1. Physical Chemistry 2nd Edition
Chapter 19
The Vibrational and Rotational Spectroscopy of Diatomic Molecules
Physical Chemistry 2nd Edition
Thomas Engel, Philip Reid

2. Objectives
Describe how light interacts with molecules to induce transitions between states
Discuss the absorption of electromagnetic radiation

3. Outline An Introduction to Spectroscopy
Absorption, Spontaneous Emission, and Stimulated Emission
An Introduction to Vibrational Spectroscopy
The Origin of Selection Rules
Infrared Absorption Spectroscopy
Rotational Spectroscopy

4. 19.1 An Introduction to Spectroscopy
Spectroscopy are tools chemists have to probe the species at an atomic and molecular level.
The frequency at which energy is absorbed or emitted is related to the energy levels involved in the transitions by

5. 19.1 An Introduction to Spectroscopy
19.1 Energy Levels and Emission Spectra

6. 19.1 An Introduction to Spectroscopy
During vibration, oscillator will absorb energy in both the stretching and compression.
The molecule can absorb energy from the field during oscillation.

7. Frequency and wavelength in air Example uses
Band name
Abbr
ITU band
Frequency and wavelength in air
Example uses
subHertz
subHz
< 3 Hz > 100,000 km
Natural and man-made electromagnetic waves (millihertz, microhertz, nanohertz) from earth, ionosphere, sun, planets, etc[citation needed]
Extremely low frequency
ELF 1
3–30 Hz 100,000 km – 10,000 km
Communication with submarines
Super low frequency
SLF 2
30–300 Hz 10,000 km – 1000 km
Ultra low frequency
ULF 3
300–3000 Hz 1000 km – 100 km
Communication within mines
Very low frequency
VLF 4
3–30 kHz 100 km – 10 km
Submarine communication, avalanche beacons, wireless heart rate monitors, geophysics
Low frequency LF 5
30–300 kHz 10 km – 1 km
Navigation, time signals, AM longwave broadcasting, RFID
Medium frequency MF 6
300–3000 kHz 1 km – 100 m
AM (medium-wave) broadcasts
High frequency HF 7
3–30 MHz 100 m – 10 m
Shortwave broadcasts, amateur radio and over-the-horizon aviation communications, RFID
Very high frequency
VHF 8
30–300 MHz 10 m – 1 m
FM, television broadcasts and line-of-sight ground-to-aircraft and aircraft-to-aircraft communications. Land Mobile and Maritime Mobile communications
Ultra high frequency
UHF 9
300–3000 MHz 1 m – 100 mm
Television broadcasts, microwave ovens, mobile phones, wireless LAN, Bluetooth, GPS and two-way radios such as Land Mobile, FRS and GMRS radios
Super high frequency
SHF 10
3–30 GHz 100 mm – 10 mm
Microwave devices, wireless LAN, most modern radars
Extremely high frequency
EHF 11
30–300 GHz 10 mm – 1 mm
Radio astronomy, high-frequency microwave radio relay
Terahertz
THz 12
300–3,000 GHz 1 mm – 100 μm
Terahertz imaging – a potential replacement for X-rays in some medical applications, ultrafast molecular dynamics, condensed-matter physics, terahertz time-domain spectroscopy, terahertz computing/communications

9. 19.2 Absorption, Spontaneous Emission, and Stimulated Emission
The 3 basic processes by which photon-assisted transitions occur are absorption, spontaneous emission and stimulated emission.

10. 19.2 Absorption, Spontaneous Emission, and Stimulated Emission
In absorption, the incident photon induces a transition to a higher level.
In emission, a photon is emitted as an excited state relaxes to one of lower energy.
Spontaneous emission is a random event and its rate is related to the lifetime of the excited state.

11. 19.2 Absorption, Spontaneous Emission, and Stimulated Emission
At equilibrium,
where = radiation density at frequency ν
= rate coefficient
Einstein concluded that

12. Example 19.1
Derive the equations
using these two pieces of information: (1) the overall rate of transition between levels 1 and 2 is zero at equilibrium, and (2) the ratio of N2 to N1 is governed by the Boltzmann distribution.

13. Solution
The rate of transitions from level 1 to level 2 is equal and opposite to the transitions from level 2 to level 1. This gives the equation
The Boltzmann distribution function states that

14. Solution
In this case These two equations can be solved for , giving Planck has showed that
For these two expressions to be equal

15. 19.3 An Introduction to Vibrational Spectroscopy
The vibrational frequency depends on two identity vibrating atoms on both end of the bond.
This property generates characteristic frequencies for atoms joined by a bond known as group frequencies.

16. Example 19.2
A strong absorption of infrared radiation is observed for 1H35Cl at 2991 cm-1.
a. Calculate the force constant, k, for this molecule.
b. By what factor do you expect this frequency to shift if deuterium is substituted for hydrogen in this molecule? The force constant is unaffected by this substitution.

17. a. We first write . Solving for k,
Solution
a. We first write Solving for k,
b. The vibrational frequency for DCl is lower by a substantial amount.

18. 19.3 An Introduction to Vibrational Spectroscopy
19.2 The Morse Potential
The bond energy D0 is defined with respect to the lowest allowed level, rather than to the bottom of the potential.
The energy level is

19. 19.3 An Introduction to Vibrational Spectroscopy
Parameters for selected model are shown.

20. 19.4 The Origin of Selection Rules
The transition probability from state n to state m is only nonzero if the transition dipole moment satisfies the following condition:
where x = spatial variable μx = dipole moment along the electric field direction

21. 19.5 Infrared Absorption Spectroscopy
Atoms and molecules possess a discrete energy spectrum that can only be absorbed or emitted which correspond to the difference between two energy levels.
Beer-Lambert law states that
where I(λ) = intensity of light leaving the cell
I0(λ) = intensity of light passing dl distance l = path length
ε(λ) = molar absorption coefficient

22. Example 19.4
The molar absorption coefficient for ethane is 40 (cm bar)-1 at a wavelength of 12 μm. Calculate in a 10-cm-long absorption cell if ethane is present at a contamination level of 2.0 ppm in one bar of air. What cell length is required to make ?

23. Solution
Using
This result shows that for this cell length, light absorption is difficult to detect. Rearranging the Beer-Lambert equation, we have

24. 19.5 Infrared Absorption Spectroscopy
Coupled system has two vibrational frequencies: the symmetrical and antisymmetric modes.
For symmetrical and asymmetrical, the vibrational frequency is

25. 19.6 Rotational Spectroscopy
19.3 Normal Modes for H2O
19.4 Normal Modes for CO2
19.5 Normal Modes for NH3
19.6 Normal Modes for Formaldehyde

26. The notation is used for the preceding functions.
Example 19.5
Using the following total energy eigenfunctions for the three-dimensional rigid rotor, show that the J=0 → J=1 transition is allowed, and that the J=0 → J=2 transition is forbidden:
The notation is used for the preceding functions.

27. Solution
Assuming the electromagnetic field to lie along the
zaxis, , and the transition dipole moment takes the form
For the J=0 → J=1 transition,

28. Solution
For the J=0 → J=2 transition,
The preceding calculations show that the J=0 → J=1 transition is allowed and that the J=0 → J=2 transition is forbidden. You can also show that is also zero unless MJ=0 .

29. 19.6 Rotational Spectroscopy
For vibrational spectroscopy, we have to change the symbol for the angular momentum quantum number from l to J.
Thus the dependence of the rotational energy on the quantum number is given by
where rotational constant is

30. 19.6 Rotational Spectroscopy
We can calculate the energy corresponding to rotational transitions

31. Example 19.5
Because of the very high precision of frequency measurements, bond lengths can be determined with a correspondingly high precision, as illustrated in this example. From the rotational microwave spectrum of 1H35Cl, we find that B= cm-1. Given that the masses of 1H and 35Cl are and amu, respectively, determine the bond length of the 1H35Cl molecule.

32. Solution
We have

33. 19.6 Rotational Spectroscopy
To excite various transitions consistent with the selection rule , we have

35. 19_16fig_PChem.jpg

36. P,Q,R branches of rotational spectrum
(vibrational ).
P,Q,R branches of rotational spectrum

37. 19.6 Rotational Spectroscopy
19.7 Rotational Spectroscopy of Diatomic Molecules
19.8 Rotational-Vibrational Spectroscopy of Diatomic Molecules
The ratio for value of J relative to the number in the ground state (J=0) can be calculated using the Boltzmann distribution:

38. Rotational Raman Spectra
The molecule can be made
anisotropically polarized and
Raman active.
Selection Rules:
Linear rotors
Symmetrical rotors

39. Proof of Rotational Raman Selection Rules

40. A Typical Rotational Raman Spectrum (Linear rotors)
Stokes lines
Anti-Stokes lines

41. 19_21fig_PChem.jpg
Vibrational Raman effect, Δ n=+1,-1