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Rotational and Vibrational Levelsof Molecules
Lecture 23
www.physics.uoguelph.ca/~pgarrett/Teaching.html
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Review of L-21
• Beer-Lambert law
• Transmittance
• Absorbance
• Extinction coefficient
xCx eIeII 00−− ==
)(
)(
0 λλ
I
IT =
xI
IA 4343.0
)(
)(log 0 =��
�
����
�=λλ
4343.0=
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Rotations and vibrations
• We have examined electronic transitions in molecules– But they can also rotate and vibrate
– ex. O2
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Rotations and vibrations
• Become more complicated for more complex molecules– ex. H2O rotational modes
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Rotations and vibrations
• Become more complicated for more complex molecules– ex. H2O vibrational modes
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Vibrations
• Just like the electrons, molecular motion is governed by quantum mechanics– Energies due to rotation and vibration are quantized
• Molecular vibrations– Chemical bond acts like a spring and can display SHM– Have an effective spring constant k for the bond involved and
effective mass meff
– Angular frequency
– Energy of vibration
– ½�ω comes from quantum mechanics and represents zero-point energy
effm
k=
( ) ( )hfvvEv 21
21 +=+= �
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Vibrations
• Vibrational energy
• Vibrational quantum number v = 0,1,2,3,… • The zero point energy ½�ω implies molecule never stops
vibrating, even when its in the v = 0 state!– Zero point energy cannot be harvested or extracted– Still exists at absolute zero
• All molecules are then in v = 0 state
• Energy levels are equally spaced with separation �ω• Obey selection rule ∆v = ±1 if no accompanying electronic
transition– Otherwise can be anything
( ) ( )hfvvEv 21
21 +=+= �
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Molecular vibrations
• For diatomic molecule with mass M1 and M2, effective mass meff can take the simple form
• Energy scale for molecular vibrations is much less than for electronic excitations
• Excitation energies correspond to IR region of the spectrum– Typical wavelengths are 2 – 50 µm = 2000 – 50000 nm for organic
molecules
• Vibrational levels are built on electronic states – each electronic state will host the whole range of vibrational states
21
21
MM
MMmeff +
=
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Vibrational excitation and de-excitation
v
3 2 10
π electronic state n = 2
.
.
.
v
3 2 10
.
.
.
IR radiation
visible radiation
At normal temperatures, most of the
molecules will be in the v = 0
state
Probability distribution for which v state is
populated during the ∆n
transitionvisible radiation
IR radiationFundamental IR transition
π electronic state n = 1
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Molecular rotations
• In quantum mechanics, the rigid rotor has energy levels
where ℑ is the moment of inertia (PHY1080), J is the angular momentum, J = 0,1,2,3,…
• The quantity is called the rotational parameter
• Moment of inertia, hence rotational parameter, can be different for each rotation axis
• Excitation energies correspond to the microwave region
• Energy scale for rotations << vibrations– Each vibrational level has rotational bands built on it
• Selection rule ∆J = ±1
( )12
2
+ℑ
= JJEJ
�
ℑ2
2�
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Rotational levels
J
3 2 10
vibrationalstate v = 1
.
.
.
J
3 2 10
.
.
.
IR radiation
At normal temperatures,
molecules will have a distribution
amongst the J states
Two types of transitions, J
increasing, and Jdecreasing, populated
during the ∆v transition
microwave radiation
vibrationalstate v = 1
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Vibrational-rotational IR spectrum
• HCl
1→→→→22→→→→3
3→→→→4
4→→→→5
5→→→→6
6→→→→7
7→→→→8
8→→→→9
9→→→→10
10→→→→11
11→→→→12
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Energy levels
• Taking rotations, vibrations, and electronic excitation into account
• If the measuring instrument has very good resolution, it is possible to see the discrete transitions
• Complex molecules may have many vibrational modes, rotational modes, etc. The combination of these different modes leads to a “smearing” of the discrete spectrum (temp. effects too) so that broad bumps appear rather than discrete lines
( ) ( )122
2
21
2
22
,,
,,
+ℑ
+++=
++=
JJvm
hnE
EEEE
eJvn
JvnJvn
��
�
Ring molecule
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Water absorption spectrum
IR Radiowave
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