The ideal gas law can be recast in terms of the density of a gas. (a) Use
dimensional analysis to find an expression for the density \(\rho\) of a gas in
terms of the number of moles \(n\), the volume \(V\), and the molecular weight \(M\)
in kilograms per mole. (b) With the expression found in part (a), show that
$$
P=\frac{\rho}{M} R T
$$
for an ideal gas. (c) Find the density of the carbon dioxide atmosphere at the
surface of Venus, where the pressure is \(90.0 \mathrm{~atm}\) and the
temperature is \(7.00 \times 10^{2} \mathrm{~K}\). (d) Would an evacuated steel
shell of radius \(1.00 \mathrm{~m}\) and mass \(2.00 \times 10^{2} \mathrm{~kg}\)
rise or fall in such an atmosphere? Why?