Gay Lussac's Law
The Pressure-Temperature Relationship at Constant Volume.
What happens when you heat a gas, but force it to stay in a rigid container of fixed volume? French chemist Joseph Louis Gay-Lussac formalized this relationship in 1808. Understanding this law is essential for predicting how gases behave in sealed environments, from aerosol cans to automobile tires.
1. Statement & Mathematical Formula
Gay-Lussac's Law states: At constant volume, the pressure of a fixed mass of a gas is directly proportional to its absolute temperature.
As temperature increases, the kinetic energy of the gas molecules increases. Since the volume of the container cannot expand (unlike in Charles's Law), these faster-moving molecules collide with the rigid walls more frequently and with greater force, resulting in a higher pressure.
$$\frac{P}{T} = k \quad (\text{where } k \text{ is a constant})$$
Just like Charles's Law, Temperature ($T$) must ALWAYS be in Kelvin ($K$). Do not use Celsius when multiplying or dividing in these formulas!
The Problem-Solving Equation:
If a gas undergoes a change in conditions from state 1 to state 2 inside a rigid container:
2. Graphical Representation (Isochores)
Graphs drawn at a constant volume are called Isochores (iso = same, chore = volume/space). The relationship $P = kT$ is a straight line passing through the origin.
Pressure vs Temp (Kelvin)
Straight line through origin.
V₂ > V₁ (Higher Volume = Lower slope)
Pressure vs Temp (Celsius)
Extrapolates to -273.15°C at P=0.
3. Real-Life Applications
Because many rigid containers exist in the real world, Gay-Lussac's Law is highly visible in our daily lives.
- Automobile Tires in Summer: In hot summer weather, the temperature of the air inside the tire increases significantly due to ambient heat and road friction. Because the tire's volume is relatively fixed, the air pressure increases sharply. If the pressure exceeds the tire's tolerance, it can lead to a dangerous blowout. This is why tire pressure should ideally be checked when tires are "cold."
- Pressure Cookers: A pressure cooker is a sealed, rigid container. As heat is applied, the temperature of the trapped steam increases, which exponentially increases the pressure inside. This high pressure forces water to boil at a much higher temperature (above $100^\circ C$), cooking food much faster.
- Aerosol Cans (Deodorants, Spray Paint): Aerosol cans display prominent warning labels: "Do not store near heat or incinerate." Because the can's volume is fixed, throwing it into a fire will drastically increase the internal gas pressure according to Gay-Lussac's Law, eventually causing a violent explosion.
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