Download Rotational and Vibrational Levels of Molecules

Rotational and Vibrational Levels
of Molecules
Lecture 23
www.physics.uoguelph.ca/~pgarrett/Teaching.html
Review of L-21
• Beer-Lambert law
I = I 0e
• Transmittance
• Absorbance
− Cx
= I 0e
− x
I (λ )
T=
I 0 (λ )
I 0 (λ )
= 0.4343 x
A = log
I (λ )
• Extinction coefficient
= 0.4343
Rotations and vibrations
• We have examined electronic transitions in molecules
– But they can also rotate and vibrate
– ex. O2
Rotations and vibrations
• Become more complicated for more complex molecules
– ex. H2O rotational modes
Rotations and vibrations
• Become more complicated for more complex molecules
– ex. H2O vibrational modes
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
k
=
meff
– Energy of vibration
Ev = (v + 1 2 )
= (v + 1 2 )hf
– ½ ω comes from quantum mechanics and represents
zero-point energy
Vibrations
• Vibrational energy
Ev = (v + 1 2 )
= (v + 1 2 )hf
• 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
Molecular vibrations
• For diatomic molecule with mass M1 and M2, effective mass
meff can take the simple form
M 1M 2
meff =
M1 + M 2
• 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
Vibrational excitation and de-excitation
.
.
.
v
3
2
1 IR radiation
0
π electronic
state n = 2
visible
radiation
v
3
2
1
0
Fundamental IR transition
IR radiation
.
.
.
π electronic
state n = 1
visible
radiation
Probability
distribution for
which v state is
populated
during the ∆n
transition
At normal
temperatures,
most of the
molecules will
be in the v = 0
state
Molecular rotations
• In quantum mechanics, the rigid rotor has energy levels
EJ =
2
J (J + 1)
2ℑ
where ℑ is the moment of inertia (PHY1080), J is the angular momentum,
J = 0,1,2,3,…
2
• The quantity
2ℑ
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
Rotational levels
.
.
.
J
3
2
1
0
vibrational
state v = 1
Two types of
transitions, J
increasing, and J
decreasing, populated
during the ∆v transition
IR radiation
At normal
temperatures,
molecules will have
a distribution
amongst the J states
3
2
1
0
microwave
radiation
vibrational
state v = 1
J
.
.
.
Vibrational-rotational IR spectrum
• HCl
2→
→3
1→
→2
3→
→4
4→
→5
5→
→6
6→
→7
7→
→8
8→
→9
9→
→10
10→
→11
11→
→12
Energy levels
• Taking rotations, vibrations, and electronic excitation into account
E n ,v , J = E n + E v + E J
E n ,v , J
n 2h 2
=
+ (v + 1 2 )
2
2me
+
2
2ℑ
J (J + 1)
Ring molecule
• 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
Water absorption spectrum
Radiowave
IR