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Using the concept of electron transitions, Bohr was able to reproduce
Rydberg’s (Balmer’s) equation:
Energy of initial energy level = Ei
Energy of final energy level = Ef
RH
RH
Ei =  
Ef =  
2
ni
nf2
ni = principal quantum number
for the initial energy level
nf = principal quantum number
for the final energy level
RH
RH
Energy of the emitted photon = h = Ei  Ef = ( )  ( )
ni2
nf2
RH
h = 
nf2
Recall:
RH
1
1
  = RH (    )
ni2
nf2
ni2
c=
c
= 

and therefore
c
1
1
1
RH
1
1
h  = RH (   ) ; By rearangement:  =  (    )

nf2
ni2

hc
nf2
ni2
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1
RH
1
1
 =  (    )

hc
nf2
ni2
By substituting:
RH = 2.179 x 1018 J
h = 6.63 x 1034 J.s
c = 3.00 x108 m/s
RYDBERG’S GENERALIZED
EQUATION IS OBTAINED.
1
1
1
 = 1.097 x 107    m 1

nf2
ni2
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1.
SAMPLE PROBLEMS
An electron in a hydrogen atom in the level n = 5 undergoes a
transition to level n = 3.
What is the frequency of the emitted radiation ?
Energy of the emitted photon = h  = Ei  Ef = E5 – E3
 RH
 RH
E5 =  = 
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 RH
E5 – E3 = 
25
 RH
 
9
 RH
 RH
E3 =  = 
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9
25 RH - 9 RH
16 RH
=  = 
225
225
16 RH
E5 – E3 = h  = 
225
16 RH
 = 
225 h
16 RH
16
2.179 x 10-18 J
 =  =  x  = 2.34 x 1014 s1
225 h
225
6.63 x 1034 J . s
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2.
What is the difference in energy between the two levels responsible
for the violet emission line of the calcium atom at 422.7 nm ?
1m
? m = 422.7 nm x  = 4.227 x 107 m
109 nm
c
3.00 x 108 m/s
 =  =  = 7.097 x 1014 s1

4.227 x 107 m
E = h  = (6.63 x 1034 J . s) (7.097 x1014 s1 ) = 4.71 x 1019 J
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QUANTUM MECHANICS
is a theory that applies to extremely small particles, such as electrons.
DUAL NATURE OF MATTER
Einstein: postulated that light has a dual nature:
Wave Properties
characterized by:
frequency and wavelength
c=
Particle Properties
a particle of light, called a photon has:
Energy = E = h 
and
Momentum = mass x speed = m c
Louis de Broglie reasoned:
- If Light
(traditionally
considered
a Wave)
exhibits Particle Properties
then
- Matter
(traditionally
considered
made of Particles)
exhibits Wave Properties
This implies, that for a particle of matter:
ENERGY of Particle of Matter = ENERGY of Wave of Matter
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Eparticle of matter
=
Ewave of matter
Eparticle = mc2 (Einstein)
Ewave
It follows:
mc2
=
hc


For Light Particles
h
 = 
mc
hc
=h = 

(Planck)
h
mc = 

For any kind of particles
h
 = 
mv
speed of light
speed of particles
NOTE:
1. Wave properties of common forms of matter are not observed because
their relatively large mass results in a very short wavelengths, which
cannot be detected. (in the range of 1034 m)
2. Electrons, with a very small mass produce longer wavelengths which
can be detected (in the range of 109)
CONCLUSION: THE ELECTRON HAS DUAL NATURE:
The electron has both particle and wave properties
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SAMPLE PROBLEMS
1.
What is the de Broglie wavelength of a 145-g baseball travelling at
30.0 m/s ?
m = 145 g
v = 30.0 m/s
=?
h
(6.63 x1034 J . s)
 =  = 
mv
m
145 g) (30.0 
s
Note that some units do not cancel!
Recall:
kg . m2
J = 
s
This requires that the mass be expressed in kg
=
kg . m2
6.63 x1034 
s

m
0.145 kg) (30.0 
s
=
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1.52 x 10 34 m
cannot be detected! (too short)
2. At what speed must an electron travel to have a wavelength of
1.00 x 1011 m ?
me = 9.11 x 1031kg
h
 = 
mv
kg . m2
h = 6.63x1034 
s
h
v = 
m
 = 1.00 x 1011 m
v=?
kg . m2
6.63x1034 
s
v = 
(9.11 x 1031 kg)( 1.00 x 1011 m )
V = 7.27 x 107 m/s
(about 4 times slower than the speed of
light)
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WAVE FUNCTIONS
Erwin Schrodinger (1926)
1. Based on de Broglie’s work devised a theory that could be used to find the wave
properties of electrons
2. Established the basis of quantum mechanics (the branch of physics that mathematically
describes the wave properties of submicroscopic particles)
Motion is viewed differently by Classical Mechanics and by Quantum Mechanics;
Motion in Classical Mechanics:
Motion in Quantum Mechanics:
(for example: the path of a thrown ball) (for example: the motion of an electron in an atom)
the path of the ball is given by
- the electron is moving so fast and it has such a
its position and its velocity at
small mass, that its path cannot be predicted
various times
we think of the ball as moving
along a continuous path
Heisenberg’s Uncertainty Principle states that for particles of very small mass and moving at
high speeds, it is impossible to predict: - the exact location of the particle at any particular time,
- the direction in which the particle is moving
In Bohr’s theory, the electron was thought of as orbiting around the nucleus, in the way the
earth orbits the sun.
Quantum Mechanics completely invalidates this view of the motion of the electron
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MOTION OF THE ELECTRON AS VIEWED BY QUANTUM MECHANICS
1. We cannot describe the electron in an atom as moving in a definite orbit.
2.
We can obtain the probability of finding the electron at a certain point in a H atom;
we can say that the electron is likely (or not likely) to be at this position.
3.
Information about the probability of finding the electron at a certain point is given by a
mathematical expression called a wave function.
4.
The wave function indicates that the probability of finding the electron at a certain position
is high at some distance away from the nucleus
5.
A wave function for an electron in an atom is called an atomic orbital.
CONCLUSION
AN ATOMIC ORBITAL IS A REGION IN SPACE WHERE THE PROBABILITY OF
FINDING THE ELECTRON IS HIGH.
(a region in space where the electron is most likely to be found)
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