First slide

Hydrogen -like ion

Question

The difference between excitation energy and binding energy of electron in a ${He}^{+}$ ion is 42.31 eV. The energy quantum number of the excited state is

Moderate

A

3
B

2
C

4
D

6

Solution

$E_{0} =$ Energy of electron in the ground state
$= - 13.6 Z^{2} e V$
E = Energy in excited state $= - \frac{13.6 Z^{2}}{n^{2}} e V$
Binding energy in the excited state $= + \frac{13.6 Z^{2}}{n^{2}} e V$
Excitation energy $= - \frac{13.6 Z^{2}}{n^{2}} - (- 13.6 Z^{2}) e V = 13.6 Z^{2} (1 - \frac{1}{n^{2}}) e V$
$∴$ Difference $= 13.6 Z^{2} (1 - \frac{1}{n^{2}}) - 13.6 \frac{Z^{2}}{n^{2}} = 13.6 Z^{2} (1 - \frac{2}{n^{2}}) e V$
$\begin{array}{l} ∴ 13.6 \times 2^{2} \times (1 - \frac{2}{n^{2}}) = 42.31 \\ \Rightarrow n = 3 \end{array}$

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