Physics 6C
Energy Levels
Bohr Model of the Atom
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Bohr’s Model of Hydrogen
•Positively charged nucleus at center
•Electrons occupy circular orbits around the nucleus
•Angular momentum of electrons is quantized (has to be a multiple of ħ)
•Radius values and energy levels are also quantized
This is the formula for calculating hydrogen energy levels.
The ‘Ground State’ is n=1.
This is the closest orbit and the lowest energy level.
e-
E 1   13 . 6 eV
En 
rn
E1
n
2
+
For larger values of n, the electron is farther
from the nucleus, and the energy is closer to
zero. The electron is ‘free’ when its energy is
zero (or positive).
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Example: Suppose a hydrogen atom is in its ground state. If a photon is absorbed, exciting
the atom to the n=3 state, what is the wavelength of the photon?
Prepared by Vince Zaccone
For Campus Learning
Assistance Services at UCSB
Example: Suppose a hydrogen atom is in its ground state. If a photon is absorbed, exciting
the atom to the n=3 state, what is the wavelength of the photon?
The diagram shows the photon approaching,
and the electron jumping to the excited state
after the photon’s energy is absorbed.
n=3
n=1
+
e-
e-
photon
Prepared by Vince Zaccone
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Example: Suppose a hydrogen atom is in its ground state. If a photon is absorbed, exciting
the atom to the n=3 state, what is the wavelength of the photon?
The diagram shows the photon approaching,
and the electron jumping to the excited state
after the photon’s energy is absorbed.
n=3
The photon energy is the difference between
the energy levels.
n=1
E photon  E 3  E 1
+
E3 
 13 . 6 eV
3
2
e-
e-
photon
  1 . 5 eV
E photon  (  1 . 5 eV )  (  13 . 6 eV )  12 . 1 eV
Prepared by Vince Zaccone
For Campus Learning
Assistance Services at UCSB
Example: Suppose a hydrogen atom is in its ground state. If a photon is absorbed, exciting
the atom to the n=3 state, what is the wavelength of the photon?
The diagram shows the photon approaching,
and the electron jumping to the excited state
after the photon’s energy is absorbed.
n=3
The photon energy is the difference between
the energy levels.
n=1
E photon  E 3  E 1
+
E3 
 13 . 6 eV
3
2
e-
e-
photon
  1 . 5 eV
E photon  (  1 . 5 eV )  (  13 . 6 eV )  12 . 1 eV
Now we can solve for the wavelength:
E photon 
hc
12 . 1 eV 

( 4 . 14  10
 15
eV  s )( 3  10
8 m
s
)

  103 nm
Note that this photon is in the ultraviolet range
Prepared by Vince Zaccone
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Example: Suppose a hydrogen atom is in its ground state. If a photon is absorbed, exciting
the atom to the n=3 state, what is the wavelength of the photon?
Now suppose the atom emits a photon and
drops down to the n=2 state. What is the
wavelength of this emitted photon?
Again, the photon energy is the
difference between the energy levels.
n=3
n=2
n=1
ee-
+
Emitted
photon
Prepared by Vince Zaccone
For Campus Learning
Assistance Services at UCSB
Example: Suppose a hydrogen atom is in its ground state. If a photon is absorbed, exciting
the atom to the n=3 state, what is the wavelength of the photon?
Now suppose the atom emits a photon and
drops down to the n=2 state. What is the
wavelength of this emitted photon?
Again, the photon energy is the
difference between the energy levels.
n=3
n=2
n=1
ee-
+
Emitted
photon
E photon  E 2  E 3
E2 
 13 . 6 eV
2
2
  3 . 4 eV
E photon  (  3 . 4 eV )  (  1 . 5 eV )  1 . 9 eV
Prepared by Vince Zaccone
For Campus Learning
Assistance Services at UCSB
Example: Suppose a hydrogen atom is in its ground state. If a photon is absorbed, exciting
the atom to the n=3 state, what is the wavelength of the photon?
Now suppose the atom emits a photon and
drops down to the n=2 state. What is the
wavelength of this emitted photon?
Again, the photon energy is the
difference between the energy levels.
n=3
n=2
n=1
ee-
+
Emitted
photon
E photon  E 2  E 3
E2 
 13 . 6 eV
2
2
  3 . 4 eV
E photon  (  3 . 4 eV )  (  1 . 5 eV )  1 . 9 eV
Now we can solve for the wavelength:
E photon 
1 . 9 eV 
hc

( 4 . 14  10
 15
eV  s )( 3  10
8 m
)
s

  654 nm
This one is visible, so we would see this in an emission spectrum.
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