The vibrational frequency of the \(\mathrm{I}_{2}\) molecule is \(f=6.42 \times 10^{12} \mathrm{~s}^{-1}\). The vibrational temperature is \(\theta_{1_{2}}^{\text {vib }}=\frac{h f}{k_{B}}=308 \mathrm{~K}\). The rotational temperature is \(\theta_{1_{2}}^{\text {rot }}=0.0538 \mathrm{~K}\). Consider a gas of \(N\) \(\mathrm{I}_{2}\) molecules at temperature \(T=300 \mathrm{~K}\). (a) What fraction of the molecules is in the vibrational ground state and what fraction have one vibrational quantum of energy? (b) What percentage of the total internal energy of the gas is: (1) translational?; (2) vibrational?; (3) rotational?

The concept of vibrational temperature is pivotal when dealing with diatomic molecules. It is a measure of the energy required to occupy different vibrational states. For the molecule \(\mathrm{I}_{2}\), the vibrational temperature \(\theta_{1_{2}}^{\text {vib}}=308 \mathrm{~K}\) is determined by the formula: \(\theta_{vib} = \frac{hf}{k_B}\)
Here, \(h\) is Planck's constant, \(f\) is the vibrational frequency, and \(k_B\) is Boltzmann's constant. This value helps in calculating how molecules distribute among various vibrational states at a given temperature. A higher vibrational temperature indicates that it takes more energy for molecules to jump to higher vibrational states.

The Boltzmann distribution describes how particles distribute among available energy levels at thermal equilibrium. For vibrational states, it’s given by \[ \frac{N_v}{N} = \frac{e^{- \frac{v hf}{k_B T}}}{Z_v} \]
where \(N_v\) is the number of molecules in vibrational state \(v\), \(hf\) is the vibrational energy, \(k_B T\) is the thermal energy, and \(Z_v\) is the partition function. The partition function summarizes all possible states: \[ Z_v = \frac{1}{1 - e^{- \frac{hf}{k_B T}}} \]
The Boltzmann distribution helps in finding fractions of molecules in different vibrational quantum states by considering the exponentials of energy divided by thermal energy.

Rotational temperature, denoted as \(\theta_{1_{2}}^{\text {rot}}\), represents the energy required for a molecule to change its rotational state. For \(\mathrm{I}_{2}\), it’s \(0.0538 \, \mathrm{K}\). The lower value signifies that rotational state changes need relatively less energy compared to vibrational changes. The formula used here is \(\theta_{rot} = \frac{h^2}{8 \pi^2 k_B I}\), where \(I\) is the moment of inertia. At room temperature (300K), many rotational states are populated due to the small energy differences between levels, making rotational contributions significant in total energy calculations.

Translational energy describes the kinetic energy of molecules moving in space. For gases, the translational energy per molecule in 3D is: \[ E_{trans} = \frac{3}{2} k_B T \]
where \(k_B\) is Boltzmann's constant and \(T\) is the temperature. With \(N\) molecules, the total translational energy is: \[ E_{trans,total} = N \frac{3}{2} k_B T \]
This form of energy is significant as it accounts for straightforward movements of the molecules and contributes a large portion to the gas’s internal energy, especially at higher temperatures.

Vibrational quantum states specify the discrete energy levels accessible by molecules due to vibrational motion. The ground state is the lowest energy state, while higher states require more energy. The energy difference between two consecutive vibrational states is given by \(hf\). For \(\mathrm{I}_{2}\), the fraction of molecules in the ground state (\(v=0\)) and the first excited state (\(v=1\)) helps in understanding the distribution of vibrational energy. Using the Boltzmann factor, we calculate these fractions, illustrating how most molecules remain in lower energy states at common temperatures.