Question

Energy and radius of first Bohr orbit of He$^+$ and Li$^2+$ are:
Given: $ R_H = 2.18 \times 10^{-18} \, \text{J}, a_0 = 52.9 \, \text{pm} $

• A. \( E_n (\text{Li}^{2+}) = -8.72 \times 10^{-18} \, \text{J}, r_n (\text{Li}^{2+}) = 26.4 \, \text{pm}, E_n (\text{He}^{+}) = -19.62 \times 10^{-18} \, \text{J}, r_n (\text{He}^{+}) = 9.6 \, \text{pm} \)
• B. \( E_n (\text{Li}^{2+}) = -19.62 \times 10^{-16} \, \text{J}, r_n (\text{Li}^{2+}) = 17.6 \, \text{pm}, E_n (\text{He}^{+}) = -8.72 \times 10^{-16} \, \text{J}, r_n (\text{He}^{+}) = 26.4 \, \text{pm} \)
• C. \( E_n (\text{Li}^{2+}) = -8.72 \times 10^{-16} \, \text{J}, r_n (\text{Li}^{2+}) = 17.6 \, \text{pm}, E_n (\text{He}^{+}) = -19.62 \times 10^{-16} \, \text{J}, r_n (\text{He}^{+}) = 26.4 \, \text{pm} \)
• D. \( E_n (\text{Li}^{2+}) = -19.62 \times 10^{-18} \, \text{J}, r_n (\text{Li}^{2+}) = 17.5 \, \text{pm}, E_n (\text{He}^{+}) = -8.72 \times 10^{-18} \, \text{J}, r_n (\text{He}^{+}) = 26.4 \, \text{pm} \)

Hint:

For hydrogen-like ions, the energy and radius of the first Bohr orbit depend on the atomic number \( Z \). The energy is proportional to \( Z^2 \), and the radius is inversely proportional to \( Z \).