Q8. Which level of the doubly-ionized lithium (Li⁺⁺) ion has the same energy as the ground
state energy of the hydrogen atom?
 

Dear Student,

The energy of n^th orbit of a Hydrogen like atom is given by, E_n = $-\frac{13.6Z^2}{n^2}$, where Z is the atomic number
and the ground state energy of Hydrogen atom is, E₁ = -13.6 eV   (Z=1,n=1).

A doubly ionized Lithium ion (Li⁺⁺) is iso electronic with Hydrogen atom having Z=3.
For Li⁺⁺ to have the same energy of Hydrogen atom's ground state

E_n(Li⁺⁺) = E₁(H) 

$$\begin{gathered} \frac{-13.6\left(3^2\right)}{n^2}=-13.6 \\ Thus n^2 = 3^2 or n = 3 \end{gathered}$$
 
The energy level (orbit) of Li⁺⁺ should be 3 for having the same energy as the ground state energy of the Hydrogen atom



Regards