Bohr Atom
I.
Bohr Atom
A.
Bohr's Postulates
1.
An electron moves in a circular orbit around the nucleus due to the
Coulomb force.

Z
Z ee
F   k 2 r̂   - ke 2  r̂

 r2
r
2.
Electrons can only exist in orbits where the magnitude of its angular
momentum is a positive integer of Planck's constant divided by 2.
L  mvr  n
3.
The electrons do not radiate electromagnetic energy while they travel in
their circular orbits. Thus, each electron orbit is a state of constant energy.
Violates Classical Electromagnetism
4.
Electromagnetic radiation is emitted (absorbed) when an electron jumps
from one energy state to a lower (higher) energy state. The energy of the
emitted (absorbed) photon is equal to the energy lost by the electron.
 E
 i
 E f   h ν  hc

λ
Bohr's postulate are strange and difficult to accept. However, it is more amazing
that with just these four postulates Bohr was able to reproduce the Ritz-Rydberg
equation and determine the size of the hydrogen atom!
B.
Size of the Atom
We will now use classical physics and Bohr's postulates to find the size of an
atom. We will assume that the nucleus is stationary. To correct for the finite mass
of the nucleus, you simply replace the mass of the electron by the reduced mass of
the electron-nucleus system.
r
Ze
The electron is in uniform circular motion due to the Coulomb force between the
electron and the nucleus.
2




Z
e
e
m
v
Fk

r
r2
r
m v2 r 2
k Z e 2 


2


m
v
r
r
 ke 2  Z m




2 2
L
c
r
 ke 2  Z m c 2






2 2 2
h c2
n

c
n 
r
 
 ke 2  Z E
 Z  4 π 2  ke 2  E



o 
o
2




For hydrogen (Z=1) ground state (n = 1), we have
ao 

1240 eV  nm 2

 0.0529 nm
2
2
2
5


4 π  ke  E o 4 π 1.44 eV  nm  5.11x 10 eV 


h c2


Thus, the nth orbit of an atom of atomic number Z is given by
rn 
C.
n2 ao
Z
Energy Levels
We can now us our previous results to calculate the energy levels of the atom. The
electron has rotational kinetic energy and electric potential energy.
2
k Ze2
L
EK U 
r
2I
Using the moment of inertia for a point charge, we have
L2  k Z e 2
r
2mr2
E
Using
Z k e2
 m v 2  2 K that we found in the previous section, we have
r
2
2
2
E  L  m v2  L  L
2mr2
2mr2 mr2
2
E  L
2mr2
n2 2
n 2  2 Z2

2
2 m n 4a o 2
2 

a n 
o
E 
2m


Z
2



 Z  1
 

 n   2E
o

E   






 hc

 2πa
o






2
For the ground state of hydrogen (n = 1, Z=1), we have
EH 



2

1
x 5.11x 105 eV





 1240 eV  nm 




 2 π x 0.0529 nm 
2
 13.6 eV
Thus, the nth energy level of an atom with atomic number Z is given by
Z2 E H
En 
n2
II.
Ritz-Rydberg Formula
We can now use our results from the Bohr atom to develop the Ritz-Rydberg
formula. If an electron drops from the mth energy level in the hydrogen atom to
the lower energy nth level, it will emit a photon of energy E given by
E  hc    E n  E m 


λ
Substituting in the Bohr energy state relationship for the hydrogen atom, we have
hc   E  1  1 
H 2

λ
m2 
n
1   - E H   1  1 
λ  h c   n 2 m 2 
1   13.6 eV   1  1 
λ  1240 eV  nm   n 2 m 2 
1   0.019677nm  1   1  1 


λ 
  n 2 m 2 
1  1.9677m  7   1  1   R  1  1 


H 2
λ 
  n 2 m 2 
m 2 
n
For an an atom of atomic number Z, the Bohr atom predicts that the
spectral lines will be given by
1  Z 2 R  1  1 
H 2

λ
m2 
n
III,
Reduced Mass
Spectroscopic data is so precise that one must account for the finite mass of the
nucleus. In the hydrogen atom, the nucleus and electron orbit around their center
of mass and not around the center of nucleus since the nucleus has a finite mass.
However, we can convert this two body problem into a single body problem by
reducing the mass of the electron as shown below:

m
r
r
X C.M.
M
Two Bodies Rotating
About The Center of Mass
Equivalent One Body
Problem
Reduced Mass Formula
μ
mM
mM
Since the mass of the electron is much less than the mass of the nucleus, the
reduced mass is approximately the mass of the electron although slightly lower.
Furthermore, the reduced mass increases slightly for higher Z atoms which
changes the Rydberg constant for higher Z atoms.
Example:
A)
Using the Bohr theory do the following:
Compute the first four energy levels of hydrogen
4
3
2
1
B)
Compute the two longest wavelengths in the Balmer series
IV.
Franck and Hertz Experiment
A.
Experimental Setup
Vc
Mercury
Gas
A
+
Vacc
B.
Vr
Results
I
Vacc
Data indicates quantization of energy levels of mercury. This was further
confirmed by observing a photon emission from mercury that matched the energy
lost by the electrons!
Example:
In a Franck-Hertz type of experiment atomic hydrogen is bombarded with electrons, and
excitation potentials are found at 10.21 V and 12.10 V.
A.
Explain the observation that three different lines of spectral emission accompany
these excitations.
B.
Find the wavelengths of the three spectral lines observed during the experiment.
Example:
Assume the angular momentum of the earth of mass 6.0x1024 kg due to its motion around
the sun at radius 1.5x1011 m to be quantized according to Bohr's relationship.
A.
What is the value of the quantum number n?
B.
What is the fractional energy difference (E/E) between the orbital energy levels
of the earth?