A hydrogen atom in state n = 6 makes two successive transitions and reaches the ground state. In the first transition a photon of 1.13 eV is emitted. (a) Find the energy of the photon emitted in the second transition. (b) What is the value of n in the intermediate state?

Hint:We find the energy difference between the successive transitions to evaluate the energy of the intermediate state. By using the energy formula for the nth energy state in a hydrogen like atom we can determine the state of the intermediate state of the transition.

Formula used:
\[{E_n} = \dfrac{{ - 13.6eV}}{{{n^2}}}\]
Where \[{E_n}\] is the energy of the nth state.

Complete step by step solution:
(a) It is given that the electron is initially in state \[n = 6\]. The electron makes two successive transitions to reach the ground state. In the first transition the energy released is 1.13 eV. As we know, the energy of the ground state in a hydrogen atom is -13.6 eV. Let the energy of the state between the ground state and the state \[n = 6\] is \[{E_n}\]. The energy of the state \[n = 6\] will be,
\[{E_6} = \dfrac{{ - 13.6eV}}{{{{\left( 6 \right)}^2}}} \\ \]
\[\Rightarrow {E_6} = - 0.378\,eV\]

The energy released in the first transition is given as 1.13 eV
\[1.13eV = {E_6} - {E_n}\]
\[\Rightarrow {E_n} = \left( { - 0.378eV} \right) - 1.13\,eV\]
\[\Rightarrow {E_n} = - 1.508\,eV\]
So, the energy released in the second transition is,
\[\Delta {E_2} = {E_n} - {E_0}\]
\[\Rightarrow \Delta {E_2} = - 1.508eV - \left( { - 13.6eV} \right)\]
\[\Rightarrow \Delta {E_2} = 12.1eV\]

Therefore, the energy released in the second transition is 12.1 eV.

(b) The energy of the intermediate state is -1.508 eV. The energy of nth state is given as,
\[{E_n} = \dfrac{{ - 13.6eV}}{{{n^2}}}\]
Putting the values, we get
\[ - 1.508\,eV = \dfrac{{ - 13.6eV}}{{{n^2}}} \\ \]
\[\Rightarrow {n^2} = \dfrac{{ - 13.6eV}}{{ - 1.508eV}} \\ \]
\[\therefore n = \sqrt {9.018} \]
As the energy state is represented with the whole number, so \[n = 3\].


Therefore, the value of n for the intermediate state is 3.

Note: From the emission spectrum of atomic hydrogen, numerous spectral series' wavelengths have been calculated using the Rydberg formula. The electron transitions between these observed spectral lines' two energy levels occur inside of atoms. According to the principle of energy conservation, the radiation's energy should be equal to the energy disparity between the levels of energy.