How this instrument works
Nothing visibly moves inside a sealed steel bottle as you heat it, yet its gauge climbs steadily. Kinetic theory splits that into two halves: mean molecular speed grows with √T, so every molecule hits a wall harder, and it also comes back to hit again sooner. Both factors carry one square root apiece, their product carries a full power of T, and pressure therefore rises lockstep with kelvin. Take dry air from 273 K up to 546 K inside rigid walls and its pressure doubles — nitrogen, argon or helium alike, since dilute molecules ignore one another and only wall collisions count.
Guillaume Amontons had this by about 1702, using a rigid air bulb whose mercury column he kept adjusting so gas volume stayed put while temperature changed — an early constant-volume gas thermometer. He noticed pressure falling toward zero at a definite cold point and put that point somewhere near −240 °C, long before anyone spoke of absolute zero. Joseph Louis Gay-Lussac's 1802 memoir on gas dilation was far more careful, and his name attached itself to this constant-volume version too; French and older British texts call it Amontons' law, which is fairer. His instrument outlived both men: constant-volume gas thermometry remained a primary way to realise the kelvin right up to 2019, when that definition moved instead onto a fixed Boltzmann constant of 1.380649 × 10⁻²³ J/K.
Three assumptions hold all of it together, and each one fails somewhere useful. Volume must stay fixed, which no real vessel quite manages: steel gains roughly 36 parts per million of volume per kelvin, so a cylinder heated 200 K swells nearly 0.7% and its pressure lands that much low. Amount of gas must stay fixed too, meaning relief valves, weeping seals and leaky Schrader cores all void this arithmetic. Finally, your gas must actually be gas — which is where people come unstuck. A propane cylinder holds liquid, so its pressure obeys a vapour-pressure curve instead, near 8.4 bar absolute at 20 °C and near 19 bar at 55 °C, climbing far more steeply than any ratio of temperatures predicts.
- Initial pressure wants an absolute figure, counted up from vacuum rather than from atmosphere. Its unit list runs Pa, kPa, bar, atm and psi; sea-level air sits at 101325 Pa.
- Give Initial absolute temperature (K) your starting kelvin value — add 273.15 to any Celsius reading before typing. Anything at or under zero gets refused, because dividing by absolute zero has no meaning.
- Final absolute temperature (K) takes wherever your gas ends up, kelvin once more. Only its ratio against that first reading drives the result.
- Final pressure comes back in pascals; swap its unit menu for bar or psi when the digit count turns silly. Knock off 101325 Pa to compare with a workshop dial.
Worked example — a steel sphere, ice point to 546 K
A thick-walled steel sphere gets filled with dry air in an ice bath, then sealed: Initial pressure 101325 Pa, Initial absolute temperature (K) 273.15. It goes into a furnace holding Final absolute temperature (K) at 546.30 — which is 273.15 °C, past the ignition point of paper yet nowhere near glowing. Substituting: P₂ = 101325 × 546.30 ⁄ 273.15 = 101325 × 2 = 202650 Pa. Two atmospheres on the nose, rendered by that unit menu as 2.0265 bar or 29.39 psi.
Any dial screwed into that sphere would show 101325 Pa gauge rather than 202650, worth remembering before sizing any burst disc. A real sphere also cheats slightly: 273 K of heating swells its steel by about 1% in volume, pulling true pressure down near 200700 Pa. Doubling absolute temperature doubles pressure; doubling a Celsius reading does nothing of the kind, which is precisely why 546.30 K rather than 546.30 °C belongs in that field.
Questions
Why does my tyre pressure fall in cold weather?
Because pressure follows absolute temperature and a tyre's volume barely shifts. Take 32 psi gauge on a warm 25 °C afternoon: that is 46.7 psi absolute at 298.15 K. Park overnight at 0 °C and P₂ = 46.7 × 273.15 ⁄ 298.15 = 42.8 psi absolute, which your gauge reports as 28.1 psi. Four psi gone with no leak anywhere. A working rule drops out of that arithmetic — near 1 psi for every 6 °C of cooling, close enough to the garage version of 1 psi per 10 °F.
Can I leave both temperatures in Celsius, since both sides use it?
No, and this is where most wrong answers come from. A ratio needs a scale whose zero is a true zero. Warming a rigid vessel from 20 °C to 40 °C looks like doubling; in kelvin it runs 293.15 to 313.15, a rise of 6.8%, so 2 bar becomes 2.14 bar rather than 4. Convert before typing anything: kelvin = °C + 273.15, and from Fahrenheit, kelvin = (°F + 459.67) ⁄ 1.8. Pressure needs identical treatment for identical reasons — counted from vacuum, not from atmosphere.
Why doesn't my propane or butane cylinder obey this?
Because liquid is sitting in it. Wherever liquid remains, pressure is set by vapour pressure at that temperature and rises roughly exponentially, not by any ratio of temperatures. Propane sits near 8.4 bar absolute at 20 °C and near 19 bar at 55 °C: a 12% rise in absolute temperature more than doubles pressure. Gay-Lussac's law governs cylinders whose contents are wholly gaseous — compressed air, nitrogen, argon, oxygen, helium. Once a part-empty propane bottle has boiled its last drop away, it rejoins this arithmetic.
How does it relate to Boyle's and Charles's laws?
All three are single slices through PV = nRT, taken two states at a time. Hold temperature and you get Boyle's law, P₁V₁ = P₂V₂. Hold pressure and you get Charles's law, V₁ ⁄ T₁ = V₂ ⁄ T₂. Hold volume — a sealed rigid tank — and you get this one. Let two quantities move at once and you need a combined form, P₁V₁ ⁄ T₁ = P₂V₂ ⁄ T₂. Setting before against after pays off practically: amount of gas, volume and R all cancel out, so none of them ever need measuring.
How hot can a real gas get before it stops obeying?
Heat rarely breaks it; density does. Warming pushes a gas further from condensation, so ideal behaviour tends to improve as things get hotter, and dry air at a few bar tracks this formula inside roughly 0.1%. Trouble starts with high initial pressure instead. A 200-bar nitrogen cylinder left in sunlight already sits outside ideal territory, its compressibility factor a few per cent above one. Past a few thousand kelvin, molecules also begin dissociating, and then even amount of gas stops being constant.
Does expansion of the container itself matter?
Usually less than your measurement error, though never quite zero. Steel gains about 36 parts per million of volume per kelvin, aluminium nearer 69. Heat a steel cylinder 100 K and it grows 0.36%, so true pressure lands roughly 0.36% under whatever this sheet returns — around 7 kPa on a 2 MPa reading. Precision gas thermometry corrects for exactly that, plus dead volume in connecting tubing. For tyres, plumbing and shop-floor work, ignore it.