Problem Set #8 Answers
Note: Relevant Chapters in Mortimer are Chapters 19 and 20.
Please note that when is expressed in cm-1; and when is expressed in Hz.
Rotating Diatomic Molecules
1. For a diatomic molecule in a electronic state, we observe a microwave transition from J = 1 to J = 2 in the presence of an electric field. How many lines will appear in the spectrum?
The selection rules for rotational transitions in a electronic state are and , where there are values of (i.e. ). The following transitions are allowed:
This is a total of nine lines.
2. Which of the following diatomic molecules have a rotational microwave spectrum: IF, , KCl, .
A pure rotational spectrum will be observed only for those molecules that contain a permanent dipole moment. Therefore, spectra will be observed only for IF and KCl.
3. Calculate the bond length of using and the reduced mass is .
Vibrating Diatomic Molecules
4. Which of the following vibrational transitions will be observed for a diatomic molecule (treated as a harmonic oscillator): v = 1 to v = 3; v = 2 to v = 3; v = 5 to v = 4.
The selection rules for vibrational translations is . Therefore, the allowed transitions are v = 2 to v = 3; and v = 5 to v = 4.
5. Calculate the frequency of the J = 3 to J = 4 transition in the pure rotational absorption spectrum of . The equilibrium bond length is 115 pm. Assume no centrifugal distortion. The mass of a nitrogen atom is 14.003 amu; the mass of an oxygen atom is 15.995 amu; and the conversion factor is .
The frequency is
Rotation of Polyatomic Molecules
6. Identify the molecules that will exhibit a pure rotational absorption microwave spectrum:
The molecules that have a permanent dipole moment will have a rotational spectrum:
7. What information about the molecular geometry for can be determined from knowing that a pure rotational absorption spectrum is observed for this molecule?
There are four possible arrangements for the atoms in : linear N-N-O; linear N-O-N; bent N-N-O; and bent N-O-N. The linear N-O-N hypothesis can be eliminated as a possibility because this molecule would not have a permanent dipole moment and would not exhibit a pure rotational spectrum.
8. The moment of inertia about an axis perpendicular to the principal axis for is . There are different types of rigid rotors: linear, spherical top, prolate symmetric top, oblate symmetric top, asymmetric top. Which type of rotor is ? Calculate the separation (expressed in ) of the pure rotational spectrum lines for . Hint: The moment of inertia about the principal axis is given by , where the mass of a hydrogen atom = mH = ; the N-H bond length = R = ; and the bond angle is 106.78° . The moment of inertia about the principal axis is .
In order to determine if is an oblate or a prolate symmetric rotor, we need to compare and (the moment of inertia about the principal axis).
is greater than (which equals). Therefore, is an oblate symmetric top.
The rotational line separations are
9. The molecule is a prolate symmetrical top with and . Calculate the energy corresponding to J = 2 and K = ± 1.
Vibration of Polyatomic Molecules
10. Consider the vibrational mode that corresponds to the uniform expansion of the benzene ring. Is it infrared active?
This vibration does not change the molecular dipole moment. Therefore, the mode is infrared inactive.
11. Explain the difference between Stokes and anti-Stokes lines in Raman Spectroscopy.
Anti-Stokes lines correspond to transitions from higher to lower energy levels. In this case, the molecule makes a transition with and the scattered photon emerges with increased energy (and therefore higher frequency than the incident radiation).
Spectral lines corresponding to transitions from a lower to a higher molecular energy levels are Stokes lines. The molecules makes a transition with and the lines appear at lower frequency than the incident radiation.