Referring to problems 1 and 2 and Eq. (6-2), construct an energy level diagram for the H2...

Referring to problems 1 and 2 and Eq. (6-2), construct an energy level diagram for the H2 molecule that shows the first vibrational and rotational states associated with the ground electronic state of the molecule. (Hint: The molecule can be vibrating and rotating at the same time.)

Problems 1–2

1. The allowed rotational energies of a diatomic molecule are given by

In this expression l is the rotational quantum number and can take the values l = 0, 1, 2 …; I is the rotational inertia of the molecule about an axis through its center of mass; and ℏ= h/2π. The equilibrium separation of the two atoms in a diatomic hydrogen molecule H2 is about 0.074 nm. The mass of each hydrogen atom is about 1.67 × 10−27 kg.

a. Show that the rotational inertia of the hydrogen molecule about an axis through its center of mass is about I =4.6× 10−48 kg · m2.

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b. Find the difference in energy between the first excited rotational energy stale and the ground rotational state. That is, find . Express the answer in both J and eV.

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c. Find the relative likelihood Pl = 1/Pl = 0 that a hydrogen molecule will be in its first excited rotational state in thermal equilibrium at room temperature, T = 293 K. (Ignore possible state degeneracies.)

2. The allowed energies associated with the vibration of a diatomic molecule are given by

Here, k is the vibrational quantum number and can take the values k = 0, 1, 2 … and f is the resonant frequency of the vibration. In a simple model of diatomic hydrogen H2, the resonant vibration frequency can be taken as f = 1.3 × 1014 Hz.

a. Find the difference in energy between the first excited vibrational energy state and the ground vibrational state of diatomic hydrogen. That is, find . Express the answer in both J and eV.

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b. Find the relative likelihood Pk =1/Pk= 0 that a hydrogen molecule will be in its first excited vibrational state in thermal equilibrium at room temperature, T = 293 K.