The Gas Laws - Physics for Pilot Assessments

The gas laws say what happens to a sealed quantity of gas when you change its pressure, its volume or its temperature, and they are the most calculation-heavy part of this module. By the end of this lesson you should be able to state each of the three laws, work out any one of the three quantities when the others change, combine all three into a single relationship, explain why the temperature has to be in kelvin every single time, and say how the pressures of the gases in a mixture add together.

Three quantities, one fixed mass of gas

Everything in this lesson concerns a fixed mass of gas — a sealed cylinder, a closed syringe, a sealed can — so that nothing gets in or out and the number of particles stays the same. Under that condition the gas has three things worth measuring: its pressure, which from the kinetic theory lesson is the force of its particle impacts spread over the area of the container walls; its volume, meaning the space it occupies; and its temperature, which measures the average kinetic energy of its particles and must be quoted in kelvin. Those three are not independent. Fix any one of them and the other two are locked together, and each of the three gas laws is the statement of one of those pairings.

Boyle's law: pressure against volume, at constant temperature

Hold the temperature steady and squeeze the gas into a smaller space, and the pressure rises — in exact inverse proportion. The mechanism is the one worked through in the kinetic theory lesson, and it is worth having in mind rather than treating the law as a bare fact: the particles are the same particles and, since the temperature has not changed, they are moving at the same speeds, so each impact on the wall is exactly as hard as it was. What has changed is how far each particle travels between one wall and the next. Halve the volume and each particle covers half the distance, so it arrives twice as often, so the wall is struck twice as often, so the pressure doubles. Nothing was added and nothing was made stronger; the impacts simply became more frequent.

Boyle's law — at constant temperature

p_1 V_1 = p_2 V_2

$p_1$: the pressure before the change, in pascals (Pa) or any consistent unit
$V_1$: the volume before the change, in cubic metres (m³) or any consistent unit
$p_2$: the pressure after the change, in the same unit as p₁
$V_2$: the volume after the change, in the same unit as V₁

Because the two sides are equal, the product of pressure and volume is a constant for that gas at that temperature, which is the most compact way to hold the law: whatever you do to one, the other must change so as to leave the product alone. Note that the units cancel out of both sides, so pressures may be in kilopascals or hectopascals and volumes in litres, provided you use the same unit on both sides. That is a genuine convenience and it does not extend to the temperature, as the next two laws will show.

Count the particles: there are eight in each cylinder, moving at the same speeds. Only the space they have has changed, so the same particles in half the space strike each wall twice as often — which is why the little outward arrows, each one a push on the wall, are packed twice as densely along every face.

Charles's law: volume against temperature, at constant pressure

Now let the gas push a piston that is free to slide, so its pressure stays equal to whatever is outside, and warm it. The particles speed up, so they strike the piston harder and more often and momentarily push it outwards; the piston moves out until the gas has spread over enough extra volume for its impact rate to fall back to the value that balances the outside pressure again. So at constant pressure the volume rises with the temperature, and it does so in direct proportion to the absolute temperature: double the temperature in kelvin and the volume doubles.

Charles's law — at constant pressure

\frac{V_1}{T_1} = \frac{V_2}{T_2}

$V_1$: the volume before the change, in cubic metres (m³) or any consistent unit
$T_1$: the absolute temperature before the change, in kelvin (K)
$V_2$: the volume after the change, in the same unit as V₁
$T_2$: the absolute temperature after the change, in kelvin (K)

This is the law the Kelvin scale came out of. Plot volume against temperature in kelvin and you get a straight line heading for the origin, so the graph says that the volume would reach zero at zero kelvin — which is where the temperature lesson found absolute zero from the same experiment. Plot the identical measurements against Celsius and you get a straight line that crosses zero volume at −273 °C instead, hitting the vertical axis at some arbitrary height on the way. Same data, and only one of the two graphs makes the proportionality visible.

The pressure law: pressure against temperature, at constant volume

The third pairing holds the volume fixed — a rigid sealed container that cannot expand — and warms the gas. Now the particles cannot spread out, so everything the extra energy buys goes into the impacts: they become both harder and more frequent, and the pressure rises in direct proportion to the absolute temperature. This one was worked out in detail in the kinetic theory lesson, where doubling the temperature raised the particle speeds by a factor of about 1.41 and multiplied both the impact rate and the force per impact by that factor, giving a doubled pressure overall.

The pressure law — at constant volume

\frac{p_1}{T_1} = \frac{p_2}{T_2}

$p_1$: the pressure before the change, in pascals (Pa) or any consistent unit
$T_1$: the absolute temperature before the change, in kelvin (K)
$p_2$: the pressure after the change, in the same unit as p₁
$T_2$: the absolute temperature after the change, in kelvin (K)

This is the law behind the warning printed on every aerosol can, and behind the fact that a tyre's pressure reads higher after a long drive than it did cold: the volume is essentially fixed, the temperature has risen, so the pressure has risen with it.

Why every gas law needs kelvin, never Celsius

Two of the three laws are proportionalities involving temperature, and a proportionality is a statement about ratios — it says that if you double this quantity you double that one. A ratio is meaningless on a scale whose zero was chosen arbitrarily, and Celsius has exactly that problem: 20 °C is not twice as hot as 10 °C in any sense a gas can respond to, because the two temperatures differ by less than four per cent in the quantity that actually matters, which is the particle energy. Put the Celsius numbers into Charles's law and you predict that warming a gas from 10 °C to 20 °C doubles its volume; put the kelvin numbers in, 283 K and 293 K, and you correctly predict a rise of about three and a half per cent. The error is not small, and it does not shrink — near 0 °C it becomes infinite, since dividing by zero is undefined, and below 0 °C the Celsius figures go negative and predict negative volumes.

So the working rule is unconditional: convert every temperature to kelvin before it goes anywhere near a gas law, using the relationship from the first lesson of this module — add 273 to the Celsius figure. Boyle's law is the only one of the three that does not contain a temperature, and even there the constant temperature has to be genuinely constant for the law to apply.

The combined gas law

In practice a change rarely leaves one of the three quantities untouched: compressing a gas usually warms it, and warming a gas in a flexible container changes both its volume and its pressure. The three laws fold into one relationship that covers every case, and the tidy way to see it is that pressure times volume divided by temperature is a constant for a fixed mass of gas.

The combined gas law

\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}

$p_1$: the pressure before the change, in pascals (Pa) or any consistent unit
$V_1$: the volume before the change, in cubic metres (m³) or any consistent unit
$T_1$: the absolute temperature before the change, in kelvin (K)
$p_2$: the pressure after the change, in the same unit as p₁
$V_2$: the volume after the change, in the same unit as V₁
$T_2$: the absolute temperature after the change, in kelvin (K)

The three separate laws are all special cases of it, which is worth checking once so that you only have to remember one thing. Hold the temperature constant and the two temperatures cancel, leaving Boyle's law. Hold the pressure constant and the two pressures cancel, leaving Charles's law. Hold the volume constant and the volumes cancel, leaving the pressure law. Memorise the combined form and the other three come out of it on demand.

Mixtures: Dalton's partial pressures

One extra idea, briefly, because it is genuinely useful. Most gases you meet are mixtures, and the particles of the different gases in a mixture take no notice of one another — each kind rattles around the whole container as though the others were not there, and each contributes its own share of the impacts on the walls. The pressure a single component would produce if it occupied the container alone is called its partial pressure, and the total pressure is simply the sum of the partial pressures of everything present.

Dalton's law of partial pressures

p_\text{total} = p_1 + p_2 + p_3 + \dots

$p_\text{total}$: the pressure the mixture exerts, in pascals (Pa) or any consistent unit
$p_1, p_2, p_3$: the partial pressure of each component gas — what it alone would exert in the same container, in the same unit

Dry air is about 78 per cent nitrogen and 21 per cent oxygen by the number of particles, with about 1 per cent of everything else, so at a total pressure of 1013 hectopascals the partial pressure of oxygen is about 21 per cent of 1013, which is roughly 213 hectopascals. The proportions of the mixture stay essentially constant throughout the lower atmosphere; what changes is the total. So when the total pressure falls, the partial pressure of oxygen falls in step with it, even though the air is still 21 per cent oxygen.

Example 1 — Boyle's law, both ways round

A sealed cylinder holds gas at 100 kilopascals in a volume of 2 cubic metres. The gas is compressed to 1 cubic metre at the same temperature. What is the new pressure? Then: what volume would it have to be compressed to in order to reach 400 kilopascals?

Boyle's law says pressure × volume is the same before and after: $100 \times 2 = p_2 \times 1$
The left-hand side is 200, so p₂ × 1 = 200, and dividing both sides by 1 gives p₂ = 200 kilopascals. Halving the volume doubled the pressure, as the particle picture said it must.
For the second part the product is still 200, so 200 = 400 × V₂.
Divide both sides by 400: $V_2 = \frac{200}{400} = 0.5\ \text{m}^3$
Check the direction of the answer before accepting it. Four times the original pressure should need a quarter of the original volume, and 0.5 is indeed a quarter of 2.

Example 2 — the pressure law, and the Celsius trap

A rigid sealed container holds gas at 100 kilopascals and 300 K. It is heated to 600 K. What is the new pressure? Then work the same problem written as 27 °C heated to 327 °C, and check that the two agree.

The volume is fixed, so use $\frac{p}{T}$ is constant: $\frac{100}{300} = \frac{p_2}{600}$
Multiply both sides by 600. On the right the 600s cancel and leave p₂; on the left, $\frac{100}{300} = \frac{1}{3}$ , and one third of 600 is 200. So p₂ = 200 kilopascals.
Now the Celsius version. Convert both temperatures first: 27 + 273 = 300 K and 327 + 273 = 600 K — the same two figures, so the answer is the same 200 kilopascals.
See what would have happened without converting. Using 27 and 327 directly gives $p_2 = \frac{100 \times 327}{27}$ , which is about 1211 kilopascals — six times too big.
The reason the error is so large is that the Celsius numbers get their ratio badly wrong: $\frac{327}{27}$ is about 12, while the true ratio of absolute temperatures is $\frac{600}{300} = 2$ .

Example 3 — Charles's law, both ways round

A gas occupies 2 cubic metres at 300 K, in a cylinder closed by a piston that slides freely so the pressure stays constant. It is warmed to 450 K. What is the new volume? Then: to what temperature would it have to be cooled to occupy 1 cubic metre?

At constant pressure, $\frac{V}{T}$ is constant: $\frac{2}{300} = \frac{V_2}{450}$
Multiply both sides by 450, which cancels on the right: $V_2 = \frac{450 \times 2}{300}$ Take it as $\frac{900}{300} = 3$ cubic metres.
Sense-check the ratio: the temperature went up by a factor of $\frac{450}{300} = 1.5$ , so the volume should too, and 2 × 1.5 = 3.
For the second part: $\frac{2}{300} = \frac{1}{T_2}$ Cross-multiplying gives 2 × T₂ = 300 × 1, so $T_2 = \frac{300}{2} = 150$ K.
Halving the volume halved the absolute temperature. Note that 150 K is −123 °C, which is a reminder of why the working is done in kelvin: the Celsius figure gives no hint that the temperature has been halved.

Example 4 — the combined law, all three changing

A sealed cylinder holds gas at 100 kilopascals, 3 cubic metres and 300 K. It is compressed to 1 cubic metre and at the same time warmed to 400 K. What is the new pressure?

Use the combined form: $\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}$ , so $\frac{100 \times 3}{300} = \frac{p_2 \times 1}{400}$ .
Work out the left-hand side: $100 \times 3 = 300 \quad\text{and}\quad \frac{300}{300} = 1$
So $1 = \frac{p_2}{400}$ , and multiplying both sides by 400 gives p₂ = 400 kilopascals.
Check it by taking the two effects separately, which is often quicker. Cutting the volume from 3 to 1 multiplies the pressure by 3, giving 300 kilopascals. Raising the temperature from 300 K to 400 K multiplies it by $\frac{400}{300}$ , which is 4/3, and 300 × 4/3 = 400 kilopascals. The same answer.
Both effects pushed the pressure the same way here, which is why the answer is four times the original. Had the gas been cooled instead, the two would have partly cancelled.

Why it matters in aviation

These relationships tie several aircraft systems together, because each system is the same statement applied to a different container. A pressurised cabin is a vessel of essentially fixed volume held at a controlled difference between the pressure inside it and the much lower pressure outside, while the air within it is continually warmed, cooled and replaced — so all three quantities are being managed at once rather than one at a time.

A tyre is a closed volume of gas inflated on a warm apron and then cold-soaked for hours at altitude: its pressure falls as its temperature falls and comes back as it warms, which is the pressure law and is why an inflation figure means little without the temperature at which it was set. A stored gas cylinder behaves the same way, reading lower in the cold and higher standing in the sun with exactly the same quantity of gas inside it. In every one of those cases the temperature that goes into the working is the absolute one, so a cold-soak problem has to be converted to kelvin even though the figures arrive in degrees Celsius.

What the test asks

One distractor dominates this topic and it is the Celsius one: an item quotes temperatures in degrees Celsius and offers the answer you get by putting those figures straight into the law. As Example 2 showed, the wrong answer is not marginally wrong, so the giveaway is scale — if one option is several times another, suspect the conversion. The rule is mechanical: convert first, always, even when the temperatures look convenient. The second trap is applying the wrong law, usually by not reading which quantity is being held fixed; the words rigid or sealed container tell you the volume is constant, while a freely sliding piston tells you the pressure is. Expect also an inverse-proportion item that offers the direct-proportion answer, so that halving the volume is claimed to halve the pressure. And on partial pressures, the reliable distractor states that the percentage of oxygen falls with altitude — it does not, it stays at about 21 per cent, and it is the total pressure and therefore oxygen's share of it in absolute terms that fall.

Three quantities, one fixed mass of gas

Boyle's law: pressure against volume, at constant temperature

Charles's law: volume against temperature, at constant pressure

The pressure law: pressure against temperature, at constant volume

Why every gas law needs kelvin, never Celsius

The combined gas law

Mixtures: Dalton's partial pressures

Why it matters in aviation

What the test asks

Check you understood