A gray gas with an absorption coefficient of \(\kappa\) is contained between two plates separated by a distance \(L\). Using the diffusion approximation for radiation, derive the relation

\[\begin{equation*}Q_{1}=\frac{1}{3 \kappa L / 4+1 / \epsilon_{1}+1 / \epsilon_{2}-1}\left(E_{\mathrm{b} 1}-E_{\mathrm{b} 2}\right) \tag{19.31}\end{equation*}\]

for the radiation exchanged from plate 1 to plate 2 .