Question

(a) (5 pts.) What are the three lowest energies of the singly ionized He-atom according to the Bohr model? (b) (5 pts.) Calculate the energies of the photons emitted when electronic transitions take place between all possible states.

The excited levels of Hydrogen have lifetimes of order 10^-8 s. In very highly excited states (large n), the states get closer and closer together.   At what value of n do the spacings of the energy levels become comparable to the energy uncertainty imposed by the Heisenberg uncertainty principle, so that individual levels can no longer be resolved?

Answer 1

Lowest Energies of a singly inonized He - atom according to Bohr's Model :-

Lets use the Bohr's postulates to calculate the same :-

Assuming the nucleus has a positive charge Ze. For a He+ ion, Z = 2.

The expression of En = -RhcZ^2/n^2 ............................(1)

R = 1.0973 x 10^7 m-1

Substituting for the constants in eqn 1 we get :-

Energy in the first excited state = E1 = -13.6 eV.

Now we have the expression En = E1/n^2 ...........................(2) [ Energy in the nth state ]

Therefore energy in the second state = E2 = -13.6/4 = -3.4 eV

Energy in the third state = E3 = E1/9 = -13.6/9 = -1.5 eV

Therefore the three lowest energies according to Bohr's models are :-

1.) -13.6 eV

2.) - 3.4 eV

3.) -1.5 eV

Energies of photons between all possible state :-

We have E1 = -13.6 eV

E2 = -3.4 eV

E3 = -1.5 eV

E4 = -0.85 eV

E5 = -0.54 eV

E6 = -0.37 eV

E7 = -0.27 eV

E8 = -0.21 eV

E9 = -0.16 eV

Possible states - Energy emitted by photons during electronic transitions

Energy emitted (state 1 to 2) = E2 - E1 = 13.6 - 3.4 eV = 10.2 eV

Energy emitted (state 2 to 3) = E3 - E2 = 3.4 - 1.5 eV = 1.9 eV

Energy emitted (state 3 to 4) = E4 - E3 = 1.5 - 0.85 eV = 0.65 eV

Energy emitted (state 4 to 5) = E5 - E4 = 0.85 - 0.54 eV = 0.31 eV

Energy emitted (state 5 to 6) = E6 - E5 = 0.54 - 0.37 eV = 0.14 eV

Energy emitted (state 6 to 7) = E7 - E6 = 0.37 - 0.27 eV = 0.10 eV

Energy emitted (state 7 to 8) = E8 - E7 = 0.27 - 0.21 eV = 0.06 eV

Energy emitted (state 8 to 9) = E9 - E8 = 0.21 - 0.16 eV = 0.05 eV