Using the equation of state pV = nRT; show

The equation of state is given by,

pV = nRT ……….. (i)

Where,

p → Pressure of gas

V → Volume of gas

n→ Number of moles of gas

R → Gas constant

T → Temperature of gas

From equation (i) we have,

$\frac{n}{V}=\frac{p}{\mathrm{RT}}$

Replacing $n$ with $\frac{m}{M}$, we have

$\frac{m}{M V}=\frac{p}{\mathrm{R} T}$......(ii)

Where,

m → Mass of gas

M → Molar mass of gas

But, $\frac{m}{V}=d(d=$ density of gas $)$

Thus, from equation (ii), we have

$\frac{d}{M}=\frac{p}{\mathrm{RT}}$

$\Rightarrow d=\left(\frac{M}{\mathrm{RT}}\right) p$

Molar mass $(M)$ of a gas is always constant and therefore, at constant temperature $(T), \frac{M}{R T}=$ constant.

$d=($ constant $) p$

$\Rightarrow d \propto p$

Hence, at a given temperature, the density (d) of gas is proportional to its pressure (p)