Boyle's law, P₁V₁ = P₂V₂, is the oldest of the classic gas laws and the simplest to picture: squeeze a fixed amount of gas into a smaller space, at the same temperature, and its pressure rises by exactly the same factor the volume shrinks. This article explains where that relationship comes from, why it only holds at constant temperature, and where it shows up in everyday life.
Where P₁V₁ = P₂V₂ comes from
Robert Boyle described the relationship in 1662 after compressing air trapped in a J-shaped tube: doubling the pressure on a fixed amount of gas halved its volume, tripling the pressure cut the volume to a third, and so on. On the molecular level, gas pressure comes from molecules colliding with the container walls. At constant temperature the molecules' average speed doesn't change, so squeezing them into a smaller volume simply packs more collisions into the same wall area per second — pressure rises in exact proportion to how much the volume shrank. That's why the product P × V stays constant: it's Boyle's law, and it is also the special case of the ideal gas law PV = nRT when both the amount of gas (n) and the temperature (T) are held fixed, so nRT collapses to a single constant k.
Why temperature has to stay constant
Boyle's law is strictly isothermal. If temperature changes along with pressure and volume, the simple inverse relationship breaks — you need the combined gas law (P₁V₁/T₁ = P₂V₂/T₂) or the full ideal gas law instead. In practice, Boyle's law is a good approximation whenever a gas is compressed or expanded quickly enough, or in a small enough space, that heat has no time to escape or the surroundings hold the temperature steady — a piston moved by hand, a syringe, or a diver's lungs near a constant body temperature. When in doubt, check whether the process is genuinely isothermal before applying P₁V₁ = P₂V₂ on its own.
Keeping units consistent
Boyle's law only compares a ratio, so any pressure unit works as long as both P₁ and P₂ use it, and any volume unit works as long as both V₁ and V₂ use it — you don't need SI units the way you do for the full ideal gas law. This calculator still converts everything to a common internal unit automatically, so you're free to enter P₁ in atm and P₂ in psi, or V₁ in litres and read V₂ back in gallons, and get a correct answer either way.
Real-world examples
Boyle's law shows up anywhere a fixed pocket of gas changes volume. A bicycle pump compresses air to inflate a tire — the smaller the stroke volume gets, the higher the pressure climbs. A scuba diver's lungs and any air spaces feel rising pressure as they descend, which is why divers must equalize their ears and never hold their breath while ascending: as pressure drops on the way up, trapped air expands and can injure the lungs if it can't escape. A weather balloon does the reverse, expanding as it rises into the lower-pressure upper atmosphere until it eventually bursts. Even breathing itself works on the same principle: expanding your rib cage lowers the pressure inside your lungs relative to the outside air, drawing air in.
Limits: real gases and the ideal gas law
Boyle's law describes an ideal gas — one whose molecules are treated as point particles with no attraction to each other. Real gases deviate from this at very high pressure (where molecules are packed close enough that their own volume matters) or near the temperature where they start to condense into a liquid. For everyday pressures and temperatures, the deviation is small enough to ignore. And because Boyle's law is just the constant-temperature slice of PV = nRT, it connects directly to the ideal gas law calculator when you also need to track changing amount or temperature.