Gas Density and Molar Volume

Finds the volume of one mole of gas as RT ÷ P, and its density as the molar mass ÷ that volume. At 0 °C and one atmosphere the molar volume is 22.4 L.

Explanation

The volume occupied by one mole of gas is its molar volume. Setting $n = 1$ in $PV = nRT$ gives it immediately.

V_m = \dfrac{RT}{P} \qquad \rho = \dfrac{M}{V_m}

Here $V_m$ is the molar volume in litres per mole, $M$ the molar mass in grams per mole, and $\rho$ the density in grams per litre.

The molar volume does not depend on which gas it is. At a given temperature and pressure, a mole of oxygen and a mole of hydrogen fill exactly the same space. Every difference in density is therefore a difference in molar mass.

Example

Find the density of air, whose average molar mass is 28.8 g/mol, at 0 °C and 101.325 kPa.

V_m = \dfrac{8.314 \times 273.15}{101325} = 0.02241\ \text{m}^3 = 22.41\ \text{L/mol}

\rho = \dfrac{28.8}{22.41} = 1.285\ \text{g/L}

A litre of air weighs about 1.3 g. The air inside a classroom of 200 m³ comes to some 250 kg.

Why balloons float

At equal volumes, a difference in density is nothing but a difference in molar mass.

Helium — 4 g/mol, giving 0.18 g/L
Air — 28.8 g/mol, giving 1.29 g/L
Carbon dioxide — 44 g/mol, giving 1.96 g/L

Helium is a seventh as dense as air and rises; carbon dioxide is heavier and pools at floor level.

Two standard states

Beware: "standard conditions" means two different things.

0 °C and 1 atm (101.325 kPa) — molar volume 22.4 L/mol, the one used in school chemistry
25 °C and 100 kPa — molar volume 24.8 L/mol, the current IUPAC recommendation

Check which convention a calculation is using before trusting the number.