Question

In $L{i^{ + + }}$, electrons in the first Bohr orbit are excited to a level by radiation of wavelength $\lambda$. When the ion gets deexcited to the ground state in all possible ways (including intermediate emissions), a total of 6 spectral lines are observed. What is the value of $\lambda$?
(Given: $h = 6.63 \times {10^{34}}Js$; $c = 3 \times {10^8}m/s$)
(a) 9.4 nm
(b) 12.3 nm
(c) 10.8 nm
(d) 11.4 nm

Answer 1

The spectral lines observed can be used to find the transition level of the atom. As transition energies are fixed for energy levels, this can be used to determine $\lambda$.

Formula used:
Number of spectral lines is given by:
No. of spectral lines$= n\dfrac{{(n - 1)}}{2}$ ……(1)
where n is the transition level.
Change in energy for transition between energy levels ${n_1}$ and ${n_2}$ is given by:
$\Delta E = 13.6 \times {Z^2}(\dfrac{1}{{n_1^2}} - \dfrac{1}{{n_2^2}})$ ……(2)
where Z is the atomic number of the atom.
Energy is related to wavelength by:
$\Delta E = hc/\lambda$ ……(3)

Complete step-by-step answer:

Given:
1. Atom is $L{i^{ + + }}$ with $Z = 3$.
2. No. of spectral lines = 6.

To find: The wavelength $\lambda$ which excited the atom.

Step 1 of 3:
Find the energy level transition from no. of spectral lines using eq (1):
$\begin{gathered} 6 = n\dfrac{{(n - 1)}}{2} \\\ {n^2} - n - 12 = 0 \\\ (n - 4)(n + 3) = 0 \\\ n = 4, - 3 \\\ \end{gathered}$
As n cannot be negative, n is 4. This means the transition occurred from ${n_1} = 1$ to ${n_2} = 4$.

Step 2 of 3:
Find $\Delta E$ for transition from ${n_1} = 1$ to ${n_2} = 4$ using eq (2):
$\begin{gathered} \Delta E = 13.6 \times {3^2}(\dfrac{1}{{{1^2}}} - \dfrac{1}{{{4^2}}}) \\\ \Delta E = 13.6 \times 9 \times \dfrac{{15}}{{16}} \\\ \Delta E = 114.75eV \\\ \end{gathered}$

Step 3 of 3:
Find $\lambda$ by putting $hc = 1240eV.nm$ in eq (3):
$\begin{gathered} 114.75eV = 1240eV.nm/\lambda \\\ \lambda = 10.8nm \\\ \end{gathered}$

Correct Answer:
The wavelength $\lambda$ which excited the atom: (c) 10.8 nm

Note: In questions like these first find the transition level. Then find the corresponding energy for this transition. From this energy, we can find the wavelength.