Ideal Gas Law

$PV = nRT$ says that for a gas, four quantities are not independent: fix any three and the fourth is decided. Squeeze it and the pressure rises; warm it and it pushes harder or swells; add more of it and both go up. What makes the equation remarkable is not that those things are true — anyone with a bicycle pump knows them — but that one constant serves every gas. Helium, nitrogen and steam all obey it with the same $R = 8.314$ J/(mol·K), which is a strong hint that pressure has nothing to do with what the molecules are and everything to do with how many there are and how fast they are moving.

A worked case in units you would actually read off a gauge. A 20 L cylinder sits at 150 kPa absolute on a 20 °C morning. Rearranged for amount, $n = PV/RT = (150\,000 \times 0.020)/(8.314 \times 293.15) = 3000/2437 \approx 1.23\ \text{mol}$. Note what had to happen before the arithmetic: pascals not kilopascals, cubic metres not litres, and kelvin not Celsius. This page converts your entries for you, but the discipline is worth keeping in your head, because a scrap of paper will not.

The law arrived in pieces. Boyle established $PV$ constant at fixed temperature in 1662; Charles and Gay-Lussac tied volume and pressure to temperature around 1800; Avogadro proposed in 1811 that equal volumes of gases hold equal numbers of particles. Émile Clapeyron folded them into a single expression in 1834. Kinetic theory later derived the whole thing from mechanics: treat molecules as point masses that bounce elastically and never attract one another, average over their collisions with the walls, and $PV = nRT$ falls out — with $RT$ revealed as a measure of the average kinetic energy per mole.

Those two assumptions are also the fine print, and here a common textbook line deserves correcting. The ideal gas law is a limit, not a fact about gases. Real molecules do occupy volume and do attract each other, so the equation is exact only as pressure approaches zero and the gas gets out of its own way. Near condensation it fails plainly: at 100 atm, or anywhere close to the boiling point, the error runs to tens of percent and you want van der Waals or a compressibility factor. Under ordinary room conditions the error is well under 1%, which is why the approximation earns its keep.

Two errors account for most wrong answers on this page, and both are unit errors rather than physics errors. The first is feeding in Celsius. Doubling a gas from 20 °C to 40 °C does not double anything — in kelvin that is 293 to 313, a rise of 7%, and a calculation that used 20 and 40 would be wrong by a factor of nearly two. The second is feeding in a gauge pressure. A tire gauge reading 220 kPa means 321 kPa absolute; $P$ here is absolute pressure, measured from vacuum, because the equation counts molecular impacts and vacuum is where there are none. A third, quieter trap: the familiar 22.4 L per mole belongs to 0 °C and 1 atm. IUPAC redefined standard pressure to 100 kPa in 1982, and at that pressure the molar volume is 22.71 L. Both numbers circulate, and quoting one against the other's conditions is a 1.3% error hiding inside a memorised constant.