Step-by-step Solution

1. Recall the Ideal Gas Law: The ideal gas equation is given by $PV = nRT$, where $P$ is the pressure, $V$ is the volume, $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is the absolute temperature.

2. Express the Ideal Gas Law in terms of density: The number of moles $n$ can be written as the ratio of the total mass of the gas $m$ to its molar mass $M$: $n = m/M$. Substituting this into the ideal gas equation gives: $PV = \frac{m}{M}RT$ Density, $\rho$, is defined as mass per unit volume, $\rho = m/V$. We can rearrange the equation to solve for density: $\frac{m}{V} = \frac{PM}{RT}$ So, the density of an ideal gas is given by: $\rho = \frac{PM}{RT}$

3. Set up the ratio of densities for the two gases: Let the two gases be Gas 1 and Gas 2. Their properties will be denoted by subscripts 1 and 2, respectively. The density of Gas 1 is $\rho_1 = \frac{P_1 M_1}{RT}$. The density of Gas 2 is $\rho_2 = \frac{P_2 M_2}{RT}$. Note that both gases are in the same vessel at a constant temperature, so the volume $V$, temperature $T$, and the gas constant $R$ are the same for both.

   The ratio of their densities is: $\frac{\rho_1}{\rho_2} = \frac{\frac{P_1 M_1}{RT}}{\frac{P_2 M_2}{RT}} = \frac{P_1 M_1}{P_2 M_2}$ This can be rewritten as: $\frac{\rho_1}{\rho_2} = \left(\frac{P_1}{P_2}\right) \left(\frac{M_1}{M_2}\right)$

4. Substitute the given values into the ratio equation: From the problem statement, we have:

   • The ratio of partial pressures: $\frac{P_1}{P_2} = \frac{4}{3}$
   • The ratio of atomic masses (and thus molar masses for monoatomic gases): $\frac{M_1}{M_2} = \frac{2}{3}$

   Substituting these values into the density ratio equation: $\frac{\rho_1}{\rho_2} = \left(\frac{4}{3}\right) \left(\frac{2}{3}\right)$

5. Calculate the final ratio of densities: $\frac{\rho_1}{\rho_2} = \frac{4 \times 2}{3 \times 3} = \frac{8}{9}$ So, the ratio of the densities of the two gases is 8 : 9.

6. Compare with the given options: The calculated ratio 8 : 9 matches option D.

   A: 1 : 4 B: 1 : 2 C: 6 : 9 (which simplifies to 2 : 3) D: 8 : 9

   The correct option is D.