Question

Selection rules and allowed transitions: Electron in a hydrogen atom is in 4p state. The energy level diagram is shown below. Note the energies are given to first decimal place only. 

(a) Draw arrows showing all the allowed transitions the 4p electron can make that lead to the emission of photon. Include all the transitions until the electron reaches the ground state. 

(b) What are the kinetic and potential energies of the electron in the 4p state? 

(c) Calculate the wavelengths (in nm) of the photons associated with the transitions in (a). 

(d) What is the ionization energy of the 4p state? 

(e) What is the degeneracy of the 4p state, including the intrinsic spin of the electron?

Answer 1

(a) Selection rules give us the allowed transitions of electrons in an atom. For the case of a hydrogen atom, an allowed transition must satisfy the relation  where l is the angular momentum of the electron. In our case, the electron is initially in the 4p state, where  l=1. From this state, the allowed transitions are to  = 0 (3s) and =2 (3d). From these two states, the only allowed transition is to l=1 (2p). From 2p, the only transition possible is to the ground state 1s. All these transitions are shown below in the diagram:

(b) The total energy of the electron in the 4p state is given by

In the hydrogen atom, the total energy, potential energy and kinetic energy satisfy:

  E = T + V = V/2

Thus, the potential energy is twice of the total energy, V = -1.7 eV, and the kinetic energy, T = 0.85 eV.

(c) The energy differences will determine the energy of the photon, and hence the wavelength of the photon. We have

Using the above formula, we immediately find that the photon has a wavelength of 1874.6 nm for the 4p --> 3s or  4p --> 3d transition, 656.1 nm for the 3s --> 2p or 3d --> 2p transition and 121.5 nm for the 2p --> 1s transition.

(d) The ionisation energy for the 4p state is 0.85 eV, which is the minimum energy required to remove the electron.

(e) In the 4p state, the angular momentum quantum number can take the values -1,0,1 (corresponding to px, py and pz orbitals). And the spin quantum number can be +1/2 or -1/2. So the total degeneracy in the 4p state is 6.