Gay-Lussac's Law Calculator – Pressure and Temperature in a Sealed, Rigid Container
Gay-Lussac's law describes a fixed amount of gas trapped in a container that cannot expand. Heat it and the pressure rises; cool it and the pressure falls, in direct proportion to the absolute temperature: P₁ / T₁ = P₂ / T₂. This calculator solves for any one of the four values, accepts gauge or absolute pressure, works in kelvin, Celsius or Fahrenheit, and shows what the two most common mistakes would have given you.
The law and its names
The French physicist Guillaume Amontons noticed the link between the pressure and temperature of air around 1700, which is why the relation is also called Amontons's law or simply the pressure law. It is usually credited to Joseph Louis Gay-Lussac, who studied gas expansion in the early 1800s. Names vary between textbooks: some use "Gay-Lussac's law" for the volume–temperature relation (the Charles's Law Calculator on this site), and others for his law of combining volumes of reacting gases. Here it always means pressure versus temperature at constant volume. Pressure versus volume at constant temperature is Boyle's law. If the volume changes too, use the Combined Gas Law Calculator.
Why temperature must be in kelvin
Pressure is proportional to temperature measured from absolute zero. Draw the line on the pressure–temperature chart and extend it to the left: it reaches zero pressure at −273.15 °C, which is 0 K. Celsius and Fahrenheit put their zeros somewhere else, so plugging them in breaks the proportion. Take a flask sealed at 1 atm in an ice bath (0 °C) and moved to boiling water (100 °C). In kelvin the answer is 1 × 373.15 / 273.15 = 1.366 atm. In Celsius you would divide by zero.
Gauge pressure versus absolute pressure
Tyre, scuba and most cylinder gauges read gauge pressure, the pressure above the surrounding air. The law needs absolute pressure, so the calculator adds the atmospheric pressure first (P_abs = P_gauge + P_atm) and takes it away again at the end. A tyre at 32 psi gauge is really at about 46.7 psi absolute. Warming it from 20 °C to 40 °C gives 35.19 psi gauge. Treating the gauge reading as absolute gives 34.18 psi, about 1 psi too low. For a 200 bar scuba tank, one extra bar of atmosphere hardly matters, and the same mistake costs less than 0.1 %.
Tyre pressure and temperature: the rule of thumb
Drivers often hear "about 1 psi per 10 °F". For a rigid container at typical car pressures, the ideal-gas figure is a bit lower: about 0.9 psi per 10 °F (0.885 psi at 32 psi gauge and 20 °C), or roughly 0.16 psi per °C. A real tyre stretches slightly as it warms, so its real change is a little smaller again. Check pressures cold, before driving, for this reason. For tyre unit conversions, see the Tire Pressure Converter.
Scuba tanks that read low after cooling
A cylinder filled quickly gets warm. Filled to 200 bar gauge at 40 °C and then cooled to 15 °C water, it reads about 184 bar: the gas is all still there, but it pushes on the walls less hard. That is why fill stations fill slowly, cool the tank in water, or fill a little above the target. At these pressures the gas is no longer ideal, so treat the figure as an estimate; the Ideal Gas Law Calculator has a van der Waals mode for a real-gas check.
The molecular picture
Gas pressure is the combined push of molecules hitting the walls. In a rigid box the number of molecules and the distance they travel between walls are fixed. Raising the temperature makes them move faster, with an average speed proportional to √T. Faster molecules hit the walls more often, and each hit is harder. The two effects multiply to give a pressure proportional to T. The container diagram shows this: same box, same molecules, longer motion streaks, and pressure and temperature bars that grow by the same factor.
P ∝ T. Never use this calculator to decide whether such a container is safe to heat.