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Step-by-Step Solution
Step 1: Write down the ideal gas equation
The ideal gas equation is given by
$PV = nRT$,
where:
$P$ is the pressure of the gas.
$V$ is the volume occupied by the gas.
$n$ is the number of moles of the gas.
$R$ is the universal gas constant.
$T$ is the absolute temperature.
Step 2: Relate number of moles to total mass and molar mass
The number of moles $n$ can be written as
$n = \frac{\text{mass of gas}}{\text{molar mass}}.$
From the ideal gas equation, we rewrite $n$ as:
$$
n = \frac{PV}{RT}.
$$
Step 3: Express density in terms of mass and volume
Density $\rho$ is defined as
$$
\rho = \frac{\text{mass of gas}}{\text{volume}}.
$$
Since the molar mass $M$ of the gas can be expressed as
$M = m \, N_A,
$
where $m$ is the mass of each molecule and $N_A$ is Avogadro's number.
Step 4: Substitute relations to find density
Using the ideal gas equation and the fact that
$R = N_A \, k,
$
where $k$ is the Boltzmann constant, we get:
From $PV = nRT,$
$$
n = \frac{PV}{RT}.
$$
Total mass of the gas $= n \times \text{(molar mass)} = \frac{PV}{RT} \times (m\, N_A).$
Therefore,
$$
\rho = \frac{\text{mass of gas}}{V}
= \frac{\left(\frac{PV}{RT} \times m\,N_A\right)}{V}
= \frac{m\,N_A \,P}{R\,T}.
$$
Since $R = N_A\, k,$
$$
\rho = \frac{m\,N_A \,P}{(N_A\,k)\,T} = \frac{m\, P}{k T}.
$$
Step 5: State the final result
Hence, the density of the gas is
$$
\rho = \frac{P m}{k T}.
$$
This corresponds to Option (2) among the choices given.
