(a) Briefly explain why \(C_{v}\) rises with increasing temperature at temperatures near \(0 \mathrm{~K}\). (b) Briefly explain why \(C_{v}\) becomes virtually independent of temperature at temperatures far removed from \(0 \mathrm{~K}\).

At extremely low temperatures near 0K, most of the atoms or molecules in a substance have their lowest energy states, known as the ground state. As temperature increases, some of these particles gain enough thermal energy to access higher energy states (known as excited states), leading to a higher heat capacity. The increase in heat capacity (Cv) at constant volume with temperature near 0K can be described by the Debye's law: Cv = AT^3 for T -> 0. Here, A is a constant. According to this law, the heat capacity at constant volume increases with the cube of temperature close to absolute zero.

At higher temperatures far from 0K, a significant proportion of the atoms or molecules occupy higher energy states. As a result, the energy gap between the ground state and higher energy states is no longer significant when compared to the total thermal energy of the system. In this scenario, the heat capacity at constant volume (Cv) becomes virtually independent of temperature. This behavior can be explained by Dulong-Petit's law for most solids (excluding some low-dimensional systems or strongly anharmonic systems), which states: Cv ≈ 3R (for T -> infinity) Here, R is the gas constant. Dulong-Petit's law indicates that, at high temperatures, the heat capacity at constant volume becomes virtually independent of temperature and approaches a constant value of approximately three times the gas constant.