Answer
$v_{rms}=\sqrt{\frac{3P}{\rho}} $.
Work Step by Step
The rms speed, equation 13–9 is:
$$v_{rms}=\sqrt{\frac{3kT}{m}} $$
From the ideal gas law, PV=NkT, we see that kT=PV/N.
$$v_{rms}=\sqrt{\frac{3PV}{Nm}} $$
The total mass M of the gas is the mass of a molecule, m, multiplied by the number of molecules, N.
$$v_{rms}=\sqrt{\frac{3PV}{M}} $$
Finally, the density of the gas,$\rho$, is the mass M divided by the volume V.
$$v_{rms}=\sqrt{\frac{3P}{\rho}} $$
This was the relationship to be shown.
