Applications of the Ideal Gas Law: Molar Mass, Density, and Volume

At standard temperature and pressure (STP)—0°C or 273 K and 1 atm—one mole of any ideal gas occupies approximately 22.4 liters. This standard molar volume applies regardless of the gas's chemical identity. Using the ideal gas law and rearranging for volume, you can calculate that 0.5 moles occupies 11.2 liters and 2 moles occupies 44.8 liters at STP.

Molar volume is inversely related to temperature. At higher temperatures and lower pressures, gases expand and their molar volume increases beyond the standard 22.4 liters. Conversely, at lower temperatures and higher pressures, molar volume decreases. This relationship is fundamental to understanding gas behavior and is directly derived from the ideal gas law equation.

Rearranging the ideal gas law and substituting the definition of molar mass (mass divided by moles) yields the density equation: d = PM/RT. This shows density is directly proportional to molar mass. Helium balloons float because helium's molar mass is much less than air's, making helium less dense. Carbon dioxide from fire extinguishers is denser than air because CO₂ has a higher molar mass than nitrogen and oxygen.

The density equation d = PM/RT shows that density is inversely proportional to temperature. Heating gas molecules increases their speed and causes expansion, reducing density. Hot air balloons exploit this principle: turning on the burner heats air inside the balloon, lowering its density below surrounding air, causing ascent. Turning off the burner allows warm air to escape, increasing density and causing descent.

Rearrange the ideal gas law to isolate molar mass: M = dRT/P, where density equals mass divided by volume. Given an unknown gas's mass, volume, pressure, and temperature, calculate its density first. Convert temperature to Kelvin, then substitute all values into the equation. For example, a 12.5-gram gas sample occupying 6.08 liters at 1.2 atm and 40°C yields a molar mass of 44 g/mol, identifying it as carbon dioxide.

The ideal gas law, PV = nRT, relates four fundamental variables: pressure (P), volume (V), number of moles (n), and temperature (T), with R as the ideal gas constant (0.08206 L·atm/mol·K). Rearranging this equation allows you to calculate any one variable if the other three are known. This universal relationship applies to all ideal gases regardless of their chemical identity, making it essential for solving gas behavior problems.

Turning on the burner heats air molecules inside the balloon, causing them to move faster and the pressure to increase. Some air escapes through the balloon's design, making the air inside less dense than the surrounding atmosphere. This density difference creates buoyancy, allowing the balloon to ascend. Conversely, turning off the burner and opening the vent allows warm air to escape, increasing internal density to match surroundings, causing descent.