d
\(\mathrm{dQ}=\mathrm{du}+\mathrm{dW}\)

\(\mathrm{CdT}=\mathrm{C}_{\mathrm{V}} \mathrm{dT}+\mathrm{PdV}\)  \(....(1)\)

\(\therefore \quad \mathrm{PV}^2=\mathrm{RT}\)

\(\quad \mathrm{P}=\text { constant }\)

\(\quad \mathrm{P}(2 \mathrm{VdV})=\mathrm{RdT}\)

\(\quad \mathrm{PdV}=\frac{\mathrm{RdT}}{2 \mathrm{~V}}\)

Put in equation \((1)\)

\(\mathrm{C}=\mathrm{C}_{\mathrm{V}}+\frac{\mathrm{R}}{2 \mathrm{~V}}\)