Each possible normal mode of oscillation of a distributed system constitutes a mechanical degree of freedom of the system. According to the classical equipartition-of-energy theorem, each oscillatory degree of freedom is to be assigned an average thermal energy of \(k T\), where \(k\) is Boltzmann's constant and \(T\) is the absolute temperature. Hence the internal thermal energy per unit volume of a monatomic gas should be \(E_{v}=n k T\), with \(n\) given by (5.7.11). For a mole of monatomic gas, kinetic theory tells us that the internal energy is \(E_{\mu}=\frac{3}{2} R T\), where \(R\) is the gas constant per mole. We can bring the two viewpoints into agreement by choosing a suitable value of \(\omega_{\max }\). Compute this value and show that the minimum wavelength is of the order of the mean spacing of the atoms in the gas.

The Equipartition of Energy Theorem is a fundamental principle in classical thermodynamics that relates the temperature of a system to its total internal energy. Its most important claim is that each degree of freedom that contributes to the system's energy will, on average, receive an equal fraction of the total thermal energy.

This theorem is especially informative for systems at thermal equilibrium. Because it assumes that energy is shared equally among all degrees of freedom, it predicts that for a one-atom molecule gas (monatomic gas), the energy is distributed equally among the translational motions (moving in three perpendicular directions). In mathematical terms, the theorem can be summarized by:
\[ E = \frac{f}{2} kT \]
where \( E \) is the average energy per molecule associated with each degree of freedom, \( k \) is Boltzmann's constant, \( T \) is the absolute temperature, and \( f \) is the number of degrees of freedom.

For a monatomic gas, which has only translational degrees of freedom, \( f = 3 \), representing motion in three-dimensional space. Thus, the average energy per molecule is \( \frac{3}{2} kT \), and the internal energy for a mole of such gas can be given by \( E_{\text{thermal}} = \frac{3}{2}N_AkT \), where \( N_A \) is the Avogadro's number. This formula plays a crucial role in connecting microscopic properties like degrees of freedom to macroscopic observables such as temperature and heat capacity.

The concept of oscillatory degrees of freedom becomes relevant when discussing molecular vibrations and the modes of oscillation within physical systems. In essence, an oscillatory degree of freedom corresponds to a mode in which a system can store thermal energy by oscillating about an equilibrium position.

Take for example a diatomic molecule, which in addition to translational motion, can also rotate and vibrate. Each of these motions constitutes a degree of freedom and according to the Equipartition Theorem, each degree of freedom would receive an average energy contribution of \( \frac{1}{2}kT \).

In the scenario given within the monatomic gas exercise, each normal mode of oscillation is considered an oscillatory degree of freedom. As the system reaches thermal equilibrium, each mode will be occupied by energy in the amount of \( kT \), as per the theorem. However, in a monatomic gas, the atoms do not have any vibrational or rotational degrees of freedom as they are single atoms and cannot vibrate or rotate in the same way molecules do. Therefore, in this context, we attribute all the energy to translational motion, hence preserving the energy across the three available translational degrees of freedom.

One of the pillars of statistical mechanics and thermodynamics, Boltzmann's constant \( k \) is the bridge between the macroscopic and microscopic worlds. It is a critical factor when linking the energy scale of individual particles (like atoms) to the temperature scale measured on thermometers.

Boltzmann's constant has the value of \( 1.380649 \times 10^{-23} \) joules per kelvin (\( J/K \)) and essentially defines the energy amount per temperature interval per particle. It can be derived from the ideal gas law constant \( R \), which is applicable to moles of gas, and Avogadro's number \( N_A \) , which is the number of particles in a mole, by the relation \( k = \frac{R}{N_A} \).

This constant is instrumental in statistical physics for it allows a transition from discussing energy in terms of moles (using \( R \)) to discussing energy in terms of individual particles (using \( k \)). Through Boltzmann's constant, the exercise in question establishes a connection between the energy of a single atom and that of a mole of gas. It aligns the two viewpoints of internal energy—per mole and per unit volume—by using \( k \) while addressing the relations between the thermal energy, absolute temperature, and the specific heat capacity of monatomic gases.