An ideal gas expands according to the law $P{V}^{3/2}=$  constant. Then we conclude that :

It is not given that the expansion is adiable, (i.e) no heat enters or leaves the system. Hence,$dQ\ne 0$  . Let C be the molar specific heat during the process. Then from $dQ=dU+dW$  have for the process.  $CdT=C_{\upsilon }dT+P.dV\Rightarrow (C-C_{\upsilon })dT=P.dV$
For the process,  $P{V}^{3/2}=$   Constant.
Hence  $\frac{dP}{P}+\frac{3}{2}\frac{dV}{V}=0\mathrm{.........}\left(i\right)$
Also from ideal gas law 
 $PV=RT,\frac{dP}{P}+\frac{dV}{V}=\frac{dT}{T}\mathrm{.........}\left(ii\right)$
From eq. (i) and (ii) 
 $\Rightarrow \frac{dT}{T}+\frac{1}{2}\frac{dV}{V}=0$
 $\Rightarrow dV=-\frac{2V}{T}dT=-\frac{2R}{P}dT$
Hence  $(C-C_{ \upsilon })dT=PdV=-2RdT\Rightarrow C=(C_{\upsilon }-2R)$