Transcript Document

Bohr model of the atom
Announcements:
• Problem solving sessions
M3-5 and T3-5.
Niels Bohr 1885 – 1965
http://www.colorado.edu/physics/phys2170/
Physics 2170 – Spring 2009
1
Summary of atomic energy levels
1) Electrons in atoms only found in specific energy levels.
2) Different set of energy levels for different atoms.
3) 1 photon emitted per electron jump down between energy
levels. Photon color determined by energy difference.
4) Electron spends very little time (10-8 s) in excited state before
hopping back down to lowest unfilled level.
5) An electron not stuck in an atom is free; can have any energy.
Hydrogen
Lithium
Energy
Electron energy levels in 2
different atoms …
Levels have different spacing.
Atoms with more than one
electron … lower levels filled.
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Set frequency to DA
Clicker question 1
A partial energy level spectrum of element X is shown below.
n=6
n=5
n=4
n=3
-1eV
-2.5eV
-5.0eV
-10eV
n=2
-20eV
n=1
-50eV
Of the transitions listed, which one
produces the shortest wavelength photon?
A.
43
5 eV photon
B.
62
19 eV photon
C.
65
1.5 eV photon
D.
21
30 eV photon
E.
41
45 eV photon
The energy of the photon emitted is equal to the
energy difference of the levels involved in the
transition.
hc
E  E  hf 
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
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Clicker question 2
Set frequency to DA
In 1885, Balmer noticed
the Hydrogen wavelengths
followed a pattern:
  91.18 nm
1  1
22 n 2
where n =
3,4,5,6, …
410.3 486.1
434.0
656.3 nm
As n gets larger, what happens to wavelengths of emitted light?
A. gets larger and larger without limit
B. gets larger and larger, but approaches a limit
C. gets smaller and smaller without limit
D. get smaller and smaller, but approaches a limit
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4
Balmer series
In 1885, Balmer noticed the Hydrogen
wavelengths followed a pattern:
  91.18 nm
1  1
22 n 2
where n =
3,4,5,6, …
So this gets smaller
Balmer predicted
additional spectral
lines which were
quickly discovered.
410.3 486.1
434.0
  91.18 nm
656.3 nm
where n = 3,4,5,6, ….
1  1
22 n2 gets smaller as n increase
gets larger as n increases,
but no larger than 1/4
limit  4  91.18nm  364.7nm
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Hydrogen atom - Balmer series
Generalizing the formula correctly predicts more hydrogen lines.
Balmer’s general formula
Hydrogen energy levels
  91.18 nm
1  1
n2'22 n2
Predicts  of nn′ transition:
n
(n>n′)
n′
(n′=1,2,3..)
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Hydrogen atom - Balmer series
  91.18 nm
1  1
n'2 n2
Hydrogen energy levels
where
n′ = 1,2,3…
and n > n′
Balmer had a formula but no
idea where it came from or
how to apply it to other atoms.
Physics strives for not just a
description but for understanding
and predictive power.

Note: we often write this
formula as: 1
1
 R 2  12

n' n
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
where R is the Rydberg
constant: R  0.0110 nm 1
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Rutherford Solar System Model of the Atom
After discovering the
nucleus, Rutherford
proposed the solar
system model.
Electrons circle the
nucleus like planets
circling the sun.
Big problem: electrons should radiate
energy, spiraling into the nucleus
Incidentally, planets radiate
gravitational radiation and
are spiraling into the sun.
Elapsed time: ~10-11 seconds
For the earth, it will take 1022 years so the sun will be long gone.
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Reading quiz 1
Set frequency to DA
Please answer this question on your own.
No discussion until after.
In the Bohr model for hydrogen, quantized energy levels for
the electrons arise…
A.
B.
C.
D.
E.
because Bohr already knew the level are quantized and he
put in the values by hand.
as a natural consequence of classical mechanics and
quantized charge.
from an assumption of quantized electron position.
from an assumption of quantized photon energy.
from an assumption of quantized angular momentum.
The Bohr model is the next step toward understanding the atom.
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Bohr model
Bohr model background:
1. 1/ = R (1/n′2 - 1/n2) – from Balmer
2. Gravity –Gm1m2/r2 force between planets and sun gives orbits.
Coulomb −ke2/r2 force between electron and proton could be
expected to give orbits as well.
3. Classical EM says electron going in circle should radiate
energy, and spiral in (accelerating charge radiates).
proton
Bohr’s additional postulate:
+
-
Electrons orbit at particular radii
which have particular energies.
But WHY?!
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Clicker question 3
-
+
v
Set frequency to DA
If the electron orbits the proton at a
constant speed, the magnitude of
the net force on the electron is…
A. mv
This force comes
B. mv2/r
from the Coulomb
C. v2/r2
force:
kq1q2
FC  2
D. mvr
r
E. ½mv2
r
Setting the net force (from Coulomb)
equal to the mass times acceleration
(mv2/r) for circular motion gives us:
mv2  ke2
r
r2
which we can also write as:
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ke
mv 
r
2
11
What does this say about total energy?
v
F
r
+
We now put together three pieces:
1. mv2 = ke2/r (just derived)
2. electrostatic potential energy is U = -ke2/r
3. nonrelativistic kinetic energy is K = ½mv2
This means K = -½U and the total energy is
2
1
ke
E U  K U  U  U  
2 r
1
2
0
potential
energy
distance from proton
1
2
The total energy, radius, and velocity
are all related. Knowing just one of
the three determines the other two!
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Set frequency to DA
Clicker question 4
Nucleus
Higher
Energy
-
Energy
levels
Electron
++
++
2
ke
mv 
r
E U  K
2
A force is applied to the electron
bringing it to a larger radius orbit.
1
2
K

mv
2
What can we say about the energy?
2
ke
A. Total, potential, and kinetic energy increase U   r
2
1
ke
E
B. Total, potential, and kinetic energy decrease
2 r
C. Total and potential energy decrease, kinetic energy increases
D. Total and potential energy increase, kinetic energy decreases
E. Some other combination
Increasing r increases potential and total
(less negative) but lowers kinetic (smaller v).
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Bohr model energy levels
0 distance from proton
Electron
energy
levels
3rd excited level
2nd excited level
1st excited level
potential
energy
ground level
In the Bohr model, each energy level
corresponds to a certain radius and velocity.
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Physics 2170 – Spring 2009
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1
ke
E
2 r
2
ke
mv 
r
2
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Bohr model energy levels
r
+
v
F
Only certain energy levels exist.
Electron can hop down energy levels,
releasing a photon on each hop.
0 distance from proton
potential
energy
But what determines these
“special” energies?
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Why are only certain energy levels allowed?
Bohr supposed that electrons could only be in certain energy
levels but then he needed to justify this in some way.
Bohr postulated that angular momentum was quantized
Remember angular momentum is
  
Lrp
L  mevr
L  mevr  n where   h / 2
For electron at radius r the angular momentum is
Quantizing, Bohr found:
Quantizing angular momentum leads
to a quantization of radius: r  n2a
n
B
2

aB 
 0.053 nm
2
meke
is the Bohr radius
Quantizing radius leads to a quantization of energy:
2
ke
1  13.6 eV / n2
En  
2aB n2
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