Understanding the Pressure–Temperature Relationship
Gay-Lussac's law (also called the pressure law) describes an isochoric process: one where the volume of a closed system remains fixed while pressure and temperature change. Unlike open systems where gases can expand freely, a sealed rigid container forces all thermal energy to translate directly into pressure increase.
The fundamental principle states that absolute pressure is directly proportional to absolute temperature. This relationship holds only for ideal gases and requires:
- A rigid, closed container with unchanging volume
- An ideal gas (real gases approximate this at moderate pressures)
- Temperature measured in Kelvin, not Celsius
- Sufficient time for thermal equilibrium
Practical systems like tires, aerosol cans, and sealed pressure vessels all obey this law. When you heat a tire in summer, internal pressure climbs. When winter arrives and temperature drops, pressure falls—even if no air escapes.
The Mathematical Form of Gay-Lussac's Law
The core relationship expresses pressure and temperature as a constant ratio in a sealed container:
p₁ ÷ T₁ = p₂ ÷ T₂
p₁ ÷ p₂ = T₁ ÷ T₂
p = n × R × T ÷ V
p₁, p₂— Initial and final absolute pressure (in pascals or bar)T₁, T₂— Initial and final absolute temperature (in Kelvin)n— Number of moles of gasR— Universal gas constant (8.3145 J mol⁻¹ K⁻¹)V— Volume of the container (in cubic meters or liters)
Working Through a Practical Example
Imagine a sealed metal can containing 500 ml of air at room temperature (20°C, or 293.15 K) and atmospheric pressure (101.3 kPa). You place it on a stove and heat it until the temperature reaches 150°C (423.15 K). What is the final pressure inside?
Using the rearranged formula:
- Start with: p₁ ÷ T₁ = p₂ ÷ T₂
- Rearrange: p₂ = p₁ × (T₂ ÷ T₁)
- Substitute: p₂ = 101.3 × (423.15 ÷ 293.15) = 101.3 × 1.443 ≈ 146.2 kPa
The pressure increased by roughly 44% because temperature rose by the same percentage on the absolute scale. This demonstrates why sealed containers become dangerous when heated—pressure can build rapidly and rupture weak walls.
Key Caveats and Common Pitfalls
Understanding these practical limitations ensures accurate predictions and safer handling of sealed gas systems.
- Always use absolute temperature (Kelvin) — Converting Celsius to Kelvin is non-negotiable. A 10°C rise is not proportional to pressure change; only the ratio of absolute temperatures matters. Room temperature (20°C = 293.15 K) and 30°C (303.15 K) differ by ~3.4%, not 50%.
- Real gases deviate at high pressure — The law assumes ideal gas behavior. At pressures above ~10 bar or very low temperatures, molecular size and intermolecular forces become significant. Predict slightly more pressure than the formula suggests for compressed air or liquefied gases.
- Thermal equilibrium takes time — Pressure changes only after the entire gas mass reaches the new temperature. Measure pressure after waiting—reading immediately after applying heat gives misleading results. Larger containers with poor insulation equilibrate more slowly.
- Account for container expansion — Real materials expand slightly when heated, slightly increasing volume and reducing the observed pressure rise. For metals at modest temperature changes this effect is negligible, but for precise work account for thermal expansion coefficients.
Real-World Applications and Safety
Pressure cookers exploit Gay-Lussac's law intentionally. By sealing the lid, volume stays constant. As heating increases temperature, pressure rises above atmospheric, raising water's boiling point above 100°C and accelerating cooking. Most cookers vent excess pressure at 1–2 bar above atmosphere—about 120–130°C—to prevent rupture.
Aerosol cans demonstrate the reverse: depressing the valve reduces internal pressure, and the gas cools as it expands during discharge. Cans left in fire are hazardous because intense heat raises internal pressure until the can ruptures explosively.
Tire pressure monitoring systems rely on this principle. A tire inflated to 32 psi at 20°C will read roughly 35 psi at 50°C, assuming no air leakage. Winter tire pressure drops noticeably because absolute temperature falls—not a sign of a puncture.