Question

Sol. [Key Concept: Energy of a hydrogen like ion]

For a one electron ion, Eₙ is proportional to −Z²/n². For He⁺ in n = 1, Z²/n² = 2²/1² = 4. For Be³⁺ in n = 2, Z²/n² = 4²/2² = 4. The two energies are equal, so the required energy is −x J, option (3).

$$E_{n} = - R_{H}\left( \frac{Z^{2}}{n^{2}} \right)J $$For ${He}^{+}(n = 1)$,

$$\begin{matrix} & E_{n} = - x = - R_{H}\left( \frac{2^{2}}{1^{2}} \right) = - 4R_{H} \\ \therefore & R_{H} = \frac{x}{4} \end{matrix}$$

For ${Be}^{3 +}(n = 2)$,

$$\begin{matrix} & E_{n} = - R_{H}\left( \frac{Z^{2}}{n^{2}} \right)J \\ & \ = - \frac{x}{4} \times \left( \frac{4 \times 4}{2 \times 2} \right) = - x\text{ }J \end{matrix}$$