PHYS 3310 Module 10.2

Molecular Spectra

Recommended Reading

10.2 Molecular Spectra

Learning Objectives

  • Use the concepts of vibrational and rotational energy to describe energy transitions in a diatomic molecule
  • Explain key features of a vibrational-rotational energy spectrum of a diatomic molecule
  • Estimate allowed energies of a rotating molecule
  • Determine the equilibrium separation distance between atoms in a diatomic molecule from the vibrational-rotational absorption spectrum

Energy of a Molecule

Rotational Energy

Practice!

Practice 10.2.1
A gas of identical diatomic molecules absorbs electromagnetic radiation over a wide range of frequencies. Molecule 1 is in the ℓ = 0 rotation state and makes a transition to the ℓ = 1 state. Molecule 2 is in the ℓ = 2 state and makes a transition to the ℓ = 3 state. The ratio of the frequency of the photon that excited molecule 2 to that of the photon that excited molecule 1 is equal to:
A. 1
B. 2
C. 3
D. 4
E. impossible to determine
Practice 10.2.2
The moment of inertia of a CO molecule is 1.46 x 10–46 kg · m2. What is the wavelength of the photon emitted if a rotational transition occurs from the ℓ = 3 to the ℓ = 2 state?
A. 430 µm
B. 870 µm
C. 1740 µm
D. 550 µm
E. 290 µm
Practice 10.2.3
A molecule makes a transition from the ℓ = 1 to the ℓ = 0 rotational energy state. The wavelength of the emitted photon is 2.6 x 10–3 m. What is the moment of inertia of the molecule?
A. 2.9 x 10–46 kg·m2
B. 5.7 x 10–45 kg·m2
C. 1.1 x 10–44 kg·m2
D. 1.5 x 10–46 kg·m2
E. 9.1 x 10–46 kg·m2
Practice 10.2.4
An oxygen molecule has a moment of inertia of 5 x 10–46 kg·m2. Calculate the bond length. Recall that the atomic mass of oxygen is 16 u (1 u = 1.66 x 10–27 kg).
A. 0.3 nm
B. 0.1 nm
C. 0.2 nm
D. 0.4 nm
E. 0.5 nm

Discuss!

Calculate the energies and wavelengths of the three lowest rotational radiations emitted by molecular H2. For H2, the reduced mass is equal to half the mass of a hydrogen atom and r0 = 0.074 nm.

Vibrational Energy

Practice!

Practice 10.2.5
Which is easier to excite in a diatomic molecule, rotational or vibrational motion?
A. rotational motion
B. vibrational motion
C. both the same
Practice 10.2.6
The fundamental frequency of the diatomic molecule CO is 6.42 x 1013 Hz. If the atomic masses are 12 u and 16 u (1 u = 1.66 x 10–27 kg), find the force constant for the diatomic molecule.
A. 970 N/m
B. 1530 N/m
C. 1850 N/m
D. 480 N/m
E. 47 N/m
Practice 10.2.7
The fundamental frequency of the diatomic molecule HF is 8.72 x 1013 Hz. The energy associated with a transition from the 10th to the 9th vibrational quantum number (in eV) is
A. 3.6 eV
B. 0.36 eV
C. 0.06 eV
D. 0.6 eV
E. 0.18 eV

Molecular Spectroscopy

Practice!

Practice 10.2.8
The energy levels of a diatomic molecule are denoted by a vibrational quantum number n and a rotational quantum number ℓ. Transitions from one level to a lower level involves the emission of a photon. Which of the following transitions are allowed?
A. n = 2, ℓ = 1 to n = 1, ℓ = 0
B. n = 3, ℓ = 1 to n = 2, ℓ = 2
C. n = 4, ℓ = 2 to n = 3, ℓ = 0
D. Both A and B are allowed.
E. All three of A, B, and C are allowed.
Practice 10.2.9

This diagram shows the vibrational and rotational energy levels of a diatomic molecule. Consider two possible transitions for this molecule:
 I. n = 2, ℓ = 5 to n = 1, ℓ = 4
 II. n = 2, ℓ = 1 to n = 1, ℓ = 0
 The energy change of the molecule is
A. greater for transition I.
B. greater for transition II.
C. the same for both transitions.