Import Question JSON

$\text{Hint: } \text{PV} = \text{nRT}$ \\ $\text{Step: Find the density of an ideal gas. We start with the ideal gas equation:}$ \\ $\text{PV} = \text{nRT}$ \\ $\text{We know that the number of moles } \text{n} \text{ is equal to the total mass } \text{W} \text{ divided by the molar mass } \text{M}.$ \\ $\text{n} = \frac{\text{W}}{\text{M}}$ \\ $\text{Substitute this into the ideal gas equation:}$ \\ $\text{PV} = \frac{\text{W}}{\text{M}}\text{RT}$ \\ $\text{Rearrange the equation to isolate the term for density } (\rho = \frac{\text{W}}{\text{V}})$ \\ $\frac{\text{P}\text{M}}{\text{RT}} = \frac{\text{W}}{\text{V}} = \rho$ \\ $\Rightarrow \rho = \frac{\text{P}\text{M}}{\text{RT}}$ \\ $\text{We are given that the mass of a single molecule is } m \text{. The molar mass } \text{M} \text{ is the mass of one mole of molecules, which is } m \text{ multiplied by Avogadro's number } \text{N}_A.$ \\ $\text{M} = m \times \text{N}_A$ \\ $\text{Substitute this into the density equation:}$ \\ $\rho = \frac{\text{P} \times m \times \text{N}_A}{\text{RT}}$ \\ $\text{Rearrange the equation to group constants:}$ \\ $\rho = \text{P} \times m \frac{\text{N}_A}{\text{RT}}$ \\ $\text{The ratio } \frac{\text{R}}{\text{N}_A} \text{ is equal to Boltzmann's constant } k.$ \\ $\frac{\text{R}}{\text{N}_A} = k \Rightarrow \frac{1}{k} = \frac{\text{N}_A}{\text{R}}$ \\ $\text{So, } \frac{\text{N}_A}{\text{R}} \text{ can be written as } \frac{1}{k}.$ \\ $\text{Therefore, } \frac{\text{N}_A}{\text{RT}} = \frac{1}{kT}.$ \\ $\text{Substitute this into the density equation:}$ \\ $\Rightarrow \rho = \frac{\text{P}m}{kT}$ \\ $\text{Hence, option (4) is the correct answer.}$