What Boyle's law states
For a fixed quantity of gas at constant temperature, pressure and volume are inversely proportional. Halve the volume and the pressure doubles; triple the volume and the pressure falls to a third. The product of the two is a constant, so P₁V₁ = P₂V₂ relates any two states of the same gas sample.
The molecular picture makes the inverse relationship obvious. Pressure is the rate at which molecules strike unit area of the container wall. Squeeze the same number of molecules into half the space and each one, moving at the same average speed because the temperature has not changed, reaches a wall twice as often. Twice the collision rate is twice the pressure.
Two conditions are load-bearing. The amount of gas is fixed — a leaking or refilling vessel obeys no such rule. And the temperature is constant, which in practice means the change is slow enough for heat to flow in or out and keep it so. A rapid compression heats the gas and the pressure rises by more than Boyle's law predicts, which is exactly why a bicycle pump gets warm.
Using the equation, and the pressure trap
Rearranging is trivial: V₂ = P₁V₁ ÷ P₂, P₂ = P₁V₁ ÷ V₂, and the same in reverse for the initial state. Because both sides carry the same units, any consistent pair works — kPa with litres, atmospheres with millilitres, psi with cubic feet. Only the two pressures must share a unit, and the two volumes must share a unit.
Pressures must be absolute. This is the error that ruins more Boyle's law calculations than all others combined. A tyre gauge, a manifold gauge and a dive gauge read gauge pressure — the excess over the surrounding atmosphere — and Boyle's law is written in absolute terms. Add 101.325 kPa (14.696 psi, 1 atm) to a gauge reading at sea level to convert. A tyre at 32 psi gauge is at 46.70 psi absolute. To see the size of the error, take a gas at 200 kPa gauge compressed until the gauge reads 500 kPa: in absolute terms that is 301.3 kPa to 601.3 kPa, so the volume falls to 301.3 ÷ 601.3 = 50.1% of its original. The gauge numbers give 200 ÷ 500 = 40.0% — a compression a fifth larger than the real one. Subtracting a constant from both pressures always pushes their ratio further from 1, so a gauge reading overstates the volume change in both directions.
Volume, by contrast, needs no such correction, because volume has a natural zero. That asymmetry is worth internalising: the same issue reappears with temperature in Charles's law, where kelvin is mandatory for exactly the same reason.
Plotted, the law is a hyperbola — P against V curves steeply near small volumes and flattens as the gas expands. Plot P against 1/V instead and you get a straight line through the origin whose slope is the constant. That linear form is how the relationship is usually verified in a teaching laboratory, since a straight line is far easier to judge by eye than a curve.
Worked example: a diver's lungful at 10 metres
A diver takes a full breath of 12.0 L at the surface, where the absolute pressure is 101.325 kPa, then descends to 10 m. Seawater adds roughly one atmosphere for every 10 m of depth, so the pressure there is about 202.65 kPa. What happens to the volume?
- Identify the terms. P₁ = 101.325 kPa, V₁ = 12.0 L, P₂ = 202.65 kPa, and V₂ is unknown.
- Compute the constant. P₁V₁ = 101.325 × 12.0 = 1215.9 kPa·L.
- Divide by the new pressure. V₂ = 1215.9 ÷ 202.65 = 6.00 L.
- Check. The pressure doubled, so the volume halved. The compression ratio V₁/V₂ = 2.00, and it equals P₂/P₁ = 2.00 as it must.
Now reverse the problem, which is the one that matters for safety. A diver at 10 m fills their lungs to 6.0 L from a regulator and ascends holding their breath. At the surface that gas expands back to 12.0 L — twice the lung's capacity. This is why the first rule taught in scuba training is to breathe continuously during ascent, and why the risk is greatest in the last ten metres, where the relative pressure change is largest: from 2 atm to 1 atm doubles the volume, whereas from 5 atm to 4 atm expands it by only 25%.
Note that the calculation used 101.325 kPa at the surface, not zero. A dive computer reading "0 bar" at the surface is a gauge reading, and using it directly would make the equation collapse.
When the answer can be trusted
Boyle's law is the isothermal special case of the ideal gas law, PV = nRT. It therefore inherits every limitation of the ideal model, and those become visible in two regimes.
High pressure. Above roughly 50 bar the finite size of molecules and the attractions between them matter, and the pressure-volume product stops being constant. The deviation is captured by a compressibility factor Z = PV/nRT, which is 1 for an ideal gas, dips below 1 at moderate pressures where attractions dominate, and rises above 1 at very high pressures where molecular volume dominates. A 200 bar scuba cylinder holds meaningfully less gas than the ideal law predicts.
Near condensation. Close to the boiling point at the working pressure, compressing the gas produces liquid rather than a pressure rise, and Boyle's law fails completely. This is the flat section of an isotherm on a phase diagram.
Between those extremes — a few atmospheres, well above the condensation temperature — the law is accurate to better than a percent for air, nitrogen, oxygen and the noble gases, which covers almost every practical use.
One more check worth doing: confirm the temperature really was constant. A compression done in seconds is closer to adiabatic than isothermal, and for a diatomic gas the adiabatic relation PV^1.4 = constant gives a substantially higher final pressure than PV = constant.
Volume of a 12.0 L gas sample at different absolute pressures
| Absolute pressure | In atm | Volume (L) | Compression ratio | Equivalent seawater depth |
|---|---|---|---|---|
| 50.66 kPa | 0.50 | 24.00 | 0.50× | — |
| 101.33 kPa | 1.00 | 12.00 | 1.00× | surface |
| 151.99 kPa | 1.50 | 8.00 | 1.50× | 5 m |
| 202.65 kPa | 2.00 | 6.00 | 2.00× | 10 m |
| 303.98 kPa | 3.00 | 4.00 | 3.00× | 20 m |
| 405.30 kPa | 4.00 | 3.00 | 4.00× | 30 m |
| 506.63 kPa | 5.00 | 2.40 | 5.00× | 40 m |
| 1013.25 kPa | 10.00 | 1.20 | 10.00× | 90 m |
Depths use the common approximation of one additional atmosphere per 10 m of seawater. Every row has the same pressure-volume product, 1215.9 kPa·L.
Mistakes that give a wrong volume or pressure
- Using gauge pressure. Add atmospheric pressure to any gauge reading first. This is the dominant error, and it always understates the pressure and so overstates the volume change.
- Letting the temperature change. A fast compression heats the gas. If the temperature moves, use the combined gas law P₁V₁/T₁ = P₂V₂/T₂ with temperatures in kelvin.
- Mixing pressure units between the two sides. The units cancel only if both pressures share one. Litres against millilitres does the same damage on the volume side.
- Applying it when gas can enter or leave. A cylinder being filled or a leaking vessel changes n, and Boyle's law assumes n is fixed.
- Using it at very high pressure. Beyond about 50 bar the ideal approximation degrades and you need a compressibility factor.
- Forgetting that a gas near its condensation point does not obey it at all. Compressing a vapour at its saturation pressure produces liquid, not a pressure rise.
Where Boyle's law sits among the gas laws
Robert Boyle published the relationship in 1662, in an appendix to New Experiments Physico-Mechanical, from measurements made with a J-shaped tube in which mercury trapped a column of air. Edme Mariotte reached the same result independently a decade later and added the crucial qualification that the temperature must be held constant, which is why continental texts call it Mariotte's law.
It is one of three empirical laws that combine into the ideal gas law. Charles's law holds pressure constant and makes volume proportional to absolute temperature. Gay-Lussac's law holds volume constant and makes pressure proportional to absolute temperature. Avogadro's law makes volume proportional to amount. Multiply them together and you get PV = nRT, of which Boyle's law is the slice at fixed n and T.
In practice, if anything other than pressure and volume changes, step up to the combined gas law P₁V₁/T₁ = P₂V₂/T₂, remembering that temperatures must be absolute — kelvin, not celsius — for exactly the same reason pressures must be absolute here. If the amount of gas also changes, only the full ideal gas law will do.
The law's practical reach is wide: it explains breathing, since the diaphragm increases thoracic volume and so lowers lung pressure below atmospheric; it sizes syringes, vacuum systems and gas-handling manifolds; and it underlies the pressure-volume work integral for isothermal expansion, W = nRT ln(V₂/V₁), which is the starting point of the Carnot cycle. For gas-phase reaction quantities rather than pure P–V behaviour, convert to moles with the ideal gas law first and then use the mole-ratio calculator or the limiting reagent calculator.
Key terms
- Absolute pressure
- Pressure measured from a perfect vacuum. Equal to gauge pressure plus the surrounding atmospheric pressure, about 101.325 kPa at sea level.
- Isothermal
- At constant temperature. Requires heat to flow into or out of the gas as it expands or is compressed.
- Adiabatic
- With no heat exchange. A rapid compression is closer to adiabatic, and for a diatomic gas follows PV^1.4 = constant instead.
- Compressibility factor
- Z = PV/nRT. Exactly 1 for an ideal gas; the extent of its departure from 1 measures how far a real gas deviates.