Gas Density from Molar Mass

Start from $PV = nRT$, substitute $n = m/M$, and rearrange for mass over volume: $\rho = PM/RT$. What the result says is that a gas has no density of its own. Unlike a solid or a liquid, whose density is close enough to a fixed property to tabulate, a gas takes whatever density its pressure and temperature impose, and the only thing the substance itself contributes is $M$. Squeeze it and it gets denser in exact proportion; warm it and it thins in inverse proportion.

Air at 101.325 kPa and 20 °C, using $M = 0.028964$ kg/mol: $\rho = (101\,325 \times 0.028964)/(8.314 \times 293.15) = 1.204\ \text{kg/m}^3$ — the figure every ventilation calculation starts from. Helium at the same conditions, with $M = 0.0040026$, comes to 0.166 kg/m³. Subtract, and a cubic metre of helium lifts about 1.04 kg. That is the entire physics of a party balloon, and it explains why a balloon large enough to lift a person has to be the size of a house.

Rearranged for molar mass the equation becomes a measurement rather than a prediction, and a historically important one. Weigh a bulb of known volume empty, fill it with a vapour at measured temperature and pressure, weigh it again, and $M = \rho RT/P$ hands you the molar mass of an unknown. This is the Dumas method, and through the middle of the nineteenth century it was one of the few routes to a molecular formula. It also settled arguments: measured vapour densities are what showed that many elemental gases travel as diatomic molecules rather than lone atoms.

The dominant error here is the molar mass unit, and it is a clean factor of a thousand. With $R = 8.314$ J/(mol·K) the equation demands $M$ in kilograms per mole. Air is 0.029 kg/mol. Enter 29 and the answer comes back as 1204 kg/m³ — air denser than water — which at least announces itself. Enter 0.029 when the calculation wanted grams and you get the mirror error. The usual companions apply too: $P$ must be absolute, not a gauge reading, and $T$ must be in kelvin.

Two conceptual notes. There is no such thing as "the molar mass of air" in the strict sense — air is a mixture, and 28.96 g/mol is a mole-weighted average of nitrogen, oxygen and argon. It works precisely because an ideal gas is indifferent to what its neighbours are; only the total count matters. That same indifference produces a result most people find backwards: humid air is lighter than dry air. Water is 18 g/mol against air's 29, so at a given pressure and temperature every water molecule that joins the mixture has displaced a heavier one. Muggy days are low-density days, which is why aircraft performance charts include humidity and why a hot, humid runway is a long takeoff.