Compute the heat capacity at constant magnetic field \(C_{H, n}\), the susceptibilities \(X_{T}, \mathrm{n}\) and \(X_{S, n}\) ' and the thermal expansivity \(\alpha_{H, n}\) for a magnetic system, given that the mechanical equation of state is \(M=\mathrm{n} D H / T\) and the heat capacity is \(C_{M, n}=\pi c_{\prime}\) where \(M\) is the magnetization, \(H\) is the magnetic field, \(\mathrm{n}\) is the number of moles, \(D\) is a constant, \(c\) is the molar heat capacity, and \(T\) is the temperature.

When we talk about heat capacity in magnetic systems, it's essential to understand how it changes under different conditions. For our exercise, we look at the heat capacity at constant magnetic field, denoted as \(C_{H,n}\).
Given the heat capacity at constant magnetization, \(C_{M,n}=\frac{\beta}{n}\),
where \(\beta\) is some constant, we can relate these two heat capacities using thermodynamic relations.
The heat capacity at constant magnetic field considers the energy absorbed by the system when the magnetic field is kept constant. This specifically influences the internal energy, changing how the system reacts to temperature changes.

Susceptibility measures how a material reacts to an external magnetic field. In our exercise, we have two types:

• Magnetic susceptibility at constant temperature, \(\text{X}_{T,n}\)
• Isothermal susceptibility, \(\text{X}_{S,n}\)

Using the given mechanical equation of state, \(M=\frac{nDH}{T}\), where \(M\) is magnetization, \(H\) is the magnetic field, \(n\) is the number of moles, \(D\) is a constant, and \(T\) is the temperature,
We can derive both susceptibilities based on how they modify the magnetization and field interactions.
For magnetic susceptibility at constant temperature:\( \text{X}{T,n}=nD/T\).
For isothermal susceptibility,\(\text{X}_{S,n}=nD\)
Each susceptibility sheds light on the material's responsiveness to changes in magnetic field under different thermal conditions.

Thermal expansivity, denoted \(\text{alpha}{H,n}\), describes how much a material tends to expand when heated, specifically under a constant magnetic field.
From our mechanical equation of state, \(M=\frac{nDH}{T}\), we can understand how the magnetization and temperature affect the material's expansion.
Thermal expansivity is crucial in understanding the interplay between a material's volume and temperature during thermal processes in magnetic fields.
Using our given data, we differentiate with respect to temperature to derive the thermal expansivity:
\(\text{alpha}{H,n} = \frac{\text{d}V}{V\text{d}T}\),which helps us see the proportional change in volume with temperature.

The mechanical equation of state for our system is \(M=\frac{nDH}{T}\).
This relation connects magnetization \(M\), the number of moles\(n\), the magnetic field\(H\), a constant \(D\), and the temperature \(T\).
Understanding this equation is vital as it explains how magnetization varies under different field strengths and temperatures.
From this, we can derive other properties like susceptibilities and thermal expansivity by manipulating and differentiating with respect to the necessary variables.
By knowing the mechanical equation of state, we obtain a foundational understanding of how the physical properties of magnetic systems behave under varying thermal and magnetic conditions.