Atomic Structure 135
Example 4.2
An electron collides with a hydrogen atom in its ground state and excites it to a state of n
3. How much energy was given to the hydrogen atom in this inelastic (KE not conserved)
collision?
Solution
From Eq. (4.15) the energy change of a hydrogen atom that goes from an initial state of quan-
tum number nito a final state of quantum number nfisEEfEiE (^1)
Here ni1, nf3, and E 1 13.6 eV, so
E13.6 eV12.1 eV
Example 4.3
Hydrogen atoms in states of high quantum number have been created in the laboratory and
observed in space. They are called Rydberg atoms.(a) Find the quantum number of the Bohr
orbit in a hydrogen atom whose radius is 0.0100 mm. (b) What is the energy of a hydrogen
atom in this state?
Solution
(a) From Eq. (4.14) with rn1.00 10 ^5 m,
n 435
(b) From Eq. (4.15),
En7.19 10 ^5 eV
Rydberg atoms are obviously extremely fragile and are easily ionized, which is why they are
found in nature only in the near-vacuum of space. The spectra of Rydberg atoms range down
to radio frequencies and their existence was established from radio telescope data.
Origin of Line Spectra
We must now confront the equations developed above with experiment. An especially
striking observation is that atoms exhibit line spectra in both emission and absorption.
Do such spectra follow from our model?
The presence of discrete energy levels in the hydrogen atom suggests the connec-
tion. Let us suppose that when an electron in an excited state drops to a lower state,
the lost energy is emitted as a single photon of light. According to our model, elec-
trons cannot exist in an atom except in certain specific energy levels. The jump of an
electron from one level to another, with the difference in energy between the levels
being given off all at once in a photon rather than in some more gradual manner, fits
in well with this model.
13.6 eV
(435)^2
E 1
n^2
1.00 10 ^5 m
5.29 10 ^11 m
rn
a 0
1
12
1
32
1
n^2 i
1
n^2 f
E 1
n^2 i
E 1
n^2 f
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