Push the plunger of a sealed syringe and you feel the air pushing back harder the farther you go. Robert Boyle measured this in 1662 and found a simple inverse relationship: at constant temperature, a gas’s pressure times its volume stays the same. This calculator solves Boyle’s law for whichever of the four values you are missing, in any pressure and volume units, draws the pressure–volume curve through both states and reports the work involved.
How to use the Boyle’s law calculator
- Under Solve for, choose the initial pressure, initial volume, final pressure or final volume.
- Enter the known values. Pressures accept Pa, kPa, MPa, bar, mbar, atm, psi, Torr, mmHg and inHg; volumes accept L, mL, m³, cm³, ft³, in³ and US gallons. Initial and final values can use different units.
- Choose the unit for the answer under Show the result in.
- Read the result with its conversions, the constant product P × V, the compression or expansion ratio and the work for a slow, isothermal change. The diagram shows both states on the same isotherm.
Boyle’s law formula
The product PV has units of energy (1 Pa·m³ = 1 J). For an ideal gas it equals nRT, which is why it stays constant when the amount of gas and the temperature do. Plotted on a pressure–volume graph, all the states at one temperature lie on a hyperbola called an isotherm.
Worked example
Compressing a gas sample
6.0 L of air at 1.0 atm is compressed slowly, at constant temperature, until the pressure reaches 3.0 atm.
V2 = 1.0 atm × 6.0 L ÷ 3.0 atm = 2.0 L
The product PV stays at 6 atm·L (608 J). The compression ratio is 3 : 1, and the work done on the gas is 101,325 × 0.006 × ln 3 = 668 J.
A syringe. Seal a syringe holding 60 mL of air at 14.7 psi and push the plunger to 20 mL. Solve for final pressure: P2 = 14.7 × 60 ÷ 20 = 44.1 psi, or about 29.4 psi above the outside air on a gauge.
Diving. At 20 m in seawater the absolute pressure is about 2.98 atm. A breath-hold diver who fills 6 L of lung air at the surface will have it squeezed to 6 ÷ 2.98 ≈ 2.0 L at that depth.
Pressure and volume pairs at constant temperature
| Pressure | Volume (starting from 1 atm, 6 L) |
|---|---|
| 0.5 atm | 12 L |
| 1 atm | 6 L |
| 2 atm | 3 L |
| 3 atm | 2 L |
| 6 atm | 1 L |
Each row multiplies to the same 6 atm·L. That shape, steep at small volumes and flat at large ones, is why the last bit of compression in a pump always takes the most force.
When Boyle’s law fails
Real gases depart from Boyle’s law at high pressure, where molecules take up a noticeable fraction of the volume, and near condensation, where they attract each other. Carbon dioxide at 20 °C liquefies once it is compressed to about 57 bar (56 atm); from then on the pressure stops rising while the volume keeps shrinking. Fast compressions are not isothermal: a bicycle pump warms up because the work done on the air raises its temperature.
If the temperature changes as well, use the combined gas law calculator. For volume changes with temperature at constant pressure, see the Charles’s law calculator, and to relate pressure, volume, temperature and moles in one state, the ideal gas law calculator.
Frequently asked questions
What does Boyle's law say?
At constant temperature, the pressure of a fixed amount of gas is inversely proportional to its volume: P₁V₁ = P₂V₂. Halve the volume and the pressure doubles; triple the volume and the pressure falls to a third.
Do I need to convert units?
No. Pressures and volumes each have their own unit menus, from pascals and atmospheres to psi and mmHg, and from milliliters to cubic feet. The calculator converts to SI internally and shows the result in the unit you choose.
Why does Boyle's law require constant temperature?
Compressing a gas quickly heats it, and expanding it quickly cools it. If the temperature changes, the pressure–volume product changes too. Boyle's law describes slow changes, or changes measured after the gas returns to its starting temperature. For changing temperature, use the combined gas law.
How does Boyle's law affect divers?
Water pressure rises by about 1 atm every 10 m of seawater. A breath-hold diver's 6 L of lung air at the surface shrinks to about 2 L at 20 m, where the absolute pressure is roughly 3 atm. Scuba divers must never hold their breath while ascending, because the air in their lungs would expand by the same ratio.
What is the work done in a Boyle's law process?
For a slow, isothermal change, W = P₁V₁ ln(V₂/V₁). Compressing 6 L of gas at 1 atm down to 2 L takes about 668 J of work on the gas, which leaves as heat to keep the temperature constant.