$C_{V}=\left.\frac{\partial U}{\partial T}\right|_{V=\text { constann }}$

$\text { or } \quad C_{V}=\frac{d U}{d T}$

Also, for $1$ mole of gas,

$C_{V}=\frac{f}{2} \cdot R$

where, $f=$ degree of freedom.

Hence, we have

$\frac{f}{2} R=\frac{d U}{d T}$

Here, $U=\frac{5}{2} p V+C=\frac{5}{2} R T+C$

[ $\because$ one mole of gas is considered]

So, $\quad \frac{f}{2} R=\frac{d}{d T}\left(\frac{5}{2} R T+C\right)$

$\Rightarrow \quad \frac{f}{2} R=\frac{5}{2} R \Rightarrow f=5$

Now, using $y=1+\frac{2}{f}$

where, $\gamma=$ ratio of specific heats

$=$ adiabatic index.

We have, $\gamma=1+\frac{2}{5} \Rightarrow \gamma=\frac{7}{5}$

So, equation of adiabats can be written as $p V^{\gamma}=$ constant $\Rightarrow p V^{7 / 5}=$ constant $\Rightarrow p^{5} V^{7}=$ constant