# Why is Molar Mass of a gas twice the vapour density?

I've seen this relation at a number of places, and I can't get why. A simple Google search doesn't give me a simple answer either.

Also, are there any exceptions/conditions for this relation to be true?

## 1 Answer

Vapour density $$(\rho_\mathrm{v})$$ of a gas is not the absolute density, such as mass density $$m/V$$ or number density $$N/V$$. It is in fact relative density, compared to the density of some other, reference gas at the same temperature and pressure: $$\rho_\mathrm{v} = \frac{\rho}{\rho_\text{ref}}.\tag{1}$$ It is a dimensionless quantity (just a number, without a unit). The choice of the reference gas is arbitrary and should be clearly stated. Usually, either hydrogen $$(\ce{H2}),$$ oxygen $$(\ce{O2}),$$ or air is chosen.

If the reference gas is hydrogen $$(M(\ce{H2})\approx \pu{2 g mol^-1}),$$ and we use the ideal gas approximation $$(\rho \propto M),$$ we can derive the following relation: $$\rho_\mathrm{v} = \frac{\rho}{\rho(\ce{H2})} = \frac{M}{M(\ce{H2})} \approx \frac{M_\mathrm r}{2} \implies \boxed{M_\mathrm r \approx 2\,\rho_\mathrm{v}},\tag{2}$$ where $$M_\mathrm r$$ is the relative molecular mass of gas.

If the reference gas is oxygen, then the relation is $$M_\mathrm r \approx 16\,\rho_\mathrm{v}$$.

The relationship holds only approximately, and it becomes more wrong as we deviate from the ideal gas conditions (e.g. low pressure, high temperature).

• Congratulations on your first answer, very well written. A couple of nitpicks regarding typesetting and notations. First, using complex abbreviations in math mode is a bad idea: $VD$ reads as $V$ times $D.$ A single letter with an index is preferred, like $\rho_\mathrm{v}$. Second, don't leave orphaned braces around math mode, enclose them too with $(…)$. This way they have appropriate height and won't be left hanging at the EoL (see my edit for examples). Mar 9, 2022 at 19:59
• @andselisk, thank you for your comments and your edit! I do not like abbreviated symbol names either, but I used $VD$ as I couldn't find any other more standard symbol for vapour density, and $VD$ appeared several times online. Mar 9, 2022 at 20:07