The value of $\gamma =C_P/C_V$ is 4/3 for an adiabatic process of an ideal gas for which internal energy $\mathrm{U}=\mathrm{K}+\mathrm{nPV}$. The value of $\mathrm{n} \left(\mathrm{K}\text{ is a constant }\right)$ is ___.

${\mathrm{PV}}^{\mathrm{\gamma }}=\text{ constant }$

${\mathrm{PV}}^{4/3}=\text{ constant }$

In $\left({\mathrm{V}}^4\right)+\mathrm{In}\left({\mathrm{P}}^3\right)=\text{ constant }$

$4\left(\frac{\mathrm{dV}}{\mathrm{V}}\right)=-3\left(\frac{\mathrm{dP}}{\mathrm{P}}\right)$

$4\mathrm{P}\left(\mathrm{dV}\right)+3\mathrm{V}\left(\mathrm{dP}\right)=0$

$3(\mathrm{PdV}+\mathrm{VdP})+\mathrm{PdV}=0$          …(i)

For adiabatic process, $\mathrm{dQ}=0=\mathrm{dU}+\mathrm{dW}$

$=\mathrm{dU}+\mathrm{PdV}$                           …(ii)

Compare Eqs. (i) and (ii), $\mathrm{dU}=3(\mathrm{PdV}+\mathrm{VdP})$

$\mathrm{U}=\mathrm{K}+3\mathrm{PV}$

So, n = 3