Density of a Gas

Number Density

Number density is the "easy" density to calculate for a gas. You just rearrange the ideal gas law in the following way.

\[{n\over V}={P\over {RT}}\]

The left side is number density which is in moles-per-liter. So you just need the pressure and temperature of the gas to calculate it (and R of course). Number density is a good way to think about gas pressures. High pressures have a lot of moles of gas per liter, as opposed to low pressures which have very few moles of gas per liter.

Consider the following diagram... it consists of 3 identical containers (equal volumes) that each contain identical numbers of gas molecules (also assume the temperature is constant). This is illustrated by showing the same number of circles representing the gas molecules.

Gas model image of 10 small dots representing gas molecules in a given volume.
Gas model image of 10 medium sized dots representing gas molecules in a given volume.
Gas model image of 10 larger dots representing gas molecules in a given volume.

Even though the particle sizes are different for each case (think about a real case with helium, neon, and argon), the pressures would all be the same. Why? Because number density is what governs pressure for a gas system where temperature is constant. The number densities here are all the same, 10 particles per unit of volume. However, the particle masses are going to be different and therefore the mass density (see upcoming section) will be different for each gas. Specifically, the larger or more massive the particle, the larger the mass density. So, the largest mass density is on the right end and the smallest mass density is on the left end.

It is important to always know whether someone is speaking about number density (mol/L) or mass density (g/mL or g/L). They are definitely related, but they are not the same.

Mass Density of a Gas

The mass density (shown as greek rho, ρ) of a gas is typically just called the "density".  This is the mass of the gas relative to the volume of the gas.

density =    mass  volume=mv=  ρ

Because gases that are behaving ideally under the same conditions (temperature, pressure) all have the same number density (see previous section), they will all have different mass densities since different gases have different masses per particle.

The formula for mass density can be derived from the number density formula by simply multiplying by the molar mass of the gas (shown as just M with units of g/mol). Remember that moles (n) times molar mass (M) is equal to mass (m).

\[\eqalign{ {n\over V} &= {P \over RT}\cr M\left({n\over V}\right) &= \left({P \over RT}\right)M\cr {m\over V} &= M\left({P \over RT}\right)\cr \rho &= M\left({P \over RT}\right)\cr }\]

This equation can be easily rearranged so that we are solving to find the molar mass of the gas based on the pressure and temperature.

\[ M = {\rho RT \over P}\]

Remembering again that M here is the molar mass in units of g/mol. So now we can either find the density of a gas given its molar mass (and the conditions) or use the density (or mass and volume) to find the molar mass.

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