I want to ask a question about Van Der Waal's equation for real gases.
I was shown the following formula for real gases (VDW) which accounts for the liquification of water vapour which is not possible with an ideal gas model.
$$P = \frac{NkT}{V-Nb} - \frac{aN^2}{V^2} = \frac{\rho RT}{1-\rho b} - a \rho^2$$
which can be expressed using densities by making the denominators equal first:
$$P = \frac{NkT}{V-Nb} - \frac{aN^2}{V^2} = \frac{NkT}{V-Nb}\left(\frac{V}{V}\right) - \frac{aN^2}{V^2} = \frac{\frac{NkT}{V}}{\frac{V-Nb}{V}} - \frac{aN^2}{V^2} $$
but I appear to yield the result
$$P = \frac{\rho kT}{1-\rho b} - a \rho^2$$
where Boltzmann's constant $k$ has been switched for the gas constant, $R$. I can't find any sources that explain this derivation apart from "Molecular Driving Forces" by Dill, and he doesn't do a thorough conversion either.
How do I convert the numerator with Boltzmann's constant $k$ to the gas constant, $\rho$ which expresses VDW's equation involving densities?