Ever seen the warning label on an aerosol can that says “do not expose to temperatures above 120 F”? That warning exists because of Gay-Lussac’s law. Heat a sealed, rigid container and the pressure inside rises. Heat it enough and the container explodes.
The Law
At constant volume and constant amount of gas, pressure and temperature are directly proportional.
The Graph
A plot of P vs. T (in Kelvin) produces a straight line through the origin, just like Charles’s law but with pressure on the y-axis instead of volume. The slope depends on the amount of gas and the volume of the container.
Why It Works (Molecular Level)
In a rigid container, the volume cannot change. When you increase the temperature, molecules gain kinetic energy - they move faster and collide with the walls more forcefully and more frequently. Since the walls cannot move outward to accommodate this increased molecular activity, the pressure increases.
The Combined Gas Law
Boyle’s, Charles’s, and Gay-Lussac’s laws are all special cases of the combined gas law:
If constant…
Cancels to…
Law
T
P₁V₁ = P₂V₂
Boyle’s
P
V₁/T₁ = V₂/T₂
Charles’s
V
P₁/T₁ = P₂/T₂
Gay-Lussac’s
Worked Example
A sealed steel tank contains gas at 300 K and 2.0 atm. The tank is heated to 600 K. What is the new pressure?
P₁/T₁ = P₂/T₂
2.0 atm / 300 K = P₂ / 600 K
P₂ = 2.0 x (300600) = 4.0 atm
Temperature doubled, so pressure doubled. The volume of the rigid tank stays the same throughout.
A rigid container of gas has a pressure of 3.0 atm at 400 K. If the temperature is decreased to 200 K, what is the new pressure?
Click to reveal answer
1.5 atm. P₁/T₁ = P₂/T₂. Temperature halved (400 K to 200 K), so pressure halves (3.0 to 1.5 atm). Gay-Lussac's law is a direct proportion between P and T at constant volume.
Which gas law would you use to solve this: a gas at 1.5 atm and 4.0 L at 300 K is changed to 2.0 atm at 400 K. Find the new volume.
Click to reveal answer
The combined gas law: P₁V₁/T₁ = P₂V₂/T₂. Since P, V, and T are all changing, you need the combined gas law. (1.5)(4.0)/300 = (2.0)(V₂)/400. V₂ = (1.5 x 4.0 x 400)/(300 x 2.0) = 4.0 L. None of the individual gas laws (Boyle's, Charles's, or Gay-Lussac's) work here because no variable is held constant.
