Boyle's Law Calculator

What the Boyle's law calculator does

Squeeze a fixed amount of gas at a steady temperature and its pressure and volume trade off against each other in a precise way. This calculator applies that rule, Boyle's law, letting you enter three of the four quantities, the starting and ending pressures and volumes, and solve for the fourth.

Below is what Boyle's law is, the equation behind it, why pressure and volume move in opposite directions, and a worked example.

How to use it

1. Choose which quantity to find: the initial pressure or volume, or the final pressure or volume.
2. Enter the three known values, each with its unit.
3. Press Calculate for the missing one, or Reset to clear it.

What Boyle's law is

Boyle's law describes how the pressure and volume of a gas are related when the temperature and the amount of gas are held fixed. It says that pressure and volume are inversely proportional: as one goes up, the other goes down, in exact step. Compress a gas into half the space and its pressure doubles; let it expand into twice the room and its pressure halves. Named after Robert Boyle, who established it in the seventeenth century, it was one of the first quantitative laws of gas behaviour.

It is really a special case of the broader ideal gas law, the slice of it you get when temperature and amount stay constant and only pressure and volume are free to change. That focus makes it especially handy, because so many practical situations involve squeezing or expanding a fixed pocket of gas at roughly steady temperature. This calculator captures exactly that trade-off.

The equation it uses

Boyle's law says the product of pressure and volume stays the same before and after a change, so:

P₁ × V₁ = P₂ × V₂

The initial pressure times the initial volume equals the final pressure times the final volume. From this, the calculator solves for whichever quantity you leave out, for instance finding the final pressure as P₂ = P₁V₁ ÷ V₂. The key feature is that the product P times V is a constant for the gas, so any change in one must be matched by an opposite change in the other to keep that product fixed.

The inverse relationship

Because the product of pressure and volume is constant, the two quantities are locked in an inverse dance. This is different from a simple proportion, where doubling one doubles the other. Here, doubling the volume halves the pressure, and cutting the volume to a third triples the pressure. They always move in opposite directions, and by reciprocal amounts.

The physical reason is intuitive once you picture the gas as countless molecules bouncing around inside a container, their impacts on the walls creating the pressure. Shrink the container and the same molecules strike the walls more often in the smaller space, so the pressure rises. Enlarge it and the impacts spread thinner, so the pressure falls. The molecules themselves have not changed, only how crowded they are, and that crowding is what pressure measures.

Boyle's law around you

This trade-off shows up constantly. Pressing the plunger of a syringe with the tip blocked squeezes the trapped air into less space, and you feel the pressure build under your thumb, that is Boyle's law resisting you. Your own breathing works by it too: your diaphragm enlarges your chest cavity, lowering the pressure in your lungs so that air flows in, then shrinks it to push air back out.

It matters in less gentle settings as well. A scuba diver's lungs and air spaces must adjust as water pressure changes the volume of the gas they hold, which is why divers are taught never to hold their breath while ascending, since the expanding gas could injure the lungs. Bubbles rising in water swell as the pressure around them drops near the surface, for the same reason. Wherever a fixed amount of gas is squeezed or released at steady temperature, Boyle's law is at work, and this calculator quantifies it.

Units and precision

The calculator works in SI units underneath, converting pressures to pascals and volumes to cubic metres, then reporting the answer across a range of pressure or volume units so you can read it however suits you. Because both sides of the equation use the same kind of quantity, the units cancel cleanly and only the ratio matters. The relationship is exact for a fixed amount of gas at constant temperature.

A worked example

Suppose a gas starts at a pressure of 100,000 pascals in a volume of 2 cubic metres, and you compress it into 1 cubic metre at the same temperature.

The final pressure is P₂ = P₁V₁ ÷ V₂ = (100,000 × 2) ÷ 1 = 200,000 pascals. Halving the volume has exactly doubled the pressure, just as Boyle's law predicts, because the product of pressure and volume must stay the same.

Questions people ask

What is the formula for Boyle's law?

P₁V₁ = P₂V₂: the initial pressure times the initial volume equals the final pressure times the final volume, at constant temperature and amount of gas.

What is the relationship between pressure and volume?

They are inversely proportional. As volume increases, pressure decreases, and as volume decreases, pressure increases, so that their product stays constant.

What is held constant in Boyle's law?

The temperature and the amount of gas. Only the pressure and volume are allowed to change. This is what makes Boyle's law a special case of the ideal gas law.

Why does compressing a gas raise its pressure?

The same molecules are confined to a smaller space, so they strike the container walls more often. More frequent impacts mean higher pressure, even though the gas itself is unchanged.

References

A quick note on where the physics comes from. Boyle's law and its place within the ideal gas law are standard thermodynamics, set out in OpenStax's Chemistry and in Georgia State University's HyperPhysics. The SI units follow the US National Institute of Standards and Technology. The HyperPhysics link is worth a quick click to confirm it lands where you expect.

1. OpenStax, Chemistry 2e, Section 9.2, Relating Pressure, Volume, Amount, and Temperature: The Ideal Gas Law. https://openstax.org/books/chemistry-2e/pages/9-2-relating-pressure-volume-amount-and-temperature-the-ideal-gas-law
2. HyperPhysics, Ideal Gas Law and Boyle's Law. http://hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/idegas.html
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811