The value of is $\mathrm{\gamma }=\frac{{\mathrm{C}}_{\mathrm{P}}}{{\mathrm{C}}_{\mathrm{V}}}=4/3$ for an adiabatic process of an ideal gas for which internal energy $\mathrm{U}=\mathrm{k}+\mathrm{nPV}$.  Find the value of n (k is a constant).

$\begin{array}{l}{\mathrm{pv}}^{\mathrm{\gamma }}=\text{ constant }\\ {\mathrm{pv}}^{4/3}=\text{ constant }\end{array}$
$\ln {\mathrm{v}}^4+\mathrm{In}{\mathrm{p}}^3=\text{ constant }$

$4\frac{\mathrm{dv}}{\mathrm{v}}=-3\frac{\mathrm{dp}}{\mathrm{p}}$

$4\mathrm{p}\left(\mathrm{dv}\right)+3\mathrm{vdp}=0$

$3(\mathrm{pdv}+\mathrm{vdp})+\mathrm{pdv}=0$

For Adiabatic process, $\mathrm{dQ}=0$

$\mathrm{du}+\mathrm{dw}=0$

$\mathrm{du}+\mathrm{pdv}=0$

$\mathrm{du}=3(\mathrm{pdv}+\mathrm{vdp})$

$\mathrm{U}=\mathrm{k}+3\mathrm{pv}$

$\mathrm{n}=3$