The combined gas law describes what happens to a sealed, fixed amount of gas as its pressure, volume, and temperature all change together — heating a tire on a summer drive, compressing air in a scuba tank, or watching a weather balloon expand as it rises. Rather than juggling three separate named laws, one equation, P₁V₁/T₁ = P₂V₂/T₂, covers every combination of change as long as the moles of gas stay constant. This article walks through where the formula comes from, why the temperature has to be in Kelvin, how it collapses into Boyle's, Charles's, and Gay-Lussac's laws as special cases, and where its ideal-gas assumption starts to break down.
Where P₁V₁/T₁ = P₂V₂/T₂ comes from
The combined gas law falls directly out of the ideal gas law, PV = nRT. Rearranging gives PV/T = nR, and since n (the amount of gas) and R (the gas constant) don't change for a sealed sample, PV/T is constant across any two states of that same gas. Writing that constant equality for state 1 and state 2 gives P₁V₁/T₁ = P₂V₂/T₂ — no need to know n or R at all, since they cancel out of the ratio.
Why Kelvin, not Celsius or Fahrenheit
The law is a direct proportion involving T itself, not a difference from some reference point. Celsius and Fahrenheit both allow zero and negative readings at ordinary conditions (0 °C, or well below 0 °F), and dividing by zero — or flipping the sign of a ratio — produces nonsense. The Kelvin scale starts at absolute zero, the coldest physically possible temperature, so a real gas sample's Kelvin temperature is always a positive number greater than zero. This calculator converts any °C or °F entry to Kelvin internally before doing the division, and rejects any input that converts to zero or below.
The three special cases
Hold any one variable constant and the combined gas law simplifies to a named 19th-century law. Constant temperature gives Boyle's law, P₁V₁ = P₂V₂ — pressure and volume move inversely. Constant pressure gives Charles's law, V₁/T₁ = V₂/T₂ — volume grows directly with Kelvin temperature. Constant volume gives Gay-Lussac's law, P₁/T₁ = P₂/T₂ — pressure grows directly with Kelvin temperature. The Special Cases tab on this page walks through all three, with links to the dedicated Boyle's Law and Charles's Law calculators.
Worked walkthrough
Suppose 2 L of gas at 1 atm and 300 K is compressed to 2 atm while being heated to 350 K. Solving for the new volume: V₂ = P₁V₁T₂ / (T₁P₂) = (1 atm × 2 L × 350 K) / (300 K × 2 atm) ≈ 1.167 L. Notice both a pressure change and a temperature change are folded into a single calculation — that's the entire point of the combined form over using Boyle's or Charles's law alone.
Limits: real gases and the ideal gas law
The combined gas law assumes an ideal gas — point-like particles with no intermolecular attraction — and a fixed amount of gas (no leaks, no reactions consuming or producing gas). Real gases deviate from ideal behavior at very high pressure or very low temperature, where molecular volume and intermolecular forces start to matter. If the amount of gas itself changes between the two states, or a fully independent P, V, T, n calculation is needed, use the full ideal gas law calculator (PV = nRT) instead of the two-state combined form.