Question

If the first excitation energy for an H-like (hypothetical) sample is 24 eV, then the separation energy of an electron in the 3rd excited state is:

• A. 2 eV
• B. 3 eV
• C. 4 eV
• D. 5 eV

Answer 1

The correct option is A 2 eV

First excitation energy $n_1\to n_2$
$\Delta E_1=R_H[\frac{1}{1^2}-\frac{1}{2^2}]z^2=\frac{3}{4}{z}^2{R}_H$
Seperation energy from 3 rd excited state ($n_4\to \infty$)
$E_1=R_H[\frac{1}{4^2}-\frac{1}{\infty }]z^2=\frac{1}{16}{z}^2{R}_H$
Taking ratio of $E_{1}and{E}_2$ we get,
$\frac{\Delta E_1}{\Delta E_2}=\frac{\frac{3}{4}{z}^2{R}_H}{\frac{1}{16}{z}^2{R}_H}$
Given $\Delta E_1=24eV$
$\Delta E_2=\frac{\Delta E_1}{12}=\frac{24eV}{12}=2eV$