Boyle's Law Calculator
Find the final pressure or final volume of a gas when temperature and amount stay constant using P1V1 = P2V2.
Isothermal Gas State Transformation (Constant Temperature: T₁ = T₂)
P₂ = (P₁ × V₁) / V₂ P₂ = (1.0 atm × 4.0 L) / 2.0 L = 2 atm This Boyle's law calculator finds the final pressure or final volume of a gas when temperature and the amount of gas stay constant. Enter three of the four values (P1, V1, P2, V2), leave the unknown blank, and the tool solves P1V1 = P2V2 with full steps. It supports atm, kPa, mmHg, torr, bar, and psi for pressure, and L, mL, and m³ for volume, so you can work in whatever units your problem uses.
Quick Reference
| Item | Value |
|---|---|
| Formula | P1V1 = P2V2 |
| Relationship | Pressure and volume are inversely proportional |
| Held constant | Temperature and amount of gas (moles) |
| Solves for | P1, V1, P2, or V2 |
| Temperature needed | No, but it must not change |
What Is Boyle's Law?
Boyle's law is a gas law stating that the pressure of a fixed amount of gas is inversely proportional to its volume when temperature stays constant. If you squeeze a gas into half the space, its pressure doubles. If you let it expand to twice the space, its pressure drops to half.
The law is named after Robert Boyle, the Anglo-Irish chemist who published it in 1662 after experiments with air trapped in a J-shaped glass tube sealed with mercury. His assistant, Robert Hooke, helped build the apparatus. The French physicist Edme Mariotte discovered the same relationship independently in 1676, so in many countries it is called the Boyle-Mariotte law.
Boyle's law is one of the three classic gas laws, along with Charles's law and Gay-Lussac's law. Together they form the combined gas law, and all of them follow from the ideal gas law.
Why the Law Works
According to the kinetic molecular theory, gas pressure comes from particles colliding with the walls of their container. When you reduce the volume, the same number of particles hit a smaller wall area more often, so pressure rises. Because temperature is constant, the average speed of the particles does not change, which is why volume and pressure trade off in a simple inverse way.
Boyle's Law Formula
The formula is:
You can also write it as PV = k, where k is a constant for a given gas sample at a given temperature.
Where the Formula Comes From
Start with the ideal gas law:
When the amount of gas (n) and temperature (T) are constant, and R is the universal gas constant, the right side nRT is a constant. That means PV is constant, so the product in the initial state equals the product in the final state.
Variable Table
| Variable | Meaning | Common Units |
|---|---|---|
| P1 | Initial pressure | atm, kPa, mmHg, torr, bar, psi |
| V1 | Initial volume | L, mL, m³ |
| P2 | Final pressure | atm, kPa, mmHg, torr, bar, psi |
| V2 | Final volume | L, mL, m³ |
How to Use This Boyle's Law Calculator
Select what to solve for
Choose P1, V1, P2, or V2.
Enter the three known values
Fill in the initial pressure, initial volume, and one final value.
Pick the units
Choose atm, kPa, mmHg, torr, bar, or psi for pressure, and L, mL, or m³ for volume.
Click Calculate
The tool rearranges P1V1 = P2V2 and returns the missing value.
Read the steps
Check the substituted equation to see how the answer was found.
Run a sense check
If pressure went up, volume must have gone down, and the other way around.
What Can This Calculator Calculate?
Final volume (V2)
How much space a gas occupies after its pressure changes.
Final pressure (P2)
The pressure of a gas after it is compressed or expanded.
Initial volume (V1)
The starting volume when you know the final state.
Initial pressure (P1)
The starting pressure when you know the final state.
If all three of pressure, volume, and temperature change together, use the Combined Gas Law Calculator. If pressure and temperature change at constant volume, use the Gay-Lussac's Law Calculator. If volume and temperature change at constant pressure, use the Charles's Law Calculator.
Boyle's Law Rearranged Formulas
Each variable can be isolated with one step of algebra.
| Solve For | Formula |
|---|---|
| Final volume | V2 = (P1 × V1) / P2 |
| Final pressure | P2 = (P1 × V1) / V2 |
| Initial volume | V1 = (P2 × V2) / P1 |
| Initial pressure | P1 = (P2 × V2) / V1 |
Boyle's Law Examples
Solve for Final Volume (V2)
Problem: A gas has a volume of 4.0 L at 1.0 atm. The pressure is increased to 2.5 atm at constant temperature. Find the new volume.
Step 1: Write the formula. V2 = (P1 × V1) / P2
Step 2: Substitute. V2 = (1.0 × 4.0) / 2.5
Pressure increased by a factor of 2.5, so volume shrank by the same factor.
Solve for Final Pressure (P2)
Problem: A syringe holds 500 mL of air at 101.3 kPa. The plunger is pushed until the volume is 200 mL. Find the final pressure, assuming constant temperature.
Step 1: Write the formula. P2 = (P1 × V1) / V2
Step 2: Substitute. P2 = (101.3 × 500) / 200
The volume dropped to 40% of the original, so pressure rose to 2.5 times the original.
Solve for Initial Volume (V1)
Problem: A gas at 3.0 atm is allowed to expand until its pressure is 1.0 atm and its volume is 6.0 L. What was the initial volume?
Step 1: Write the formula. V1 = (P2 × V2) / P1
Step 2: Substitute. V1 = (1.0 × 6.0) / 3.0
Mixed Units (mmHg and atm)
Problem: A 2.0 L gas sample is at 760 mmHg. The pressure changes to 1.50 atm at constant temperature. Find the final volume.
Step 1: Convert to the same pressure unit. 1.50 atm × 760 mmHg/atm = 1140 mmHg
Step 2: Write the formula. V2 = (P1 × V1) / P2
Step 3: Substitute. V2 = (760 × 2.0) / 1140
Pressure units must match before you solve. This calculator converts them for you.
Diver's Lungs
Problem: A diver takes in 6.0 L of air at the surface, where the pressure is 1.0 atm. At a depth of about 10 m, the absolute pressure is about 2.0 atm. What volume would that air occupy at depth?
Solution: V2 = (1.0 × 6.0) / 2.0 = 3.0 L
This is why divers must breathe normally and never hold their breath while ascending. Air in the lungs expands as pressure falls, which can cause serious lung injury.
Boyle's Law Units
Pressure
| Unit | Equivalent |
|---|---|
| 1 atm | 101.325 kPa |
| 1 atm | 760 mmHg |
| 1 atm | 760 torr |
| 1 atm | 1.01325 bar |
| 1 atm | 14.696 psi |
Use the same pressure unit for P1 and P2. Use absolute pressure, not gauge pressure. A tire gauge reads pressure above the atmosphere, so add about 14.7 psi (1 atm) to convert it to absolute.
Volume
| Unit | Equivalent |
|---|---|
| 1 L | 1000 mL |
| 1 mL | 1 cm³ |
| 1 L | 1 dm³ |
| 1 m³ | 1000 L |
Use the same volume unit for V1 and V2. Because the law is a ratio, units cancel, and any consistent pair works.
Temperature: Temperature is not in the formula, but it must stay constant. This is called an isothermal process.
Boyle's Law Graph: The Inverse Relationship
Plotting pressure against volume gives a curve called a hyperbola. As volume increases, pressure falls along the curve, and the two never reach zero. Each curve is called an isotherm because it represents one fixed temperature. A higher temperature gives a curve farther from the axes.
If you plot pressure against 1/V, the result is a straight line through the origin. This confirms that P is proportional to 1/V, which is the mathematical meaning of "inversely proportional."
| Volume Change | Pressure Change |
|---|---|
| Volume halves | Pressure doubles |
| Volume doubles | Pressure halves |
| Volume becomes one third | Pressure triples |
| Volume increases by 4 times | Pressure drops to one fourth |
Real-World Examples of Boyle's Law
Breathing
The diaphragm contracts and increases lung volume, which lowers pressure inside the lungs, and air flows in.
Syringes
Pulling the plunger increases volume and lowers pressure, which draws fluid in.
Scuba diving
Air spaces in the lungs and equipment change volume with depth.
Bicycle pumps
Pushing the piston compresses air and raises its pressure.
Weather balloons
A balloon expands as it rises because outside pressure falls.
Soda bottles
Bubbles grow when you open a bottle and pressure drops.
Ears on airplanes
Cabin pressure changes cause air in the middle ear to expand or compress.
When Can You Use Boyle's Law?
Use Boyle's law when all of the following are true:
Constant temperature
The process is isothermal.
Constant amount of gas
No gas is added, removed, or lost, and nothing reacts.
A gas sample
The law applies to gases, not liquids or solids.
Two states
You are comparing an initial state and a final state of the same sample.
Boyle's Law vs Other Gas Laws
| Gas Law | Formula | Held Constant | Relationship |
|---|---|---|---|
| Boyle's law | P1V1 = P2V2 | Temperature, moles | Pressure and volume are inversely related |
| Charles's law | V1/T1 = V2/T2 | Pressure, moles | Volume and temperature are directly related |
| Gay-Lussac's law | P1/T1 = P2/T2 | Volume, moles | Pressure and temperature are directly related |
| Combined gas law | P1V1/T1 = P2V2/T2 | Moles | All three change together |
| Ideal gas law | PV = nRT | None | Links P, V, T, and moles |
Common Boyle's Law Mistakes
| Mistake | Why It Fails | Fix |
|---|---|---|
| Letting temperature change | Boyle's law needs constant temperature | Use the combined gas law instead |
| Unit mismatch | atm and kPa in the same equation break the ratio | Convert to one unit first |
| Using gauge pressure | Ignores atmospheric pressure | Add atmospheric pressure to get absolute |
| Treating the relationship as direct | Pressure and volume move in opposite directions | Remember: one goes up, the other goes down |
| Swapping P1 and P2 | Puts the initial value in the final slot | Label all four values before solving |
| Changing the amount of gas | Moles must stay constant | Use the ideal gas law |
| Skipping the sense check | Wrong answers go unnoticed | Check that pressure and volume moved oppositely |
Boyle's Law FAQs
Frequently asked questions about isothermal gas behavior, pressure-volume relationships, scuba diving, and units.
What is Boyle's law?
Boyle's law states that the pressure of a fixed amount of gas is inversely proportional to its volume at constant temperature. The formula is P1V1 = P2V2.
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.
How do you use a Boyle's law calculator?
Enter any three of the four values (P1, V1, P2, V2), choose your units, leave the unknown blank, and click Calculate. The tool returns the missing value with steps.
Does Boyle's law need temperature in Kelvin?
Temperature is not in the formula, so no conversion is needed. It only has to stay constant. If temperature changes, use the combined gas law, which does require Kelvin.
What does inversely proportional mean in Boyle's law?
It means that when one quantity increases, the other decreases by the same factor. Doubling pressure halves volume.
What units can I use for Boyle's law?
Any pressure unit and any volume unit work if they match between the initial and final states. Common choices are atm, kPa, mmHg, liters, and milliliters.
Who discovered Boyle's law?
Robert Boyle published it in 1662. Edme Mariotte found the same relationship independently in 1676, so it is also called the Boyle-Mariotte law.
What is an isothermal process?
An isothermal process is one in which temperature stays constant. Boyle's law describes gas behavior during an isothermal change.
Is Boyle's law accurate for real gases?
It is accurate for real gases at low to moderate pressure and temperatures well above the boiling point. At very high pressure or near condensation, real gases deviate from the law.
What is the difference between Boyle's law and the ideal gas law?
Boyle's law compares two states of a gas at constant temperature and moles. The ideal gas law (PV = nRT) describes a single state and lets you solve for moles or temperature.
Why does pressure increase when volume decreases?
Gas particles hit the container walls more often when they are confined to a smaller space, which raises the pressure.
How is Boyle's law related to the combined gas law?
Boyle's law is the special case of the combined gas law where temperature is constant. Setting T1 = T2 in P1V1/T1 = P2V2/T2 gives P1V1 = P2V2.