Combined Gas Law Calculator

Boyle's Law Calculator

Solve P₁V₁ = P₂V₂ for any missing pressure or volume at constant temperature. Leave one field blank, enter the other three, and get an instant result with full step-by-step working. No formula rearranging required.

Boyle's Law Solver

What Is Boyle's Law?

Boyle's Law, published by Robert Boyle in 1662, describes one of the simplest and most reliable relationships in gas behavior: at constant temperature, the pressure and volume of a fixed quantity of gas are inversely proportional. Squeeze a gas into half its volume and its pressure doubles; let it expand to twice the volume and pressure falls by half.

Mathematically, this is written as P₁V₁ = P₂V₂, where the subscript 1 marks the initial state and 2 marks the final state. Because the product of pressure and volume stays constant, a plot of pressure against volume traces a hyperbola, the classic "PV isotherm" shown in the diagram alongside this text.

Boyle's Law is a special case of the combined gas law, P₁V₁/T₁ = P₂V₂/T₂, that applies specifically when temperature does not change. If your problem also involves a temperature change, use the full combined gas law calculator instead.

Pressure vs. Volume at Constant T 1 2 Volume (V) Pressure (P)

As volume increases, pressure falls along a smooth curve so that P·V never changes.

Real-World Examples of Boyle's Law

Syringe Compression Low pressure large volume High pressure small volume

Pushing a syringe's plunger shrinks the trapped air's volume, so its pressure climbs.

Boyle's Law shows up constantly outside the classroom. A syringe with its tip blocked feels stiffer to push the further in the plunger goes, because the trapped air is being squeezed into an ever-smaller volume and pushing back with rising pressure.

Scuba divers rely on Boyle's Law for safety: as a diver descends, surrounding water pressure rises and compresses the air in their lungs and equipment; on ascent, that same air expands again, which is why divers are trained to ascend slowly and exhale continuously to avoid lung overexpansion injury.

A bicycle pump works the same way in reverse of the syringe, each stroke reduces the air's volume inside the pump barrel, raising its pressure until it exceeds the tire's internal pressure and flows in. In every case, the same rule governs the outcome: P₁V₁ = P₂V₂, as long as temperature does not change much during the process.

How the Calculator Works

1. Pick the unknown
Select P₁, V₁, P₂, or V₂ as the value you want solved.
2. Fill in the rest
Enter the other three values in any supported unit. The tool converts automatically.
3. Get instant results
The answer and a full substitution walkthrough appear immediately, with no submit button needed.

Worked Example: Boyle's Law in Practice

Problem: A diver's lungs hold 6.00 L of air at 3.00 atm (20 m depth). What volume would that air occupy at the surface (1.00 atm) if exhaled fully during ascent?

Given: P₁ = 3.00 atm, V₁ = 6.00 L, P₂ = 1.00 atm. Find: V₂. Boyle's Law: P₁V₁ = P₂V₂ → V₂ = P₁V₁ / P₂ V₂ = (3.00 × 6.00) / 1.00 V₂ = 18.0 L

That same lungful of air would triple in volume by the time it reached the surface. Exactly why divers are trained to exhale continuously and never hold their breath while ascending. Try changing the depth (pressure) in the calculator above to see how dramatically volume responds near the surface, where the pressure ratio changes fastest.

Common Mistakes When Applying Boyle's Law

The single most frequent error is forgetting that Boyle's Law only applies when temperature truly stays constant. Students often apply P₁V₁ = P₂V₂ to a problem where a gas is both compressed and heated, getting a plausible-looking but wrong answer because the formula silently ignores the temperature term. Before reaching for Boyle's Law, always check the problem statement for any mention of temperature change. If there is one, you need the combined gas law instead.

A second common mistake is mixing pressure units mid-calculation, for instance, entering P₁ in atmospheres and P₂ in kPa without converting one to match the other first. Because Boyle's Law is a direct algebraic relationship between the two pressures, both must be expressed in the same unit before you multiply or divide; the same rule applies to V₁ and V₂. This calculator handles that conversion automatically, but if you're solving by hand, always convert to a common unit as your first step.

A third, subtler mistake is applying Boyle's Law to a gas sample where the amount of gas itself is changing. For example, gas escaping slowly from a leaking container while it is also being compressed. Boyle's Law assumes a fixed, sealed quantity of gas; if moles are changing too, you need the ideal gas law's more general treatment instead.

Boyle's Law FAQ

What is Boyle's Law?
Boyle's Law states that for a fixed amount of gas held at a constant temperature, pressure and volume are inversely proportional: P₁V₁ = P₂V₂. If you compress the gas into a smaller volume, its pressure rises by the same factor; if you let it expand, its pressure falls. The law only holds while temperature and the amount of gas (moles) stay fixed. It is the constant-temperature special case of the combined gas law.
What is the formula for Boyle's Law?
The formula is P₁V₁ = P₂V₂, where P₁ and V₁ are the initial pressure and volume, and P₂ and V₂ are the final pressure and volume. Rearranged for any unknown: P₂ = P₁V₁/V₂, or V₂ = P₁V₁/P₂. All you need is any three of the four values. This calculator solves for whichever one you leave blank.
Does Boyle's Law require the temperature to stay exactly constant?
Yes: Boyle's Law is only valid for an isothermal process, meaning temperature (and the number of moles of gas) must not change between state 1 and state 2. If temperature also changes, you need the full combined gas law, P₁V₁/T₁ = P₂V₂/T₂, which this calculator's parent tool on the homepage solves directly.
What units does this Boyle's Law calculator support?
Pressure can be entered in Pa, kPa, MPa, bar, mbar, atm, mmHg, torr, psi, or inHg. Volume can be entered in m³, L, mL, cm³, dm³, ft³, in³, or gal (US). You can mix units freely: for example, enter P₁ in atm and P₂ in kPa. The calculator converts everything internally before solving.
Why does pressure increase when volume decreases?
At the molecular level, gas pressure comes from gas molecules colliding with the walls of their container. Squeezing the same number of molecules into a smaller volume packs them closer together, so they strike the walls more often per second, and pressure rises. Boyle's Law is simply the macroscopic, measurable expression of that microscopic crowding effect.
Can Boyle's Law be used for real gases, or only ideal gases?
Boyle's Law is derived from the ideal gas model and is most accurate at low to moderate pressures and temperatures well above a gas's boiling point, where intermolecular forces are negligible. Real gases deviate from it at very high pressure or near condensation, but for typical classroom, laboratory, and engineering problems the inverse P–V relationship holds closely enough to be treated as exact.
How is Boyle's Law used in scuba diving safety calculations?
Dive tables and dive computers use Boyle's Law to calculate how a diver's lung volume and any trapped air spaces will change with depth, since water pressure increases by roughly 1 atm every 10 meters of depth. This is the physical basis for the rule that divers must never hold their breath while ascending. Trapped air expanding under Boyle's Law inside sealed lungs can cause serious injury.
How do I know if a problem needs Boyle's Law or the full combined gas law?
If the problem states or implies that temperature stays the same, or simply never mentions a temperature change, Boyle's Law is sufficient and simpler. If temperature also changes between the two states, you need the combined gas law instead, since Boyle's Law would silently ignore that effect and give a wrong answer.

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