Question

A sample of gas is allowed to expand adiabatically. As a consequence its volume increases from $1{\text{m}}^3$ to $32{\text{m}}^3$ and temperature drops from $320\text{K}$ to $80\text{K}$. How many degree of freedom are associated with the molecules of gas.

Answer 1

For Adiabatic Proces :-
$P{V}^{\gamma }=$ constant ... (1)
For an ideal gas,
$PV=nRT$
$P=\frac{nRT}{V}$
Put value of $‘P^{\prime }$ in equation (1),
$\left[\frac{nRT}{V}\right]V^{\gamma }=C$
$T{V}^{\gamma -1}=$ constant
$T_1{V}_1^{\gamma -1}=T_2{V}_2^{\gamma -1}$
$\left[\frac{T_1}{T_2}\right]={\left[\frac{V_2}{V_1}\right]}^{\gamma -1}$
$\left[\frac{320}{80}\right]={\left[\frac{32}{1}\right]}^{\gamma -1}$
$[2{]}^2=[2{]}^{5\gamma -5}$
On comparing
$5\gamma -5=2$
$\gamma =\frac{7}{5}$
For Degree of freedom,
$\gamma =1+\frac{2}{f}$
$\frac{7}{5}=1+\frac{2}{f}$
$\frac{2}{5}=\frac{2}{f}$
$f=5$
Sample-gas is diatomic.