At low temperatures a solid's specific heat is approximately proportional to the cube of the absolute temperature; for copper \(c=31(T / 343 \mathrm{K})^{3} \mathrm{J} / \mathrm{g} \cdot \mathrm{K} .\) Integrate Equation 16.3 in differential form \((d Q=m c d T)\) to find the heat required to bring a \(40-g\) sample of copper from \(10 \mathrm{K}\) to \(25 \mathrm{K}\)

Low temperature physics, also known as cryophysics, is a branch of physics that deals with the behavior of materials and phenomena that occur below the freezing point of water, particularly those found near absolute zero. At such low temperatures, solids exhibit unique thermal properties that differ significantly from their behavior at higher temperatures. For instance, the specific heat of a solid at low temperatures doesn't increase linearly with temperature but instead follows a power law, as represented in the exercise by the specific heat capacity formula for copper: \(c=31(T / 343 \textrm{K})^{3} \textrm{J} / \textrm{g} \textrm{K}\).

As temperature approaches absolute zero, certain quantum mechanical effects become more pronounced. A well-known example is superconductivity, where certain materials exhibit zero electrical resistance and the expulsion of magnetic fields. Understanding these phenomena is crucial for advancing technologies such as magnetic resonance imaging (MRI), superconducting magnets for particle accelerators, and quantum computing.

Copper is a metal with a high thermal conductivity, which means it is very efficient in transferring heat. However, at low temperatures, the thermal properties of copper change. According to the Debye model, as the temperature decreases, the specific heat capacity of copper does not remain constant but instead is proportional to the cube of the temperature, given by the specific heat capacity formula in the exercise. This is a result of the quantization of vibrational energy levels in the copper's lattice structure.

The formula \(c=31(T / 343 \textrm{K})^{3} \textrm{J} / \textrm{g} \textrm{K}\) shows how the specific heat capacity at low temperatures is much lower than at room temperature, reflecting how much less energy is required to raise the temperature of the copper as it is cooled towards absolute zero. Understanding these properties is not only academically interesting but also has practical implications in industries that require precise temperature control and in designing materials for low-temperature applications.

Integration is a core mathematical tool in physics, particularly when dealing with variables that are continuously changing, like in the case of thermal energy transfer. In the context of our exercise, to find the total amount of heat required to change the temperature of a substance, we need to integrate the heat capacity over the temperature range of interest. The specific heat capacity is not constant and is dependent on temperature, which makes the integration necessary.

In our example, you're asked to integrate the formula for specific heat over the given temperature range to find the total heat required to raise the temperature of the copper sample. The integration process turns the variable specific heat formula into a cumulative measure of heat energy over the entire range of temperatures, from the initial 10K to the final 25K. Performing this integration gives physicists and engineers a practical way to calculate the thermal energy required in processes involving heat transfer in materials like copper, which is an essential step in many applications of thermal physics.