Question

The number of spectral lines that are possible when electrons in 7th shell in different hydrogen atoms return to the second shell is:
$
  A)12 \\
  B)15 \\
  C)14 \\
  D)10 \\
 $

Answer 1

Hint: We can define a spectral line as a light or dark line in a uniform and continuous spectra that results from light emission (or) light absorption in a frequency range compared with the adjacent frequencies. A spectral line could be either an emission line (or) absorption line.
We can calculate the total number of spectral lines of hydrogen atoms from higher energy level orbital to the lower energy level orbital.

Complete step by step answer:
We can calculate the number of spectral lines using the formula,
Spectral lines=$\dfrac{{\left( {{n_2} - {n_1}} \right)\left( {{n_2} - {n_1} + 1} \right)}}{2}$
Here, ${n_1}$ is the lower energy level, and ${n_2}$ is the highest energy level.
Complete step by step answer:
We have to calculate the number of possible spectral lines, when an electron from 7th shell returns to the 2nd shell.
We can calculate this using the formula,
Spectral lines=$\dfrac{{\Delta n\left( {\Delta n + 1} \right)}}{2}\,\,\,\,\,\,\,\,\,\,\,\left( 1 \right)$
The value of $\Delta n$ is the difference between highest energy level and lowest energy level.
We can represent the highest energy level as ${n_2}$ and the lowest energy level as ${n_1}$.
Therefore, we can write the equation of $\Delta n$ as,
$\Delta n = {n_2} - {n_2}\,\,\,\,\,\,\,\,\left( 2 \right)$
We can now substitute $\Delta n$ in equation (1), and we will get the final equation as,
Spectral line=$\dfrac{{\left( {{n_2} - {n_2}} \right)\left( {{n_2} - {n_1} + 1} \right)}}{2}$
So now, we can find the number of spectral lines using the above formula.
We have to substitute the values of ${n_2}$ and ${n_1}$ in the final equation to calculate the number of spectral lines.
The value of lowest energy level ${n_1}$ is 2.
The value of the highest energy level ${n_2}$ is 7.
Let us now substitute the known values of ${n_2}$ and ${n_1}$ in the final equation.
Spectral lines=$\dfrac{{\left( {{n_2} - {n_2}} \right)\left( {{n_2} - {n_1} + 1} \right)}}{2}$
Spectral lines=$\dfrac{{\left( {7 - 2} \right)\left( {7 - 2 + 1} \right)}}{2}$
Spectral lines=$\dfrac{{\left( 5 \right)\left( 6 \right)}}{2}$
Spectral lines=$\dfrac{{30}}{2}$
Spectral lines=$15$
The number of spectral lines that are possible when electrons in the 7th shell in different hydrogen atoms return to the second shell is 15.
$\therefore $Option (B) is correct.

Note:
We can use spectral lines to recognize the atoms and molecules. The fingerprints could be compared to the before collected fingerprints of molecules and atoms. Thus, it would be useful to recognize the atomic and molecular components present in stars, and planets. Atoms such as thallium, helium and caesium were discovered by spectroscopic means. Spectral lines are atom-specific in nature.