Consider an ideal Bose gas consisting of molecules with internal degrees of freedom. Assuming that, besides the ground state \(\varepsilon_{0}=0\), only the first excited state \(\varepsilon_{1}\) of the internal spectrum needs to be taken into account, determine the condensation temperature of the gas as a function of \(\varepsilon_{1}\). Show that, for \(\left(\varepsilon_{1} / k T_{c}^{0}ight) \gg 1\),

\[
\frac{T_{c}}{T_{c}^{0}} \simeq 1-\frac{\frac{2}{3}}{\zeta\left(\frac{3}{2}ight)} e^{-\varepsilon_{1} / k T_{c}^{0}}
\]

while, for \(\left(\varepsilon_{1} / k T_{c}^{0}ight) \ll 1\),

\[
\frac{T_{c}}{T_{c}^{0}} \simeq\left(\frac{1}{2}ight)^{2 / 3}\left[1+\frac{2^{4 / 3}}{3 \zeta\left(\frac{3}{2}ight)}\left(\frac{\pi \varepsilon_{1}}{k T_{c}^{0}}ight)^{1 / 2}ight]
\]