Question

A. Derive an expression for the rotational partition function in the "high-temperature" limit where qrot can be approximated as an integral. Remember that the rotational energies as a function of rotational quantum number j are given by: ϵ (j) = B j (j + 1) where B is called the “rotational constant” B = ℏ2 /2µ r 2 , and the degeneracy of each "j" state is D(j) = 2j + 1.

B. What is the average rotational energy in the “high-temperature” limit ?

C. For diatomic molecules, vibrational transitions from v=0 → v= 1 are accompanied by rotational transitions with Δj = ± 1. This gives rise to “R-branch” transitions with Δj = +1 (jinitial = 0, 1, 2, 3, ...), and “P-branch” transitions with Δj = -1, (jinitial = 1, 2, 3, 4, ...). In the rigid-rotor/harmonic oscillator approximation, the R-branch transitions occur with energies ℏω + 2B(1 + jinitial) and the P-branch transitions occur with energies ℏω - 2B jinitial . The intensities of these rotational transitions depend on the temperature.

(1) Given B = 0.002 eV, make a table of rotational state populations from j = 0 → 15 at T= 300 K.

(2) Given ℏω = 0.15 eV, make a “stick-spectrum” of the rotationally resolved vibrational spectrum where the “height” of each stick mimics the intensity (population of jinitial). You should get something that approximates the HCl absorption spectrum that you may have looked at in the PChem lab.

Answer 1