The Combined Gas Law — The Complete Physics Guide
Long before anyone understood atoms or molecules, scientists like Robert Boyle, Jacques Charles, and Joseph Gay-Lussac discovered — through careful experiment — that a fixed amount of gas obeys remarkably simple mathematical relationships between its pressure, volume, and temperature. The combined gas law merges their three separate discoveries into a single, unified equation, letting you predict exactly how a gas will respond when any of these three properties changes, as long as the amount of gas stays fixed.
This single relationship — P₁V₁/T₁ = P₂V₂/T₂ — is one of the most practically useful equations in all of introductory physics and chemistry, describing everything from a scuba tank's behaviour underwater to why a sealed bag of chips puffs up at high altitude.
What is the Combined Gas Law?
The combined gas law describes how pressure (P), volume (V), and temperature (T) of a fixed amount of gas relate to each other between any two states. It merges three individual laws: Boyle's Law (at constant temperature, P and V are inversely proportional — squeeze a gas and its pressure rises), Charles's Law (at constant pressure, V and T are directly proportional — heat a gas and it expands), and Gay-Lussac's Law (at constant volume, P and T are directly proportional — heat a sealed container and pressure rises). Each of these is really just a special case of the combined gas law with one variable held fixed.
The combined gas law is itself a special case of the more complete ideal gas law, PV = nRT, valid when the amount of gas (n, the number of moles) doesn't change between the two states being compared — which is true for the vast majority of everyday gas problems, like a sealed container being heated or a fixed quantity of air being compressed.
The Formula Explained
P₁, V₁, T₁ describe the gas's initial state; P₂, V₂, T₂ describe its final state, after some change has occurred. Pressure and volume can be expressed in any units, as long as the same unit is used consistently for both states (the units cancel out in the ratio). Temperature, however, must always be in kelvin — using Celsius or Fahrenheit will give badly wrong answers, since the gas law relationships only hold true when temperature is measured from absolute zero. Convert with K = °C + 273.15 before substituting into the formula.
How to Use This Calculator
Fill in any five of the six fields and leave the sixth blank — the calculator automatically detects which value you're solving for and computes it. This works for any of the six unknowns, whether that's finding a final pressure after heating, an initial volume before compression, or any other combination the combined gas law can address.
Worked Example 1 — Heating a Sealed Container
Problem: A gas at 100 kPa, 2 L, and 300 K is heated to 350 K in a container that also allows the pressure to rise to 150 kPa. Find the new volume.
V₂ = P₁V₁T₂/(T₁P₂) = (100)(2)(350) / [(300)(150)]
V₂ = 70,000/45,000 = 1.56 L
Worked Example 2 — Scuba Tank at Depth
Problem: A diver's air at the surface (101 kPa, 293 K) has volume 12 L in their lungs. If they ascend too quickly while holding their breath and surface pressure conditions apply at 101 kPa but the temperature stays the same, this models the reverse: air compressed at depth (303 kPa, 293 K, unknown volume) will expand to what volume at the surface (101 kPa, 293 K)?
V₂ = P₁V₁T₂/(T₁P₂) = (303)(V₁)(293)/[(293)(101)] = 3V₁
The air triples in volume as it rises to the surface — precisely why divers are trained to never hold their breath while ascending, since rapidly expanding air can cause serious lung injury.
Common Mistakes
Using Celsius instead of Kelvin: the single most common error in gas law problems. The relationship only works with absolute temperature — always convert with K = °C + 273.15 before substituting.
Mixing pressure or volume units between states: P₁ and P₂ must use the same unit (both in kPa, or both in atm, for example) — mixing units within the same ratio invalidates the calculation, even though the units themselves cancel algebraically.
Applying the law when the amount of gas changes: the combined gas law assumes a fixed, sealed quantity of gas. If gas is added, removed, or leaks between the two states, the full ideal gas law (accounting for a changing number of moles) must be used instead.
Real-World Applications
Scuba diving: understanding how gas volume changes with pressure and depth is essential for diver safety, governing everything from buoyancy control to the risk of decompression sickness and lung overexpansion injuries.
Weather balloons: balloons expand dramatically as they rise into the low-pressure upper atmosphere, engineered with enough slack material to accommodate this predictable volume increase without bursting prematurely.
Aerosol cans and pressurised containers: warning labels advising against storing aerosol cans in hot cars directly reflect Gay-Lussac's Law — heating a sealed, fixed-volume container raises internal pressure, potentially beyond the container's safe limit and risking rupture or explosion, which is precisely the physics manufacturers account for when setting maximum safe storage temperatures.
Why the Gas Laws Work — Kinetic Theory
The gas laws aren't arbitrary empirical rules — they emerge directly from the kinetic theory of gases, which models a gas as vast numbers of tiny particles in constant, random motion, colliding with each other and with the walls of their container. Pressure arises from the cumulative force of countless molecular collisions against the container walls; temperature is a direct measure of the average kinetic energy of these molecules. Compressing a gas into a smaller volume forces molecules to collide with the walls more frequently, raising pressure — precisely Boyle's Law. Heating a gas increases average molecular speed and collision force, raising pressure or volume depending on which is free to change — precisely Charles's and Gay-Lussac's Laws.
This microscopic picture explains why the gas laws break down under extreme conditions: at very high pressure, molecules are packed close enough that their own finite size becomes significant (violating the kinetic theory's assumption of point-like particles), and at very low temperature, intermolecular attractive forces become significant enough to affect collision behaviour (violating the assumption of no forces between particles except during collisions). Under ordinary laboratory and everyday conditions, however, these idealisations hold remarkably well.
Standard Temperature and Pressure (STP)
Because gas volume depends so strongly on both temperature and pressure, scientists needed an agreed reference point to compare gas quantities meaningfully — this is standard temperature and pressure (STP), currently defined by IUPAC as 0°C (273.15 K) and 100 kPa (though older definitions using 101.325 kPa, one standard atmosphere, are still widely used and worth checking carefully in any given textbook or dataset). At STP, one mole of any ideal gas occupies almost exactly 22.4 litres — a foundational reference value used constantly throughout chemistry.
Comparing gas volumes always requires stating the conditions they were measured under, precisely because volume alone is meaningless without knowing the temperature and pressure — a fact the combined gas law formalises directly, letting any measurement be converted to any other set of conditions for fair comparison.