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1. a) Derive an expression for the value of J corresponding to the most highly populated rotational energy level of a spherical rotator (max). For spherical rotator each energy level is (2J + 1)2 - fold degenerate and energy is given by E, = hcĒJ( + 1). The population N, of molecules in energy level J is given by the Boltzmann expression N, = Ng,e-Ey/KT where N is the total number of molecules in the sample, g, is the degeneracy of level J, k is the Boltzmann constant, and T is the absolute temperature. b) Evaluate Imax for CH4if B = 5.24 cm-1 at 25 °C. 2. The total tt -electron energy of naphthalene is given by En = 10a + 13.68B Calculate the delocalization energy of naphthalene. 3. Predict the shape of nitronium ion, No7, from its Lewis structure and VSEPRT model. It has one Raman active vibrational mode at 1400 cm-1, two strong IR active modes at 2360 and 540 cm-1, and one weak IR mode at 3735 cm-1. Are these data consistent with the predicted shape of the molecule? Assign the vibrational wavenumbers to the modes from which they arise. 4. In the group theoretical language, a spherical rotator is a molecule that belongs to a cubic or icosahedral point group, a symmetric rotator is a molecule with at least a threefold axis of symmetry, and an asymmetric rotator is a molecule without a threefold (or higher) axis. Linear molecules are linear rotators. Classify each of the following molecules as a spherical, symmetric, linear, or asymmetric rotator and justify your answers. (a) CH4, (b) CH3CN, (C) CO2, (d) CH3OH, (e) C6H6 (benzene), (f) Identify molecules that are rotationally Raman inactive. 5. Confirm that a Morse Oscillator has a finite number of bound states, the states with V
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Explanation & Answer

Hello there, here you have your answers! :)

Answer 1:
(a) We can write the probability of population as a function of temperature as:
(

(

)

)

The rotational quantum number (J) for the associated to the most highly populated
rotational energy level will correspond to the derivate of the equation above with respect to
J, equal to zero. Because is when probability reaches its maximum value:
(

(

)

(

)

) (

[(

(

(

)

(

)

(

)

)

(



(

))

(

)

(

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)

)

]



(b) Jmax for CH4 at 25 oC =298 K is equal to:


(



)(

(

) (

)
) (

)

Answer 2:
Naphthalene is an aromatic molecule as shown in the figure below, it has 5 pair of pi
electrons that can move through the molecule by resonance.

This conjugation gives additional stabilization energy (E D) to the total  energy (E). To
calculate this delocalization energy the localization  energy (EL) must be subtracted from
the total  energy (E). It’s well-known that applying the secular determinant on the easier
organic system (ethylene) with 2  electrons, the E= 2 + 2as ethylene hasn’t
delocalization, this magnitude of energy can be considered as the localization energy per
pair of  electrons. So in the case of naphthalene, it has 5 pair...


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