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Physical Chemistry 2^nd Edition

Thomas Engel, Philip Reid

Chapter 19

The Vibrational and Rotational Spectroscopy �of Diatomic Molecules

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Objectives

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

Physical Chemistry 2^nd Edition

Chapter 19: The Vibrational and Rotational Spectroscopy of Diatomic Molecules

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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

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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

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19.1 An Introduction to Spectroscopy

19.1 Energy Levels and Emission Spectra

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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.

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Band name	Abbr	ITU band	Frequency�and�wavelength in air	Example uses
subHertz	subHz	0	< 3 Hz�> 100,000 km	Natural and man-made electromagnetic waves (millihertzNatural and man-made electromagnetic waves (millihertz, microhertzNatural 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	Communication with submarines
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 beaconsSubmarine communication, avalanche beacons, wireless heart rate monitorsSubmarine communication, avalanche beacons, wireless heart rate monitors, geophysics
Low frequency	LF	5	30–300 kHz�10 km – 1 km	NavigationNavigation, time signalsNavigation, time signals, AM longwaveNavigation, 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	ShortwaveShortwave broadcasts, amateur radioShortwave broadcasts, amateur radio and over-the-horizon aviation communications, RFID
Very high frequency	VHF	8	30–300 MHz�10 m – 1 m	FMFM, 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	TelevisionTelevision broadcasts, microwaveTelevision broadcasts, microwave ovens, mobile phonesTelevision broadcasts, microwave ovens, mobile phones, wireless LANTelevision broadcasts, microwave ovens, mobile phones, wireless LAN, BluetoothTelevision 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	MicrowaveMicrowave devices, wireless LANMicrowave devices, wireless LAN, most modern radars
Extremely high frequency	EHF	11	30–300 GHz�10 mm – 1 mm	Radio astronomyRadio astronomy, high-frequency microwave radio relay
Terahertz	THz	12	300–3,000 GHz�1 mm – 100 μm	TerahertzTerahertz imaging – a potential replacement for X-rays in some medical applications, ultrafast molecular dynamics, condensed-matter physicsTerahertz imaging – a potential replacement for X-rays in some medical applications, ultrafast molecular dynamics, condensed-matter physics, terahertz time-domain spectroscopy, terahertz computing/communications

Physical Chemistry 2^nd Edition

Chapter 19: The Vibrational and Rotational Spectroscopy of Diatomic Molecules

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Physical Chemistry 2^nd Edition

Chapter 19: The Vibrational and Rotational Spectroscopy of Diatomic Molecules

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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.

Physical Chemistry 2^nd Edition

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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.

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19.2 Absorption, Spontaneous Emission, and Stimulated Emission

At equilibrium,

where = radiation density at frequency ν

= rate coefficient

Einstein concluded that

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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.

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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

Physical Chemistry 2^nd Edition

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Solution

In this case . These two equations can be solved for , giving . Planck has showed that

For these two expressions to be equal

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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.

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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.

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Solution

a. We first write . �Solving for k,

b. The vibrational frequency for DCl is lower by a substantial amount.

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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

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19.3 An Introduction to Vibrational Spectroscopy

Parameters for selected model are shown.

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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 � μ_x = dipole moment along � the electric field � direction

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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

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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 � ?

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Solution

Using

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

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19.5 Infrared Absorption Spectroscopy

Coupled system has two vibrational frequencies: the symmetrical and antisymmetric modes.
For symmetrical and asymmetrical, the vibrational frequency is

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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

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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.

Physical Chemistry 2^nd Edition

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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,

Physical Chemistry 2^nd Edition

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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 .

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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

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19.6 Rotational Spectroscopy

We can calculate the energy corresponding to rotational transitions

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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=10.59342cm^-1. Given that the masses of ¹H and ₃₅Cl are 1.0078250 and 34.9688527 amu, respectively, determine the bond length of the ¹H₃₅Cl molecule.

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Solution

We have

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19.6 Rotational Spectroscopy

To excite various transitions consistent with the selection rule , we have

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Physical Chemistry 2^nd Edition

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19_16fig_PChem.jpg

Physical Chemistry 2^nd Edition

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P,Q,R branches of rotational spectrum

, R:

, Q:

(vibrational

).

Physical Chemistry 2^nd Edition

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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:

Physical Chemistry 2^nd Edition

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Rotational Raman Spectra

The molecule can be made

anisotropically polarized and

Raman active.

Selection Rules:

Linear rotors

Symmetrical rotors

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Proof of Rotational Raman Selection Rules

Selection rules

Physical Chemistry 2^nd Edition

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A Typical Rotational Raman Spectrum (Linear rotors)

(Linear rotors)

Stokes lines

Anti-Stokes lines

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19_21fig_PChem.jpg

Vibrational Raman effect, Δ n=+1,-1

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Physical Chemistry 2^nd Edition

Chapter 19: The Vibrational and Rotational Spectroscopy of Diatomic Molecules