Gay-Lussac's Law: pressure rises with temperature
Gay-Lussac's Law states that at constant volume, the pressure of a fixed amount of gas is directly proportional to its absolute temperature. In equation form: P₁/T₁ = P₂/T₂. Heat a sealed container of gas and the pressure rises; cool it and the pressure drops. Because volume is fixed, the molecules cannot expand outward, so the increased kinetic energy translates entirely into higher pressure.
Joseph Louis Gay-Lussac published this relationship in 1802, the same paper in which he reported Charles's Law. Gay-Lussac was a careful experimentalist. Working with the physicist John Dalton's earlier observations about thermal expansion, Gay-Lussac used a gas thermometer of his own design to measure the relationship across a wide temperature range, establishing the linearity that earlier observers had only approximated.
The sealed container
Gay-Lussac's Law applies whenever a gas is confined in a rigid container. Unlike Boyle's Law (flexible container, constant temperature) or Charles's Law (flexible container, constant pressure), Gay-Lussac's Law is the constant-volume case. This makes it relevant to any situation where a gas cannot expand: sealed tanks, pressure cookers, aerosol cans, and rigid piping systems.
The practical consequence: never heat a sealed container of gas unless it is rated for the resulting pressure increase. An aerosol can at 20 °C and 3 atm internal pressure would rise to about 3.6 atm at 80 °C — which is why aerosol cans carry explicit temperature warnings. A sealed container of liquid nitrogen warming to room temperature experiences an enormous pressure increase and must have a pressure-relief valve.
Worked examples
Example 1: Heating a sealed flask. A sealed glass flask contains gas at 1.00 atm and 25 °C. It is heated to 200 °C. Final pressure? P₂ = P₁T₂/T₁ = 1.00 × 473.15/298.15 = 1.59 atm.
Example 2: Pressure cooker. A pressure cooker at 1.00 atm and 100 °C with a sealed lid heats further to 121 °C. P₂ = 1.00 × 394.15/373.15 = 1.06 atm gauge, or about 2.06 atm absolute. The elevated pressure raises the boiling point, which cooks food faster.
Example 3: Tire in summer. A tire at 2.20 atm (gauge) at 10 °C reaches 35 °C on a hot road. At constant volume, P₂ = (2.20 + 1.01) × 308.15/283.15 - 1.01 = 2.50 atm gauge, a 0.3 atm increase.
Gay-Lussac's Law in engineering safety
Pressure vessels — from propane tanks to industrial autoclaves — are designed around Gay-Lussac's Law. A propane tank rated for 250 psi at 70 °F could reach 300 psi at 130 °F (the approximate temperature inside a car parked in summer sun). All pressure vessels incorporate safety factors to account for this, and many include pressure relief valves that vent gas before dangerous pressures are reached.
The BLEVE (Boiling Liquid Expanding Vapor Explosion) is the catastrophic failure mode that occurs when a pressure vessel containing liquid above its boiling point is exposed to fire. The liquid boils rapidly, the pressure rises according to Gay-Lussac's Law, and the vessel ruptures if the relief valve capacity is exceeded. This is why propane tanks, CNG cylinders, and chemical reactor vessels must be protected from fire exposure. The physics is straightforward: heat a sealed container of gas, and the pressure rises. Contain it beyond its design pressure, and it fails.
The discovery: Gay-Lussac's 1802 paper
In 1802, Joseph Louis Gay-Lussac published "On the Expansion of Gases by Heat" in the Annales de Chimie. Using a gas thermometer of his own design — essentially a glass bulb connected to a mercury manometer — he measured the pressure of air, hydrogen, oxygen, nitrogen, and several other gases across a range of temperatures from 0 °C to 100 °C. His key finding: all gases expanded by the same fraction per degree of temperature increase.
Gay-Lussac reported the expansion coefficient as 1/266.66 per °C. The modern value is 1/273.15 per °C. The 2.4% discrepancy arose from Gay-Lussac's thermometer calibration and the trace water vapor in his samples. Correcting for water vapor brings his measurements within 0.5% of modern values — remarkable precision for 1802.
Gay-Lussac was also a pioneering balloonist. In 1804, he ascended to 7,016 meters (23,018 feet) in a hydrogen balloon, a record that stood for over 50 years. He collected air samples at altitude to study the composition of the upper atmosphere. His combined interests in gas behavior and high-altitude exploration made him one of the most versatile scientists of his era.
More worked examples
Example 4: Autoclave. A medical autoclave operates at 121 °C and 2.0 atm. If starting from 25 °C at 1.0 atm in a sealed chamber, P₂ = P₁T₂/T₁ = 1.0 × 394.15/298.15 = 1.32 atm gauge. The additional pressure comes from the steam generated during heating.
Example 5: Propane tank. A propane tank at 20 °C reads 120 psi. In direct sun, the tank reaches 60 °C. New pressure (absolute) = (120 + 14.7) × 333.15/293.15 - 14.7 = 138 psi gauge. An 18 psi increase purely from solar heating.
Frequently asked questions
Pressure vessels — from propane tanks to industrial autoclaves — are designed around Gay-Lussac's Law. A propane tank rated for 250 psi at 70 °F could reach 300 psi at 130 °F (the approximate temperature inside a car parked in summer sun). All pressure vessels incorporate safety factors to account for this, and many include pressure relief valves that vent gas before dangerous pressures are reached.
The BLEVE (Boiling Liquid Expanding Vapor Explosion) is the catastrophic failure mode that occurs when a pressure vessel containing liquid above its boiling point is exposed to fire. The liquid boils rapidly, the pressure rises according to Gay-Lussac's Law, and the vessel ruptures if the relief valve capacity is exceeded. This is why propane tanks, CNG cylinders, and chemical reactor vessels must be protected from fire exposure. The physics is straightforward: heat a sealed container of gas, and the pressure rises. Contain it beyond its design pressure, and it fails.
In 1802, Joseph Louis Gay-Lussac published "On the Expansion of Gases by Heat" in the Annales de Chimie. Using a gas thermometer of his own design — essentially a glass bulb connected to a mercury manometer — he measured the pressure of air, hydrogen, oxygen, nitrogen, and several other gases across a range of temperatures from 0 °C to 100 °C. His key finding: all gases expanded by the same fraction per degree of temperature increase.
Gay-Lussac reported the expansion coefficient as 1/266.66 per °C. The modern value is 1/273.15 per °C. The 2.4% discrepancy arose from Gay-Lussac's thermometer calibration and the trace water vapor in his samples. Correcting for water vapor brings his measurements within 0.5% of modern values — remarkable precision for 1802.
Gay-Lussac was also a pioneering balloonist. In 1804, he ascended to 7,016 meters (23,018 feet) in a hydrogen balloon, a record that stood for over 50 years. He collected air samples at altitude to study the composition of the upper atmosphere. His combined interests in gas behavior and high-altitude exploration made him one of the most versatile scientists of his era.
Example 4: Autoclave. A medical autoclave operates at 121 °C and 2.0 atm. If starting from 25 °C at 1.0 atm in a sealed chamber, P₂ = P₁T₂/T₁ = 1.0 × 394.15/298.15 = 1.32 atm gauge. The additional pressure comes from the steam generated during heating.
Example 5: Propane tank. A propane tank at 20 °C reads 120 psi. In direct sun, the tank reaches 60 °C. New pressure (absolute) = (120 + 14.7) × 333.15/293.15 - 14.7 = 138 psi gauge. An 18 psi increase purely from solar heating.
What is Gay-Lussac's Law? At constant volume, P₁/T₁ = P₂/T₂. Heating a sealed gas container increases its pressure.
Why do aerosol cans warn about heat? Because heating increases internal pressure via Gay-Lussac's Law. Above a safe temperature, the pressure can exceed the can's burst strength.
How is this different from Charles's Law? Charles's Law holds pressure constant and lets volume change. Gay-Lussac's holds volume constant and lets pressure change.