Gay-Lussac’s Law: Pressure and Temperature

Gay-Lussac’s law states that for a fixed amount of gas at constant volume, the pressure is directly proportional to the absolute temperature in kelvins: P₁/T₁ = P₂/T₂. In brief: a sealed container of gas at 2.00 atm and 300 K will have a pressure of 2.40 atm when heated to 360 K.

This law explains why a sealed aerosol can can explode in a fire and why tire pressure changes with the seasons. It describes how heat affects a gas that cannot expand. This article gives the formula, the method, worked examples, a data table and the history, followed by common mistakes.

What does Gay-Lussac’s law say?

If the volume and the amount of a gas stay fixed, pressure and absolute temperature increase together:

P / T = k, or P1/T1 = P2/T2

The molecular explanation is straightforward. At a higher temperature the particles move faster, so they strike the walls more often and with greater force. Because the container cannot expand, the pressure rises. Temperature must always be in kelvins; a graph of P against T in kelvins is a straight line through the origin.

How do you solve Gay-Lussac’s law problems?

1. Check that the gas is in a rigid, sealed container.
2. Convert temperatures to kelvins (K = °C + 273.15).
3. Rearrange: P₂ = P₁ × T₂ / T₁.
4. Substitute values and calculate.
5. Check: heating must increase the pressure, and cooling must reduce it.

Worked example 1: a heated cylinder

A steel cylinder holds oxygen at 150 kPa and 20 °C. What is the pressure after it is heated to 85 °C?

T₁ = 293 K and T₂ = 358 K. P₂ = 150 kPa × 358 / 293 = 183 kPa.

Worked example 2: a cooled tire

A car tire is filled to an absolute pressure of 320 kPa at 25 °C. After a night at −5 °C, what is the pressure? Assume the volume is unchanged.

T₁ = 298 K and T₂ = 268 K. P₂ = 320 × 268 / 298 = 288 kPa. That is a drop of 32 kPa, or about 10 %, which is enough to trigger a low-pressure warning in some cars.

Worked example 3: finding the temperature

A sealed flask of gas has a pressure of 1.00 atm at 300 K. At what temperature will the pressure reach 1.50 atm?

T₂ = 300 K × 1.50 / 1.00 = 450 K, or 177 °C.

Pressure and temperature data table

Where do you see Gay-Lussac’s law in daily life?

• Aerosol cans: the warnings against heating them are based on this law, since the pressure inside can exceed the strength of the can.
• Pressure cookers: a sealed pot raises the pressure and with it the boiling point of water (although the main effect there involves vapor pressure).
• Tires: pressure falls by roughly 1 psi for each 10 °F drop in temperature, which is why checks are recommended in autumn.
• Autoclaves and sterilizers: sealed vessels operate at higher pressures at elevated temperature.
• Gas thermometers: constant-volume gas thermometers measure temperature through the pressure of a gas.

Who was Gay-Lussac?

Joseph Louis Gay-Lussac (1778–1850) was a French chemist and physicist. In 1802 he published measurements of the thermal expansion of gases that confirmed the work of Jacques Charles. In 1808 he stated the law of combining volumes, which says that gases react in simple whole-number volume ratios. Guillaume Amontons had described the pressure–temperature relationship much earlier, in the late 1600s, so the law is sometimes called Amontons’s law. The name Gay-Lussac’s law is applied to the pressure law in most English-language textbooks, but in some countries it refers to the volume–temperature relationship.

Which mistakes are common?

• Using Celsius: doubling the temperature from 20 °C to 40 °C does not double the pressure; use kelvins.
• Forgetting that the volume must be constant: if the container can expand, Charles’s law applies instead.
• Using gauge pressure: use absolute pressure in the calculation, then convert back if needed.
• Switching the subscripts: P₂ belongs with T₂.
• Ignoring leaks: a law that assumes a fixed amount of gas cannot be applied to a leaking container.