To solve this problem, we need to determine the energy levels of an electron in a hydrogen atom according to Bohr's model.$\\$The energy of an electron in the $n^{th}$ energy level of a hydrogen atom is given by:$\\ E_n = -\frac{13.6}{n^2}$ eV$\\$where $n$ is the principal quantum number.$\\$For the first excited state, $n = 2: \\ T_1 = E_2 = -\frac{13.6}{2^2} = -\frac{13.6}{4}$ eV$\\$For the second excited state, $n = 3: \\ T_2 = E_3 = -\frac{13.6}{3^2} = -\frac{13.6}{9}$ eV$\\$Now, we find the ratio $T_1 : T_2: \\ T_1 : T_2 = \left(-\frac{13.6}{4}\right) : \left(-\frac{13.6}{9}\right) \\$Simplify the ratio:$\\ T_1 : T_2 = \frac{13.6}{4} \times \frac{9}{13.6} \\ T_1 : T_2 = \frac{9}{4} \\$Therefore, the ratio $T_1 : T_2$ is $9 : 4. \\$This corresponds to Option 4.