Question

A hydrogen atom in the ground state is excited by an electron beam of $ 12.5eV $ energy. Find out the maximum number of lines emitted by the atoms from its excited state.

Answer 1

Hint: We can find the energy of a hydrogen atom in the ground state using Bohr’s equation. According to Bohr’s model electrons are revolving around a specific orbit or shells with fixed radius. Electrons would absorb energy and get excited to the higher energy state. The excited electron is less stable so they emit a photon and come back to a lower stable energy level.

Complete step by step solution:
Energy in the ground state = $ - \dfrac{{13.6}}{{{n^2}}} $
 $ \Rightarrow {E_1} = - 13.6eV $
 $ \Rightarrow {E_n} = - 13.6 + 12.5 $
 $ \Rightarrow {E_n} = - 1.5eV $
 $ {E_n} = \dfrac{{{E_1}}}{{{n^2}}} $
 $ \Rightarrow - 1.5 = \dfrac{{ - 13.6}}{{{n^2}}} $
 $ \Rightarrow {n^2} = \dfrac{{13.6}}{{1.5}} $
 $ \Rightarrow n = 3 $
For the electron to transit from ground state to excited state $ \Delta n = 3 - 1 $
 $ \Delta n = 2 $
Maximum number of lines emitted by the lines $ N = \sum {\Delta n} $
 $ N = 3 $ .

Additional Information:
Limitations of Bohr’s model are,
-It didn’t work well for atoms which were complex.
-It couldn’t give an explanation about the difference in the intensity of spectral lines.
-Failed to give the explanation of splitting of spectral lines in the presence of magnetic fields.
-Bohr’s idea of electrons is contradicted by Heisenberg’s uncertainty principle.
Another model of atom is Rutherford’s model which states that
-Atoms have a lot of empty space inside it since most of the alpha particles passed without deviation.
-Since some of the alpha particles were deviated by an angle of $ {180^ \circ } $ the entire positive charge and mass of the atom is confined to the centre. They called it a nucleus.

Notes
Some phenomena like interference, diffraction is explained on the basis of electromagnetic wave nature only, where interaction of radiation takes place within itself whereas photoelectric effect, Compton effect can be explained on the basis of quantum nature of radiation because the interaction takes place with matter. Some other phenomena like reflection and refraction can be explained on the basis of either nature of radiation because neither of the interaction takes place here.