QB_PH_SEC6_009: Atomic Spectra & Multi-Electron Atoms

What is the ionization energy (in eV) of a singly ionized helium ion ($ ext{He}^+$) in its ground state? Take the ground-state ionization energy of hydrogen to be $13.6\text{ eV}$. (Give your answer to one decimal place)

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Hydrogen-like Ion Energy**: $E_n(Z) = -\frac{13.6\, Z^2}{n^2}\text{ eV}$ 2. **Parameter Substitution**: - For helium ion ($ ext{He}^+$), nuclear charge $Z = 2$. - Ground state: $n = 1$. $E_1 = -13.6 \times \frac{2^2}{1^2} = -13.6 \times 4 = -54.4\text{ eV}$ 3. **Ionization Energy Definition**: $E_{\text{ion}} = E_\infty - E_1 = 0 - (-54.4\text{ eV}) = 54.4\text{ eV}$ 4. **Official Value**: 54.4 eV.