Transcript Lecture_19-Energy Levels in the Bohr model of the atom

Still have a few registered iclickers (3 or 4 ?) that need to be mapped
to names
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Today’s program
Spectral lines and Bohr model bootcamp.
Clicker questions to test your understanding of the
Bohr model.
BTW today, October 7th, is Niels Bohr’s Birthday.
(He would be 131 years old).
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Review: Breakdown of classical physics (Crisis)
• Rutherford’s experiment
suggested that electrons orbit
around the nucleus like a
miniature solar system.
• However, classical physics
predicts that an orbiting
electron (accelerating charge)
would emit electromagnetic
radiation and fall into the
nucleus. So classical physics
could not explain why atoms
are stable.
There is a ground
state energy level
Question: What is the
solution to this crisis ?
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Quantization of atomic energy levels (Experimental)
Three classes of
spectral features:
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Quantization of atomic energy levels (visual evidence)
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Quantization of atomic energy levels
• Niels Bohr explained atomic line spectra and the stability of
atoms by postulating that atoms can only be in certain discrete
energy levels. When an atom makes a transition from one
energy level to a lower level, it emits a photon whose energy
equals that lost by the atom.
• An atom can also absorb a photon, provided the photon energy
equals the difference between two energy levels.
Insert Figure 39.16
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Quantization of atomic energy levels
• An atom can also absorb a photon, provided the photon energy
equals the difference between two energy levels.
The master equation for the photon
energy in these transitions is
hf = h
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c
l
= Ei - E f
Next exercise
Are you ready for Bohr Model Bootcamp ?
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The Bohr model of hydrogen (original argument)
• Bohr explained the line
spectrum of hydrogen with a
model in which the single
hydrogen electron can only be
in certain definite orbits.
• In the nth allowed orbit, the
electron has orbital angular
momentum nh/2π (see Figure
on the right).
• Bohr proposed that angular
momentum is quantized (this
will turn out to be correct in
general in quantum mechanics
but is not right for the hydrogen
atom).
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Ln=rp=m vn rn
The Bohr model of hydrogen
Let’s use a different argument based on
deBroglie waves to obtain the same conclusions.
Think of a standing wave with wavelength λ
that extends around the circle.
2p rn = nln
Q: How is the momentum of the atomic electron
related to its wavelength ? (remember the Prince)
h
2p rn = nln = n
mvn
h
mvn rn = n
2p
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Same as the Bohr
quantization
condition
The Bohr model of hydrogen
Now let’s use a Newtonian
argument for a planetary
model of the atom but use
the Bohr quantization
condition. (A little hooky).
h
mvn rn = n
2p
e
mvn
Fe =
= Fc =
2
4pe 0 rn
rn
2
(The mass m is that
of the electron.)
2
Balance electrostatic
and centripetal forces
mvn 2 rn 2 (mvn rn )2
e2
(nh / 2p )2
=
=
Þ
=
4pe 0
rn
rn
4pe 0
rn
e2
Here we used the Bohr quantization condition
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The Bohr model of hydrogen (Bohr radius)
e 0 (nh)
rn =
p me2
2
e 0 (nh)2
2
rn =
Þ
r
=
n
a0
n
2
p me
Here n is the “principal quantum number” and a0 is the “Bohr
radius”, which is the minimum radius of an electron orbital.
e 0h
2
a0 =
Þ
r
=
n
a0
n
2
p me
2
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a0 = 5.29 ´10-11 m
The Bohr model of hydrogen (Energy levels, derivation)
e2
Þ vn =
e 0 (2nh)
h
mvn rn = n
2p
4
1
me
K n = mvn 2 = 2 2
2
8n h
e 0 (nh)
rn =
p me2
-e
2
-me
Un =
= 2 2 2
4pe 0 rn e 0 4n h
2
4
-me4
En = K n +Un = 2 2 2
e 0 8n h
Note that E and U are
negative (1/8-1/4=-1/8)
This expression for the allowed energies can be rewritten and used
to predict atomic spectral lines !
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The Bohr model of hydrogen (Energy levels)
-me4
-hcR
me4
En = 2 2 2 = 2 ; R =
e 0 8n h
n
8 e 0 2 h 3c
-hcR
En = 2
n
Here R is the “Rydberg
constant”, R=1.097 x 107 m-1
Also hcR = 13.60 eV is a useful result.
Question: How can we find the energies of
photon transitions between atomic levels ?
hc
l
= EUpper - ELower = hcR(
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1
n
2
Upper
-
1
n
2
Lower
)
The Bohr model of hydrogen (It works)
1
l
= R(
1
n
2
Upper
-
1
n
2
)
Lower
Here R is the Rydberg
constant, R=1.097 x 107 m-1
If nupper=3, nlower=2, let’s calculate the
wavelength.
1
1 1
1
7
-1 1
= R( 2 - 2 ) = (1.097 ´10 m )( - ) = 656.3nm
l
2 3
4 9
Balmer Hα line, agrees with experiment within 0.1%
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Hydrogen spectrum (also has other spectral lines)
•
The line spectrum at the bottom of the previous slide is not the entire
spectrum of hydrogen; it is just the visible-light portion.
•
Hydrogen also has series of spectral lines in the infrared and the
ultraviolet.
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Hydrogen-like atoms
•
The Bohr model can be applied to any atom with a single electron. This
includes hydrogen (H) and singly-ionized helium (He+). See the Figure
below.
-hcR
me4
En = 2 ; R =
n
8 e 0 2 h 3c
Question: How should this
formula be modified for
singly-ionized helium ?
Ans: He has 2p. If an
atom is singly ionized,
then rnrn/ZEnZ2E
Ze2
(nh / 2p )2
=
4pe 0
rn
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But the Bohr model does not
work for other atoms; need QM