Question

Which energy state of doubly ionized lithium has same energy as that of first excitation state of hydrogen? (given $\mathrm{z}$ for lithium $=3$)

Answer 1

Step 1: Formula for the energy of hydrogen and hydrogen-like species

$\text{Energy}=\frac{-13.6z^2}{n^2}$

Step 2: Calculate the energy state of doubly ionized lithium:

Let nth level of ${\mathrm{Li}}^{2+}$ has the same energy as the ground state of Hydrogen atom as $\mathrm{z}=3$ for $\mathrm{Li}$

${\mathrm{E}}_{\mathrm{n}}\left({\mathrm{Li}}^{2+}\right)={\mathrm{E}}_1\left(\mathrm{H}\right)$

$$\begin{gathered} -13.6\frac{3^2}{\mathit{}{n}^2}=-13.6\times \frac{1}{1} \\ n=3 \end{gathered}$$

Hence, $n=3$ is a state of doubly ionized lithium has same energy as that of first excitation state of hydrogen.