1.3.1 Molecular motion, temperature and gas pressure

Gas molecules

Definition

Closed system

A system that no matter can enter or leave, so its total mass stays constant.

A gas is made of tiny molecules that are very far apart compared with their own size.
The molecules move randomly in all directions at high speed, travelling in straight lines between collisions.
At room temperature a typical air molecule moves at about $m/s500\ \text{m/s}$ , which is faster than a passenger jet.
The molecules do not all move at the same speed, because collisions keep sharing energy between them, so a gas always contains a range of speeds.
Each molecule changes direction every time it collides with another molecule or with a wall of its container.
Because the molecules are moving, they store energy in their kinetic energy stores.
In this topic a gas is treated as a closed system, so no molecules enter or leave its container.
With the number of molecules fixed, any change in the gas has to come from a change in how the existing molecules move.

Gas molecules moving in random directions inside a closed box, with starbursts marking collisions with the walls. Each of these collisions pushes outwards on the wall, and together they produce the pressure of the gas.

Motion and temperature

Definition

Temperature

A measure of how hot or cold an object is, which depends on the average kinetic energy of its particles.

The temperature of a gas is linked to the average kinetic energy of its molecules.
The faster the molecules move on average, the greater their average kinetic energy and the higher the temperature of the gas.
Heating a gas transfers energy to the kinetic energy stores of its molecules, so they speed up and the temperature rises.
Cooling a gas transfers energy away from its molecules, so they slow down and the temperature falls.
Temperature describes the average motion, so at any instant some molecules are moving faster than average and some are moving slower.
Temperature is measured in degrees Celsius, $∘C^\circ\text{C}$ , using a thermometer.

Motion and pressure

Definition

Pressure

The force acting at right angles to a surface on each unit area of that surface.

Definition

Gas pressure

The force per unit area that gas molecules exert on the walls of their container when they collide with them.

Definition

Pascal

The unit of pressure, equal to a force of one newton acting at right angles on an area of one square metre.

When a molecule hits a wall it bounces off in a new direction, and at the same moment it pushes on the wall, so every collision exerts a tiny force on the wall.
One collision is far too small to notice, but billions of collisions every second add up to a steady force on every part of the wall.
Pressure is the force acting at right angles on each unit area of a surface.
Gas pressure is therefore the total force from the collisions on a wall divided by the area of that wall.
Pressure is measured in pascals, $Pa\text{Pa}$ , where $N/m21\ \text{Pa} = 1\ \text{N/m}^2$ .
The molecules move randomly in every direction, so they strike all the walls of a container equally often and the gas pushes with the same pressure on every wall.
The size of the pressure depends on two features of the collisions:
1. how often the molecules hit each square metre of wall every second;
2. how hard each molecule hits the wall.
Faster molecules cross the container in less time, so they reach the walls more often.
Faster molecules also hit the walls harder, so each collision exerts a larger force.
The speed of the molecules is therefore the link between the two quantities: it sets the temperature of the gas and it controls the pressure the gas exerts.
Slower molecules hit the walls less often and less hard, so a gas whose molecules slow down exerts a lower pressure on any wall that cannot move away from it.

Practical

Heating and cooling trapped air

Aim: to show how heating and cooling a fixed amount of trapped air changes the motion of its molecules and the pressure they exert, first with balloons and then with a manometer.
Apparatus: two identical balloons, a piece of string and a ruler, a freezer, a warm radiator or a bowl of water at about $∘C50\ ^\circ\text{C}$ , a conical flask with a one-holed bung and glass tube, rubber tubing, a glass U-tube clamped in front of a metre rule, coloured water, two large troughs, a kettle, ice, petroleum jelly and eye protection.
Method, the balloons:
- Inflate both balloons to the same size, tie them, and measure the circumference of each around its widest part using the string and ruler.
- Put one balloon in a freezer for about $20$ minutes and hold the other close to a warm radiator, or float it in the warm water, for the same time.
- Take each balloon out and measure its circumference again straight away.
- Squeeze each balloon gently and compare how firm it feels.
- Leave both balloons at room temperature for another $20$ minutes and measure them again to check that the changes reverse.
Method, the manometer:
- Pour coloured water into the U-tube until both arms are about half full and the two levels are equal.
- Smear the bung with petroleum jelly, push it firmly into the flask, and connect the glass tube to one arm of the U-tube with the rubber tubing, leaving the other arm open to the air.
- Check that the two levels are still equal, which shows that the trapped air starts at the same pressure as the air in the room.
- Fill one trough with hot water from the kettle and hold the flask in it so that the whole flask is under the surface.
- Record the difference in height between the two liquid levels with the metre rule every $30$ seconds until it stops changing.
- Move the flask into a trough of ice and water and record the level difference again as the trapped air cools.
Results: the frozen balloon shrinks and feels softer, while the warmed balloon grows slightly and feels firmer.
Results: when the flask is heated, the liquid is pushed down in the arm joined to the flask and rises in the open arm, and when the flask is cooled the liquid moves the opposite way.
Explanation: heating gives the air molecules more kinetic energy, so they move faster and hit the rubber or the liquid surface more often and harder, which raises the pressure.
Explanation: cooling slows the molecules down, so they hit less often and less hard, and the pressure falls.
Reading the manometer: a bigger difference between the two levels means a bigger difference between the pressure of the trapped air and the pressure of the room.
Watch out: a leaking bung lets air escape, so the gas is no longer a closed system and the level difference is smaller than it should be.
Watch out: read each level at eye level from the bottom of the curved meniscus, so that every reading is taken in the same way.
Safety: wear eye protection, use heatproof gloves near the hot water, and never heat the sealed flask directly over a Bunsen burner, because the pressure inside could force out the bung or crack the glass.

Common Mistake

Do not say that the molecules get bigger when a gas is heated, because the molecules stay the same size and only their speed changes.
Do not say that heating produces more molecules, because in a closed system the number of molecules never changes.
Do not write "more collisions" on its own; say the molecules collide with the walls more often, because the change is in the number of collisions each second.
Do not say gas molecules vibrate, because they move freely through the whole container.

Exam technique

Explaining gas pressure

Write the explanation as a chain: the molecules move, they collide with the walls, each collision exerts a force, and the force on each unit area is the pressure.
When the molecules speed up, mention both effects on the collisions, saying the molecules hit the walls more often and with more force, because each effect usually carries its own mark.
Say the molecules gain kinetic energy rather than writing "heat energy", which earns no credit.
Use either "molecules" or "particles" and stick with it, rather than switching to "atoms".

Self review

Describe how the molecules in a gas move.
State what the temperature of a gas tells you about its molecules.
Explain how molecules colliding with a wall produce gas pressure.
Name the two features of the collisions that decide the size of the gas pressure.
State the unit of pressure and what it is equal to in newtons and square metres.

1.3.2 Gas pressure at constant volume

A fixed volume of gas

Definition

Gas pressure

The force per unit area that gas molecules exert on the walls of their container when they collide with them.

A gas has a constant volume when it is sealed inside a rigid container whose walls cannot move, such as a steel gas cylinder, a sealed metal can or a stoppered glass flask.
Because the container is sealed, the number of molecules inside stays the same as well.
With the volume and the number of molecules both fixed, the only way the pressure can change is through a change in the speed of the molecules.
The speed of the molecules is set by the temperature, so at constant volume the pressure of a gas depends on its temperature.

Gas molecules trapped in a rigid box. The walls cannot move, so any change in how fast the molecules travel changes how often and how hard they strike the walls, and so changes the pressure.

Raising the temperature

Definition

Temperature

A measure of how hot or cold an object is, which depends on the average kinetic energy of its particles.

Heating the container transfers energy to the gas and increases the kinetic energy of its molecules.
The molecules move faster on average, which is what a rise in temperature means.
Faster molecules cross the container in less time, so each one hits the walls more often.
Faster molecules also hit the walls harder, so each collision exerts a greater force.
More frequent collisions and a greater force per collision together increase the total force on each square metre of wall.
The pressure of the gas therefore increases, even though its volume and the number of molecules have not changed.
You only need to explain this relationship in words, so no calculation is needed for a change of temperature at constant volume.

Lowering the temperature

Cooling the container transfers energy away from the gas, so its molecules move more slowly on average.
Slower molecules hit the walls less often and less hard.
The total force on each square metre of wall falls, so the pressure decreases.
Warming the gas back to its starting temperature returns the pressure to its starting value, because nothing has left the sealed container.
The pattern holds in both directions: at constant volume, a higher temperature gives a higher pressure and a lower temperature gives a lower pressure.

Constant volume in everyday life

A car tyre holds a fixed amount of air in a casing that barely stretches, so its pressure rises during a long motorway drive as the tyre warms up.
This is why drivers are told to check tyre pressures when the tyres are cold, because a reading taken straight after a journey is higher than normal.
Aerosol cans warn you not to leave them in temperatures above $∘C50\ ^\circ\text{C}$ , because heating the sealed can raises the pressure inside and could make it burst.
A gas cylinder for a barbecue is stored in the shade for the same reason.
A plastic bottle sealed on a warm day and then put in a fridge is squashed slightly, because the pressure of the cooled air inside falls below the pressure of the air outside.

Example

Heating a sealed can

A sealed metal can of air at $∘C20\ ^\circ\text{C}$ is left on a hot radiator and warms to $∘C60\ ^\circ\text{C}$ .
The can is rigid and sealed, so the volume of the air and the number of molecules stay the same.
Energy is transferred from the radiator to the air, so the air molecules gain kinetic energy and move faster.
The faster molecules strike the inside of the can more often and with a greater force.
The force on each square metre of the can increases, so the pressure of the air inside rises.
If the heating continued, the pressure could keep rising until the can bulged or split along its weakest seam.

Common Mistake

Do not say the molecules expand when the gas is heated, because the molecules stay the same size and the volume of the container is fixed.
Do not say the pressure rises because there are more molecules, because the container is sealed and the number of molecules has not changed.
Do not explain the rise only by "more collisions"; say the collisions happen more often, and add that each one exerts a greater force.
Do not use this explanation for a balloon or a syringe, because their volume can change and the gas can expand instead.

Exam technique

Writing the full chain

A typical three-mark explanation rewards three linked points: the molecules move faster, they collide with the walls more often or harder, and so the pressure increases.
State that the volume stays the same, because that is why the faster molecules cannot simply spread out.
If the question mentions energy, say the molecules gain kinetic energy rather than "heat".
For a cooling question, write the same chain in reverse instead of saying "the opposite happens".

Self review

State what is meant by a gas at constant volume.
Explain why the pressure of a gas in a sealed rigid container rises when it is heated.
Describe what happens to the pressure of the gas when the container is cooled.
Explain why drivers should check tyre pressures before a journey rather than straight after one.

1.3.3 Compressing and expanding gases

Why gases can be compressed

Definition

Compression

A decrease in the volume of a substance caused by a force or pressure acting on it.

Definition

Expansion

An increase in the volume of a substance.

The molecules in a gas are very far apart, with mostly empty space between them.
Pushing on a gas forces its molecules closer together, so the same amount of gas fits into a smaller volume.
This decrease in volume is called compression.
The molecules themselves are not squashed, because only the spaces between them get smaller.
If the push on a gas is reduced, its molecules spread out into a larger volume, which is called expansion.
Liquids and solids are very hard to compress, because their particles are already touching and there is almost no space left to close up.
Gases can be compressed into a much smaller volume, which is how a diver's tank holds enough air for a long dive.

Pressure changes and volume

A gas changes its volume until the pressure it exerts balances the pressure acting on it from outside.
If the pressure outside a gas increases, the gas is compressed until its own pressure rises to match.
If the pressure outside a gas decreases, the gas expands until its own pressure falls to match.
Pushing the plunger of a sealed syringe increases the pressure on the trapped air, so the air is compressed into a smaller volume.
A sealed packet of crisps puffs up on an aeroplane, because the air pressure in the cabin is lower than on the ground and the air inside the packet expands.
A partly blown-up balloon placed in a sealed bell jar grows larger as a vacuum pump removes air from the jar, because the pressure around the balloon falls.
When air is let back into the jar, the pressure outside the balloon rises again and the balloon shrinks to its original size.

Force at right angles to a surface

Definition

Pressure difference

The difference between the pressures on the two sides of a surface, which produces a net force on the surface towards the side at lower pressure.

Gas pressure pushes on every surface the gas touches, including the walls of its container and the face of a piston or plunger.
The force produced by pressure always acts at right angles to the surface, which is also called normal to the surface.
This happens because molecules strike the surface from every direction, so on average their sideways pushes along the surface cancel and only the push straight into the surface remains.
On the flat face of a plunger, the force from the trapped gas therefore acts straight along the barrel of the syringe.
On the curved inside of a balloon, the force acts outwards at right angles to the rubber at every point, which is why a balloon inflates into a rounded shape.
Most surfaces have gas pressing on both sides, so each side receives a force at right angles to it.
If the pressures on the two sides are equal, the two forces cancel and there is no net force on the surface.
If the pressures are different, there is a pressure difference across the surface, and the force from the higher-pressure side is larger.
This gives a net force at right angles to the surface, directed towards the side with the lower pressure.
A syringe plunger stays still while the pressure of the trapped air equals the pressure of the air outside, and it moves as soon as one of these pressures changes.

Example

Squeezing a sealed syringe

A syringe holds $cm350\ \text{cm}^3$ of air with its nozzle sealed, and the plunger is pushed in until the air fills only $cm330\ \text{cm}^3$ .
The push on the plunger increases the pressure acting on the trapped air.
The air is compressed, because its molecules are pushed closer together into the smaller space.
The trapped air now pushes back on the plunger with a larger force at right angles to its face, which you feel as resistance.
When the plunger is released, the pressure inside is greater than the pressure outside, so the net force pushes the plunger back out until the two pressures balance again.

Practical

Compressing sand, water and air

Aim: to compare how easily a gas, a liquid and a solid in grains can be compressed, using three identical sealed syringes.
Apparatus: three identical $cm360\ \text{cm}^3$ plastic syringes without needles, three syringe caps, dry sand, water, a clamp stand with a boss and clamp, a $kg1\ \text{kg}$ slotted mass with a hanger and eye protection.
Method:
- Pull the plunger of the first syringe back to the $cm350\ \text{cm}^3$ mark so that it fills with air, then push a cap firmly onto the nozzle.
- Fill the second syringe with water to the $cm350\ \text{cm}^3$ mark, tap out any air bubbles, and cap it.
- Fill the third syringe with dry sand to the $cm350\ \text{cm}^3$ mark, tap it gently so that the sand settles, and cap it.
- Clamp the first syringe upright with its nozzle at the bottom, and read the starting volume at eye level.
- Rest the $kg1\ \text{kg}$ mass on top of the plunger and read the new volume once the plunger stops moving.
- Lift the mass off and record whether the plunger returns to its starting position.
- Repeat with the water and sand syringes, then repeat each test three times and work out a mean change in volume for each material.
Control variables: use identical syringes, the same starting volume and the same mass each time, so that the only thing that changes is the material inside.
Results: the air is compressed by a large amount and the plunger springs back when the mass is removed, the water hardly moves at all, and the sand is compressed slightly but does not spring back.
Explanation: the air molecules are far apart, so there is plenty of space for them to be pushed closer together.
Explanation: the water particles are already touching, so there is almost no space to close up and the volume barely changes.
Explanation: the sand grains themselves are not compressed, because the small change happens as the grains shift into a tighter arrangement and squeeze the air in the gaps between them.
Watch out: an air bubble left in the water syringe makes the water seem compressible, and a loose cap lets air escape so the air seems more compressible than it is.
Safety: wear eye protection, because a cap can fly off a strongly compressed syringe, and wipe up spilt water and sand straight away so that no one slips.

Common Mistake

Do not say the gas molecules get smaller when a gas is compressed, because it is the spaces between them that shrink.
Do not say a low pressure "sucks" a surface, because the net force comes from the higher pressure on the other side pushing it.
Do not draw the force from a gas along a surface or at an angle to it, because pressure always produces a force at right angles to the surface.
Do not describe sand as easily compressed, because only the gaps between the grains close up.

Exam technique

Describing compression and expansion

Link a change in volume to a change in pressure in the same sentence, such as "the pressure outside increases, so the gas is compressed".
When asked why a gas is easier to compress than a liquid, refer to the spacing of the particles, not to their size or mass.
When you draw or describe a pressure force, show it at right angles to the surface.
For a net force question, compare the pressures on the two sides of the surface and say which side is higher.

Self review

Explain why a gas can be compressed but a liquid can hardly be compressed at all.
State what happens to a sealed volume of gas when the pressure around it decreases.
State the direction of the force that gas pressure produces on a surface.
Explain why a syringe plunger moves when the pressure inside differs from the pressure outside.
Explain why sand in a sealed syringe can be compressed slightly.

1.3.4 Gas volume and pressure at constant temperature

A fixed temperature

Definition

Gas pressure

The force per unit area that gas molecules exert on the walls of their container when they collide with them.

Picture a fixed amount of gas trapped in a cylinder by a movable piston, with the temperature of the gas held constant.
A constant temperature means the molecules keep the same average speed and the same average kinetic energy throughout.
Each collision with a wall is therefore just as hard as before, whatever happens to the volume.
The pressure can still change, because the volume decides how often the molecules reach the walls.

Increasing the volume

Pulling the piston out gives the same number of molecules a larger volume to move around in.
The molecules are now more spread out, so there are fewer molecules in each cubic centimetre of gas.
Each molecule has further to travel between one collision with a wall and the next.
The walls also have a larger area, so the collisions are shared out over more surface.
As a result, each square metre of wall is hit less often every second.
Each collision exerts the same force as before, because the molecules are moving at the same speed.
Fewer collisions per second on each square metre give a smaller force per unit area, so the pressure decreases.
Pushing the piston in has the opposite effect: the molecules are crowded into a smaller volume, hit each square metre of wall more often, and the pressure increases.

Pressure and volume

Definition

Inversely proportional

A relationship between two quantities in which their product stays constant, so doubling one of them halves the other.

For a fixed mass of gas at constant temperature, pressure and volume are inversely proportional.
Doubling the volume halves the pressure, and halving the volume doubles the pressure.
The relationship is $pressure×volume=constant\text{pressure} \times \text{volume} = \text{constant}$ $pressure \times volume = constant$ , or $pV=constantpV = \text{constant}$ $p V = constant$ .
1. $p$ is the pressure in pascals, $Pa\text{Pa}$ .
2. $V$ is the volume in cubic metres, $m3\text{m}^3$ .
If the gas starts at pressure $p_1$ and volume $V_1$ and changes to pressure $p_2$ and volume $V_2$ , the product stays the same, so $p_1 V_1 = p_2 V_2$ .
Any units can be used in $p_1 V_1 = p_2 V_2$ , such as $cm3\text{cm}^3$ or $kPa\text{kPa}$ , as long as both volumes are in the same unit and both pressures are in the same unit.
You do not need to memorise this equation, but you must be able to choose it from a list of equations and use it.
The equation only works if the temperature stays constant and no gas leaks in or out.
A graph of pressure against volume is a curve that falls steeply at first and then levels off, getting closer to the axes without ever touching them.
A graph of pressure against $1V\dfrac{1}{V}$ is a straight line through the origin, which is the clearest test that pressure and volume are inversely proportional.
Compressing a gas quickly makes it warm up, so in an experiment the volume must be changed slowly to keep the temperature constant.

Example

Compressing air in a cylinder

A cylinder holds $m30.060\ \text{m}^3$ of air at a pressure of $Pa100\,000\ \text{Pa}$ .
The piston is pushed in slowly until the volume is $m30.020\ \text{m}^3$ , and the temperature does not change.

p_1 V_1 = p_2 V_2

100\,000 \times 0.060 = p_2 \times 0.020

p_2 = \dfrac{100\,000 \times 0.060}{0.020} = 300\,000\ \text{Pa}

The volume has been divided by $3$ , so the pressure has been multiplied by $3$ , which matches the inverse relationship.

Example

Finding a new volume

A syringe contains $cm340\ \text{cm}^3$ of air at $kPa100\ \text{kPa}$ .
The plunger is pulled out slowly until the pressure falls to $kPa80\ \text{kPa}$ , at constant temperature.
Rearrange $p_1 V_1 = p_2 V_2$ to make $V_2$ the subject.

V_2 = \dfrac{p_1 V_1}{p_2}

V_2 = \dfrac{100 \times 40}{80} = 50\ \text{cm}^3

Both pressures are in kilopascals and the volume is in $cm3\text{cm}^3$ , so the answer comes out in $cm3\text{cm}^3$ with no conversion needed.

Practical

Pressure and volume of trapped air

Aim: to measure how the pressure of a fixed mass of air changes as its volume is changed at constant temperature.
Apparatus: a $cm360\ \text{cm}^3$ plastic syringe, a short length of thick-walled rubber tubing, a pressure sensor connected to a data logger or a pressure gauge reading in $kPa\text{kPa}$ , a clamp stand with a boss and clamp, and a thermometer.
Method:
- Set the plunger at the $cm350\ \text{cm}^3$ mark and connect the nozzle to the pressure sensor with the shortest possible length of tubing.
- Clamp the syringe so that its scale can be read at eye level, then record the starting volume, pressure and room temperature.
- Push the plunger in slowly to $cm345\ \text{cm}^3$ and hold it still for about $30$ seconds so that the air returns to room temperature.
- Record the pressure, then repeat in steps of $cm35\ \text{cm}^3$ down to about $cm325\ \text{cm}^3$ .
- Pull the plunger back out through the same volumes, recording the pressure again at each step, and work out a mean pressure for each volume.
- Multiply each mean pressure by its volume, then plot one graph of pressure against volume and another of pressure against $1V\dfrac{1}{V}$ .
Variables: the independent variable is the volume, the dependent variable is the pressure, and the control variables are the temperature and the mass of trapped air.
Results: the product $p V$ comes out roughly the same for every reading, the graph of $p$ against $V$ is a falling curve, and the graph of $p$ against $1V\dfrac{1}{V}$ is a straight line through the origin.
Watch out: the air inside the tubing is not shown on the syringe scale, so every volume reading is slightly too small, which is a systematic error that short tubing keeps small.
Watch out: pushing the plunger too quickly warms the air and raises the pressure above its true value, so always wait before taking a reading.
Watch out: a leak lets air escape at high pressure and makes $p V$ drift downwards, so check that every joint is tight before you start.
Safety: do not compress the air beyond the range of the sensor or gauge, and keep your face away from the joints in case the tubing slips off.

Common Mistake

Do not say the pressure falls because the molecules slow down, because the temperature is constant and their average speed does not change.
Do not say the extra space fills up with air or anything else, because the same molecules simply spread out through a larger volume.
Do not use $p_1 V_1 = p_2 V_2$ if the temperature changes or if gas escapes.
Do not mix units, because if $V_1$ is in $cm3\text{cm}^3$ then $V_2$ must be in $cm3\text{cm}^3$ too.

Exam technique

Explaining a pressure change at constant temperature

Say that the molecules are more spread out or travel further between collisions, then link this to fewer collisions with the walls each second.
State that the speed of the molecules does not change because the temperature is constant, since this is the step that weaker answers leave out.
In calculations, write $p_1 V_1 = p_2 V_2$ , substitute both pairs of values, and only then rearrange, so that each step can earn credit.
Check that the answer makes sense, because a smaller volume must always give a larger pressure.

Self review

Explain why increasing the volume of a gas at constant temperature decreases its pressure.
State what happens to the speed of the molecules when a gas is compressed slowly at constant temperature.
Write the relationship between pressure and volume for a fixed mass of gas at constant temperature.
A gas at $kPa200\ \text{kPa}$ occupies $cm330\ \text{cm}^3$ ; calculate its pressure when its volume is increased to $cm360\ \text{cm}^3$ at the same temperature.
Describe the shape of a graph of pressure against volume for a gas at constant temperature.

1.3.5 Doing work on a gas

Work and energy

Definition

Work done

The energy transferred when a force moves an object through a distance in the direction of the force.

Definition

Internal energy

The total energy stored by the particles of a system in their kinetic and potential energy stores.

Work is done whenever a force moves an object through a distance in the direction of the force.
Doing work is one way of transferring energy, and the energy transferred is equal to the work done.
Work done is measured in joules, $J\text{J}$ , the same unit as energy.
Work is done on a gas when a force pushes a piston or plunger into it and makes its volume smaller.
The energy transferred by this work goes into the internal energy of the gas.
In a gas, the internal energy is almost entirely the kinetic energy of its fast-moving molecules.

Why the temperature rises

Definition

Temperature

A measure of how hot or cold an object is, which depends on the average kinetic energy of its particles.

As the piston moves inwards, molecules keep colliding with its moving face.
A molecule that hits a piston moving towards it rebounds faster than it arrived, in the same way that a tennis ball leaves a racket faster when the racket is swung into it.
Each of these collisions transfers energy from the piston to a molecule, so the molecules gain kinetic energy.
The average kinetic energy of the molecules therefore increases.
Temperature depends on the average kinetic energy of the molecules, so the temperature of the gas rises.
The energy comes from whatever pushes the piston, such as the chemical energy store of your muscles when you work a pump.
The faster the gas is compressed, the larger the rise in temperature, because there is less time for energy to be transferred to the surroundings by heating.
If the gas is compressed very slowly, energy leaves through the walls almost as fast as it arrives, so the temperature hardly changes.
The reverse also happens: a gas that expands and pushes a piston outwards does work on its surroundings, so its molecules lose kinetic energy and the gas cools.

Examples of doing work on a gas

Bicycle pump: pushing the handle down quickly compresses the air in the barrel, so after a few strokes the barrel feels warm, especially near the valve end where the air is squeezed most.
Some of that warmth comes from friction between the plunger seal and the barrel, but the work done compressing the air adds to it.
Fire piston: slamming the plunger of a sealed, clear tube down very quickly raises the temperature of the trapped air by several hundred degrees, enough to set a small piece of cotton wool alight.
Diesel engine: the air in each cylinder is compressed so strongly and so quickly that it becomes hot enough to ignite the fuel sprayed into it, so no spark plug is needed.

Example

Pumping up a bicycle tyre

You push the pump handle down, and the piston moves through a distance in the direction of your push, so you do work on the air inside the barrel.
Energy is transferred mechanically from the chemical energy store of your muscles to the trapped air.
Air molecules rebound faster from the advancing piston, so their average kinetic energy increases.
The internal energy of the air increases and its temperature rises.
Energy is then transferred by heating from the warm air to the barrel, so the barrel feels warm in your hand.
Pumping in short, fast strokes makes the barrel warmer than slow strokes, because less energy escapes during each stroke.

Common Mistake

Do not say the gas gains "heat energy", because the energy is transferred to the internal energy store, or the kinetic energy stores, of the molecules.
Do not explain the temperature rise only by saying the molecules are closer together, because being closer together does not change their speed.
Do not call this heating, because the energy arrives by a force doing work, not because of a temperature difference.

Exam technique

Explaining a temperature rise from work

Start with the cause: a force moves the piston, so work is done on the gas.
Follow the energy through: it is transferred to the gas, the molecules gain kinetic energy and move faster, and the temperature rises.
State the link between the average kinetic energy of the molecules and the temperature, because this is the step most often missed.
If the question says the pump gets hot, finish by saying energy is transferred from the warm air to the pump by heating.

Self review

State what is meant by doing work on a gas.
Explain why molecules gain kinetic energy when they collide with a piston moving towards them.
Explain why compressing a gas quickly raises its temperature.
Explain why a bicycle pump becomes warm when a tyre is pumped up.
State why compressing a gas very slowly causes little change in its temperature.

1.3.6 The Earth's atmosphere and atmospheric pressure

A simple model of the atmosphere

Definition

Atmosphere

The layer of gases surrounding the Earth, held in place by the Earth's gravity.

The Earth is surrounded by a thin layer of air called the atmosphere, which is held in place by the Earth's gravity.
Air is a mixture of gases, mainly nitrogen and oxygen, and like all matter it has mass.
Compared with the size of the Earth the atmosphere is very thin, because most of the air lies within about $km10\ \text{km}$ of the ground while the Earth's radius is about $km6400\ \text{km}$ .
A scientific model simplifies a real system so that its main behaviour can be explained.
In the simple model, the atmosphere is treated as a layer of air with a uniform density, so every cubic metre holds the same mass of air from the ground to the top.
The model also gives the atmosphere a definite top, above which there is no air.
With a uniform density of about $kg/m31.2\ \text{kg/m}^3$ , the model atmosphere would need to be about $km8\ \text{km}$ deep to produce the pressure measured at sea level.
The real atmosphere gets less dense with height and fades away gradually rather than stopping at a sharp edge, so the uniform model is an approximation.

Atmospheric pressure

Definition

Atmospheric pressure

The pressure exerted on a surface by the air of the atmosphere, caused by the weight of the column of air above that surface.

Definition

Weight

The force acting on an object because of gravity, measured in newtons.

Gravity pulls on the air, so every column of air above the ground has weight.
Picture a column of air standing on $m21\ \text{m}^2$ of ground and reaching all the way to the top of the atmosphere.
That column has a mass of about $kg10\,000\ \text{kg}$ , so its weight is about $N100\,000\ \text{N}$ .
This weight pressing down on $m21\ \text{m}^2$ gives an atmospheric pressure at sea level of about $Pa100\,000\ \text{Pa}$ , which is $1.0 \times 10^{5}\ \text{Pa}$ or $kPa100\ \text{kPa}$ .
The same pressure can be described in terms of molecules, because air molecules constantly collide with every surface and each collision exerts a tiny force on it.
The molecules move randomly in all directions, so atmospheric pressure acts in every direction, on the top, the sides and the underneath of objects alike.
You do not feel crushed because the fluids inside your body push outwards with a matching pressure, so the forces on the two sides of your skin balance.

Effects of atmospheric pressure

Definition

Pressure difference

The difference between the pressures on the two sides of a surface, which produces a net force on the surface towards the side at lower pressure.

Atmospheric pressure only has a noticeable effect when there is a pressure difference across a surface.
If the pressure on one side of a surface is lowered, the atmosphere on the other side pushes harder, giving a net force towards the low-pressure side.
A suction cup stays on a smooth tile because pressing it flat pushes most of the air out from underneath, so the atmosphere outside pushes the cup against the tile.
Drinking through a straw works because you lower the pressure inside your mouth, and atmospheric pressure acting on the surface of the drink pushes the liquid up the straw.
When you breathe in, your chest expands and lowers the pressure inside your lungs, so atmospheric pressure pushes air in through your nose and mouth.
In every case the object or liquid is pushed by the higher pressure, because a lower pressure cannot pull.

Practical

The collapsing can

Aim: to show the size of the force that atmospheric pressure exerts when the pressure inside a container is lowered.
Apparatus: an empty, clean aluminium drinks can, a Bunsen burner, a tripod and gauze, a pair of tongs, a large bowl of cold water with some ice in it, heatproof gloves and eye protection.
Method:
- Pour about $cm310\ \text{cm}^3$ of water into the can.
- Stand the can on the gauze and heat it until the water boils strongly and a steady stream of steam has been leaving the opening for about $30$ seconds.
- The steam drives most of the air out of the can, so the can is now filled with steam.
- Grip the can firmly with the tongs and, in one quick movement, turn it upside down and dip its opening just under the surface of the cold water.
- Watch the can closely from a safe distance.
Results: the can is crushed almost instantly, often with a loud crack.
Explanation: the cold water cools the steam, which condenses into a few drops of liquid water, leaving very little gas inside the can.
Explanation: the pressure inside the can falls far below atmospheric pressure, and the water seals the opening so air cannot rush back in.
Explanation: the atmosphere pushes on the outside of the can with a total force of more than $N1000\ \text{N}$ , far more than the thin walls can withstand, so the can collapses inwards.
Watch out: if the water is not boiled for long enough, too much air stays in the can and it only dents.
Safety: wear eye protection, handle the hot can only with tongs and heatproof gloves, and keep your face and hands away from the steam, which scalds.

Common Mistake

Do not say the can is "sucked in" by a vacuum, because it is pushed in by the greater atmospheric pressure outside.
Do not say there is no pressure on you because you cannot feel it, because it is balanced by the pressure inside your body.
Do not say atmospheric pressure acts only downwards, because air molecules collide with surfaces from every direction.

Exam technique

Describing the model and explaining effects

When describing the model, give its two key features: a layer of air around the Earth held by gravity, and a uniform density.
Link the weight of the air above a surface to the pressure on that surface.
For any effect caused by a pressure difference, name the side with the higher pressure and say that it pushes.
Avoid "suck" and "suction" as explanations, because they are not given credit.

Self review

Describe the simple model of the Earth's atmosphere.
Explain what causes atmospheric pressure.
State the approximate value of atmospheric pressure at sea level.
Explain why a heated can collapses when its opening is dipped into cold water.
Explain how atmospheric pressure helps you drink through a straw.

1.3.7 Atmospheric pressure and height

Pressure falls with height

Definition

Altitude

The height of a point above sea level.

Definition

Atmospheric pressure

The pressure exerted on a surface by the air of the atmosphere, caused by the weight of the column of air above that surface.

Atmospheric pressure is greatest at sea level and decreases as altitude increases.
At the top of Ben Nevis, about $m1350\ \text{m}$ above sea level, the pressure is roughly $Pa86\,000\ \text{Pa}$ , compared with about $Pa100\,000\ \text{Pa}$ at sea level.
At the cruising height of an airliner, about $km11\ \text{km}$ , it is only about $Pa23\,000\ \text{Pa}$ , which is why aircraft cabins are pressurised.
The atmospheric pressure at any point is caused by the weight of the air above that point.
Moving upwards leaves part of the atmosphere below you, so the column of air above you is shorter.
A shorter column of air has less mass and so a smaller weight.
A smaller weight of air pressing on each square metre means a lower pressure.

The effect of density

Definition

Density

The mass per unit volume of a material.

Air is a gas and can be compressed, so the air near the ground is squashed by the weight of all the air above it.
This makes the air at low altitude denser, with more molecules packed into each cubic metre.
Higher up there is less air above to do the squashing, so the air is less dense.
With fewer molecules in each cubic metre, fewer molecules collide with each square metre of a surface every second.
Fewer collisions per second give a smaller force on each unit area, so the pressure is lower.
In the simple uniform-density model, the pressure falls by the same amount for every metre you climb and reaches zero at the top of the model atmosphere.
In the real atmosphere the density also falls with height, so the pressure drops quickly near the ground and more slowly higher up, fading gradually instead of stopping at a sharp edge.
About half of the air lies below a height of roughly $km5.5\ \text{km}$ , so the pressure there is about half of its sea-level value.

Effects of lower pressure at altitude

A sealed crisp packet carried up a mountain or onto an aeroplane swells, because the pressure of the air inside it is now greater than the pressure outside.
Your ears pop in a lift, a plane or a car climbing a hill, because the air trapped behind each eardrum briefly has a higher pressure than the air outside until the two pressures equalise.
Climbers high on Everest often use bottled oxygen, because each breath of the low-density air contains far fewer oxygen molecules than a breath at sea level.

Example

Climbing a hill

A walker climbs from a car park at sea level to a summit $m1000\ \text{m}$ higher.
At the summit, $m1000\ \text{m}$ of the atmosphere now lies below the walker, so the column of air above is shorter.
The shorter column has a smaller weight, so it presses on each square metre with a smaller force.
The air at the summit is also less compressed and less dense, so fewer molecules hit each square metre every second.
Both ideas lead to the same result: the atmospheric pressure at the summit is lower, at about $Pa90\,000\ \text{Pa}$ .

Common Mistake

Do not say pressure falls with height because there is no gravity higher up, because gravity at the top of a mountain is almost exactly as strong as at sea level.
Do not say there is no air at high altitude, because there is still air, it is just less dense.
Do not say air is "sucked" out of a crisp packet, because the packet swells as the air inside pushes outwards harder than the air outside pushes in.

Exam technique

Explaining a change with height

Start from the weight of the air above the point: higher up there is less air above, so its weight is smaller.
Add the particle reason: the air is less dense, so there are fewer collisions per second on each square metre.
Finish by naming the result clearly, that the atmospheric pressure is lower.
Use comparative words such as "less", "fewer" and "lower", because a statement about only one height does not answer a comparison question.

Self review

State how atmospheric pressure changes as altitude increases.
Explain, in terms of the air above, why atmospheric pressure is lower at the top of a mountain.
Explain, in terms of molecules, why less dense air exerts a lower pressure.
Explain why a sealed crisp packet swells on an aeroplane.

1.3.8 Floating and sinking

Weight and upthrust

Definition

Weight

The force acting on an object because of gravity, measured in newtons.

Definition

Upthrust

The upward force that a fluid exerts on an object that is partly or fully submerged in it.

Definition

Fluid

A substance that can flow, which means any liquid or gas.

Any object in a fluid, which can be a liquid or a gas, has two vertical forces acting on it.
Its weight acts vertically downwards, caused by gravity pulling on its mass.
The upthrust from the fluid acts vertically upwards.
If the weight is greater than the upthrust, the resultant force is downwards and the object sinks.
If the upthrust is greater than the weight, the resultant force is upwards and the object rises.
If the upthrust equals the weight, the forces are balanced, so the object floats at the surface or stays still within the fluid.
A floating object sinks into the fluid only until the upthrust has grown to match its weight, and then it stays at that level.

Upthrust and displaced fluid

Definition

Displaced fluid

The fluid pushed aside by an object placed in it, which has the same volume as the part of the object below the surface.

An object placed in a fluid pushes some of the fluid out of the way, and this is the displaced fluid.
The upthrust on an object is equal to the weight of the fluid it displaces.
The more of an object that sits below the surface, the more fluid it displaces and the larger the upthrust.
Once an object is completely under the surface it cannot displace any more fluid, so the upthrust stops increasing.
An object therefore floats only if it can displace its own weight of fluid before it is completely submerged.
A floating ship displaces a weight of water exactly equal to its own weight, which is why a ship's mass is often quoted as its displacement.

Comparing densities

Definition

Density

The mass per unit volume of a material.

Definition

Average density

The total mass of an object divided by its total volume, including any air spaces inside it.

Whether an object floats or sinks depends on its average density compared with the density of the fluid.
If the object's average density is less than the fluid's density, it floats with only part of it below the surface.
If the object's average density is greater than the fluid's density, it sinks, because even when fully submerged it displaces less than its own weight of fluid.
If the two densities are equal, the object neither rises nor sinks and stays at whatever depth it is placed.
Water has a density of $kg/m31000\ \text{kg/m}^3$ , so oak at about $kg/m3750\ \text{kg/m}^3$ floats and steel at about $kg/m37900\ \text{kg/m}^3$ sinks.
The closer an object's density is to the fluid's density, the more of it lies below the surface, which is why about nine-tenths of an iceberg is hidden under the sea.

Factors that affect floating

Shape: a solid lump of steel sinks, but the same steel shaped into a hollow hull encloses a large volume of air, so the ship's average density is less than that of water.
The wide hull also lets the ship displace a large volume of water, giving enough upthrust to balance its weight.
Load: adding cargo increases a ship's weight, so it settles lower in the water until it displaces enough extra water to balance the extra weight.
The Plimsoll line painted on a ship's hull marks the lowest safe level, because an overloaded ship has too little hull above the water and can be swamped by waves.
Density of the fluid: a denser fluid gives more upthrust for the same displaced volume, so objects float higher in sea water than in fresh water.
Swimmers float with almost no effort in the Dead Sea, whose water is about a quarter denser than fresh water.
Changing average density: a submarine floods its ballast tanks with sea water to increase its average density and dive, and blows the water out with compressed air to decrease its average density and surface.
Floating also happens in gases, because a helium balloon rises when the weight of air it displaces is greater than the total weight of the balloon, the helium and the string.

Example

Why a steel ship floats

A $kg1\ \text{kg}$ block of solid steel has a volume of about $cm3127\ \text{cm}^3$ , so when it is submerged it displaces only $g127\ \text{g}$ of water.
The upthrust, equal to the weight of $g127\ \text{g}$ of water, is far smaller than the weight of $kg1\ \text{kg}$ of steel, so the block sinks.
The same $kg1\ \text{kg}$ of steel pressed into a bowl shape that encloses $cm32000\ \text{cm}^3$ can displace up to $kg2\ \text{kg}$ of water.
The bowl settles with $cm31000\ \text{cm}^3$ below the surface, displacing $kg1\ \text{kg}$ of water, where the upthrust equals its weight.

\text{average density} = \dfrac{1000\ \text{g}}{2000\ \text{cm}^3} = 0.5\ \text{g/cm}^3

The bowl's average density is half that of water, so it floats with half of its volume below the surface.

Common Mistake

Do not say heavy objects sink and light objects float, because a huge ship floats and a small coin sinks.
Do not say a floating object has no weight, because its weight is balanced by an equal upthrust.
Do not compare an object's mass with the fluid's density, because you must compare density with density, or weight with upthrust.

Exam technique

Describing floating and sinking

Name both forces and their directions, then say which is larger or that they are equal.
Link the comparison to the outcome, such as "the upthrust equals the weight, so the resultant force is zero and the boat floats".
When density is involved, compare the object's average density with the density of the fluid it is in, which is not always water.
For a ship or a submarine, explain how its shape or its ballast changes its average density.

Self review

Name the two vertical forces acting on an object in a fluid and give their directions.
State the condition for an object to float.
State what the upthrust on an object is equal to.
Explain why a steel ship floats although a steel block sinks.
Explain how a submarine dives and surfaces.

1.3.9 Pressure in liquids and upthrust

Pressure increases with depth

Definition

Liquid pressure

The pressure at a point in a liquid caused by the weight of the liquid above that point.

Definition

Depth

The vertical distance of a point below the surface of a liquid.

A liquid has weight, so it presses on everything beneath it and on the walls of its container.
A point deep in a liquid has a taller column of liquid above it than a point near the surface.
A taller column has more mass and so a greater weight pressing on each square metre beneath it.
The pressure in a liquid therefore increases with depth.
Doubling the depth doubles the height of the liquid column above a point, so it doubles the pressure caused by the liquid.
At any one depth, the pressure acts equally in all directions: upwards, downwards and sideways.
The force caused by liquid pressure always acts at right angles to the surface it presses on.
The pressure depends only on the vertical depth, not on the width or shape of the container.
This is why water spurts out hardest from the lowest hole in a tall container, and why a dam is built much thicker at its base than at its top.
Divers feel the pressure on their ears grow as they swim down, because the water pushing on the outside of each eardrum presses harder with depth.

A tall container of water with holes at three different depths. The jet from the lowest hole is pushed out hardest and travels furthest, showing that the water pressure is greatest at the greatest depth.

Pressure increases with density

Definition

Density

The mass per unit volume of a material.

A denser liquid has more mass in every cubic metre.
At the same depth, a column of a denser liquid is heavier, so it presses with a greater force on each square metre.
The pressure in a liquid therefore increases with density.
At the same depth, sea water at about $kg/m31030\ \text{kg/m}^3$ exerts a slightly greater pressure than fresh water at $kg/m31000\ \text{kg/m}^3$ .
Mercury is $13.6$ times denser than water, so at the same depth it exerts $13.6$ times the pressure.

How pressure produces upthrust

Definition

Upthrust

The upward force that a fluid exerts on an object that is partly or fully submerged in it.

A partially submerged object has its bottom surface under the liquid and its top surface in the air.
The liquid presses upwards on the bottom surface, because liquid pressure acts in all directions, including upwards.
The air presses down on the top of the object, but it also presses on the surface of the liquid, which adds the same amount of pressure underneath, so the effect of the air cancels out.
The liquid also presses sideways on the vertical sides of the object, but the forces on opposite sides are equal and opposite, so they cancel.
The upward push of the liquid on the bottom surface is left unbalanced, giving a resultant force upwards, and this force is the upthrust.
Pushing the object further down makes its bottom surface deeper, so the pressure there increases and the upthrust grows.
For a fully submerged object, liquid presses down on the top surface as well, but the bottom surface is always deeper, so the upward force on the bottom is larger than the downward force on the top.
The difference between these two forces is the upthrust, and it stays the same however deep the object goes, because the difference in depth between its top and bottom does not change.
A denser liquid gives a bigger pressure difference between the top and bottom of an object, so the upthrust is greater, which is why objects float higher in sea water.

Example

Lowering a block into water

A wooden block hangs from a newton meter and is lowered slowly into a tank of water, bottom face first.
When the bottom face is $cm2\ \text{cm}$ below the surface, the water pushes up on it with a small pressure, so there is a small upthrust and the newton meter reading falls slightly.
At $cm4\ \text{cm}$ the bottom face is twice as deep, so the pressure on it and therefore the upthrust have both doubled.
If the block is free to float, it settles at the depth where the upthrust on its bottom face exactly balances its weight.
If the block is held fully under the water, the water also presses down on its top face, and the upthrust is the difference between the upward force on the bottom and the downward force on the top.

Practical

Water jets at different depths

Aim: to show how the pressure in a liquid changes with depth.
Apparatus: a tall $2$ litre plastic bottle, a drawing pin, a ruler, a marker pen, a strip of sticky tape, a jug of water, a large tray or sink, and a metre rule.
Method:
- Mark three points in a vertical line on the side of the bottle at $cm5\ \text{cm}$ , $cm15\ \text{cm}$ and $cm25\ \text{cm}$ above the base.
- Push the drawing pin through at each mark to make three holes of the same size, then cover all three with one strip of sticky tape.
- Stand the bottle at the edge of the tray and fill it with water to just below the neck.
- Pull the tape off in one movement while a partner tops up the water from the jug, so that the water level stays the same throughout.
- Observe the three jets and use the metre rule to measure how far from the bottle each one lands.
Results: the jet from the lowest hole leaves the bottle fastest and travels furthest outwards, while the jet from the top hole leaves slowly and drops close to the bottle.
Explanation: the lowest hole has the deepest water above it, so the pressure pushing water out through it is the greatest.
Watch out: the jets start at different heights above the tray, so judge how hard each jet leaves its hole as well as where it lands.
Watch out: holes of different sizes give an unfair comparison, and if the water level falls during the test, the pressure at every hole falls too.
Safety: take care when pushing the drawing pin through the plastic, and wipe up any spilt water straight away so that no one slips.

Common Mistake

Do not say liquid pressure acts only downwards, because at any point it acts equally in all directions.
Do not say upthrust happens because the liquid "wants" to push objects out, because it is caused by the pressure on the bottom of the object being greater than the pressure on the top.
Do not say the pressure depends on the total volume of liquid in the container, because only the vertical depth and the density matter.

Exam technique

Explaining upthrust

Begin with the cause: pressure in a liquid increases with depth because of the weight of liquid above.
Compare the surfaces: the bottom of the object is deeper than the top, so the pressure on it is greater.
Turn pressure into force: the greater pressure on the bottom gives a greater upward force than any downward force on the top.
End with the result, which is a resultant upward force called upthrust.

Self review

Explain why the pressure in a liquid increases with depth.
State how the pressure at a given depth changes if the liquid is replaced by a denser one.
Describe the directions in which liquid pressure acts at a point.
Explain how liquid pressure produces an upward force on a partially submerged object.
Explain why water spurts out hardest from the lowest hole in a tall container.

1.3.10 Pressure differences with depth

The liquid pressure equation

Definition

Liquid pressure

The pressure at a point in a liquid caused by the weight of the liquid above that point.

Definition

Gravitational field strength

The gravitational force acting on each kilogram of mass, which is $N/kg10\ \text{N/kg}$ near the Earth's surface.

The pressure caused by a column of liquid depends on its height, the density of the liquid and the gravitational field strength.
The equation is $strength\text{pressure due to a column of liquid} = \text{height of column} \times \text{density of liquid} \times \text{gravitational field strength}$ .
In symbols, this is $p=hρgp = h\rho g$ $p = h ρ g$ .
1. $p$ is the pressure due to the liquid in pascals, $Pa\text{Pa}$ .
2. $h$ is the height of the liquid column, or the depth, in metres, $m\text{m}$ .
3. $ρ\rho$ is the density of the liquid in kilograms per cubic metre, $kg/m3\text{kg/m}^3$ .
4. $g$ is the gravitational field strength in newtons per kilogram, $N/kg\text{N/kg}$ .
Near the Earth's surface, $10\ \text{N/kg}$ .
You do not need to memorise this equation, but you must be able to choose it from a list of equations and use it.
The equation comes from the weight of the liquid column: a column of height $h$ standing on an area $A$ has a volume of $h A$ , a mass of $hAρhA\rho$ and a weight of $hAρghA\rho g$ .
Dividing this weight by the area $A$ that it rests on gives the pressure, $h\rho g$ , so the area cancels out.
This is why the pressure depends only on depth and density, not on the width or shape of the container.
The equation gives the pressure due to the liquid alone, because the total pressure on a submerged object also includes the atmospheric pressure acting on the liquid surface.

Water jets from holes at three depths in a tall container. The deeper the hole, the taller the column of water above it, so the greater the pressure forcing the water out.

Pressure differences between depths

Definition

Depth

The vertical distance of a point below the surface of a liquid.

Definition

Pressure difference

The difference between the pressures on the two sides of a surface, which produces a net force on the surface towards the side at lower pressure.

The difference in pressure between two points in the same liquid depends only on the vertical distance between them.
For a difference in depth $Δh\Delta h$ , the difference in pressure is $g\Delta p = \Delta h\,\rho\,g$ .
The pressure difference does not depend on how far below the surface the two points are, only on how far apart they are vertically.
In fresh water, the pressure increases by $Pa10\,000\ \text{Pa}$ for every extra metre of depth, because $\times 1000 \times 10 = 10\,000$ .
A diver $m10\ \text{m}$ down therefore feels an extra $Pa100\,000\ \text{Pa}$ from the water, which is about the same as the whole of atmospheric pressure.
Always convert depths to metres before substituting, because a depth in centimetres gives an answer $100$ times too large.
Large pressures are often given in kilopascals, where $Pa1\ \text{kPa} = 1000\ \text{Pa}$ .
Rearranging the equation gives $Δh=Δpρg\Delta h = \dfrac{\Delta p}{\rho g}$ for a difference in depth and $g\rho = \dfrac{\Delta p}{\Delta h\,g}$ for a density.

Example

Pressure at the bottom of a pool

A swimming pool is $m2.5\ \text{m}$ deep and is filled with water of density $kg/m31000\ \text{kg/m}^3$ .

h\rho g

\times 1000 \times 10

25\,000\ \text{Pa}

The water adds $Pa25\,000\ \text{Pa}$ , or $kPa25\ \text{kPa}$ , to the atmospheric pressure acting on the surface of the pool.

Example

Difference between two depths

A diver swims down from $m3.0\ \text{m}$ to $m12.0\ \text{m}$ below the surface of sea water of density $kg/m31030\ \text{kg/m}^3$ .
Find the difference in depth first.

\Delta h = 12.0 - 3.0 = 9.0\ \text{m}

\Delta p = \Delta h\,\rho\,g = 9.0 \times 1030 \times 10

\Delta p = 92\,700\ \text{Pa} \approx 9.3 \times 10^{4}\ \text{Pa}

The pressure on the diver increases by $Pa92\,700\ \text{Pa}$ .

Example

Finding a depth

The pressure difference between the top and the bottom of a water tank is $kPa18\ \text{kPa}$ , and the water has a density of $kg/m31000\ \text{kg/m}^3$ .
Convert the pressure difference into pascals.

\Delta p = 18 \times 1000 = 18\,000\ \text{Pa}

Rearrange the equation to make the depth the subject.

\Delta h = \dfrac{\Delta p}{\rho g} = \dfrac{18\,000}{1000 \times 10} = 1.8\ \text{m}

The water in the tank is $m1.8\ \text{m}$ deep.

Common Mistake

Do not substitute a depth in centimetres, because the equation needs the depth in metres.
Do not use the depth of only one point when a question asks for a pressure difference, because you must subtract the two depths first.
Do not add atmospheric pressure unless the question asks for the total pressure.
Do not use the mass of the liquid in place of its density.

Exam technique

Liquid pressure calculations

Write $h\rho g$ before you substitute, because the equation and the substitution can each earn credit even if the arithmetic slips.
Show the difference in depth on its own line, because skipping it is a common way to lose a mark.
Give the final answer with its unit, $Pa\text{Pa}$ or $kPa\text{kPa}$ , and check that its size is sensible at about $kPa10\ \text{kPa}$ for each metre of water.
If a later part of a question uses your answer, carry it forward even if you think it may be wrong, because the method marks can still be awarded.

Self review

Write the equation that links pressure due to a column of liquid, height, density and gravitational field strength.
Calculate the pressure due to the water at a depth of $m4.0\ \text{m}$ in fresh water of density $kg/m31000\ \text{kg/m}^3$ .
Explain why the pressure due to a liquid does not depend on the width of its container.
Calculate the pressure difference between depths of $m2.0\ \text{m}$ and $m7.0\ \text{m}$ in sea water of density $kg/m31030\ \text{kg/m}^3$ .
State the value of $g$ to use near the Earth's surface.

1.3.1 Molecular motion, temperature and gas pressure

Gas molecules

Definition

Closed system

A system that no matter can enter or leave, so its total mass stays constant.

A gas is made of tiny molecules that are very far apart compared with their own size.
The molecules move randomly in all directions at high speed, travelling in straight lines between collisions.
At room temperature a typical air molecule moves at about $m/s500\ \text{m/s}$ , which is faster than a passenger jet.
The molecules do not all move at the same speed, because collisions keep sharing energy between them, so a gas always contains a range of speeds.
Each molecule changes direction every time it collides with another molecule or with a wall of its container.
Because the molecules are moving, they store energy in their kinetic energy stores.
In this topic a gas is treated as a closed system, so no molecules enter or leave its container.
With the number of molecules fixed, any change in the gas has to come from a change in how the existing molecules move.

Motion and temperature

Definition

Temperature

A measure of how hot or cold an object is, which depends on the average kinetic energy of its particles.

The temperature of a gas is linked to the average kinetic energy of its molecules.
The faster the molecules move on average, the greater their average kinetic energy and the higher the temperature of the gas.
Heating a gas transfers energy to the kinetic energy stores of its molecules, so they speed up and the temperature rises.
Cooling a gas transfers energy away from its molecules, so they slow down and the temperature falls.
Temperature describes the average motion, so at any instant some molecules are moving faster than average and some are moving slower.
Temperature is measured in degrees Celsius, $∘C^\circ\text{C}$ , using a thermometer.

Motion and pressure

Definition

Pressure

The force acting at right angles to a surface on each unit area of that surface.

Definition

Gas pressure

The force per unit area that gas molecules exert on the walls of their container when they collide with them.

Definition

Pascal

The unit of pressure, equal to a force of one newton acting at right angles on an area of one square metre.

When a molecule hits a wall it bounces off in a new direction, and at the same moment it pushes on the wall, so every collision exerts a tiny force on the wall.
One collision is far too small to notice, but billions of collisions every second add up to a steady force on every part of the wall.
Pressure is the force acting at right angles on each unit area of a surface.
Gas pressure is therefore the total force from the collisions on a wall divided by the area of that wall.
Pressure is measured in pascals, $Pa\text{Pa}$ , where $N/m21\ \text{Pa} = 1\ \text{N/m}^2$ .
The molecules move randomly in every direction, so they strike all the walls of a container equally often and the gas pushes with the same pressure on every wall.
The size of the pressure depends on two features of the collisions:
1. how often the molecules hit each square metre of wall every second;
2. how hard each molecule hits the wall.
Faster molecules cross the container in less time, so they reach the walls more often.
Faster molecules also hit the walls harder, so each collision exerts a larger force.
The speed of the molecules is therefore the link between the two quantities: it sets the temperature of the gas and it controls the pressure the gas exerts.
Slower molecules hit the walls less often and less hard, so a gas whose molecules slow down exerts a lower pressure on any wall that cannot move away from it.

Practical

Heating and cooling trapped air

Aim: to show how heating and cooling a fixed amount of trapped air changes the motion of its molecules and the pressure they exert, first with balloons and then with a manometer.
Apparatus: two identical balloons, a piece of string and a ruler, a freezer, a warm radiator or a bowl of water at about $∘C50\ ^\circ\text{C}$ , a conical flask with a one-holed bung and glass tube, rubber tubing, a glass U-tube clamped in front of a metre rule, coloured water, two large troughs, a kettle, ice, petroleum jelly and eye protection.
Method, the balloons:
- Inflate both balloons to the same size, tie them, and measure the circumference of each around its widest part using the string and ruler.
- Put one balloon in a freezer for about $20$ minutes and hold the other close to a warm radiator, or float it in the warm water, for the same time.
- Take each balloon out and measure its circumference again straight away.
- Squeeze each balloon gently and compare how firm it feels.
- Leave both balloons at room temperature for another $20$ minutes and measure them again to check that the changes reverse.
Method, the manometer:
- Pour coloured water into the U-tube until both arms are about half full and the two levels are equal.
- Smear the bung with petroleum jelly, push it firmly into the flask, and connect the glass tube to one arm of the U-tube with the rubber tubing, leaving the other arm open to the air.
- Check that the two levels are still equal, which shows that the trapped air starts at the same pressure as the air in the room.
- Fill one trough with hot water from the kettle and hold the flask in it so that the whole flask is under the surface.
- Record the difference in height between the two liquid levels with the metre rule every $30$ seconds until it stops changing.
- Move the flask into a trough of ice and water and record the level difference again as the trapped air cools.
Results: the frozen balloon shrinks and feels softer, while the warmed balloon grows slightly and feels firmer.
Results: when the flask is heated, the liquid is pushed down in the arm joined to the flask and rises in the open arm, and when the flask is cooled the liquid moves the opposite way.
Explanation: heating gives the air molecules more kinetic energy, so they move faster and hit the rubber or the liquid surface more often and harder, which raises the pressure.
Explanation: cooling slows the molecules down, so they hit less often and less hard, and the pressure falls.
Reading the manometer: a bigger difference between the two levels means a bigger difference between the pressure of the trapped air and the pressure of the room.
Watch out: a leaking bung lets air escape, so the gas is no longer a closed system and the level difference is smaller than it should be.
Watch out: read each level at eye level from the bottom of the curved meniscus, so that every reading is taken in the same way.
Safety: wear eye protection, use heatproof gloves near the hot water, and never heat the sealed flask directly over a Bunsen burner, because the pressure inside could force out the bung or crack the glass.

Common Mistake

Do not say that the molecules get bigger when a gas is heated, because the molecules stay the same size and only their speed changes.
Do not say that heating produces more molecules, because in a closed system the number of molecules never changes.
Do not write "more collisions" on its own; say the molecules collide with the walls more often, because the change is in the number of collisions each second.
Do not say gas molecules vibrate, because they move freely through the whole container.

Exam technique

Explaining gas pressure

Write the explanation as a chain: the molecules move, they collide with the walls, each collision exerts a force, and the force on each unit area is the pressure.
When the molecules speed up, mention both effects on the collisions, saying the molecules hit the walls more often and with more force, because each effect usually carries its own mark.
Say the molecules gain kinetic energy rather than writing "heat energy", which earns no credit.
Use either "molecules" or "particles" and stick with it, rather than switching to "atoms".

Self review

Describe how the molecules in a gas move.
State what the temperature of a gas tells you about its molecules.
Explain how molecules colliding with a wall produce gas pressure.
Name the two features of the collisions that decide the size of the gas pressure.
State the unit of pressure and what it is equal to in newtons and square metres.

1.3.2 Gas pressure at constant volume

A fixed volume of gas

Definition

Gas pressure

The force per unit area that gas molecules exert on the walls of their container when they collide with them.

A gas has a constant volume when it is sealed inside a rigid container whose walls cannot move, such as a steel gas cylinder, a sealed metal can or a stoppered glass flask.
Because the container is sealed, the number of molecules inside stays the same as well.
With the volume and the number of molecules both fixed, the only way the pressure can change is through a change in the speed of the molecules.
The speed of the molecules is set by the temperature, so at constant volume the pressure of a gas depends on its temperature.

Gas molecules trapped in a rigid box. The walls cannot move, so any change in how fast the molecules travel changes how often and how hard they strike the walls, and so changes the pressure.

Raising the temperature

Definition

Temperature

A measure of how hot or cold an object is, which depends on the average kinetic energy of its particles.

Heating the container transfers energy to the gas and increases the kinetic energy of its molecules.
The molecules move faster on average, which is what a rise in temperature means.
Faster molecules cross the container in less time, so each one hits the walls more often.
Faster molecules also hit the walls harder, so each collision exerts a greater force.
More frequent collisions and a greater force per collision together increase the total force on each square metre of wall.
The pressure of the gas therefore increases, even though its volume and the number of molecules have not changed.
You only need to explain this relationship in words, so no calculation is needed for a change of temperature at constant volume.

Lowering the temperature

Cooling the container transfers energy away from the gas, so its molecules move more slowly on average.
Slower molecules hit the walls less often and less hard.
The total force on each square metre of wall falls, so the pressure decreases.
Warming the gas back to its starting temperature returns the pressure to its starting value, because nothing has left the sealed container.
The pattern holds in both directions: at constant volume, a higher temperature gives a higher pressure and a lower temperature gives a lower pressure.

Constant volume in everyday life

A car tyre holds a fixed amount of air in a casing that barely stretches, so its pressure rises during a long motorway drive as the tyre warms up.
This is why drivers are told to check tyre pressures when the tyres are cold, because a reading taken straight after a journey is higher than normal.
Aerosol cans warn you not to leave them in temperatures above $∘C50\ ^\circ\text{C}$ , because heating the sealed can raises the pressure inside and could make it burst.
A gas cylinder for a barbecue is stored in the shade for the same reason.
A plastic bottle sealed on a warm day and then put in a fridge is squashed slightly, because the pressure of the cooled air inside falls below the pressure of the air outside.

Example

Heating a sealed can

A sealed metal can of air at $∘C20\ ^\circ\text{C}$ is left on a hot radiator and warms to $∘C60\ ^\circ\text{C}$ .
The can is rigid and sealed, so the volume of the air and the number of molecules stay the same.
Energy is transferred from the radiator to the air, so the air molecules gain kinetic energy and move faster.
The faster molecules strike the inside of the can more often and with a greater force.
The force on each square metre of the can increases, so the pressure of the air inside rises.
If the heating continued, the pressure could keep rising until the can bulged or split along its weakest seam.

Common Mistake

Do not say the molecules expand when the gas is heated, because the molecules stay the same size and the volume of the container is fixed.
Do not say the pressure rises because there are more molecules, because the container is sealed and the number of molecules has not changed.
Do not explain the rise only by "more collisions"; say the collisions happen more often, and add that each one exerts a greater force.
Do not use this explanation for a balloon or a syringe, because their volume can change and the gas can expand instead.

Exam technique

Writing the full chain

A typical three-mark explanation rewards three linked points: the molecules move faster, they collide with the walls more often or harder, and so the pressure increases.
State that the volume stays the same, because that is why the faster molecules cannot simply spread out.
If the question mentions energy, say the molecules gain kinetic energy rather than "heat".
For a cooling question, write the same chain in reverse instead of saying "the opposite happens".

Self review

State what is meant by a gas at constant volume.
Explain why the pressure of a gas in a sealed rigid container rises when it is heated.
Describe what happens to the pressure of the gas when the container is cooled.
Explain why drivers should check tyre pressures before a journey rather than straight after one.

1.3.3 Compressing and expanding gases

Why gases can be compressed

Definition

Compression

A decrease in the volume of a substance caused by a force or pressure acting on it.

Definition

Expansion

An increase in the volume of a substance.

The molecules in a gas are very far apart, with mostly empty space between them.
Pushing on a gas forces its molecules closer together, so the same amount of gas fits into a smaller volume.
This decrease in volume is called compression.
The molecules themselves are not squashed, because only the spaces between them get smaller.
If the push on a gas is reduced, its molecules spread out into a larger volume, which is called expansion.
Liquids and solids are very hard to compress, because their particles are already touching and there is almost no space left to close up.
Gases can be compressed into a much smaller volume, which is how a diver's tank holds enough air for a long dive.

Pressure changes and volume

A gas changes its volume until the pressure it exerts balances the pressure acting on it from outside.
If the pressure outside a gas increases, the gas is compressed until its own pressure rises to match.
If the pressure outside a gas decreases, the gas expands until its own pressure falls to match.
Pushing the plunger of a sealed syringe increases the pressure on the trapped air, so the air is compressed into a smaller volume.
A sealed packet of crisps puffs up on an aeroplane, because the air pressure in the cabin is lower than on the ground and the air inside the packet expands.
A partly blown-up balloon placed in a sealed bell jar grows larger as a vacuum pump removes air from the jar, because the pressure around the balloon falls.
When air is let back into the jar, the pressure outside the balloon rises again and the balloon shrinks to its original size.

Force at right angles to a surface

Definition

Pressure difference

The difference between the pressures on the two sides of a surface, which produces a net force on the surface towards the side at lower pressure.

Gas pressure pushes on every surface the gas touches, including the walls of its container and the face of a piston or plunger.
The force produced by pressure always acts at right angles to the surface, which is also called normal to the surface.
This happens because molecules strike the surface from every direction, so on average their sideways pushes along the surface cancel and only the push straight into the surface remains.
On the flat face of a plunger, the force from the trapped gas therefore acts straight along the barrel of the syringe.
On the curved inside of a balloon, the force acts outwards at right angles to the rubber at every point, which is why a balloon inflates into a rounded shape.
Most surfaces have gas pressing on both sides, so each side receives a force at right angles to it.
If the pressures on the two sides are equal, the two forces cancel and there is no net force on the surface.
If the pressures are different, there is a pressure difference across the surface, and the force from the higher-pressure side is larger.
This gives a net force at right angles to the surface, directed towards the side with the lower pressure.
A syringe plunger stays still while the pressure of the trapped air equals the pressure of the air outside, and it moves as soon as one of these pressures changes.

Example

Squeezing a sealed syringe

A syringe holds $cm350\ \text{cm}^3$ of air with its nozzle sealed, and the plunger is pushed in until the air fills only $cm330\ \text{cm}^3$ .
The push on the plunger increases the pressure acting on the trapped air.
The air is compressed, because its molecules are pushed closer together into the smaller space.
The trapped air now pushes back on the plunger with a larger force at right angles to its face, which you feel as resistance.
When the plunger is released, the pressure inside is greater than the pressure outside, so the net force pushes the plunger back out until the two pressures balance again.

Practical

Compressing sand, water and air

Aim: to compare how easily a gas, a liquid and a solid in grains can be compressed, using three identical sealed syringes.
Apparatus: three identical $cm360\ \text{cm}^3$ plastic syringes without needles, three syringe caps, dry sand, water, a clamp stand with a boss and clamp, a $kg1\ \text{kg}$ slotted mass with a hanger and eye protection.
Method:
- Pull the plunger of the first syringe back to the $cm350\ \text{cm}^3$ mark so that it fills with air, then push a cap firmly onto the nozzle.
- Fill the second syringe with water to the $cm350\ \text{cm}^3$ mark, tap out any air bubbles, and cap it.
- Fill the third syringe with dry sand to the $cm350\ \text{cm}^3$ mark, tap it gently so that the sand settles, and cap it.
- Clamp the first syringe upright with its nozzle at the bottom, and read the starting volume at eye level.
- Rest the $kg1\ \text{kg}$ mass on top of the plunger and read the new volume once the plunger stops moving.
- Lift the mass off and record whether the plunger returns to its starting position.
- Repeat with the water and sand syringes, then repeat each test three times and work out a mean change in volume for each material.
Control variables: use identical syringes, the same starting volume and the same mass each time, so that the only thing that changes is the material inside.
Results: the air is compressed by a large amount and the plunger springs back when the mass is removed, the water hardly moves at all, and the sand is compressed slightly but does not spring back.
Explanation: the air molecules are far apart, so there is plenty of space for them to be pushed closer together.
Explanation: the water particles are already touching, so there is almost no space to close up and the volume barely changes.
Explanation: the sand grains themselves are not compressed, because the small change happens as the grains shift into a tighter arrangement and squeeze the air in the gaps between them.
Watch out: an air bubble left in the water syringe makes the water seem compressible, and a loose cap lets air escape so the air seems more compressible than it is.
Safety: wear eye protection, because a cap can fly off a strongly compressed syringe, and wipe up spilt water and sand straight away so that no one slips.

Common Mistake

Do not say the gas molecules get smaller when a gas is compressed, because it is the spaces between them that shrink.
Do not say a low pressure "sucks" a surface, because the net force comes from the higher pressure on the other side pushing it.
Do not draw the force from a gas along a surface or at an angle to it, because pressure always produces a force at right angles to the surface.
Do not describe sand as easily compressed, because only the gaps between the grains close up.

Exam technique

Describing compression and expansion

Link a change in volume to a change in pressure in the same sentence, such as "the pressure outside increases, so the gas is compressed".
When asked why a gas is easier to compress than a liquid, refer to the spacing of the particles, not to their size or mass.
When you draw or describe a pressure force, show it at right angles to the surface.
For a net force question, compare the pressures on the two sides of the surface and say which side is higher.

Self review

Explain why a gas can be compressed but a liquid can hardly be compressed at all.
State what happens to a sealed volume of gas when the pressure around it decreases.
State the direction of the force that gas pressure produces on a surface.
Explain why a syringe plunger moves when the pressure inside differs from the pressure outside.
Explain why sand in a sealed syringe can be compressed slightly.

1.3.4 Gas volume and pressure at constant temperature

A fixed temperature

Definition

Gas pressure

The force per unit area that gas molecules exert on the walls of their container when they collide with them.

Picture a fixed amount of gas trapped in a cylinder by a movable piston, with the temperature of the gas held constant.
A constant temperature means the molecules keep the same average speed and the same average kinetic energy throughout.
Each collision with a wall is therefore just as hard as before, whatever happens to the volume.
The pressure can still change, because the volume decides how often the molecules reach the walls.

Increasing the volume

Pulling the piston out gives the same number of molecules a larger volume to move around in.
The molecules are now more spread out, so there are fewer molecules in each cubic centimetre of gas.
Each molecule has further to travel between one collision with a wall and the next.
The walls also have a larger area, so the collisions are shared out over more surface.
As a result, each square metre of wall is hit less often every second.
Each collision exerts the same force as before, because the molecules are moving at the same speed.
Fewer collisions per second on each square metre give a smaller force per unit area, so the pressure decreases.
Pushing the piston in has the opposite effect: the molecules are crowded into a smaller volume, hit each square metre of wall more often, and the pressure increases.

Pressure and volume

Definition

Inversely proportional

A relationship between two quantities in which their product stays constant, so doubling one of them halves the other.

For a fixed mass of gas at constant temperature, pressure and volume are inversely proportional.
Doubling the volume halves the pressure, and halving the volume doubles the pressure.
The relationship is $pressure×volume=constant\text{pressure} \times \text{volume} = \text{constant}$ $pressure \times volume = constant$ , or $pV=constantpV = \text{constant}$ $p V = constant$ .
1. $p$ is the pressure in pascals, $Pa\text{Pa}$ .
2. $V$ is the volume in cubic metres, $m3\text{m}^3$ .
If the gas starts at pressure $p_1$ and volume $V_1$ and changes to pressure $p_2$ and volume $V_2$ , the product stays the same, so $p_1 V_1 = p_2 V_2$ .
Any units can be used in $p_1 V_1 = p_2 V_2$ , such as $cm3\text{cm}^3$ or $kPa\text{kPa}$ , as long as both volumes are in the same unit and both pressures are in the same unit.
You do not need to memorise this equation, but you must be able to choose it from a list of equations and use it.
The equation only works if the temperature stays constant and no gas leaks in or out.
A graph of pressure against volume is a curve that falls steeply at first and then levels off, getting closer to the axes without ever touching them.
A graph of pressure against $1V\dfrac{1}{V}$ is a straight line through the origin, which is the clearest test that pressure and volume are inversely proportional.
Compressing a gas quickly makes it warm up, so in an experiment the volume must be changed slowly to keep the temperature constant.

Example

Compressing air in a cylinder

A cylinder holds $m30.060\ \text{m}^3$ of air at a pressure of $Pa100\,000\ \text{Pa}$ .
The piston is pushed in slowly until the volume is $m30.020\ \text{m}^3$ , and the temperature does not change.

p_1 V_1 = p_2 V_2

100\,000 \times 0.060 = p_2 \times 0.020

p_2 = \dfrac{100\,000 \times 0.060}{0.020} = 300\,000\ \text{Pa}

The volume has been divided by $3$ , so the pressure has been multiplied by $3$ , which matches the inverse relationship.

Example

Finding a new volume

A syringe contains $cm340\ \text{cm}^3$ of air at $kPa100\ \text{kPa}$ .
The plunger is pulled out slowly until the pressure falls to $kPa80\ \text{kPa}$ , at constant temperature.
Rearrange $p_1 V_1 = p_2 V_2$ to make $V_2$ the subject.

V_2 = \dfrac{p_1 V_1}{p_2}

V_2 = \dfrac{100 \times 40}{80} = 50\ \text{cm}^3

Both pressures are in kilopascals and the volume is in $cm3\text{cm}^3$ , so the answer comes out in $cm3\text{cm}^3$ with no conversion needed.

Practical

Pressure and volume of trapped air

Aim: to measure how the pressure of a fixed mass of air changes as its volume is changed at constant temperature.
Apparatus: a $cm360\ \text{cm}^3$ plastic syringe, a short length of thick-walled rubber tubing, a pressure sensor connected to a data logger or a pressure gauge reading in $kPa\text{kPa}$ , a clamp stand with a boss and clamp, and a thermometer.
Method:
- Set the plunger at the $cm350\ \text{cm}^3$ mark and connect the nozzle to the pressure sensor with the shortest possible length of tubing.
- Clamp the syringe so that its scale can be read at eye level, then record the starting volume, pressure and room temperature.
- Push the plunger in slowly to $cm345\ \text{cm}^3$ and hold it still for about $30$ seconds so that the air returns to room temperature.
- Record the pressure, then repeat in steps of $cm35\ \text{cm}^3$ down to about $cm325\ \text{cm}^3$ .
- Pull the plunger back out through the same volumes, recording the pressure again at each step, and work out a mean pressure for each volume.
- Multiply each mean pressure by its volume, then plot one graph of pressure against volume and another of pressure against $1V\dfrac{1}{V}$ .
Variables: the independent variable is the volume, the dependent variable is the pressure, and the control variables are the temperature and the mass of trapped air.
Results: the product $p V$ comes out roughly the same for every reading, the graph of $p$ against $V$ is a falling curve, and the graph of $p$ against $1V\dfrac{1}{V}$ is a straight line through the origin.
Watch out: the air inside the tubing is not shown on the syringe scale, so every volume reading is slightly too small, which is a systematic error that short tubing keeps small.
Watch out: pushing the plunger too quickly warms the air and raises the pressure above its true value, so always wait before taking a reading.
Watch out: a leak lets air escape at high pressure and makes $p V$ drift downwards, so check that every joint is tight before you start.
Safety: do not compress the air beyond the range of the sensor or gauge, and keep your face away from the joints in case the tubing slips off.

Common Mistake

Do not say the pressure falls because the molecules slow down, because the temperature is constant and their average speed does not change.
Do not say the extra space fills up with air or anything else, because the same molecules simply spread out through a larger volume.
Do not use $p_1 V_1 = p_2 V_2$ if the temperature changes or if gas escapes.
Do not mix units, because if $V_1$ is in $cm3\text{cm}^3$ then $V_2$ must be in $cm3\text{cm}^3$ too.

Exam technique

Explaining a pressure change at constant temperature

Say that the molecules are more spread out or travel further between collisions, then link this to fewer collisions with the walls each second.
State that the speed of the molecules does not change because the temperature is constant, since this is the step that weaker answers leave out.
In calculations, write $p_1 V_1 = p_2 V_2$ , substitute both pairs of values, and only then rearrange, so that each step can earn credit.
Check that the answer makes sense, because a smaller volume must always give a larger pressure.

Self review

Explain why increasing the volume of a gas at constant temperature decreases its pressure.
State what happens to the speed of the molecules when a gas is compressed slowly at constant temperature.
Write the relationship between pressure and volume for a fixed mass of gas at constant temperature.
A gas at $kPa200\ \text{kPa}$ occupies $cm330\ \text{cm}^3$ ; calculate its pressure when its volume is increased to $cm360\ \text{cm}^3$ at the same temperature.
Describe the shape of a graph of pressure against volume for a gas at constant temperature.

1.3.5 Doing work on a gas

Work and energy

Definition

Work done

The energy transferred when a force moves an object through a distance in the direction of the force.

Definition

Internal energy

The total energy stored by the particles of a system in their kinetic and potential energy stores.

Work is done whenever a force moves an object through a distance in the direction of the force.
Doing work is one way of transferring energy, and the energy transferred is equal to the work done.
Work done is measured in joules, $J\text{J}$ , the same unit as energy.
Work is done on a gas when a force pushes a piston or plunger into it and makes its volume smaller.
The energy transferred by this work goes into the internal energy of the gas.
In a gas, the internal energy is almost entirely the kinetic energy of its fast-moving molecules.

Why the temperature rises

Definition

Temperature

A measure of how hot or cold an object is, which depends on the average kinetic energy of its particles.

As the piston moves inwards, molecules keep colliding with its moving face.
A molecule that hits a piston moving towards it rebounds faster than it arrived, in the same way that a tennis ball leaves a racket faster when the racket is swung into it.
Each of these collisions transfers energy from the piston to a molecule, so the molecules gain kinetic energy.
The average kinetic energy of the molecules therefore increases.
Temperature depends on the average kinetic energy of the molecules, so the temperature of the gas rises.
The energy comes from whatever pushes the piston, such as the chemical energy store of your muscles when you work a pump.
The faster the gas is compressed, the larger the rise in temperature, because there is less time for energy to be transferred to the surroundings by heating.
If the gas is compressed very slowly, energy leaves through the walls almost as fast as it arrives, so the temperature hardly changes.
The reverse also happens: a gas that expands and pushes a piston outwards does work on its surroundings, so its molecules lose kinetic energy and the gas cools.

Examples of doing work on a gas

Bicycle pump: pushing the handle down quickly compresses the air in the barrel, so after a few strokes the barrel feels warm, especially near the valve end where the air is squeezed most.
Some of that warmth comes from friction between the plunger seal and the barrel, but the work done compressing the air adds to it.
Fire piston: slamming the plunger of a sealed, clear tube down very quickly raises the temperature of the trapped air by several hundred degrees, enough to set a small piece of cotton wool alight.
Diesel engine: the air in each cylinder is compressed so strongly and so quickly that it becomes hot enough to ignite the fuel sprayed into it, so no spark plug is needed.

Example

Pumping up a bicycle tyre

You push the pump handle down, and the piston moves through a distance in the direction of your push, so you do work on the air inside the barrel.
Energy is transferred mechanically from the chemical energy store of your muscles to the trapped air.
Air molecules rebound faster from the advancing piston, so their average kinetic energy increases.
The internal energy of the air increases and its temperature rises.
Energy is then transferred by heating from the warm air to the barrel, so the barrel feels warm in your hand.
Pumping in short, fast strokes makes the barrel warmer than slow strokes, because less energy escapes during each stroke.

Common Mistake

Do not say the gas gains "heat energy", because the energy is transferred to the internal energy store, or the kinetic energy stores, of the molecules.
Do not explain the temperature rise only by saying the molecules are closer together, because being closer together does not change their speed.
Do not call this heating, because the energy arrives by a force doing work, not because of a temperature difference.

Exam technique

Explaining a temperature rise from work

Start with the cause: a force moves the piston, so work is done on the gas.
Follow the energy through: it is transferred to the gas, the molecules gain kinetic energy and move faster, and the temperature rises.
State the link between the average kinetic energy of the molecules and the temperature, because this is the step most often missed.
If the question says the pump gets hot, finish by saying energy is transferred from the warm air to the pump by heating.

Self review

State what is meant by doing work on a gas.
Explain why molecules gain kinetic energy when they collide with a piston moving towards them.
Explain why compressing a gas quickly raises its temperature.
Explain why a bicycle pump becomes warm when a tyre is pumped up.
State why compressing a gas very slowly causes little change in its temperature.

1.3.6 The Earth's atmosphere and atmospheric pressure

A simple model of the atmosphere

Definition

Atmosphere

The layer of gases surrounding the Earth, held in place by the Earth's gravity.

The Earth is surrounded by a thin layer of air called the atmosphere, which is held in place by the Earth's gravity.
Air is a mixture of gases, mainly nitrogen and oxygen, and like all matter it has mass.
Compared with the size of the Earth the atmosphere is very thin, because most of the air lies within about $km10\ \text{km}$ of the ground while the Earth's radius is about $km6400\ \text{km}$ .
A scientific model simplifies a real system so that its main behaviour can be explained.
In the simple model, the atmosphere is treated as a layer of air with a uniform density, so every cubic metre holds the same mass of air from the ground to the top.
The model also gives the atmosphere a definite top, above which there is no air.
With a uniform density of about $kg/m31.2\ \text{kg/m}^3$ , the model atmosphere would need to be about $km8\ \text{km}$ deep to produce the pressure measured at sea level.
The real atmosphere gets less dense with height and fades away gradually rather than stopping at a sharp edge, so the uniform model is an approximation.

Atmospheric pressure

Definition

Atmospheric pressure

The pressure exerted on a surface by the air of the atmosphere, caused by the weight of the column of air above that surface.

Definition

Weight

The force acting on an object because of gravity, measured in newtons.

Gravity pulls on the air, so every column of air above the ground has weight.
Picture a column of air standing on $m21\ \text{m}^2$ of ground and reaching all the way to the top of the atmosphere.
That column has a mass of about $kg10\,000\ \text{kg}$ , so its weight is about $N100\,000\ \text{N}$ .
This weight pressing down on $m21\ \text{m}^2$ gives an atmospheric pressure at sea level of about $Pa100\,000\ \text{Pa}$ , which is $1.0 \times 10^{5}\ \text{Pa}$ or $kPa100\ \text{kPa}$ .
The same pressure can be described in terms of molecules, because air molecules constantly collide with every surface and each collision exerts a tiny force on it.
The molecules move randomly in all directions, so atmospheric pressure acts in every direction, on the top, the sides and the underneath of objects alike.
You do not feel crushed because the fluids inside your body push outwards with a matching pressure, so the forces on the two sides of your skin balance.

Effects of atmospheric pressure

Definition

Pressure difference

The difference between the pressures on the two sides of a surface, which produces a net force on the surface towards the side at lower pressure.

Atmospheric pressure only has a noticeable effect when there is a pressure difference across a surface.
If the pressure on one side of a surface is lowered, the atmosphere on the other side pushes harder, giving a net force towards the low-pressure side.
A suction cup stays on a smooth tile because pressing it flat pushes most of the air out from underneath, so the atmosphere outside pushes the cup against the tile.
Drinking through a straw works because you lower the pressure inside your mouth, and atmospheric pressure acting on the surface of the drink pushes the liquid up the straw.
When you breathe in, your chest expands and lowers the pressure inside your lungs, so atmospheric pressure pushes air in through your nose and mouth.
In every case the object or liquid is pushed by the higher pressure, because a lower pressure cannot pull.

Practical

The collapsing can

Aim: to show the size of the force that atmospheric pressure exerts when the pressure inside a container is lowered.
Apparatus: an empty, clean aluminium drinks can, a Bunsen burner, a tripod and gauze, a pair of tongs, a large bowl of cold water with some ice in it, heatproof gloves and eye protection.
Method:
- Pour about $cm310\ \text{cm}^3$ of water into the can.
- Stand the can on the gauze and heat it until the water boils strongly and a steady stream of steam has been leaving the opening for about $30$ seconds.
- The steam drives most of the air out of the can, so the can is now filled with steam.
- Grip the can firmly with the tongs and, in one quick movement, turn it upside down and dip its opening just under the surface of the cold water.
- Watch the can closely from a safe distance.
Results: the can is crushed almost instantly, often with a loud crack.
Explanation: the cold water cools the steam, which condenses into a few drops of liquid water, leaving very little gas inside the can.
Explanation: the pressure inside the can falls far below atmospheric pressure, and the water seals the opening so air cannot rush back in.
Explanation: the atmosphere pushes on the outside of the can with a total force of more than $N1000\ \text{N}$ , far more than the thin walls can withstand, so the can collapses inwards.
Watch out: if the water is not boiled for long enough, too much air stays in the can and it only dents.
Safety: wear eye protection, handle the hot can only with tongs and heatproof gloves, and keep your face and hands away from the steam, which scalds.

Common Mistake

Do not say the can is "sucked in" by a vacuum, because it is pushed in by the greater atmospheric pressure outside.
Do not say there is no pressure on you because you cannot feel it, because it is balanced by the pressure inside your body.
Do not say atmospheric pressure acts only downwards, because air molecules collide with surfaces from every direction.

Exam technique

Describing the model and explaining effects

When describing the model, give its two key features: a layer of air around the Earth held by gravity, and a uniform density.
Link the weight of the air above a surface to the pressure on that surface.
For any effect caused by a pressure difference, name the side with the higher pressure and say that it pushes.
Avoid "suck" and "suction" as explanations, because they are not given credit.

Self review

Describe the simple model of the Earth's atmosphere.
Explain what causes atmospheric pressure.
State the approximate value of atmospheric pressure at sea level.
Explain why a heated can collapses when its opening is dipped into cold water.
Explain how atmospheric pressure helps you drink through a straw.

1.3.7 Atmospheric pressure and height

Pressure falls with height

Definition

Altitude

The height of a point above sea level.

Definition

Atmospheric pressure

The pressure exerted on a surface by the air of the atmosphere, caused by the weight of the column of air above that surface.

Atmospheric pressure is greatest at sea level and decreases as altitude increases.
At the top of Ben Nevis, about $m1350\ \text{m}$ above sea level, the pressure is roughly $Pa86\,000\ \text{Pa}$ , compared with about $Pa100\,000\ \text{Pa}$ at sea level.
At the cruising height of an airliner, about $km11\ \text{km}$ , it is only about $Pa23\,000\ \text{Pa}$ , which is why aircraft cabins are pressurised.
The atmospheric pressure at any point is caused by the weight of the air above that point.
Moving upwards leaves part of the atmosphere below you, so the column of air above you is shorter.
A shorter column of air has less mass and so a smaller weight.
A smaller weight of air pressing on each square metre means a lower pressure.

The effect of density

Definition

Density

The mass per unit volume of a material.

Air is a gas and can be compressed, so the air near the ground is squashed by the weight of all the air above it.
This makes the air at low altitude denser, with more molecules packed into each cubic metre.
Higher up there is less air above to do the squashing, so the air is less dense.
With fewer molecules in each cubic metre, fewer molecules collide with each square metre of a surface every second.
Fewer collisions per second give a smaller force on each unit area, so the pressure is lower.
In the simple uniform-density model, the pressure falls by the same amount for every metre you climb and reaches zero at the top of the model atmosphere.
In the real atmosphere the density also falls with height, so the pressure drops quickly near the ground and more slowly higher up, fading gradually instead of stopping at a sharp edge.
About half of the air lies below a height of roughly $km5.5\ \text{km}$ , so the pressure there is about half of its sea-level value.

Effects of lower pressure at altitude

A sealed crisp packet carried up a mountain or onto an aeroplane swells, because the pressure of the air inside it is now greater than the pressure outside.
Your ears pop in a lift, a plane or a car climbing a hill, because the air trapped behind each eardrum briefly has a higher pressure than the air outside until the two pressures equalise.
Climbers high on Everest often use bottled oxygen, because each breath of the low-density air contains far fewer oxygen molecules than a breath at sea level.

Example

Climbing a hill

A walker climbs from a car park at sea level to a summit $m1000\ \text{m}$ higher.
At the summit, $m1000\ \text{m}$ of the atmosphere now lies below the walker, so the column of air above is shorter.
The shorter column has a smaller weight, so it presses on each square metre with a smaller force.
The air at the summit is also less compressed and less dense, so fewer molecules hit each square metre every second.
Both ideas lead to the same result: the atmospheric pressure at the summit is lower, at about $Pa90\,000\ \text{Pa}$ .

Common Mistake

Do not say pressure falls with height because there is no gravity higher up, because gravity at the top of a mountain is almost exactly as strong as at sea level.
Do not say there is no air at high altitude, because there is still air, it is just less dense.
Do not say air is "sucked" out of a crisp packet, because the packet swells as the air inside pushes outwards harder than the air outside pushes in.

Exam technique

Explaining a change with height

Start from the weight of the air above the point: higher up there is less air above, so its weight is smaller.
Add the particle reason: the air is less dense, so there are fewer collisions per second on each square metre.
Finish by naming the result clearly, that the atmospheric pressure is lower.
Use comparative words such as "less", "fewer" and "lower", because a statement about only one height does not answer a comparison question.

Self review

State how atmospheric pressure changes as altitude increases.
Explain, in terms of the air above, why atmospheric pressure is lower at the top of a mountain.
Explain, in terms of molecules, why less dense air exerts a lower pressure.
Explain why a sealed crisp packet swells on an aeroplane.

1.3.8 Floating and sinking

Weight and upthrust

Definition

Weight

The force acting on an object because of gravity, measured in newtons.

Definition

Upthrust

The upward force that a fluid exerts on an object that is partly or fully submerged in it.

Definition

Fluid

A substance that can flow, which means any liquid or gas.

Any object in a fluid, which can be a liquid or a gas, has two vertical forces acting on it.
Its weight acts vertically downwards, caused by gravity pulling on its mass.
The upthrust from the fluid acts vertically upwards.
If the weight is greater than the upthrust, the resultant force is downwards and the object sinks.
If the upthrust is greater than the weight, the resultant force is upwards and the object rises.
If the upthrust equals the weight, the forces are balanced, so the object floats at the surface or stays still within the fluid.
A floating object sinks into the fluid only until the upthrust has grown to match its weight, and then it stays at that level.

Upthrust and displaced fluid

Definition

Displaced fluid

The fluid pushed aside by an object placed in it, which has the same volume as the part of the object below the surface.

An object placed in a fluid pushes some of the fluid out of the way, and this is the displaced fluid.
The upthrust on an object is equal to the weight of the fluid it displaces.
The more of an object that sits below the surface, the more fluid it displaces and the larger the upthrust.
Once an object is completely under the surface it cannot displace any more fluid, so the upthrust stops increasing.
An object therefore floats only if it can displace its own weight of fluid before it is completely submerged.
A floating ship displaces a weight of water exactly equal to its own weight, which is why a ship's mass is often quoted as its displacement.

Comparing densities

Definition

Density

The mass per unit volume of a material.

Definition

Average density

The total mass of an object divided by its total volume, including any air spaces inside it.

Whether an object floats or sinks depends on its average density compared with the density of the fluid.
If the object's average density is less than the fluid's density, it floats with only part of it below the surface.
If the object's average density is greater than the fluid's density, it sinks, because even when fully submerged it displaces less than its own weight of fluid.
If the two densities are equal, the object neither rises nor sinks and stays at whatever depth it is placed.
Water has a density of $kg/m31000\ \text{kg/m}^3$ , so oak at about $kg/m3750\ \text{kg/m}^3$ floats and steel at about $kg/m37900\ \text{kg/m}^3$ sinks.
The closer an object's density is to the fluid's density, the more of it lies below the surface, which is why about nine-tenths of an iceberg is hidden under the sea.

Factors that affect floating

Shape: a solid lump of steel sinks, but the same steel shaped into a hollow hull encloses a large volume of air, so the ship's average density is less than that of water.
The wide hull also lets the ship displace a large volume of water, giving enough upthrust to balance its weight.
Load: adding cargo increases a ship's weight, so it settles lower in the water until it displaces enough extra water to balance the extra weight.
The Plimsoll line painted on a ship's hull marks the lowest safe level, because an overloaded ship has too little hull above the water and can be swamped by waves.
Density of the fluid: a denser fluid gives more upthrust for the same displaced volume, so objects float higher in sea water than in fresh water.
Swimmers float with almost no effort in the Dead Sea, whose water is about a quarter denser than fresh water.
Changing average density: a submarine floods its ballast tanks with sea water to increase its average density and dive, and blows the water out with compressed air to decrease its average density and surface.
Floating also happens in gases, because a helium balloon rises when the weight of air it displaces is greater than the total weight of the balloon, the helium and the string.

Example

Why a steel ship floats

A $kg1\ \text{kg}$ block of solid steel has a volume of about $cm3127\ \text{cm}^3$ , so when it is submerged it displaces only $g127\ \text{g}$ of water.
The upthrust, equal to the weight of $g127\ \text{g}$ of water, is far smaller than the weight of $kg1\ \text{kg}$ of steel, so the block sinks.
The same $kg1\ \text{kg}$ of steel pressed into a bowl shape that encloses $cm32000\ \text{cm}^3$ can displace up to $kg2\ \text{kg}$ of water.
The bowl settles with $cm31000\ \text{cm}^3$ below the surface, displacing $kg1\ \text{kg}$ of water, where the upthrust equals its weight.

\text{average density} = \dfrac{1000\ \text{g}}{2000\ \text{cm}^3} = 0.5\ \text{g/cm}^3

The bowl's average density is half that of water, so it floats with half of its volume below the surface.

Common Mistake

Do not say heavy objects sink and light objects float, because a huge ship floats and a small coin sinks.
Do not say a floating object has no weight, because its weight is balanced by an equal upthrust.
Do not compare an object's mass with the fluid's density, because you must compare density with density, or weight with upthrust.

Exam technique

Describing floating and sinking

Name both forces and their directions, then say which is larger or that they are equal.
Link the comparison to the outcome, such as "the upthrust equals the weight, so the resultant force is zero and the boat floats".
When density is involved, compare the object's average density with the density of the fluid it is in, which is not always water.
For a ship or a submarine, explain how its shape or its ballast changes its average density.

Self review

Name the two vertical forces acting on an object in a fluid and give their directions.
State the condition for an object to float.
State what the upthrust on an object is equal to.
Explain why a steel ship floats although a steel block sinks.
Explain how a submarine dives and surfaces.

1.3.9 Pressure in liquids and upthrust

Pressure increases with depth

Definition

Liquid pressure

The pressure at a point in a liquid caused by the weight of the liquid above that point.

Definition

Depth

The vertical distance of a point below the surface of a liquid.

A liquid has weight, so it presses on everything beneath it and on the walls of its container.
A point deep in a liquid has a taller column of liquid above it than a point near the surface.
A taller column has more mass and so a greater weight pressing on each square metre beneath it.
The pressure in a liquid therefore increases with depth.
Doubling the depth doubles the height of the liquid column above a point, so it doubles the pressure caused by the liquid.
At any one depth, the pressure acts equally in all directions: upwards, downwards and sideways.
The force caused by liquid pressure always acts at right angles to the surface it presses on.
The pressure depends only on the vertical depth, not on the width or shape of the container.
This is why water spurts out hardest from the lowest hole in a tall container, and why a dam is built much thicker at its base than at its top.
Divers feel the pressure on their ears grow as they swim down, because the water pushing on the outside of each eardrum presses harder with depth.

Pressure increases with density

Definition

Density

The mass per unit volume of a material.

A denser liquid has more mass in every cubic metre.
At the same depth, a column of a denser liquid is heavier, so it presses with a greater force on each square metre.
The pressure in a liquid therefore increases with density.
At the same depth, sea water at about $kg/m31030\ \text{kg/m}^3$ exerts a slightly greater pressure than fresh water at $kg/m31000\ \text{kg/m}^3$ .
Mercury is $13.6$ times denser than water, so at the same depth it exerts $13.6$ times the pressure.

How pressure produces upthrust

Definition

Upthrust

The upward force that a fluid exerts on an object that is partly or fully submerged in it.

A partially submerged object has its bottom surface under the liquid and its top surface in the air.
The liquid presses upwards on the bottom surface, because liquid pressure acts in all directions, including upwards.
The air presses down on the top of the object, but it also presses on the surface of the liquid, which adds the same amount of pressure underneath, so the effect of the air cancels out.
The liquid also presses sideways on the vertical sides of the object, but the forces on opposite sides are equal and opposite, so they cancel.
The upward push of the liquid on the bottom surface is left unbalanced, giving a resultant force upwards, and this force is the upthrust.
Pushing the object further down makes its bottom surface deeper, so the pressure there increases and the upthrust grows.
For a fully submerged object, liquid presses down on the top surface as well, but the bottom surface is always deeper, so the upward force on the bottom is larger than the downward force on the top.
The difference between these two forces is the upthrust, and it stays the same however deep the object goes, because the difference in depth between its top and bottom does not change.
A denser liquid gives a bigger pressure difference between the top and bottom of an object, so the upthrust is greater, which is why objects float higher in sea water.

Example

Lowering a block into water

A wooden block hangs from a newton meter and is lowered slowly into a tank of water, bottom face first.
When the bottom face is $cm2\ \text{cm}$ below the surface, the water pushes up on it with a small pressure, so there is a small upthrust and the newton meter reading falls slightly.
At $cm4\ \text{cm}$ the bottom face is twice as deep, so the pressure on it and therefore the upthrust have both doubled.
If the block is free to float, it settles at the depth where the upthrust on its bottom face exactly balances its weight.
If the block is held fully under the water, the water also presses down on its top face, and the upthrust is the difference between the upward force on the bottom and the downward force on the top.

Practical

Water jets at different depths

Aim: to show how the pressure in a liquid changes with depth.
Apparatus: a tall $2$ litre plastic bottle, a drawing pin, a ruler, a marker pen, a strip of sticky tape, a jug of water, a large tray or sink, and a metre rule.
Method:
- Mark three points in a vertical line on the side of the bottle at $cm5\ \text{cm}$ , $cm15\ \text{cm}$ and $cm25\ \text{cm}$ above the base.
- Push the drawing pin through at each mark to make three holes of the same size, then cover all three with one strip of sticky tape.
- Stand the bottle at the edge of the tray and fill it with water to just below the neck.
- Pull the tape off in one movement while a partner tops up the water from the jug, so that the water level stays the same throughout.
- Observe the three jets and use the metre rule to measure how far from the bottle each one lands.
Results: the jet from the lowest hole leaves the bottle fastest and travels furthest outwards, while the jet from the top hole leaves slowly and drops close to the bottle.
Explanation: the lowest hole has the deepest water above it, so the pressure pushing water out through it is the greatest.
Watch out: the jets start at different heights above the tray, so judge how hard each jet leaves its hole as well as where it lands.
Watch out: holes of different sizes give an unfair comparison, and if the water level falls during the test, the pressure at every hole falls too.
Safety: take care when pushing the drawing pin through the plastic, and wipe up any spilt water straight away so that no one slips.

Common Mistake

Do not say liquid pressure acts only downwards, because at any point it acts equally in all directions.
Do not say upthrust happens because the liquid "wants" to push objects out, because it is caused by the pressure on the bottom of the object being greater than the pressure on the top.
Do not say the pressure depends on the total volume of liquid in the container, because only the vertical depth and the density matter.

Exam technique

Explaining upthrust

Begin with the cause: pressure in a liquid increases with depth because of the weight of liquid above.
Compare the surfaces: the bottom of the object is deeper than the top, so the pressure on it is greater.
Turn pressure into force: the greater pressure on the bottom gives a greater upward force than any downward force on the top.
End with the result, which is a resultant upward force called upthrust.

Self review

Explain why the pressure in a liquid increases with depth.
State how the pressure at a given depth changes if the liquid is replaced by a denser one.
Describe the directions in which liquid pressure acts at a point.
Explain how liquid pressure produces an upward force on a partially submerged object.
Explain why water spurts out hardest from the lowest hole in a tall container.

1.3.10 Pressure differences with depth

The liquid pressure equation

Definition

Liquid pressure

The pressure at a point in a liquid caused by the weight of the liquid above that point.

Definition

Gravitational field strength

The gravitational force acting on each kilogram of mass, which is $N/kg10\ \text{N/kg}$ near the Earth's surface.

The pressure caused by a column of liquid depends on its height, the density of the liquid and the gravitational field strength.
The equation is $strength\text{pressure due to a column of liquid} = \text{height of column} \times \text{density of liquid} \times \text{gravitational field strength}$ .
In symbols, this is $p=hρgp = h\rho g$ $p = h ρ g$ .
1. $p$ is the pressure due to the liquid in pascals, $Pa\text{Pa}$ .
2. $h$ is the height of the liquid column, or the depth, in metres, $m\text{m}$ .
3. $ρ\rho$ is the density of the liquid in kilograms per cubic metre, $kg/m3\text{kg/m}^3$ .
4. $g$ is the gravitational field strength in newtons per kilogram, $N/kg\text{N/kg}$ .
Near the Earth's surface, $10\ \text{N/kg}$ .
You do not need to memorise this equation, but you must be able to choose it from a list of equations and use it.
The equation comes from the weight of the liquid column: a column of height $h$ standing on an area $A$ has a volume of $h A$ , a mass of $hAρhA\rho$ and a weight of $hAρghA\rho g$ .
Dividing this weight by the area $A$ that it rests on gives the pressure, $h\rho g$ , so the area cancels out.
This is why the pressure depends only on depth and density, not on the width or shape of the container.
The equation gives the pressure due to the liquid alone, because the total pressure on a submerged object also includes the atmospheric pressure acting on the liquid surface.

Water jets from holes at three depths in a tall container. The deeper the hole, the taller the column of water above it, so the greater the pressure forcing the water out.

Pressure differences between depths

Definition

Depth

The vertical distance of a point below the surface of a liquid.

Definition

Pressure difference

The difference between the pressures on the two sides of a surface, which produces a net force on the surface towards the side at lower pressure.

The difference in pressure between two points in the same liquid depends only on the vertical distance between them.
For a difference in depth $Δh\Delta h$ , the difference in pressure is $g\Delta p = \Delta h\,\rho\,g$ .
The pressure difference does not depend on how far below the surface the two points are, only on how far apart they are vertically.
In fresh water, the pressure increases by $Pa10\,000\ \text{Pa}$ for every extra metre of depth, because $\times 1000 \times 10 = 10\,000$ .
A diver $m10\ \text{m}$ down therefore feels an extra $Pa100\,000\ \text{Pa}$ from the water, which is about the same as the whole of atmospheric pressure.
Always convert depths to metres before substituting, because a depth in centimetres gives an answer $100$ times too large.
Large pressures are often given in kilopascals, where $Pa1\ \text{kPa} = 1000\ \text{Pa}$ .
Rearranging the equation gives $Δh=Δpρg\Delta h = \dfrac{\Delta p}{\rho g}$ for a difference in depth and $g\rho = \dfrac{\Delta p}{\Delta h\,g}$ for a density.

Example

Pressure at the bottom of a pool

A swimming pool is $m2.5\ \text{m}$ deep and is filled with water of density $kg/m31000\ \text{kg/m}^3$ .

h\rho g

\times 1000 \times 10

25\,000\ \text{Pa}

The water adds $Pa25\,000\ \text{Pa}$ , or $kPa25\ \text{kPa}$ , to the atmospheric pressure acting on the surface of the pool.

Example

Difference between two depths

A diver swims down from $m3.0\ \text{m}$ to $m12.0\ \text{m}$ below the surface of sea water of density $kg/m31030\ \text{kg/m}^3$ .
Find the difference in depth first.

\Delta h = 12.0 - 3.0 = 9.0\ \text{m}

\Delta p = \Delta h\,\rho\,g = 9.0 \times 1030 \times 10

\Delta p = 92\,700\ \text{Pa} \approx 9.3 \times 10^{4}\ \text{Pa}

The pressure on the diver increases by $Pa92\,700\ \text{Pa}$ .

Example

Finding a depth

The pressure difference between the top and the bottom of a water tank is $kPa18\ \text{kPa}$ , and the water has a density of $kg/m31000\ \text{kg/m}^3$ .
Convert the pressure difference into pascals.

\Delta p = 18 \times 1000 = 18\,000\ \text{Pa}

Rearrange the equation to make the depth the subject.

\Delta h = \dfrac{\Delta p}{\rho g} = \dfrac{18\,000}{1000 \times 10} = 1.8\ \text{m}

The water in the tank is $m1.8\ \text{m}$ deep.

Common Mistake

Do not substitute a depth in centimetres, because the equation needs the depth in metres.
Do not use the depth of only one point when a question asks for a pressure difference, because you must subtract the two depths first.
Do not add atmospheric pressure unless the question asks for the total pressure.
Do not use the mass of the liquid in place of its density.

Exam technique

Liquid pressure calculations

Write $h\rho g$ before you substitute, because the equation and the substitution can each earn credit even if the arithmetic slips.
Show the difference in depth on its own line, because skipping it is a common way to lose a mark.
Give the final answer with its unit, $Pa\text{Pa}$ or $kPa\text{kPa}$ , and check that its size is sensible at about $kPa10\ \text{kPa}$ for each metre of water.
If a later part of a question uses your answer, carry it forward even if you think it may be wrong, because the method marks can still be awarded.

Self review

Write the equation that links pressure due to a column of liquid, height, density and gravitational field strength.
Calculate the pressure due to the water at a depth of $m4.0\ \text{m}$ in fresh water of density $kg/m31000\ \text{kg/m}^3$ .
Explain why the pressure due to a liquid does not depend on the width of its container.
Calculate the pressure difference between depths of $m2.0\ \text{m}$ and $m7.0\ \text{m}$ in sea water of density $kg/m31030\ \text{kg/m}^3$ .
State the value of $g$ to use near the Earth's surface.

1.3 Pressure

1.3.1 Molecular motion, temperature and gas pressure

Gas molecules

Motion and temperature

Motion and pressure

Heating and cooling trapped air

Explaining gas pressure

1.3.2 Gas pressure at constant volume

A fixed volume of gas

Raising the temperature

Lowering the temperature

Constant volume in everyday life

Heating a sealed can

Writing the full chain

1.3.3 Compressing and expanding gases

Why gases can be compressed

Pressure changes and volume

Force at right angles to a surface

Squeezing a sealed syringe

Compressing sand, water and air

Describing compression and expansion

1.3.4 Gas volume and pressure at constant temperature

A fixed temperature

Increasing the volume

Pressure and volume

Compressing air in a cylinder

Finding a new volume

Pressure and volume of trapped air

Explaining a pressure change at constant temperature

1.3.5 Doing work on a gas

Work and energy

Why the temperature rises

Examples of doing work on a gas

Pumping up a bicycle tyre

Explaining a temperature rise from work

1.3.6 The Earth's atmosphere and atmospheric pressure

A simple model of the atmosphere

Atmospheric pressure

Effects of atmospheric pressure

The collapsing can

Describing the model and explaining effects

1.3.7 Atmospheric pressure and height

Pressure falls with height

The effect of density

Effects of lower pressure at altitude

Climbing a hill

Explaining a change with height

1.3.8 Floating and sinking

Weight and upthrust

Upthrust and displaced fluid

Comparing densities

Factors that affect floating

Why a steel ship floats

Describing floating and sinking

1.3.9 Pressure in liquids and upthrust

Pressure increases with depth

Pressure increases with density

How pressure produces upthrust

Lowering a block into water

Water jets at different depths

Explaining upthrust

1.3.10 Pressure differences with depth

The liquid pressure equation

Pressure differences between depths

Pressure at the bottom of a pool

Difference between two depths

Finding a depth

Liquid pressure calculations

Review 1.3 Pressure by teaching Genie

Recap your knowledge with an interactive lesson

Put it into practice with exam-style questions

Remember key concepts with flashcards

1.3 Pressure Revision Guide

1.3 Pressure

1.3.1 Molecular motion, temperature and gas pressure

Gas molecules

Motion and temperature

Motion and pressure

Heating and cooling trapped air

Explaining gas pressure