A certain magnetic salt is found to obey Curie's law, and to...

A certain magnetic salt is found to obey Curie's law, and to have a heat capacity per unit volume (at constant magnetic field) inversely proportional to the square of the absolute temperature, i.e., $\chi=b / T, c_H=\alpha V / T^2$, where $\alpha=b+a H^2, a$ and $b$ being constants, and $\chi$ is the susceptibility. A sample of this salt at temperature $T_{\mathrm{i}}$ is placed in a magnetic field of strength $H$. The sample is adiabatically demagnetised by slowly reducing the strength of the field to zero. What is the final temperature, $T$, of the salt?

Key Concepts

Thermodynamics of Magnetic Systems

The thermodynamics of magnetic systems involves the study of how magnetic properties, such as susceptibility and magnetization, interplay with other thermodynamic quantities like temperature, heat capacity, and entropy. This branch of thermodynamics is pivotal in analyzing processes where magnetic work (i.e., work done by or against magnetic forces) is significant, such as in adiabatic demagnetization. It provides the framework for understanding energy transfer, equilibrium, and phase transitions in materials subject to magnetic fields.

Entropy Conservation in Adiabatic Processes

In an adiabatic process, the system does not exchange heat with its surroundings, which implies that its entropy remains constant. This conservation of entropy provides a link between the initial and final states of the system. By applying this concept, one can derive relationships between various thermodynamic variables such as temperature and magnetic field, thereby elucidating how intrinsic properties of the system change under adiabatic conditions.

Heat Capacity Dependence on Temperature

Heat capacity measures how much heat energy is needed to change the temperature of a system. In this context, the heat capacity per unit volume is given as being inversely proportional to the square of the temperature, which highlights an unusual thermal response of the material. This temperature dependence is crucial for predicting how the system exchanges energy as its temperature varies, especially under constant magnetic field conditions.

Curie's Law

Curie's law quantifies the behavior of paramagnetic materials by stating that their magnetic susceptibility is inversely proportional to absolute temperature. This fundamental concept explains how, as the temperature increases, the alignment of magnetic moments becomes less effective in producing a net magnetization in response to an applied magnetic field, making it a cornerstone for understanding magnetic responses in thermodynamic systems.

Adiabatic Demagnetization

Adiabatic demagnetization is a process where a magnetic system is de-energized (i.e., the magnetic field is reduced) under adiabatic conditions, meaning no heat is exchanged with the environment. This leads to a drop in the system’s temperature as a result of work done by the magnetic field on the system and the constancy of entropy. This phenomenon is extensively used in magnetic refrigeration and low-temperature physics to achieve temperatures lower than those attainable by conventional cooling methods.

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