Consider a spherical shell of radius R at temperature T. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume  $u=\frac{U}{V}\propto T^4$  and pressure  $p=\frac{1}{3}\left(\frac{U}{V}\right)$. If the shell  now undergoes an adiabatic expansion the relation between T and R is:

$$\begin{gathered} \frac{U}{V}=K{T}^4 \\ P=\frac{1}{3}K{T}^4 \end{gathered}$$

For adiabatic process,   $-dV=\omega$

$$\begin{gathered} \Rightarrow d\left(KV{T}^4\right)=-PdV\Rightarrow K[4V{T}^{3}dT+T^{4}dV]=-\frac{1}{3}K{T}^{4}dV \\ \Rightarrow 4KV{T}^{3}dT=-\frac{3}{4}K{T}^{4}dV\Rightarrow \frac{dT}{T}=-\frac{1}{3}\frac{dV}{V} \\ \Rightarrow \ln T=-\frac{1}{3}\ln V\Rightarrow \ln T+\ln V^{1/3}=cons \end{gathered}$$

$\Rightarrow T{V}^{1/3}=cons.\Rightarrow T\times {\left(\frac{4}{3}\pi R^3\right)}^{1/3}=cons\tan t$

$\Rightarrow TR=cons\tan t \Rightarrow T\propto \frac{1}{R}$