Question

Separation energy of a hydrogen like ion from its third excited state is $2.25$ times the separation energy of hydrogen atom from its first excited state. Find out the atomic number of this hydrogen like ion.

Answer 1

Answer: 3

Let the separation energy of the ion in third excited state $\left(n=4\right)$ be $E$ . Then
$E=13.6\times Z^2\times \frac{1}{16}eV$
and in the first excited state $\left(n=2\right)$ , $E_H=13.6\times \frac{1}{4}eV$
$\Rightarrow \frac{E}{E_H}=Z^2\times \frac{4}{16}$
$\Rightarrow \frac{2.25E_H}{E_H}=\frac{Z^2}{4}$
$\Rightarrow Z^2=9\Rightarrow Z=3$