Gay-Lussac's Law

Seal a gas in a container that cannot change size and there is only one thing left for it to do when you heat it: push harder. Pressure then tracks absolute temperature in direct proportion, $P_1/T_1 = P_2/T_2$. The mechanism is the same one behind Charles's law with the outcome swapped. Faster molecules strike the walls both more often and with more momentum each time, and since the walls will not move aside, all of that arrives as pressure.

Here is the case everyone actually meets, worked carefully, because the careless version is the standard error. A tire is set to 220 kPa gauge on a 5 °C morning and warms to 45 °C after an hour on the highway. Convert to absolute pressure first: 220 + 101 = 321 kPa absolute. Then $P_2 = 321 \times 318.15/278.15 = 367\ \text{kPa}$ absolute, which is 266 kPa on the gauge — a rise of 46 kPa, about 6.6 psi. Run the same calculation on the gauge reading alone and you get 252 kPa, understating the rise by a third. This is why tire pressures are specified cold, and why topping up a hot tire leaves it soft in the morning.

The relation is usually credited to Gay-Lussac's 1802 paper, though Guillaume Amontons had it a century earlier: around 1702 he built an air thermometer that worked on precisely this principle, and noticed that the pressure line extrapolated toward a temperature below which it could not go. Some texts call it Amontons's law for that reason. Combine it with Boyle's law and Charles's law and you have the combined gas law; add Avogadro and you have $PV = nRT$, of which this is the constant-volume slice.

Absolute pressure is the trap here, more than absolute temperature. The ratio $P_1/P_2$ is only meaningful when both pressures are measured from vacuum, because the equation is counting molecular impacts and a gauge has quietly subtracted an atmosphere from the count. Temperature has the same requirement for the same reason — kelvin, not Celsius — and this page converts your entries on both fronts. But when you meet the equation on paper, ask twice whether the pressure in your hand is gauge or absolute. It usually is gauge; almost every instrument in a mechanical room reads that way.

The other honest limit is that constant volume is an idealisation. A tire is not rigid — it grows a little as it warms, which relieves some of the pressure rise, so the measured increase runs slightly below the calculation. A steel cylinder is much closer to the ideal, which is what makes this law genuinely dangerous rather than merely academic. An aerosol can left on a dashboard, a propane cylinder in a closed vehicle, or a sealed pressure vessel in a fire all follow this line with nothing to relieve them, and the pressure keeps climbing until something gives. Relief valves exist because the equation has no upper bound.