Problem 41
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 \((\mathrm{T}=2.75 \mathrm{K})\). a. For HCN the vibrational frequencies are \(2041 \mathrm{cm}^{-1}\) (CN stretch), \(712 \mathrm{cm}^{-1}\) (bend, doubly degenerate), and \(3669 \mathrm{cm}^{-1}\) (CH stretch). The rotational constant is \(1.477 \mathrm{cm}^{-1}\) Calculate the rotational and vibrational partition functions for \(\mathrm{HCN}\) 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 \(2024 \mathrm{cm}^{-1}(\mathrm{NC} \text { stretch }), 464 \mathrm{cm}^{-1}\) (bend, doubly degenerate), and \(3653 \mathrm{cm}^{-1} \mathrm{v}\) (NH stretch). The rotational constant is \(1.512 \mathrm{cm}^{-1}\). 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 \(\mathrm{J}=1\) to 0 transition of \(\mathrm{HNC}\) at \(90.665 \mathrm{MHz}\) Considering your values for the rotational partition functions, can you rationalize why this transition would be observed? Why not the \(\mathrm{J}=20-19\) transition?
Problem 42
Evaluate the electronic partition function for atomic Fe at \(298 \mathrm{K}\) given the following energy levels. $$\begin{array}{ccc} \text { Level }(\boldsymbol{n}) & \text { Energy }\left(\mathrm{cm}^{-1}\right) & \text {Degeneracy } \\ \hline 0 & 0 & 9 \\ 1 & 415.9 & 7 \\ 2 & 704.0 & 5 \\ 3 & 888.1 & 3 \\ 4 & 978.1 & 1 \end{array}$$
Problem 43
a. Evaluate the electronic partition function for atomic Si at \(298 \mathrm{K}\) given the following energy levels: $$\begin{array}{ccc} \text { Level }(\boldsymbol{n}) & \text { Energy }\left(\mathbf{c m}^{-1}\right) & \text {Degeneracy } \\ \hline 0 & 0 & 1 \\ 1 & 77.1 & 3 \\ 2 & 223.2 & 5 \\ 3 & 6298 & 5 \end{array}$$ b. At what temperature will the \(n=3\) energy level contribute 0.100 to the electronic partition function?
Problem 44
NO is a well-known example of a molecular system in which excited electronic energy levels are readily accessible at room temperature. Both the ground and excited electronic states are doubly degenerate and are separated by \(121.1 \mathrm{cm}^{-1}\) a. Evaluate the electronic partition function for this molecule at \(298 \mathrm{K}\) b. Determine the temperature at which \(q_{E}=3\)
Problem 45
Rhodopsin is a biological pigment that serves as the primary photoreceptor in vision (Science \(266[1994]: 422)\) The chromophore in rhodopsin is retinal, and the absorption spectrum of this species is centered at roughly 500 nm. Using this information, determine the value of \(q_{E}\) for retinal. Do you expect thermal excitation to result in a significant excitedstate population of retinal?
Problem 46
Determine the total molecular partition function for \(\mathrm{I}_{2}\) confined to a volume of \(1000 . \mathrm{cm}^{3}\) at \(298 \mathrm{K}\). Other information you will find useful is that \(B=0.0374 \mathrm{cm}^{-1},\left(\widetilde{\nu}=208 \mathrm{cm}^{-1}\right)\) and the ground electronic state is nondegenerate.
Problem 47
Determine the total molecular partition function for gaseous \(\mathrm{H}_{2} \mathrm{O}\) at \(1000 .\) K confined to a volume of \(1.00 \mathrm{cm}^{3}\) The rotational constants for water are \(B_{A}=27.8 \mathrm{cm}^{-1}\) \(B_{B}=14.5 \mathrm{cm}^{-1},\) and \(B_{C}=9.95 \mathrm{cm}^{-1} .\) The vibrational frequencies are \(1615,3694,\) and \(3802 \mathrm{cm}^{-1}\). The ground electronic state is nondegenerate.
Problem 48
The effect of symmetry on the rotational partition function for \(\mathrm{H}_{2}\) was evaluated by recognizing that each hydrogen is a spin \(1 / 2\) particle and is, therefore, a fermion. However, this development is not limited to fermions, but is also applicable to bosons. Consider \(\mathrm{CO}_{2}\) in which rotation by \(180^{\circ}\) results in the interchange of two spin 0 particles. a. Because the overall wave function describing the interchange of two bosons must be symmetric with respect to exchange, to what \(J\) levels is the summation limited in evaluating \(q_{R}\) for \(\mathrm{CO}_{2} ?\) b. The rotational constant for \(\mathrm{CO}_{2}\) is \(0.390 \mathrm{cm}^{-1}\). Calculate \(q_{R}\) at \(298 \mathrm{K} .\) Do you have to evaluate \(q_{R}\) by summation of the allowed rotational energy levels? Why or why not?
