The transition from the ground state to the first rotational excited state in diatomic oxygen \(\left(\mathrm{O}_{2}\right)\) requires \(356 \mu \mathrm{eV}\). At what temperature would the thermal energy \(k T\) be sufficient to set diatomic oxygen into rotation? Would you ever find diatomic oxygen exhibiting the specific heat of a monatomic gas at normal pressure?

Understanding Boltzmann's constant is crucial in grasping how thermal energy relates to temperature at a microscopic scale. This constant, denoted by k, is a fundamental physical constant that links an individual particle's energy with the temperature of the system. It's a key part of the equation E = kT, where E is the average energy per particle, T is the absolute temperature, and k is Boltzmann's constant.

In the context of the exercise, Boltzmann's constant is used to calculate the temperature at which the thermal energy is sufficient to excite the rotational energy levels of diatomic oxygen. With a value of approximately \(1.38 \times 10^{-23} \frac{J}{K}\), this tiny number may seem insignificant, but it is the bridge that allows us to describe the behavior of gases, relate macroscopic properties (like pressure and temperature) to microscopic properties (like energy and molecular speed), and understand phenomena in statistical mechanics.

The concept of rotational energy levels is fundamental in the study of molecular physics and quantum mechanics. Molecules can rotate about their center of mass, and these rotational motions are quantized, meaning they can only occur at specific energy levels. For a diatomic molecule like oxygen (O₂), the rotational energy levels are determined by its moment of inertia and the quantum number associated with its rotation.

When we talk about the transition from the ground state to the first excited state, it requires a specific amount of energy. In our exercise, diatomic oxygen requires \(356 \mu eV\) to jump to the next energy level, which is associated with the beginning of rotational motion. At a certain temperature, thermal energy provided by the environment is enough to populate higher rotational states. The population of these states is determined by the temperature and the energy difference between them, as given by the Boltzmann distribution.

The concept of specific heat capacity is related to the amount of heat required to raise the temperature of a substance. It indicates how much energy is needed to increase the temperature of a unit mass of a material by one degree Celsius.

For gases, the specific heat can be measured at constant volume (C_v) or constant pressure (C_p). The specific heat capacities are different because energy added at constant pressure not only increases the internal energy of the gas but also does work by expanding the volume. In the case of diatomic oxygen, the specific heat at normal pressure would reflect the contributions of translational, rotational, and vibrational motions. However, at very low temperatures as indicated in the exercise, the rotational and vibrational motions are 'frozen out,' and diatomic oxygen would behave similarly to a monatomic gas, possessing only translational kinetic energy, resulting in a lower specific heat capacity. This also aligns with the Einstein Model and Debye Model of specific heat which describe how quantum effects influence specific heat at low temperatures.