Boyle's Law: Pressure, Volume and Why There Are Two Graphs
Boyle's Law states that the volume of a given mass of gas is inversely proportional to the pressure exerted on it, provided the temperature remains constant.
Squeeze a gas into half the space and the pressure doubles. This page covers why that happens, the formula, the experiment, and the question almost every revision site skips: why does the method ask for two graphs instead of one?
Why it happens
Air is mostly empty space. The molecules are far apart, moving fast, and constantly hitting the walls of whatever contains them. Those collisions are what pressure is.
Squeeze the same air into half the room and each molecule hits the walls twice as often. Same number of molecules, same speed, half the space, so twice the pressure.
Nothing about the molecules themselves has changed. They are not smaller and they are not slower. There are simply more collisions per second on every square centimetre of wall.
That is the kinetic molecular theory explanation, and it is worth being able to say in one sentence, because "the particles are closer together" on its own is not an answer.
The formula
Inversely proportional means:
V ∝ 1 ÷ p
which rearranges to the form you will use most:
pV = k, where k is a constant
And for a gas going from one state to another at the same temperature:
p1V1 = p2V2
The constant temperature clause is not decoration. Heat a gas and it expands, so if the temperature drifts you are no longer measuring what you think you are measuring. Leave those four words out of an exam answer and it is not Boyle's Law.
Where this fits in the curriculum
| Subject | Physical Sciences |
|---|---|
| Grade | 11 |
| Topic | Ideal gases and thermal properties |
| Marks | 40 |
The experiment
Trap a fixed amount of air in a sealed syringe, stack weights on the piston, and read the volume each time.
Apparatus
| Item | Qty |
|---|---|
| Boyle's Law apparatus, simple form | 1 |
| Slotted mass set, 1 kg | 1 |
| Graph paper | 2 sheets |
That is the entire list. No chemicals, nothing that expires, nothing you have to buy in on the morning. It is the only practical in this range where the teacher supplies nothing at all.
Method
- Set the piston to a convenient starting volume with plenty of air in the syringe, and fit the sealing cap.
- Record the starting volume and the atmospheric pressure. Use 101,3 kPa unless you have a barometer.
- Place the first mass on the platform. Wait for the piston to settle, then record the new volume.
- Add the next mass. Wait, record. Keep going until the piston is near the bottom of its travel.
- Work out the total pressure for each reading.
- Plot V against p, then plot V against 1 ÷ p on a second set of axes.
Working out the pressure
The gas is pushed on by two things: the atmosphere, and the weight on the piston. Both count.
| Step | Working |
|---|---|
| Piston radius | r, in metres |
| Piston area | A = πr² |
| Weight on the platform | F = mg, in newtons |
| Extra pressure from the weight | F ÷ A |
| Total gas pressure | atmospheric + F ÷ A |
Worked through, for a piston of radius 1,5 cm carrying 20 N:
- r = 1,5 cm = 1,5 × 10-2 m
- A = πr² = 3,14 × (1,5 × 10-2)² = 7,07 × 10-4 m²
- p = 20 ÷ (7,07 × 10-4) = 28 300 Pa = 28,3 kPa
- Total = 101,3 + 28,3 = 129,6 kPa
The commonest mistake in the whole practical is squaring the radius in centimetres. That gives an area in cm², and a pressure ten thousand times too small. Convert to metres first, then square.
The second commonest is forgetting the atmosphere. 28,3 kPa is what the weight adds. It is not the pressure of the gas.
What you should see
An illustrative set, from a 1 kg mass set on a piston of about 5 cm²:
| Total pressure (kPa) | Volume (cm³) | p × V |
|---|---|---|
| 101,3 | 23,7 | 2 401 |
| 120,0 | 20,1 | 2 412 |
| 140,0 | 17,0 | 2 380 |
| 160,0 | 15,1 | 2 416 |
| 180,0 | 13,2 | 2 376 |
| 200,0 | 12,1 | 2 420 |
The last column is the result. Six different pressures, six different volumes, and pV lands on about 2 400 every time.
Expect a spread of one or two percent. Anything inside 5 % is a good class result. A column that drifts steadily in one direction is telling you something systematic is wrong, usually a sticking piston or a temperature change.
Every group will get a different k, and every one of them is right. k depends on how much air was sealed in at the start.
Why there are two graphs
This is the part almost nobody explains, and it is the whole point of the method.
| Graph | Shape | What it proves |
|---|---|---|
| V against p | A curve falling steeply then flattening, called a rectangular hyperbola | Nothing, on its own |
| V against 1 ÷ p | A straight line through the origin | V is inversely proportional to p |
A curve can be a great many things. Plenty of relationships produce a line that falls and flattens. Looking at that curve and concluding "inversely proportional" is a guess.
A straight line through the origin can only be direct proportion. So if V plotted against 1 ÷ p is straight and passes through zero, then V really is proportional to 1 ÷ p, which is exactly what inversely proportional to p means.
That is why the method asks for the reciprocal graph and not just the obvious one. And the gradient of that straight line is k, the same constant as the pV column.
Extend the line back to the origin
A best-fit line that stops at your lowest data point cannot show you whether it passes through zero, and that is the only thing the graph is for. Extend it.
The result most learners skip
pV has units, and those units are joules.
k = p × V = Pa × m³ = N·m-2 × m³ = N·m = J
So the constant in Boyle's Law is an energy. The 2 400 kPa·cm³ in the table above works out to 2,40 J.
It is a question in the textbook and it is the one most learners leave blank, because it looks like a units exercise rather than a finding. It is a finding.
No apparatus? You can still do most of this
The textbook prints a full set of results for exactly this reason and tells schools without the apparatus to use them. That is an unusual thing for a textbook to admit and it is worth taking at face value: a great many schools run this as a data-handling exercise.
You lose the measuring and you keep everything else. The pressure calculation, both graphs, the interpretation and the conclusion all survive. On our marking memo that is 32 of the 40 marks.
Or build one for the price of a syringe
The textbook's own alternative method is a large syringe with its sealing cap, a small platform on the piston, and slotted masses, stood on a bottle filled with water so it does not topple.
If your school already owns slotted masses, and most do, the whole thing costs about R29 for a 100 ml syringe.
One warning, and it is the reason the shop-bought apparatus exists. A disposable syringe piston sticks. The friction between the rubber seal and the barrel is about the same size as the force you are applying, so the piston jumps instead of sliding and the readings scatter.
Smear petroleum jelly on the seal and tap the barrel after adding each mass. Do that and it works. Skip it and you will get a graph nobody can draw a line through.
Safety
- The mass pieces are heavy and they sit on top of a piston. Load them one at a time and keep hands clear of the base
- Stop when the piston nears the bottom of its travel. Forcing it further can split the barrel or blow the seal
- Do not push the piston down by hand and hold it there. You cannot read a pressure you are applying with your arm
- Nothing here is hot, toxic or flammable. Goggles are not required
If it does not work
| What you see | What caused it |
|---|---|
| The piston jumps instead of moving smoothly | Static friction, and it is the commonest failure by far. Tap the barrel or twist the piston slightly after each mass, and grease the seal |
| pV drifts steadily downwards | The piston is sticking, so the gas is at a higher pressure than your calculation says |
| pV drifts steadily upwards | A leak past the seal. Gas is escaping, so there is less of it than you started with |
| The volume barely changes | Not enough mass for the piston area. Roughly 1 kg per 5 cm² to see a useful change |
| Volumes creep while you watch | The temperature is not constant. The apparatus may still be warming from the storeroom, or a hand is holding the barrel |
| The V against 1 ÷ p line misses the origin | Atmospheric pressure has been left out of the total. See below |
| Every group gets a different k | Correct and expected. k depends on how much air was trapped at the start |
The failure worth keeping
A group whose straight line misses the origin has almost always forgotten atmospheric pressure.
They have plotted the pressure the weights apply, not the pressure the gas is under. The intercept is the size of the mistake, and if you get them to read it off the axis they will find it comes to about 101 kPa.
That is a better lesson than a clean graph. A systematic error that shows up as an intercept, and can be measured straight off the page, is the most useful thing a Grade 11 learner can meet.
How the 40 marks are made up
| Section | Marks |
|---|---|
| Planning and variables | 8 |
| Results and calculations | 12 |
| Graphs | 12 |
| Interpretation and conclusion | 8 |
The textbook prints no mark allocation for this one. The split above is ours, weighted towards the calculations and the graphs because that is where the learning sits.
The mark most often dropped is leaving at constant temperature out of the statement of Boyle's Law. The second most often dropped is answering "as pressure increases, volume decreases", which is true of any falling curve and worth nothing. The marks are for inversely proportional.
If you have time
Put a hand around the barrel for a minute and watch the volume creep. That is the temperature clause, demonstrated in thirty seconds.
Work out k in joules. Most learners have never met a constant that turns out to be an energy.
Look ahead to Charles's Law. Boyle holds temperature constant and varies pressure. Charles holds pressure constant and varies temperature. Put them together and you have the general gas equation, which is where the chapter goes next.
Free worksheet and marking memo
Both free, no sign up, straight to the PDF.
- Learner worksheet, 40 marks, with the planning section, the pressure calculation, the results table and both graph questions set out. Works whether or not your school has the apparatus
- Marking memorandum, with the calculation worked line by line, a completed results table, the mark allocation and a note on the seven places learners most often drop marks
Related practicals
- Intermolecular forces, Grade 11. The other matter and materials practical
- Newton's Second Law, Grade 11. Prescribed, and the other big graph practical
- Ohm's Law, Grade 11. Also turns on a straight line through the origin
Buy this experiment
The Boyle's Law apparatus, simple form at R565 is the version of this practical we recommend. It is the textbook's syringe method, factory built, with the piston already lubricated, which is the one thing that stops the home-made version working.
There is also a large demonstration apparatus with a hand pump and a graduated column, which is a fine instrument for a lecture room and considerably more than most schools need for one practical a year.
You will need a slotted mass set with either one. Roughly 1 kg per 5 cm² of piston area.