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Q.
Which energy state of doubly ionised lithium has the same energy as that of the ground state of hydrogen ? Given Z for lithium = 3.
Atoms
Solution:
The energy of $n^{th}$ state of a hydrogen like atom is given as
$E_{n} = \frac{CZ^{2}}{n^{2}}$, where C is a constant.
For ground state of hydrogen atom, $Z = 1, \,n = 1$
$\therefore \quad E_{1} = \frac{C\left(1\right)^{2}}{\left(1\right)^{2}} = C$
For $n^{th}$ state of $Li^{2+} ion \left(Z = 3\right); E_{n} = \frac{C\left(3\right)^{2}}{n^{2}} = \frac{9C}{n^{2}}$
As $E_{n} = E_{1} \quad\therefore \frac{9C}{n^{2}} = C$ or $n^{2} = 9$ or $n = 3$