## College Physics (4th Edition)

$v_{rms} = \sqrt{\frac{3~R~T}{M}}$
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 $v_{rms}$: $\overline{KE} = \frac{3}{2}~k~T$ $\frac{1}{2}m~v_{rms}^2 = \frac{3}{2}~k~T$ $v_{rms} = \sqrt{\frac{3~k~T}{m}}$ $v_{rms} = \sqrt{\frac{3~nR~T}{m~N}}$ $v_{rms} = \sqrt{\frac{3~nR~T}{m~(6.022\times 10^{23})~n}}$ $v_{rms} = \sqrt{\frac{3~nR~T}{M~n}}$ $v_{rms} = \sqrt{\frac{3~R~T}{M}}$