Using the equation of state $pV = nRT$,show that at a given temperature,the density of a gas is proportional to the gas pressure $p$.

(N/A) The equation of state is given by,
$pV = nRT$ .......... $(i)$
Where,
$p \rightarrow$ Pressure of gas
$V \rightarrow$ Volume of gas
$n \rightarrow$ Number of moles of gas
$R \rightarrow$ Gas constant
$T \rightarrow$ Temperature of gas
From equation $(i)$,we have,
$\frac{n}{V} = \frac{p}{RT}$
Replacing $n$ with $\frac{m}{M}$,where $m$ is the mass and $M$ is the molar mass,we have
$\frac{m}{MV} = \frac{p}{RT}$ .......... $(ii)$
Since density $d = \frac{m}{V}$,substituting this into equation $(ii)$ gives,
$\frac{d}{M} = \frac{p}{RT}$
$\Rightarrow d = \left(\frac{M}{RT}\right) p$
Since the molar mass $M$ and gas constant $R$ are constants,at a constant temperature $T$,the term $\left(\frac{M}{RT}\right)$ is constant.
Therefore,$d = (\text{constant}) \times p$
$\Rightarrow d \propto p$
Hence,at a given temperature,the density $d$ of a gas is directly proportional to its pressure $p$.