Gas Density and Molar Mass Calculations
Gas density is the mass of a gas per unit volume, and for an ideal gas it is given by d = PM/RT, which can also be rearranged to find the molar mass of an unknown gas: M = dRT/P. In brief: carbon dioxide (M = 44.01 g/mol) at 0 °C and 1 atm has a density of 44.01 / 22.414 = 1.96 g/L, which is about 1.5 times the density of air.
The density of a gas, unlike that of liquids and solids, changes a lot with temperature and pressure. For this reason, gas densities are always quoted with conditions. Knowing the formula lets you predict whether a gas will rise or sink in air, and it provides one of the simplest ways to determine the molar mass of an unknown gas. This article derives the formula, solves typical problems and gives a density table.
How do you calculate gas density?
Start with the ideal gas law, PV = nRT. The number of moles is n = m/M, where m is the mass and M the molar mass. Substituting gives PV = (m/M)RT. Since density d = m/V, the equation becomes:
d = PM / RT
Pressure and molar mass in the numerator mean that density rises when the pressure rises or the gas is heavier. Temperature in the denominator means that density falls when the gas is heated. With P in atmospheres, M in g/mol, R = 0.08206 L·atm/(mol·K) and T in kelvins, the density is in g/L.
Densities of common gases
| Gas | Molar mass (g/mol) | Density (g/L) |
|---|---|---|
| Hydrogen, H2 | 2.016 | 0.0899 |
| Helium, He | 4.003 | 0.1786 |
| Methane, CH4 | 16.04 | 0.7156 |
| Nitrogen, N2 | 28.014 | 1.250 |
| Dry air (average) | 28.96 | 1.292 |
| Oxygen, O2 | 31.998 | 1.428 |
| Carbon dioxide, CO2 | 44.01 | 1.964 |
| Chlorine, Cl2 | 70.90 | 3.163 |
How do you find density from conditions?
Worked example 1: air at room temperature
Find the density of dry air (M = 28.96 g/mol) at 25 °C and 1.00 atm.
T = 298.15 K. d = (1.00 × 28.96) / (0.08206 × 298.15) = 28.96 / 24.466 = 1.18 g/L.
Worked example 2: effect of temperature
Find the density of the same air at 100 °C and 1.00 atm.
T = 373.15 K. d = 28.96 / (0.08206 × 373.15) = 28.96 / 30.621 = 0.946 g/L. Heating the air from 25 °C to 100 °C lowers its density by about 20 %. A hot-air balloon with an envelope of 2000 m3 therefore gets a lift of about (1.184 − 0.946) × 2000 ≈ 476 kg of mass.
How do you find molar mass from density?
Rearranging the formula gives M = dRT / P. If instead of the density you know the mass m and the volume V of a sample, use M = mRT / (PV).
Worked example 3: an unknown gas
A flask of volume 1.50 L contains 2.52 g of a gaseous hydrocarbon at 0.950 atm and 30 °C. Identify the gas.
- T = 303.15 K.
- M = mRT / (PV) = 2.52 × 0.08206 × 303.15 / (0.950 × 1.50).
- Numerator: 2.52 × 24.877 = 62.69. Denominator: 1.425.
- M = 62.69 / 1.425 = 44.0 g/mol.
The molar mass of 44.0 g/mol matches propane, C3H8 (3 × 12.011 + 8 × 1.008 = 44.10 g/mol).
Worked example 4: using the density directly
An unknown gas has a density of 2.86 g/L at 1.00 atm and 20 °C. Its molar mass is M = 2.86 × 0.08206 × 293.15 / 1.00 = 68.8 g/mol.
Why do some gases float and others sink?
At the same temperature and pressure, the density of a gas is proportional to its molar mass. Gases with a molar mass lower than the average for air, 28.96 g/mol, tend to rise. These include hydrogen, helium, methane, ammonia (17.03 g/mol) and water vapor (18.02 g/mol). Gases heavier than air, such as carbon dioxide, propane and chlorine, tend to collect near the ground. This is the reason why carbon dioxide can accumulate in cellars and wells, and why a gas leak detector for propane is placed low on the wall while one for methane is placed high.
How were gas molar masses first determined?
Jean-Baptiste Dumas developed a vapor density method in 1826. He weighed a bulb filled with the vapor of a volatile liquid at a known temperature and pressure and used Avogadro’s hypothesis to find the molar mass. The Dumas method, together with the work of Cannizzaro, helped to set up the table of atomic masses in the middle of the 19th century. The same principle, in a more modern form, is still taught in laboratories, and the density measurement of gases is applied in industrial gas analyzers.
Which mistakes are common?
- Quoting a density without conditions: the value depends strongly on T and P.
- Using Celsius: convert to kelvins first.
- Mixing units of R: 0.08206 requires atmospheres, liters and g/mol to give g/L.
- Using the atomic mass of a diatomic gas: for oxygen use 31.998 g/mol of O2, not 16.00.
- Assuming ideal behavior at high pressure: real gases may be considerably denser than the formula predicts.