Answer
$v_{rms} = \sqrt{\frac{3~R~T}{M}}$
Work Step by Step
We know that $PV = NkT = nRT$
Then $Nk = nR$ and $k = \frac{nR}{N}$
Suppose there are $n$ moles of molecules.
Then $N = (6.022\times 10^{23})~n$
Let $m$ be the mass of one molecule, and let $M$ be the molar mass. We can find an expression for $v_{rms}$:
$\overline{KE} = \frac{3}{2}~k~T$
$\frac{1}{2}m~v_{rms}^2 = \frac{3}{2}~k~T$
$v_{rms} = \sqrt{\frac{3~k~T}{m}}$
$v_{rms} = \sqrt{\frac{3~nR~T}{m~N}}$
$v_{rms} = \sqrt{\frac{3~nR~T}{m~(6.022\times 10^{23})~n}}$
$v_{rms} = \sqrt{\frac{3~nR~T}{M~n}}$
$v_{rms} = \sqrt{\frac{3~R~T}{M}}$
