Question

For emission line of atomic hydrogen from \[{n_i} = 8\;\] to \[nf{^ - }\] the plot of wave number \[(v)\] against $(\dfrac{1}{{{n^2}}})$will be:
(The Rydberg constant, \[{R_H}\] ​ is in wavenumber unit):
A.Linear with slope \[ - {R_H}\]
B.Linear with intercept \[ - {R_H}\]
C.Non linear
D.Linear with slope \[{R_H}\]

Answer 1

Hint: To answer this question, you should recall the concept of spectral lines of a hydrogen atom. Bohr’s theory and the various transition series when an excited electron comes back to the lower energy state. Once the electrons in the gas are excited, they make transitions between the energy levels.
Formula used: \[\dfrac{1}{\lambda } = {\text{R}}\left( {\dfrac{1}{{{{\text{n}}_{\text{1}}}^{\text{2}}}} - \dfrac{1}{{{{\text{n}}_{\text{2}}}^{\text{2}}}}} \right){\text{c}}{{\text{m}}^{{\text{ - 1}}}}\]where \[R\] is Rydberg constant, \[n\]is the shell and \[\dfrac{1}{\lambda }\] is the wave number

Complete step by step answer:
Emission spectrum refers to the radiations emitted. This spectrum of radiation emitted by electrons in the excited atoms or molecules is known as an emission spectrum. The emission spectrum of atoms in the gas phase do not exhibit a continuous spread of wavelength from one colour to others. Rather, the emitted light is made up of a specific wavelength having dark spaces existing between them. This form of spectra is known as atomic spectra or line spectra.
Substituting the known values of shell and substituting in the formula we have:
\[\dfrac{1}{\lambda } = {\text{R}}\left( {\dfrac{1}{{{{\text{n}}_{\text{1}}}^2}} - \dfrac{1}{{{8^2}}}} \right){\text{c}}{{\text{m}}^{{\text{ - 1}}}}\].
After solving this equation \[{\text{v}} = \dfrac{{\text{R}}}{{{{\text{n}}_{\text{1}}}^{\text{2}}}} - \dfrac{{\text{R}}}{{{8^2}}}\].
Comparing this equation with the standard equation of straight line: \[{\text{v}} = {\text{m}}\].

Hence, the correct option is option D.

Note:
You should know remember the spectral lines in the following electron energy changes:
-The transition of electron from the first shell to any other shell – Lyman series
-The transition of electron from the second shell to any other shell – Balmer series
-The transition of electron from the third shell to any other shell – Paschen series
-The transition of electron from the fourth shell to any other shell – Bracket series
-The transition of electron from the fifth shell to any other shell – Pfund series