Question

In a collection of ${{H}}$ atom, electrons make transition from ${{{5}}^{{{th}}}}$ excited state to ${{{5}}^{{{th}}}}$ excited state, then the maximum number of different types of photons observed are;
A. ${{3}}$
B. ${{4}}$
C. ${{6}}$
D. ${{15}}$

Answer 1

${{{n}}^{{{th}}}}$excited state is equal to the ${{(n + 1)}}$ transition line, first excited state can be considered as ${{{n}}_{{2}}}$ and second excited state can be considered as ${{{n}}_{{1}}}$.Substituted these values in number of photons lin formula we get maximum observed photons in the spectrum.

Complete step by step answer:
Maximum number of lines can be calculated by the following formula.
${\text{Number of lines = }}\dfrac{{{{(}}{{{n}}_{{2}}}{{ - }}{{{n}}_{{1}}}{{)(}}{{{n}}_{{2}}}{{ - }}{{{n}}_{{1}}}{{ + 1)}}}}{{{2}}}$……………………………..(1)
From the given,
${{{5}}^{{{th}}}}$ excited state = ${6^{{{th}}}}$transition line = ${{{n}}_{{2}}}$
${{{2}}^{{{nd}}}}$excited state = ${{{3}}^{{{rd}}}}$ transition line = ${{{n}}_{{1}}}$
Substitute these values in equation (1)

$$\Rightarrow \dfrac{{(6 - 3)(6 - 3 + 1)}}{2} \\\ \Rightarrow \dfrac{{(3)(4)}}{2} \\\ \Rightarrow (3) \times (2) \\\ \Rightarrow 6 \\\$$

So, the maximum number of lines of different photons will be observed six.
The number of emissions lines are as follows.
$6 \to 3,6 \to 4,6 \to 5,5 \to 3,5 \to 4,4 \to 3$
Hence, the correct option is (C).

Additional Information:
For this type of question, one should remember that they are asking the type of photons observed. If they ask about spectral lines when electron jump from ${{{n}}^{{{th}}}}$ level to the ground state then we have to use the formula of spectral lines from transition to ground state.
$\dfrac{{{{{N}}_{{2}}}{{(}}{{{N}}_{{2}}}{{ - 1)}}}}{{{2}}}$
In hydrogen atom absorption spectra is obtained by giving energy to ${{H}}$- atom and passing the released light energy through prism for obtaining spectra.

Note:
When an electron jumps from higher energy orbit to lower energy orbit, it emits radiation with various spectral lines. In a ${{H}}$-atom when an electron makes a transition from an excited state to the ground state its kinetic energy increases while potential and total energy decreases.