The molar heat capacity at constant volume of a metal at low temperatures varies with the temperature according to the equation $$ \frac{C_{V}}{n}=\left(\frac{124.8}{\Theta}\right)^{3} T^{3}+\gamma T $$ where \(\Theta\) is the Debye temperature, \(\gamma\) is a constant, and \(C_{V} / n\) is measured in units of \(\mathrm{mJ} / \mathrm{mol} \cdot \mathrm{K}\). The first term on the left is the contribution attributable to lattice vibrations and the second term is due to the contribution of free electrons. For copper, \(\Theta\) is \(343 \mathrm{~K}\) and \(\gamma\) is \(0.688 \mathrm{~mJ} / \mathrm{mol} \cdot \mathrm{K}^{2}\). How much heat per mole is transferred during a process in which the temperature changes from 2 to \(3 \mathrm{~K}\) ?

The Debye temperature, \( \Theta \), is a fundamental property that characterizes the behavior of lattice vibrations within a material. It is named after physicist Peter Debye. This temperature sets a scale for quantum mechanical effects on the heat capacity of solids. At temperatures below \( \Theta \), the specific heat capacity of a solid is significantly influenced by the quantization of its vibrational modes. In our example, copper has a Debye temperature of 343 K, which signifies the temperature below which its heat capacity starts to deviate from classical predictions. Understanding the Debye temperature is crucial because it allows us to predict how the heat capacity varies with temperature, especially at low temperatures.

Heat transfer in thermodynamics refers to the exchange of thermal energy between physical systems. This transfer can occur via conduction, convection, or radiation. In our problem, we are concerned with the heat transferred per mole of a substance as it is heated from one temperature to another. We use the equation for molar heat capacity to calculate this transfer. The molar heat capacity at constant volume for a metal is given by \[ \frac{C_{V}}{n}=\frac{124.8}{\Theta}^{3} T^{3} + \gamma T \] where \gamma is a constant. By integrating this expression between two temperatures, we find the total amount of heat transferred during the process. This integration accounts for both lattice vibrations and the contribution from free electrons.

Lattice vibrations, or phonons, are quantized modes of vibrations occurring in the crystal lattice of a material. These vibrations are crucial for understanding the thermal properties of solids. The first term in the heat capacity equation, \[ \left(\frac{124.8}{\Theta } \right)^{3} T^{3} \], represents the heat capacity due to lattice vibrations. At low temperatures, the energy levels associated with these vibrations become quantized, meaning that only certain discrete energies are allowed. This quantization explains why the heat capacity follows a \( T^3 \) dependence at low temperatures, a behavior predicted by the Debye model. Understanding lattice vibrations helps in predicting how solids conduct heat and how their heat capacity changes with temperature.

In metals, free electrons play a significant role in thermal and electrical properties. The second term in the heat capacity equation, \[ \gamma T \], accounts for the contribution from these free electrons. Electrons can move relatively freely within the metal lattice, and their energy distribution follows the Fermi-Dirac statistics. This free electron contribution to heat capacity typically varies linearly with temperature, unlike lattice vibrations. In our problem, the constant \gamma is provided as 0.688 mJ/mol·K² for copper. This value allows us to calculate the portion of the heat capacity due to free electrons, combining this with the lattice vibration term to find the total heat transferred.

Low temperature thermodynamics deals with the behavior of materials at temperatures close to absolute zero. At these temperatures, classical theories often fail to describe the thermal properties accurately, and quantum effects become significant. One of the cornerstones of low temperature thermodynamics is understanding how heat capacities deviate from classical predictions. As shown in the exercise, the heat capacity of a metal at low temperatures can be expressed as a sum of contributions from lattice vibrations and free electrons. By studying the temperature dependence of these terms, we gain insights into the quantum mechanical nature of solids. This field is crucial for applications like cryogenics and understanding superconductivity.