$\frac{\mathrm{n} \mathrm{R} \mathrm{dT}}{\gamma-1}+\mathrm{PdV}=\mathrm{dQ}$

$\frac{\mathrm{PdV}+\mathrm{VdP}}{\gamma-1}+\mathrm{PdV}=\mathrm{dQ}$

$\mathrm{dQ}=\frac{\gamma \mathrm{PdV}+\mathrm{VdP}}{\gamma-1}=\frac{\gamma\left(\frac{4 \mathrm{T}}{\mathrm{r}}\right) 4 \pi \mathrm{r}^{2} \mathrm{dr}+\frac{4}{3} \pi \mathrm{r}^{3}\left(\frac{-4 \mathrm{T}}{\mathrm{r}^{2}}\right) \mathrm{dr}}{\gamma-1}$

$=\frac{16 \pi \operatorname{dr} T\left(\gamma-\frac{1}{3}\right)}{\gamma-1}$

$\quad$ nRd $T=P d V+V d P$

$\Rightarrow$ ndT $=16 \pi r d r T \frac{2}{3 R}$

$C=\frac{d Q}{\text { ndT }}=\frac{\gamma-\frac{1}{3}}{(\gamma-1) \frac{2}{3}}=\frac{\frac{4}{3}}{\frac{2}{3} \times \frac{2}{3 R}}=3 R$