Download Bohr Model of the Atom

School of Mathematical and Physical Sciences PHYS1220
PHYS1220 – Quantum Mechanics
Lecture 3
August 22, 2002
Dr J. Quinton
Office: PG 9
ph 49-21-7025
[email protected]
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School of Mathematical and Physical Sciences PHYS1220
Early Models of the Atom
Between 450 - 410 BC, Leucippus of Miletus (now Turkey) and
Democritus of Abdera (now Thrace, Greece) postulated that
matter was made of fundamental, indivisible units called atoms
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This fact was widely accepted by 1900
The periodic table was well underway at the time
Discovery of radioactivity in mid 1890s posed a problem in that if
particles smaller than the atom existed, perhaps they were not
indivisible at all
J.J. Thomson proposed the ‘plum-pudding’
model of the atom
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comprised a uniform positively charged
sphere, with negatively charged electrons
After J.J. discovered the electron in 1897
(Nobel Prize 1906), he amended his model
to include the idea that the electrons should
be moving
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School of Mathematical and Physical Sciences PHYS1220
Rutherford’s Experiments
"All science is either physics or stamp collecting”
Rutherford was the first to experiment with the radiation from
radioactivity named a, b and g particles
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He found that a and b particles were charged
Won the 1908 (Chemistry!) Nobel Prize
Rutherford (~1911) experiments with a particles and gold foil
The result
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Majority of the alpha particles went straight through
Occasional scattering (even backward) ”as if you fired a 15 inch shell
at a sheet of tissue paper and it came back and hit you.”
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School of Mathematical and Physical Sciences PHYS1220
Rutherford’s “Planetary” Model of the Atom
Since most alpha particles were undeflected, the atom
must be mostly empty space
The only way back scattering could occur is if there is
a concentrated positive charge at the nucleus
The nucleus must therefore contain more than 0.999
of the atom’s mass
Theorized a new model of the atom
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A tiny but massive, positively charged nucleus that is
surrounded by electrons that orbit some distance away.
The electrons must be moving, otherwise they would
fall into the nucleus due to Coulomb interaction
Measured closest distance between alpha particle and
nucleus from its KE which is converted to electric PE
q1q2
1 2
(2e)( Ze)
4kZe2
14
mv  k
k
d 

3.2x10
m
2
2
r
d
mv
Nuclear radius must be smaller than this
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School of Mathematical and Physical Sciences PHYS1220
Atomic Spectra
Heated solids emit a continuous spectrum of EM radiation
rarefied gases can also be excited to emit photons
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intense heating or dielectric breakdown (low pressure gases)
emitted spectra are comprised of discrete lines
emitted wavelengths are unique to each element
Balmer (1885) studied the visible lines of hydrogen
emission, showed that they are related by
 1 1 
 R  2  2  , n  3, 4,... Balmer Series

2 n 
1
where R = 1.0974 x 107 m-1 (determined experimentally) is called the
Rydberg constant and n=3,4,5,6 for visible lines at 656, 486, 434
and 410 nm respectively
Later found to extend into UV region ending at 365nm (the
lines get too close together to distinguish). This corresponds
with the n →  case
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School of Mathematical and Physical Sciences PHYS1220
Atomic Spectra II
Later experiments with hydrogen by others showed similar
emission line series existed in the UV and IR regions, with patterns
similar to the Balmer series. The formula also fitted, but ½2 needed
to be replaced by 1/12, 1/32 etc to suit the particular series
1 1 
 R  2  2  , n  2,3,... Lyman Series (UV)

1 n 
1
1 1 
 R  2  2  , n  4,5,... Paschen Series (IR)

3 n 
1
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School of Mathematical and Physical Sciences PHYS1220
Failings of Rutherford’s Model
Rutherford’s model of the atom was superior to Thomson’s plum
pudding model, but it did have a few shortcomings
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Unable to explain why atoms emit discrete line spectra
Orbiting electrons could have any energy based upon their orbital
motion and distance from the nucleus
Orbiting charges accelerate (centripetally). They should emit radiation
like any other accelerating charge, lose energy and spiral into the
nucleus
As they spiral inward, their orbital frequency increases, thus the
frequency of photons they emit should also increase. Therefore, they
should lose even more energy and decay even more rapidly
Basically, there are two major flaws with Rutherford’s atomic model
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Continuous emission spectra should be emitted
It predicts that atoms are unstable
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School of Mathematical and Physical Sciences PHYS1220
Bohr Model of the Atom
Neils Bohr thought that Rutherford’s model had merit, but needed
to include some of the newly developing quantum theory to make
it work (Bohr studied in Rutherford’s lab in 1912)
Planck and Einstein had shown that the energy of oscillating
charges must change in discrete amounts. Einstein argued that in
changing energy states, a photon would be emitted with energy
equal to that change
Bohr (1913) argued that perhaps electrons in the atom may also
behave in this way. Electrons don’t radiate with just any energy,
but rather must do so in a quantised fashion.
He developed the following theory of the atom from these ideas
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School of Mathematical and Physical Sciences PHYS1220
Bohr Model of the Atom II
Electrons are only allowed to exist in specific circular orbits
(called stationary states) with definite energies and they do so
without emitting radiation.
Photon emission only occurs when an electron jumps
from one stationary state to another of lower energy
If the electron exists in stationary states with fixed orbits, then
the angular momentum must be quantised
h
L  mvrn  n
 n , n  1, 2,3,... Bohr’s Quantum Condition
2
where n is an integer and rn is the radius of the nth possible orbit

It’s important to note that this postulate was made by Bohr to make
the model work. He found that this condition was needed but he did
not understand why at the time
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School of Mathematical and Physical Sciences PHYS1220
Bohr Model of the Atom II
The allowed orbits are numbered 1,2,3,… etc according to the
value of n, which is called the (principle) quantum number
Consider an electron in an orbit of radius rn
Coulombic force between electron and nucleus = centripetal force
Ze2 mv 2

2
4 0 rn
rn
1
Ze2 4 2 mrn2
Ze2
rn 

2
4 0 mv
4 0 n 2 h 2
n 2 h 2 0 n 2
h 2 0
rn 
 r1 , r1 
2
 mZe
Z
 me2
The smallest orbit (n=1) for hydrogen (Z=1) is
called the Bohr Radius
r1  a0  0.529x1010 m
Radius of the
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nth
orbit
rn  n 2 r1
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School of Mathematical and Physical Sciences PHYS1220
Bohr Model of the Atom III
The total energy is sum of kinetic and potential energies
Ze2
PEn  U n  eVn  
4 0 rn
1
1 2
1 Ze2
En  mv 
2
4 0 rn
Substituting Bohr’s quantum condition for v and the expression for rn
 Z 2e4 m   1 
En    2 2   2  , n  1, 2,3,...
 8 0 h   n 
Z2
me4
En  2 E1 , E1   2 2  2.17x1018 J  13.6eV
n
8 0 h
13.6eV
E

n
For hydrogen (Z=1)
n2
E (eV)
0
-1.51
-3.40
So energy is quantised
Lowest energy level is E1, called the ground state
E2 – first excited state
E3 – second excited state
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13.6
 3.4eV
4
13.6
E3 
 1.51eV
9
E2 
-13.6
n=
n=3
n=2
n=1
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School of Mathematical and Physical Sciences PHYS1220
Bohr Model of the Atom IV
Although energies are negative, the orbit closest to the nucleus,
corresponding with n=1, has the lowest energy
For two charges, the electric PE is only zero as their separation
approaches infinity, so an electron with zero KE must have n  
to become free of the atom
An atom’s Binding energy (or ionisation energy) is the energy
required to remove a ground state electron

measured for hydrogen to be 13.6eV.
Removing an electron from the lowest state n=1 (E=-13.6eV) to
E=0 (n) requires 13.6 eV of energy and thus corresponds
completely with .
Finally, for a relaxation transition from state n to state n’, the
emitted photon will have a wavelength given by
h f
1
1  Z 2e4 m   1
1  Z 2e4 m  1
1
    En  En '     2 2   2  ' 2   2 3  ' 2  2 
 h c hc
hc  8 0 h   n (n )  8 0 h c  (n ) n 
1
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School of Mathematical and Physical Sciences PHYS1220
Bohr Model of the Atom IV
Z 2e4 m  1
1 
 2 3  ' 2 2
 8 0 h c  (n ) n 
1
Bohr’s model accurately
predicts the emission
spectrum of hydrogen
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n’=1 – Lyman series
n’=2 - Balmer series
n’=3 - Paschen series
Bohr won the Physics
Nobel Prize in 1922
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School of Mathematical and Physical Sciences PHYS1220
The Correspondence Principle
Any theory must match the well-known laws of classical physics if the
conditions match the classical case. This is known as
The Correspondence Principle
Recall from special relativity that when v<<c, the theory must
simplify to Newtonian physics
1 2
2
 eg relativistic kinetic energy KE    1 mc becomes KE 
mv
2
if you take the expansion and make v<<c
In Quantum Mechanics, the same idea applies in going from
microscopic to macroscopic situations
When a quantum number approaches , the system being described
should behave in a way that is consistent with classical physics

Eg in the Bohr model, the discrete energy levels En get closer and closer
together and as n→, they essentially become ‘continuous’
 Z 2e4 m   1 
lim En  En1  0
En    2 2   2 
n
 8 0 h   n 
The same applies for rn and L as n→. The exercise is left to the student
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School of Mathematical and Physical Sciences PHYS1220
Applying de Broglie’s Hypothesis to Atoms
Bohr did not explain why orbits were quantised in his model
de Broglie applied his hypothesis by considering the wave nature of
electrons and applied it to a circular orbit
Standing wave modes sustained on a string have nodes at the ends
de Broglie argued that the electron was a circular standing wave
that must close on itself, otherwise it would destructively interfere
with itself and die out very quickly
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School of Mathematical and Physical Sciences PHYS1220
Wave nature applied to orbits
In order for a standing wave to be sustained, there must be an
integral number of wavelengths around the circle’s circumference
2 rn  n , n  1, 2,3,...
nh
2 rn 
mv
nh
mvrn 
n
2
which is Bohr’s Quantum Condition!
The moral of the story?

the wave nature of electrons is inescapable It is integral to the nature
of electrons and electron states in the atom!
This approach began what is now Quantum Mechanics
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