Ncert Exercises & Solutions

5.3 Using the equation of state pV=nRT; show that at a given temperature density of a gas is proportional to gas pressure

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,

$\frac{n}{V} = \frac{p}{R T}$

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

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

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

$\frac{d}{M} = \frac{p}{R T}$
$\Rightarrow d = (\frac{M}{R t}) p$

Molar mass (M) of a gas is always constant and there fore, at contstant temperature

$(T), \frac{M}{R t} = c o n s \tan t$
$d = (c o n s \tan t) p$
$\Rightarrow d α p$

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