Question
Energy and radius of first Bohr orbit of ${He}^{+}$and ${Li}^{2 +}$ are
[Given $R_{H} = 2.18 \times 10^{- 18}\text{ }J,a_{0} = 52.9pm$ ]
Options
$E_{n}\left( {Li}^{2 +} \right) = - 19.62 \times 10^{- 18}\text{ }J$;
$$\begin{matrix} & r_{n}\left( {Li}^{2 +} \right) = 17.6pm \\ & E_{n}\left( {He}^{+} \right) = - 8.72 \times 10^{- 18}\text{ }J; \\ & r_{n}\left( {He}^{+} \right) = 26.4pm \end{matrix}$$
$\ E_{n}\left( {Li}^{2 +} \right) = - 8.72 \times 10^{- 18}\text{ }J$;
$$\begin{matrix} & r_{n}\left( {Li}^{2 +} \right) = 26.4pm \\ & E_{n}\left( {He}^{+} \right) = - 19.62 \times 10^{- 18}\text{ }J; \\ & r_{n}\left( {He}^{+} \right) = 17.6pm \end{matrix}$$
Solution
Sol. [Key Concept: Energy and radius of the first Bohr orbit of He⁺ and Li²⁺ (E ∝ Z², r ∝ 1/Z)] $E_{n} = \frac{- 2.18 \times 10^{- 18} \times z^{2}}{n^{2}}J;r_{n} = \frac{52.9 \times n^{2}}{z}pm$
For ${He}^{+}$
$$\begin{matrix} & E_{{He}^{+}} = - 2.18 \times 10^{- 18} \times 4 = - 8.72 \times 10^{- 18}\text{ }J \\ & r_{{He}^{+}} = \frac{52.9 \times 1}{2} = 26.45pm \end{matrix}$$
For ${Li}^{2 +}$
$$\begin{matrix} & E_{{Li}^{2 +}} = - 2.18 \times 10^{- 18} \times 9 = 19.62 \times 10^{- 18}\text{ }J \\ & r_{{Li}^{2 +}} = \frac{52.9 \times 1}{3} = 17.63pm \end{matrix}$$
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