The molar heat capacity at constant magnetic field of a paramagnetic solid at low temperatures varies with the temperature and field according to the relation $$ \alpha_{\mathcal{K}}=\frac{B+C \mathcal{f}^{2}}{T^{2}}+D T^{2} $$ where \(B, C\), and \(D\) are constants. What is the entropy change of \(n\) moles of material when the temperature changes from \(T_{i}\) to \(T_{f}\) while \(\mathcal{A}\) remains constant at the value \(\pi_{0} ?\)

The molar heat capacity of a substance refers to the amount of heat required to raise the temperature of one mole of the substance by one degree Celsius. In the context of paramagnetic solids at low temperatures, the molar heat capacity at constant magnetic field can be quite complex. It is given by the formula: \[ \alpha_{\mathcal{K}}=\frac{B+C \mathcal{f}^{2}}{T^{2}}+D T^{2} \] This equation takes into account various terms that depend on temperature and magnetic field. Here:

• B is a constant related to the electronic properties.
• C is a constant associated with the field-dependent term.
• D relates to temperature-dependent lattice vibrations.

As temperature varies, these terms contribute differently to the heat capacity, resulting in a non-linear relationship.

Thermodynamics at low temperatures involves studying the behavior of materials when they are extremely cold, often near absolute zero. In such conditions, quantum mechanical effects become significant. For instance, in paramagnetic solids, low temperatures influence how magnetic dipoles align with an external magnetic field. These effects are crucial to understand heat capacity, entropy changes, and other thermodynamic properties. At these temperatures:

• Thermal energy is very low, meaning particles have little kinetic energy.
• Quantum states are more distinct and impactful on the material's properties, leading to phenomena like superconductivity and superfluidity.

For paramagnetic materials, this means exploring how the interplay of thermal energy, magnetic energy, and lattice vibrations contribute to observable thermal properties.

Paramagnetic materials are substances that are weakly attracted by an external magnetic field. Unlike ferromagnetic materials, they do not retain magnetization in the absence of an external field. This property arises because paramagnetic materials have unpaired electrons that align with the magnetic field but do not have interactions strong enough to maintain this alignment once the field is removed. In low temperatures, their paramagnetic properties become more pronounced. Some important points include:

• Magnetic susceptibility increases as temperature decreases, following Curie's law.
• External magnetic fields align magnetic dipoles, reducing randomness and therefore, entropy.
• At extremely low temperatures, magnetic ordering could become significant, leading to changes in heat capacity.

These features highlight why studying paramagnetic materials at low temperatures is fascinating for understanding fundamental physical principles.

Heat capacity is a measure of the amount of heat required to change a material's temperature. In the context of a constant magnetic field, this concept becomes more intriguing as it affects the internal energy states of the material. For paramagnetic solids at low temperatures, the heat capacity at constant magnetic field is captured by the equation: \[ \alpha_{\mathcal{K}}=\frac{B+C \mathcal{f}^{2}}{T^{2}}+D T^{2} \] Here, the different terms represent:

• \( B / T^{2} \): A term inversely proportional to temperature squared, often associated with electronic contributions.
• \( C \mathcal{f}^{2} / T^{2} \): Another inversely proportional term, but dependent on the magnetic field strength \(\mathcal{f}\).
• \( D T^{2} \): A term directly proportional to temperature squared, typically linked to lattice vibrations.

As temperature changes, so does the contribution of each term, impacting how much heat energy the material can store and transfer under a magnetic field. Understanding this can be pivotal for applications in low-temperature physics and material science.