Question

Specify the formula for calculating the number of spectral lines in a hydrogen atom. Identify the number of spectral lines seen for a hydrogen atom when an electron jumps from \[{n_2} = 5\] and \[{n_1} = 1\] in a visible region.

Answer 1

Hint: The emission spectrum of atomic hydrogen has been divided into a number of spectral series, with wavelengths given by the Rydberg formula. These spectral series are Lyman series, Balmer series, Paschen series, Brackett series and Pfund series. Spectral emission occurs when an electron transitions, or jumps, from a higher energy state to a lower energy state.

Complete answer:
The formula to calculate the number of spectral lines in a hydrogen atom is given by:
Number of spectral lines \[ = \dfrac{{({n_2} - {n_1})({n_2} - {n_1} - 1)}}{2}\]
Where, \[{n_2} \to \] the higher orbit from which the electron jumps
     \[{n_1} \to \] the lower orbit to which the electron jumps
In the question, we are given \[{n_2} = 5\] and \[{n_1} = 1\].
Therefore, the total number of spectral lines \[ = \dfrac{{(5 - 1)(5 - 1 + 1)}}{2} = \dfrac{{4 \times 5}}{2} = 10\]
Hence, there are ten possible spectral lines that will be produced when an electron jumps from \[{n_2} = 5\] to \[{n_1} = 1\]. These ten spectral lines are:
\[5 \to 4,5 \to 3,5 \to 2,5 \to 1,4 \to 3,4 \to 2,4 \to 1,3 \to 2,3 \to 1,2 \to 1\]
In the hydrogen spectrum, when an electron jumps from any upper orbit to a lower orbit in a visible region, then that spectrum is called the Balmer series. The visible lines are obtained only when the electron jumps from a higher orbit to the second stationary orbit \[(n = 2)\].
From the above ten spectral lines, only \[5 \to 2,4 \to 2,3 \to 2\] are the spectral lines that fall in the visible region.
Hence, there will be three spectral lines seen for hydrogen atoms when an electron jumps from \[{n_2} = 5\] and \[{n_1} = 1\] in a visible region.

Note:
Lyman series lies in the ultraviolet band of spectrum while Paschen series and Brackett series lie in the infrared and far infrared band respectively of light spectrum. The Pfund series also lies in the far infrared spectrum. When the energies of each spectrum line is expressed by vertical lines on a proper scale, the diagram obtained is called the energy level diagram.