Question

Calculate the energy of a \( \text{He}^+ (Z = 2) \) in its first excited state.

Hint:

The energy of the electron in the excited state is always less negative than the ground state energy, but it still remains negative.

Answer 1

Step 1: Energy formula for hydrogen-like ions.
The energy of an electron in the \( n \)-th orbit of a hydrogen-like atom is given by the formula: \[ E_n = - \frac{13.6 \, \text{eV} \times Z^2}{n^2} \] where \( Z \) is the atomic number, \( n \) is the principal quantum number, and \( 13.6 \, \text{eV} \) is the energy of the ground state of hydrogen.
Step 2: Apply the formula for \( \text{He}^+ \).
For \( \text{He}^+ \) (with \( Z = 2 \)), the energy of the electron in the first excited state corresponds to \( n = 2 \). Substituting into the formula: \[ E_2 = - \frac{13.6 \times 2^2}{2^2} = - \frac{13.6 \times 4}{4} = - 13.6 \, \text{eV} \]
Step 3: Conclusion.
The energy of the electron in the first excited state of \( \text{He}^+ \) is \( -13.6 \, \text{eV} \).