Question

1 Molecular rotation, vibration, and spectroscopy (34 pts) For a linear molecule with moment of inertia I, the total energy for vibration and rotation is E = E_v + E_J = (v + 1/2)?? + J(J+1)?²/2I (v = 0, 1, 2, ..., J = 0, 1, 2, ... and m = 0, ±1, ..., ±J). Here we have assumed harmonic approximation, and moment of inertia does not changes with vibrational levels. The selection rule for vibrational spectroscopy is ?v = ±1 and the selection rule for rotational spectroscopy is ?J = ±1, ?m = 0, ±1. Based on the information above, a) What is the zero-point energy expression for this system? (4 pts) b) What is the energy expression of the lowest 3 energy levels, and what is the degeneracy for each of them? (6 pts) c) What is the transition energy expression for allowed pure rotational transitions? (4 pts) d) What is the transition energy expression for vibrational-rotational transitions with ?v = 1 and ?J = 1? (4 pts) e) Given the results of c), Sketch the corresponding rotational absorption (?J = +1) spectrum. Label your axes, and comment on the energy spacing between each adjacent peak. (4 pts) f) Will you be able to observe the rotational spectrum of 1. H? 2. CH? 3. HBr with regular microwave spectroscopy? Why? How about their vibrational spectra with IR spectroscopy? Why? (12 pts)

          1 Molecular rotation, vibration, and spectroscopy (34 pts)
For a linear molecule with moment of inertia I, the total energy for vibration and rotation is E = E_v + E_J = (v + 1/2)?? + J(J+1)?²/2I (v = 0, 1, 2, ..., J = 0, 1, 2, ... and m = 0, ±1, ..., ±J). Here we have assumed harmonic approximation, and moment of inertia does not changes with vibrational levels. The selection rule for vibrational spectroscopy is ?v = ±1 and the selection rule for rotational spectroscopy is ?J = ±1, ?m = 0, ±1. Based on the information above,
a) What is the zero-point energy expression for this system? (4 pts)
b) What is the energy expression of the lowest 3 energy levels, and what is the degeneracy for each of them? (6 pts)
c) What is the transition energy expression for allowed pure rotational transitions? (4 pts)
d) What is the transition energy expression for vibrational-rotational transitions with ?v = 1 and ?J = 1? (4 pts)
e) Given the results of c), Sketch the corresponding rotational absorption (?J = +1) spectrum. Label your axes, and comment on the energy spacing between each adjacent peak. (4 pts)
f) Will you be able to observe the rotational spectrum of 1. H? 2. CH? 3. HBr with regular microwave spectroscopy? Why? How about their vibrational spectra with IR spectroscopy? Why? (12 pts)

1 Molecular rotation, vibration, and spectroscopy (34 pts)
For a linear molecule with moment of inertia I, the total energy for vibration and rotation is E = Ev + EJ = (v + 1/2)?? + J(J+1)?²/2I (v = 0, 1, 2, ..., J = 0, 1, 2, ... and m = 0, ±1, ..., ±J). Here we have assumed harmonic approximation, and moment of inertia does not changes with vibrational levels. The selection rule for vibrational spectroscopy is ?v = ±1 and the selection rule for rotational spectroscopy is ?J = ±1, ?m = 0, ±1. Based on the information above,
a) What is the zero-point energy expression for this system? (4 pts)
b) What is the energy expression of the lowest 3 energy levels, and what is the degeneracy for each of them? (6 pts)
c) What is the transition energy expression for allowed pure rotational transitions? (4 pts)
d) What is the transition energy expression for vibrational-rotational transitions with ?v = 1 and ?J = 1? (4 pts)
e) Given the results of c), Sketch the corresponding rotational absorption (?J = +1) spectrum. Label your axes, and comment on the energy spacing between each adjacent peak. (4 pts)
f) Will you be able to observe the rotational spectrum of 1. H? 2. CH? 3. HBr with regular microwave spectroscopy? Why? How about their vibrational spectra with IR spectroscopy? Why? (12 pts)

Added by Fernando H.

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