Question

What is the wavelength of the \[{H_\beta }\] line of the Balmer series of hydrogen spectrum? (R= Rydberg constant)
(A) \[\dfrac{{36}}{{5R}}\]
(B) \[\dfrac{{5R}}{{36}}\]
(C) \[\dfrac{{3R}}{{16}}\]
(D) \[\dfrac{{16}}{{3R}}\]

Answer 1

Hint: Wavelength of the \[{H_\beta }\] line of the Balmer series of hydrogen spectrum can be calculated by using Rydberg formula. The Rydberg formula is used to determine the wavelength of the moving electrons. The Rydberg formula for the Balmer series is given below:
\[\dfrac{1}{\lambda } = R\left( {\dfrac{1}{{{2^2}}} - \dfrac{1}{{{n^2}}}} \right)\]
where n=3, 4, 5….

Complete Solution :
Emission spectrum of the hydrogen atom has been divided into several numbers of spectral lines. This is called Hydrogen spectral lines. The spectral lines are observed due to the transition between two energy levels in an atom, i.e. from higher energy level to the lower energy level.
- When the spectral emission of lines occurs when the electronic transition from the higher energy level to the lower energy level whose principal quantum number is 2 are referred to as the Balmer series.
- In order to find the wavelength of the \[{H_\beta }\] line of the Balmer series of the hydrogen spectrum, we have to consider the Rydberg formula. Rydberg formula can be used to determine the wavelength of light which has resulted from the electron moving between the energy levels.

- Rydberg formula is given below:
\[\dfrac{1}{\lambda } = R\left( {\dfrac{1}{{{2^2}}} - \dfrac{1}{{{n^2}}}} \right)\]
Where
\[R\]= Rydberg constant
n = 4
\[\dfrac{1}{\lambda } = R\left( {\dfrac{1}{{{2^2}}} - \dfrac{1}{{{4^2}}}} \right)\]
\[\dfrac{1}{\lambda } = R\left( {\dfrac{1}{4} - \dfrac{1}{{16}}} \right)\]
\[\dfrac{1}{\lambda } = R\left( {\dfrac{4}{{16}} - \dfrac{1}{{16}}} \right) = R\left( {\dfrac{3}{{16}}} \right)\]
\[\lambda = \dfrac{{16}}{{3R}}\]
Therefore, the wavelength of the \[{H_\beta }\] line of the Balmer series of the hydrogen spectrum is \[\dfrac{{16}}{{3R}}\].
So, the correct answer is “Option D”. \[\dfrac{{16}}{{3R}}\]

Additional information:
Let us now see what an emission spectrum is. When an electromagnetic radiation is passed through a substance, that time the radiation will interact with the electrons present in the substance. These electrons on interaction will absorb some energy and will get excited to a higher energy stable. This electron will become unstable. In order to regain its stability, the electron will jump back to the lower energy state with emission of the radiation. This spectrum of emission of radiation is called emission spectrum.

Note: The hydrogen atoms emission spectrum consists of several spectral series other than Balmer series namely
- Lyman series (n = 1 )
- Paschen series (n = 3 )
- Brackett series (n = 4 )
- Pfund series (n = 5 )
- Humphreys series (n = 6 )