= Solution
In the gas-pressure limit $\beta\to1$, the mixture becomes a monatomic <ideal gas>, so
$$
\boxed{\gamma=\Gamma_1=\Gamma_2=\Gamma_3=\frac53}.
$$
In the radiation-pressure limit $\beta\to0$,
$$
\boxed{\Gamma_1=\Gamma_2=\Gamma_3=\frac43},
$$
whereas $\chi_\rho=\beta\to0$ and therefore
$$
\boxed{\gamma=\frac{c_P}{c_V}\to\infty}.
$$
The divergence reflects the singular constant-pressure heat capacity of pure radiation: its pressure fixes its temperature independently of density.
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