Hydrogen isocyanide, HNC, is the tautomer of hydrogen cyanide (HCN). HNC is of interest as an intermediate species in a variety of chemical processes in interstellar space (T = 2.75 K).

a. For HCN the vibrational frequencies are 2041cm⁻¹ (CN stretch), 712 cm⁻¹ (bend, doubly degenerate), and 3669 cm⁻¹ (CH stretch). The rotational constant is 1.477 cm⁻¹. Calculate the rotational and vibrational partition functions for HCN in interstellar space. Before calculating the vibrational partition function, is there an approximation you can make that will simplify this calculation?

b. Perform the same calculations for HNC, which has vibrational frequencies of 2024 cm⁻¹ (NC stretch), 464 cm⁻¹ (bend, doubly degenerate), and 3653 cm⁻¹ (NH stretch). The rotational constant is 1.512 cm⁻¹.

c. The presence of HNC in space was first established by Snyder and Buhl (Bulletin of the American Astronomical Society 3 [1971]: 388) through the microwave emission of the J = 1 to 0 transition of HNC at 90.665 MHz. Considering your values for the rotational partition functions above, can you rationalize why this transition would be observed? Why not the J = 20-19 transition?