cracking, snapping, or very large forces on the supports.
The order of expansion, and what happens to mass and density: a one- or two-mark State or Compare, or a Deduce from a table of volume increases For the same temperature rise: solid expands least, liquid more, gas most. When anything expands, its mass stays the same and its density decreases.
Why a gas sealed in a rigid container does not expand: a one- or two-mark Predict about a gas heated in a closed, rigid container The volume stays the same; the pressure increases. A gas expands only if it is free to push its container outwards, as under a freely sliding piston at constant pressure.

Words to know

Thermal expansion: The increase in size of a material when its temperature rises. Its mass stays the same, so its density decreases.
Contraction: The decrease in size of a material when its temperature falls.
At constant pressure: With the pressure kept the same, so that a gas is free to push its container outwards and increase its volume, as under a freely sliding piston.
Amplitude (of vibration): The greatest distance a particle in a solid moves from its fixed position as it vibrates. Heating increases it.
Expansion joint: A gap left in a bridge, road or other structure so that it can expand without pushing on its supports.
Bimetallic strip: Two different metals fixed firmly together along their length. Because they expand by different amounts, the strip bends when its temperature changes.
Thermostat: A device that keeps a temperature near a set value, for example by switching a heater off when it gets too warm and back on when it cools.

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