Gay-Lussac's law: pressure and temperature of gases

The warning label: pressure follows temperature

Every aerosol can carries the same warning: do not expose to heat, do not puncture, do not incinerate. That label is Gay-Lussac's law in plain language. Inside the rigid can, the gas cannot expand, so every bit of added heat becomes extra pressure:

$$\frac{P_1}{T_1} = \frac{P_2}{T_2}$$

Pressure divided by absolute temperature stays constant. Like Charles's law it is a direct proportion, but with volume locked instead of pressure, and temperature must again be in kelvins.

A worked example: a spray can holds propellant at 3.00 atm at $25^\circ\text{C}$ (298 K). Forgotten in a car at $60^\circ\text{C}$ (333 K), its pressure becomes $P_2 = P_1T_2/T_1 = 3.00 \times 333/298 \approx 3.35$ atm. That is only a 12 percent rise, well within design margins. But tossed into a fire at $500^\circ\text{C}$ (773 K), it reaches $3.00 \times 773/298 \approx 7.78$ atm, beyond what thin can walls tolerate, which is why the warning says incineration, not warm afternoons.

Locked walls, harder hits: the particle view

With the walls fixed, faster particles have nowhere to go but into harder, more frequent collisions. Heating raises particle speed, and each wall impact delivers more momentum; impacts also arrive more often because each particle crosses the container sooner. Both effects push the pressure up together, producing the clean linear proportion.

Cooling reverses it: slower particles hit softly and less often, so pressure sags. This is why a basketball left in a cold garage feels flat by morning even though no air leaked out.

A worked example: a rigid 2.0 L flask holds nitrogen at 1.50 atm and $22^\circ\text{C}$ (295 K). Plunged into ice water at $0^\circ\text{C}$ (273 K), the pressure falls to $P_2 = 1.50 \times 273/295 \approx 1.39$ atm. The 22-degree cooling cost about 7 percent of the pressure. Now heat that flask to $200^\circ\text{C}$ (473 K) and the pressure climbs to $1.50 \times 473/295 \approx 2.40$ atm, a 60 percent jump from a temperature change the can barely notices structurally.

Solving P1/T1 = P2/T2 problems

Gay-Lussac problems mirror Charles's-law problems, with pressure and volume swapping roles:

1. Convert temperatures to kelvins before anything else.
2. Confirm the container is rigid: constant volume is the law's entry ticket.
3. Solve: $P_2 = P_1T_2/T_1$ or $T_2 = T_1P_2/P_1$.
4. Sanity-check the direction: hotter means higher pressure.

A worked example: a fire extinguisher is pressurized to 12.0 atm at $20^\circ\text{C}$ (293 K) and its safety rating caps at 15.0 atm. What temperature hits the cap? $T_2 = T_1P_2/P_1 = 293 \times 15.0/12.0 = 366$ K, which is $93^\circ\text{C}$. Storage rooms are kept far cooler than that, but the calculation shows why extinguishers are never stored beside furnaces: the margin is one hot industrial accident wide.

Tires, tanks, and seasonal pressure swings

Tires live this law daily. A tire is nearly rigid, so its pressure tracks the kelvin temperature of the trapped air. Drivers who check pressure on a frosty morning get a lower reading than at noon, with no leak involved.

A worked example: tires set to 32.0 psi gauge at $20^\circ\text{C}$ (293 K). Convert to absolute first: $32.0 + 14.7 = 46.7$ psia. On a $-10^\circ\text{C}$ (263 K) morning the absolute pressure falls to $46.7 \times 263/293 \approx 41.9$ psia, which reads $41.9 - 14.7 = 27.2$ psi on the gauge. Nearly 5 psi vanished into the cold, enough to trigger the dashboard warning light. Mechanics know to set winter pressures slightly high to compensate, and they always measure cold tires for a consistent baseline.

Limits: leaks, liquids, and real containers

The law holds for a fixed amount of gas in a truly rigid container at ideal conditions. Real containers flex slightly, real gases deviate near condensation, and filling or venting changes the amount of gas, breaking the 'fixed n' assumption.

Propane tanks illustrate a subtler limit. They are never filled completely with liquid; regulations leave vapor space because liquid propane expands with heat far more aggressively than the gas law predicts. The gas-phase pressure still follows Gay-Lussac's law, but a liquid-full tank heated up would burst from hydraulic pressure long before the gas law mattered.

A worked example of the fixed-amount rule: a cylinder at 100 atm and 298 K springs a slow leak overnight, dropping to 95 atm at the same temperature. Gay-Lussac's law cannot explain it because $n$ changed; the lost 5 percent of pressure means roughly 5 percent of the gas escaped. Pressure ratios at constant temperature and volume double as mole ratios, a handy leak detector and a preview of the ideal gas law.

Common mistakes

Real-world applications

• Printing heat warnings on aerosol cans and gas cylinders

• Designing pressure relief valves for storage tanks

• Predicting tire pressure swings between seasons

• Setting safe fill limits for propane tanks