Two ideal gases, helium and oxygen, are maintained in separate...

The ideal gas equation in terms of mass density $\rho$ is given by $P = \dfrac{\rho RT}{M_0}$, where $M_0$ is the molar mass of the gas.

Rearranging for density, we get $\rho = \dfrac{P M_0}{RT}$.

Since the pressure $P$ and temperature $T$ are identical for both gases, the mass density is directly proportional to the molar mass ($\rho \propto M_0$).

The molar mass of monatomic helium ($He$) is $4$ g/mol and the molar mass of diatomic oxygen ($O_2$) is $32$ g/mol.

The required ratio is $\dfrac{\rho_{He}}{\rho_{O_2}} = \dfrac{4}{32} = \dfrac{1}{8}$, which is $1 : 8$.

Using the atomic mass of oxygen ($16$ g/mol) instead of its molecular mass leads to the incorrect ratio $1 : 4$.

Answer: $1 : 8$