The Bohr Model of the Hydrogen Atom
I. There is a state of the lowest energy –
the ground state – which represents the normal
condition of the electron as it orbits the nucleus.
II. An atom can be excited when an electron is
at greater distance from the nucleus than in
the ground state. Energy of the excited state
is higher than that of the ground one.
III. The electron can exist only in
certain sharply defined energy states,
called orbitals.
IV. There is a maximum energy that the electron
can have and still be bound to the nucleus.
V. An atom is not radiating energy in the form
of light while the electron is on one of its
orbitals.
VI. Atom radiates energy only when the
electron makes a transition from one orbital
to a lower one. The radiation energy is the
difference between the energies of the
orbitals:
Erad=Ek-Ej
Examples:
1. Hydrogen atoms in states of high quantum
number have been created in a lab. (a) Find the
quantum number of the Bohr orbit in a
hydrogen atom whose radius is 0.01 mm. (b)
What is the energy of a hydrogen atom in this
state?
Solution:
5 m
r
1
.
0

10
n
 435
(a) n  a 

11
5.2910
0
E1 13.6eV
5 eV
(b) En  2 


7
.
19

10
n
(435)2
2. Find the longest wavelength present in the
Balmer series of hydrogen, corresponding to the
H line.
In the Balmer series the quantum number of the
final state is nf=2.
The longest wavelength
corresponds to the smallest energy difference
between energy levels. Hence the initial state must
be ni=3. Then
Solution:
1  R( 1  1 )  R( 1  1 )  0.139R

n2f ni2
22 32

1  6.56107 m;
0.139R
This wavelength is near the red end of the visible
spectrum.
Examples:
Find the de Broglie wavelength of (a) a 46-g golf
ball with a velocity of 30 m/s, and (b) an electron
with a velocity of 107 m/s.
Solution:
Both speeds are nonrelativistic: v«c. Hence:
34 J  s
h
6
.
63

10
 4.81034 m
(a)   mv 
(0.046kg)(30m / s)
The wavelength of the golf ball is so small
compared with its dimensions that we would not
expect to find any wave aspect in its behavior.
6.631034 J  s
11m

7
.
3

10
(9.11031kg)(107 m / s)
The dimensions of atoms are comparable with this
figure—the radius of the hydrogen atom, for
instance, is 5.31011 m. Thus, we need the wave
properties of a moving electron to understand
atomic structure and behavior.
h 
(b)   mv