Question

An ideal gas having initial pressures $P$, volume $V$ and temperature $R$ is allowed to expand adiabatically until its volume becomes $5.66V$ while its temperature falls to $T/2$. How many degrees of freedom do the gas molecules have?

Answer 1

For an adiabatic change
${\left(\frac{V^{\prime }}{V}\right)}^{(gamma-1)}=\frac{T}{T^{\prime }}$
Given $V^{\prime }=5.66V$ and $T^{\prime }=\frac{T}{2}$.
$\Rightarrow (5.66{)}^{(\gamma -1)}=2$
$(\gamma -1)\log \left(5.66\right)=\log \left(2\right)$

$\Rightarrow \gamma =1+\frac{\log \left(2\right)}{\log \left(5.66\right)}=1+\frac{0.3010}{0.7528}=1+0.4=1.4$

Since $\gamma =1.4$, the gas is diatomic. For a diatomic gas, the number of degrees of freedom of the molecules $=5$.