Derive an expression for the value of \(/\) corresponding to the most highly populated rotational energy level of a diatomic rotor at a temperature \(\mathrm{T}\) remembering that the degeneracy of each level is \(2 \mathrm{j}+1 .\) Evaluate the expression for ICI (for which \(\mathrm{B}=0.1142 \mathrm{~cm}^{-1}\) ) at \(25^{\circ} \mathrm{C}\). Repeat the problem for the most highly populated level of a spherical rotor, taking note of the fact that each level is \((2 \mathrm{j}+1)^{2}\) -fold degenerate. Evaluate the expression for \(\mathrm{CH}_{4}\) \(\left(\right.\) for which \(\mathrm{B}=5.24 \mathrm{~cm}^{-1}\) ) at \(25^{\circ} \mathrm{C}\).

The Boltzmann distribution is foundational to understanding the distribution of energy levels in molecules at thermal equilibrium. It tells us that the population of a molecule in a particular energy state is not random but influenced by the temperature and the energy of that state. At a given temperature, T, the tendency for a molecule to occupy a state with energy E is proportional to the degeneracy of that state, represented by the term \( (2J+1) \) for a diatomic molecule and \( (2J+1)^2 \) for a spherical rotor, and inversely proportional to the exponential of the energy divided by the product of Boltzmann's constant, k, and the temperature, T. This concept becomes particularly important when identifying the most populated rotational energy level, a typical task faced by students in physical chemistry.

Quantum number J, known as the rotational quantum number, is a term from quantum mechanics representing the angular momentum of a molecule. As we delve into the microscopic world, we see that angular momentum is quantized, meaning it can only take certain discrete values. Each integer value of J corresponds to a different rotational energy level. The higher the value of J, the higher the rotational energy of the molecule. Understanding the role of quantum numbers is crucial in predicting molecular behavior because it defines the overall energy level structure for rotational motion in molecules.

Degeneracy refers to the number of distinct quantum states that share the same energy level. In the case of a diatomic molecule, each rotational energy level has a degeneracy of \( 2J+1 \), reflecting the different spatial orientations the molecule can have while still maintaining the same rotational energy. For a spherical rotor, like a methane molecule, the degeneracy increases significantly to \( (2J+1)^2 \) due to its symmetrical shape allowing for even more spatial arrangements without energy change. The concept of degeneracy is critical for understanding and calculating the Boltzmann distribution, which relies on this value to predict the most populated energy level.

A diatomic molecule is a molecule composed of two atoms, which can be either identical or different. This simplest molecular system is perfect for studying rotational motion. In physical chemistry, diatomic molecules are often treated as rigid rotors to calculate rotational spectra and understand molecular motion. The rotational energy levels of such a molecule can be described with the equation \( E_j = hBJ(J+1) \), with J being the rotational quantum number. This model is a stepping-stone for understanding more complex molecular structures and behaviors.

A spherical rotor model is applied to molecules with spherical symmetry, such as methane \( (CH_4) \). In this model, we assume that the molecule can rotate with equal ease about any axis, leading to a higher degeneracy of the energy levels, \( (2J+1)^2 \). Despite having the same expression for energy as a diatomic rotor, the difference in degeneracy significantly affects the rotational spectroscopy and the Boltzmann distribution calculations of spherical rotors. Understanding this allows students to unravel the behavior of complex molecules and their interactions at various energy states.