The energy of $\mathrm{He}^{+}$ ion in its first excited state is, (The ground state energy for the Hydrogen atom is $-13.6 ~\mathrm{eV})$ :

The energy levels of a one-electron ion can be described by the formula: <br/><br/> $E_n = -\frac{Z^2}{n^2} \times E_0$ <br/><br/> where $E_n$ is the energy of the nth level, Z is the atomic number (number of protons), n is the principal quantum number, and $E_0$ is the ground state energy of the hydrogen atom (-13.6 eV). <br/><br/> For the $\mathrm{He}^{+}$ ion, the atomic number Z is 2 (since helium has 2 protons). We are looking for the energy of the first excited state, which corresponds to n = 2. Plugging these values into the formula, we get: <br/><br/> $E_2 = -\frac{2^2}{2^2} \times (-13.6 ~\mathrm{eV}) = -13.6 ~\mathrm{eV}$ <br/><br/> So, the energy of the $\mathrm{He}^{+}$ ion in its first excited state is $-13.6 ~\mathrm{eV}$.