Question

When a certain mass of an ideal gas is adiabatically compressed so that its volume is reduced to $1/32$ times, its absolute temperature is increased $4$ times. The number of atoms in a molecule (atomicity) of the gas is :

• A. $3$
• B. $4$
• C. $2$
• D. $1$

Answer 1

The correct option is C $2$

Adiabatic compression

$P{V}^{\gamma }=$ constant $PV=nRT$

$T{V}^{\gamma -1}=$ constant $P\propto \frac{T}{V}$

$\Rightarrow T_1{V}_1^{\gamma -1}=T_2{V}_2^{\gamma -1}$

$\Rightarrow T{V}^{\gamma -1}=4T{\left(\frac{V}{32}\right)}^{\gamma -1}$

$\Rightarrow \frac{1}{4}=\left(\frac{1}{2}\right)5\gamma -5$

$\Rightarrow 2=5\gamma -5$

$\Rightarrow \gamma =\frac{7}{5}=1.40$ is a diatomic molecule.