(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 very low temperatures near absolute zero, the Debye model provides a good approximation of a solid's heat capacity. The Debye model states that the heat capacity at constant volume, Cv, is proportional to the cube of the temperature (T^3). Mathematically, \[ C_v = k \cdot T^3, \] where k is a proportionality constant. As temperature increases near 0 K, the cube of temperature also increases. Due to this cubic relationship, the heat capacity (Cv) rises with increasing temperature. This is known as the Debye T^3 law, valid for low-temperature regions.

As temperature increases and moves far away from 0 K, the Debye model no longer provides an accurate approximation for heat capacity. In this higher temperature range, the heat capacity of a solid follows the classical limit or Dulong-Petit's law and becomes virtually independent of temperature. According to the Dulong-Petit's law, the heat capacity at constant volume (Cv) approaches a constant value, which is 3R, where R is the gas constant. This occurs because the vibrational energy levels of atoms in a solid become more closely spaced at higher temperatures, leading to a more continuous distribution of energy. As a result, the heat capacity plateaus at a value close to 3R and becomes virtually independent of temperature. In summary, at temperatures near 0 K, the heat capacity (Cv) rises due to the Debye T^3 law, while at high temperatures (far removed from 0 K), Cv becomes virtually independent of temperature because of the Dulong-Petit's law.