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?




[Vapour density](https://en.wikipedia.org/wiki/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).


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