Rotational Spectra Simplest Case: Diatomic or Linear Polyatomic molecule Rigid Rotor Model: Two nuclei joined by a weightless rod J = Rotational quantum

Rotational SpectraSimplest Case:Simplest Case: Diatomic or Linear Polyatomic moleculeDiatomic or Linear Polyatomic molecule

Rigid Rotor Model:Rigid Rotor Model: Two nuclei joined by a weightless rodTwo nuclei joined by a weightless rod

J = Rotational quantum number (J = 0, 1, 2, …)J = Rotational quantum number (J = 0, 1, 2, …)

I = Moment of inertia = I = Moment of inertia = rr22

= reduced mass = m= reduced mass = m11mm22 / (m / (m11 + m + m22))

r = internuclear distancer = internuclear distance

m1

m2

r

1 JJI2

E2

J

Rigid Rotor Model

In wavenumbers (cmIn wavenumbers (cm-1-1):):

1 JJIc8

h F

2J

1 JJB FJ

Separation between adjacent levels:F(J) – F(J-1) = 2BJ

Rotational Energy Levels

Selection Rules:Selection Rules:

Molecule must have aMolecule must have a permanent dipole.permanent dipole.

J = J = 11

J. Michael Hollas, J. Michael Hollas, Modern SpectroscopyModern Spectroscopy, , John Wiley & Sons, New York, 1992.John Wiley & Sons, New York, 1992.

Rotational Spectra

J. Michael Hollas, J. Michael Hollas, Modern SpectroscopyModern Spectroscopy, John Wiley & Sons, New York, 1992., John Wiley & Sons, New York, 1992.

J” → J’ F(J’)-F(J”)

3 → 4 2(1.91)(4) 15.3 cm-1

4 → 5 2(1.91)(5) 19.1 cm-1

5 → 6 2(1.91)(6) 22.9 cm-1

6 → 7 2(1.91)(7) 26.7 cm-1

7 → 8 2(1.91)(8) 30.6 cm-1

8 → 9 2(1.91)(9) 34.4 cm-1

9 → 10 2(1.91)(10) 38.2 cm-1

Intensity of Transitions

J. Michael Hollas, J. Michael Hollas, Modern SpectroscopyModern Spectroscopy, , John Wiley & Sons, New York, 1992.John Wiley & Sons, New York, 1992.

2

1

2 J

2

1

max hcB

kT

%T

cm-1

Are you getting the concept?Calculate the most intense line in the CO rotational spectrum atroom temperature and at 300 °C. The rigid rotor rotational constant is 1.91 cm-1.

Recall: k = 1.38 x 10-23 J/Kh = 6.626 x 10-34 Jsc = 3.00 x 108 m/s

The Non-Rigid RotorAccount for the dynamic nature of the chemical bond:Account for the dynamic nature of the chemical bond:

J = 0, J = 0, 11

22J 1)(JJ

hc

D 1 JJ

hc

B E

D is the centrifugal distortion constantD is the centrifugal distortion constant(D is large when a bond is easily stretched)(D is large when a bond is easily stretched)

Typically, D < 10Typically, D < 10-4-4*B and B = 0.1 – 10 cm*B and B = 0.1 – 10 cm-1-1

22J 1)(JJD 1 JJB F

k

cD

2

1 and

B4

2

3

More Complicated MoleculesStill must have a permanent dipoleStill must have a permanent dipole

J = 0, J = 0, 11

2J K B -A 1 JBJ E

K is a second rotational quantum number accounting for K is a second rotational quantum number accounting for rotation around a secondary axis A.rotation around a secondary axis A.

Vibrational TransitionsSimplest Case:Simplest Case: Diatomic MoleculeDiatomic Molecule

Harmonic Oscillator Model:Harmonic Oscillator Model: Two atoms connected by a Two atoms connected by a spring.spring.

1/2 E vv

1/2 h E vv

vv = vibrational quantum number (v = 0, 1, 2, …) = vibrational quantum number (v = 0, 1, 2, …) = classical vibrational frequency= classical vibrational frequency

2/1k

2

1

k = force constant (related to the bond order).k = force constant (related to the bond order).

in Joules

in cm-1

Vibrational Energy Levels


Selection Rules:Selection Rules:1)1) Must have a change in dipole moment (for IR).Must have a change in dipole moment (for IR).2) 2) vv = = 11

Anharmonicity

Ingle and Crouch, Ingle and Crouch, Spectrochemical AnalysisSpectrochemical Analysis

... 1/2 - 1/2 E 2e vvv

Selection Rules:Selection Rules:vv = = 1, 1, 2, 2, 3, …3, …

vv = 2, 3, … are called = 2, 3, … are called overtones.overtones.

Overtones are often weak Overtones are often weak because anharmonicity at because anharmonicity at low low vv is small. is small.

Rotation – Vibration TransitionsThe rotational selection rule during The rotational selection rule during

a vibrational transition is:a vibrational transition is:

J = J = 11Unless the molecule has an odd Unless the molecule has an odd number of electrons (e.g. NO).number of electrons (e.g. NO).

Then,Then,

J = 0, J = 0, 11

0,1,2... J and 0,1,2,... for 1 1/2 E v vJJBvvJ

Bv signifies the dependence of B on vibrational level

Rotation – Vibration

Transitions


If If J = -1 J = -1 P – BranchP – Branch

If If J = 0 J = 0 Q – BranchQ – Branch

If If J = +1 J = +1 R – BranchR – Branch

Rotation – Vibrational Spectra

Why are the intensities different?Why are the intensities different?


Are you getting the concept?In an infrared absorption spectrum collected from a mixture ofHCl and DCl, there are eight vibrational bands (with rotationalstructure) centered at the values listed below. Identify thecause (species and transition) for each band.

Band Location Species/Transition

2096 cm-1

2101 cm-1

2903 cm-1

2906 cm-1

4133 cm-1

4139 cm-1

5681 cm-1

5685 cm-1

Atomic masses

H → 1.0079 amuD → 2.0136 amu35Cl → 34.9689 amu37Cl → 36.9659 amu

Raman Spectra


Selection Rule: Selection Rule: J = 0, J = 0, 22

PolyatomicsIf linear If linear (3N – 5) vibrational modes (3N – 5) vibrational modes (N is the # of atoms)(N is the # of atoms)

If non-linear If non-linear (3N – 6) vibrational modes (3N – 6) vibrational modes

Only those that have a change in Only those that have a change in dipole moment are seen in IR.dipole moment are seen in IR.

http://jchemed.chem.wisc.edu/JCEWWW/Articles/WWW0001/index.html

http://jchemed.chem.wisc.edu/JCEWWW/Articles/WWW0001/index.html

Linear Polyatomic


How many vibrational bands do we expect to see?How many vibrational bands do we expect to see?

Nonlinear Polyatomic (Ethylene)


Infrared SpectroscopyInfrared Spectroscopy

• Near Infrared: Near Infrared: 770 to 2500 nm770 to 2500 nm

12,900 to 4000 cm12,900 to 4000 cm-1-1

• Mid Infrared: Mid Infrared: 2500 to 50,000 nm (2.5 to 50 2500 to 50,000 nm (2.5 to 50 m)m)

4000 to 200 cm4000 to 200 cm-1-1

• Far Infrared: Far Infrared: 50 to 1000 50 to 1000 mm

200 to 10 cm200 to 10 cm-1-1

Infrared Spectroscopy: Vibrational ModesInfrared Spectroscopy: Vibrational Modes


http://en.wikipedia.org/wiki/Infrared_spectroscopy

Pretsch/Buhlmann/Affolter/Pretsch/Buhlmann/Affolter/Badertscher, Badertscher, Structure Structure Determination of Organic Determination of Organic CompoundsCompounds

Group FrequenciesGroup Frequencies

2/1k

2

1

c

Estimate band location:

Are you getting the concept?Are you getting the concept?

Estimate the stretching vibrational frequency for a carbonyl Estimate the stretching vibrational frequency for a carbonyl group with a force constant, k, of 12 N/cm. If a C=S bondgroup with a force constant, k, of 12 N/cm. If a C=S bondhad the same force constant, where would its stretchinghad the same force constant, where would its stretchingband appear in the infrared absorption spectrum?band appear in the infrared absorption spectrum?

Recall:1 amu = 1.6605 x 10-27 kg1N = 1 kg*m*s-2

Atomic masses

C → 12.000 amuO → 15.995 amuS → 31.972 amu

Infrared SpectroscopyInfrared Spectroscopy

• Near Infrared: Near Infrared: 770 to 2500 nm770 to 2500 nm

12,900 to 4000 cm12,900 to 4000 cm-1-1

** OvertonesOvertones

* Combination tones* Combination tones

* Useful for quantitative measurements* Useful for quantitative measurements

• Mid Infrared: Mid Infrared: 2500 to 50,000 nm (2.5 to 50 um)2500 to 50,000 nm (2.5 to 50 um)

4000 to 200 cm4000 to 200 cm-1-1

** Fundamental vibrationsFundamental vibrations

* Fingerprint region 1300 to 400 cm* Fingerprint region 1300 to 400 cm-1-1

(characteristic for molecule as a whole)(characteristic for molecule as a whole)

• Far Infrared: Far Infrared: 2.5 to 1000 um2.5 to 1000 um

200 to 10 cm200 to 10 cm-1-1

** Fundamental vibrations of bonds with heavyFundamental vibrations of bonds with heavy

atoms (useful, e.g., for organometallics) atoms (useful, e.g., for organometallics)

Example of an OvertoneExample of an Overtone

• Wagging vibration at 920 cmWagging vibration at 920 cm-1-1..• Overtone at approximately 2 x 920 cmOvertone at approximately 2 x 920 cm-1 -1 = 1840 cm= 1840 cm-1-1.. H

H

H3C OH

H

Fermi ResonanceFermi Resonance

N.B. Colthup et al., N.B. Colthup et al., Introduction to Infrared and Raman Spectroscopy, Introduction to Infrared and Raman Spectroscopy, Academic Press, Academic Press, Boston, 1990.Boston, 1990.

Example of a Fermi ResonanceExample of a Fermi Resonance

• Stretching vibration of C-C=(O) at 875 cmStretching vibration of C-C=(O) at 875 cm-1-1..• Overtone at approximately 2 x 875 cmOvertone at approximately 2 x 875 cm-1 -1 = 1750 cm= 1750 cm-1-1

coincides with C=O stretchcoincides with C=O stretch

ClO

Cl

Light Source: GlobarLight Source: Globar

Silicon Carbide Rod (5mm diameter, 50 mm long)Silicon Carbide Rod (5mm diameter, 50 mm long)

Heated electrically to 1300 – 1500 KHeated electrically to 1300 – 1500 K

Positive temperature coefficient of resistancePositive temperature coefficient of resistance

Electrical contact must be water cooled to prevent arcingElectrical contact must be water cooled to prevent arcing


Sample Preparation for IR SpectroscopySample Preparation for IR Spectroscopy


Liquid Samples: Cell ThicknessLiquid Samples: Cell Thickness


2

nb

Window and Cell MaterialsWindow and Cell Materials


SolventsSolvents

Pretsch/Buhlmann/Affolter/Badertscher, Pretsch/Buhlmann/Affolter/Badertscher, Structure Determination of Organic CompoundsStructure Determination of Organic Compounds

Suspension Media for Solid SamplesSuspension Media for Solid Samples

Pretsch/Buhlmann/Affolter/Badertscher, Pretsch/Buhlmann/Affolter/Badertscher, Structure Determination of Organic CompoundsStructure Determination of Organic Compounds

InterferencesInterferences

Pretsch/Buhlmann/Affolter/Pretsch/Buhlmann/Affolter/Badertscher, Badertscher, Structure Structure Determination of Organic Determination of Organic CompoundsCompounds

Documents

Rotational Spectra Simplest Case: Diatomic or Linear Polyatomic molecule Rigid Rotor Model: Two nuclei joined by a weightless rod J = Rotational quantum