Combined Gas Law Calculator
Solve P₁V₁/T₁ = P₂V₂/T₂ for a gas changing pressure, volume, and temperature at once — or switch to Boyle's law (constant T) or Charles's law (constant P) alone. Two 3D diagrams compare piston compression to balloon heating, and charts show the classic hyperbolic and linear gas-law curves.
Reviewed by the ToolNestr Editorial Team — July 2026
Compression vs. heating
1. Boyle's law — piston compression
Squeezing the same gas molecules into a smaller volume raises pressure, at constant temperature.
2. Charles's law — heated balloon
Heating the same gas at constant pressure makes it expand into a larger volume.
Gas law graphs
How it works
The core idea in one line: pressure, volume, and temperature of a fixed amount of gas are all locked together — change any one of them and at least one of the others has to shift to keep P·V/T constant.
P₁V₁ = P₂V₂
Boyle's law — constant temperature
V₁/T₁ = V₂/T₂
Charles's law — constant pressure, T in kelvin
P₁V₁/T₁ = P₂V₂/T₂
combined gas law — all three variables change, moles constant
A gas sample's pressure, volume, and temperature aren't independent of each other — they're tied together by P·V/T = constant, as long as the number of gas molecules doesn't change. Squeeze a gas into a smaller volume and its pressure must rise to compensate (Boyle's law); heat a gas at constant pressure and it must expand to compensate (Charles's law); heat a gas trapped in a rigid, unchanging volume and its pressure alone must rise (Gay-Lussac's law). The combined gas law is simply the general version of all three, letting pressure, volume, and temperature all shift simultaneously while keeping their overall ratio fixed.
Worked example 1 — a balloon heated at constant pressure
Given: A balloon has a volume of 3.0 L at 20°C. What is its volume when heated to 40°C at constant pressure?
The volume increases by only about 6.8%, even though the temperature rose by 20°C — because the gas law scales with absolute (kelvin) temperature, not the Celsius change directly.
Worked example 2 — a gas compressed and cooled together
Given: A gas at P₁ = 1.0 atm, V₁ = 10.0 L, T₁ = 300 K is compressed to V₂ = 4.0 L and cooled to T₂ = 250 K. Find P₂.
This is the general case where pressure, volume, and temperature all change together — Boyle's and Charles's laws are just special cases of this same equation with one variable held fixed.
How Boyle's and Charles's laws are special cases of the combined law
Holding one variable constant in P₁V₁/T₁ = P₂V₂/T₂ collapses it into a simpler, named law.
| Law | Held constant | Equation |
|---|---|---|
| Boyle's law | Temperature | P₁V₁ = P₂V₂ |
| Charles's law ★ | Pressure | V₁/T₁ = V₂/T₂ |
| Gay-Lussac's law | Volume | P₁/T₁ = P₂/T₂ |
| Combined gas law | Moles of gas only | P₁V₁/T₁ = P₂V₂/T₂ |
★ Reference row (worked example 1). All four rows describe the exact same underlying physics — they differ only in which variable is held fixed.
Where the combined gas law actually matters
🎈 Weather balloons
As a weather balloon rises, atmospheric pressure drops sharply while temperature also falls — the combined gas law predicts how much the balloon expands, which engineers must account for so it doesn't burst before reaching its target altitude.
🚗 Tire pressure and temperature
A car tire's pressure reading changes noticeably between a cold morning and a hot afternoon, purely from Gay-Lussac's law (nearly constant volume) — this is why manufacturers specify a 'cold' tire pressure as the standard reference.
🏭 Industrial gas storage and transport
Compressed gas cylinders are designed using these laws to predict safe pressure limits across a range of storage and transport temperatures, preventing over-pressurization on hot days.
🫁 Scuba diving physiology
Boyle's law explains why a diver's lungs must never hold their breath while ascending — the pressure drop as they rise causes trapped gas to expand rapidly, which can cause serious lung injury if the diver doesn't exhale continuously.
Common misconceptions
"You can use Celsius directly in any gas law as long as you're consistent."
Charles's law and the combined gas law rely on temperature ratios (T₂/T₁), which only make physical sense on an absolute scale. Using 20°C and 40°C gives a ratio of 2.0, while the correct kelvin ratio (293.15/313.15) is only about 1.068 — a massive, invalidating error.
"The combined gas law only works when all three variables change by the same proportion."
The combined gas law works for any combination of changes to pressure, volume, and temperature — it doesn't require them to change proportionally, only that the total number of gas molecules (moles) stays fixed throughout.
"Boyle's law and Charles's law are unrelated equations that happen to both involve gases."
Both are special cases of the single combined gas law (and, more fundamentally, of the ideal gas law PV=nRT) obtained by holding one variable constant — they aren't separate physical principles, just different simplified views of the same relationship.
"Real gases always obey these laws exactly."
These are ideal-gas relationships, which assume no intermolecular forces and negligible molecular volume. Real gases deviate from them at high pressure or low temperature, where intermolecular attractions and finite molecular size actually start to matter.
Formula sources & further reading
The formulas here are standard, traceable to:
- • OpenStax, Chemistry 2e — Chapter 9, "Gases" (free, peer-reviewed). openstax.org
- • Brown, LeMay & Bursten, Chemistry: The Central Science — Chapter 10, Gases.
- • Zumdahl & Zumdahl, Chemistry — Gas laws chapter.
P₁V₁/T₁ = P₂V₂/T₂ (moles constant). Temperature must be in kelvin. Results are rounded for display.
How to use this calculator
Select your law
Choose Boyle's law, Charles's law, or the combined law depending on which variables are changing.
Enter three values
The field labels update to match your selection — fill in any three of the four to solve the fourth.
Convert to kelvin first
Add 273.15 to any Celsius temperature before entering it — the calculator flags zero kelvin as invalid.
Related tools
Frequently asked questions
What is Boyle's law?
Boyle's law states that at constant temperature, the pressure and volume of a gas are inversely proportional: P₁V₁ = P₂V₂. Doubling the pressure halves the volume.
What is Charles's law?
Charles's law states that at constant pressure, the volume of a gas is proportional to its absolute temperature: V₁/T₁ = V₂/T₂. Temperature must be in kelvin.
What is the combined gas law?
The combined gas law is P₁V₁/T₁ = P₂V₂/T₂, unifying Boyle's, Charles's, and Gay-Lussac's laws into one equation. It applies whenever the amount of gas (moles) stays constant.
Why must temperature be in kelvin?
Kelvin is an absolute temperature scale starting at absolute zero (−273.15°C). Gas laws depend on proportional relationships that only hold on an absolute scale — using Celsius directly would give badly wrong results.
What is Gay-Lussac's law?
Gay-Lussac's law states that at constant volume, pressure is proportional to temperature: P₁/T₁ = P₂/T₂ — this describes how pressure rises in a sealed, rigid container when heated.