Compute the entropy, enthalpy, Helmholtz free energy, and Gibbs free energy of a paramagnetic substance and write them explicitly in terms of their natural variables when possible. Assume that the mechanical equation of state is \(m=(D H / T)\) and that the molar heat capacity at constant magnetization is \(c_{\mathrm{m}}=c\), where \(m\) is the molar magnetization, \(H\) is the magnetic field, \(D\) is a constant, \(c\) is a constant, and \(T\) is the temperature.

Entropy is a measure of the disorder or randomness in a system. In thermodynamics, entropy is represented by the symbol \(S\). For a paramagnetic substance, we can calculate the entropy using the formula:
\[ dS = \frac{1}{T} (dU + PdV - H dm) \]
Since the volume \(V\) and pressure \(P\) remain constant, the equation simplifies to:
\[ dS = \frac{1}{T} (dU - H dm) \]
Given the mechanical equation of state, \( m = \frac{DH}{T} \), we find that:
\[ dm = d \left( \frac{DH}{T} \right) = \frac{D dH}{T} - \frac{DH dT}{T^2} \]
Substituting \(dU = c_m dT = c dT\) and the expression for \(dm\), we get:
\[ dS = \frac{c}{T} dT + \frac{DH}{T^2} dT - \frac{D}{T} dH \]
Integrating this gives us the entropy as:
\[ S = c \ln{T} + \frac{DH}{T} \]
This equation shows how entropy depends on temperature and magnetic field intensity, incorporating constants \(c\) and \(D\).

Enthalpy is a measure of the total energy of a thermodynamic system, including both internal energy and the energy required to displace its environment to make room for it. In thermodynamics, enthalpy is denoted by the symbol \(H\):
\[ H = U + PV \]
For our paramagnetic substance, the volume \(V\) and pressure \(P\) are constant, so the equation simplifies to:
\[ H = U \]
Given that \(U = cT\) from integrating the molar heat capacity, we get the enthalpy as:
\[ H = cT \]
This result shows that enthalpy is directly proportional to temperature when pressure and volume remain constant.

The Helmholtz free energy, denoted as \(F\), measures the useful work obtainable from a thermodynamic system at constant temperature and volume. It is defined as:
\[ F = U - TS \]
For our paramagnetic substance, substituting \(U = cT\) and \(S = c\ln{T} + \frac{DH}{T}\), we get:
\[ F = cT - T \left( c \ln{T} + \frac{DH}{T} \right) \]
Simplifying this expression, we find:
\[ F = cT - Tc \ln{T} - DH = cT (1 - \ln{T}) - DH \]
Thus, the Helmholtz free energy depends on temperature, magnetic field intensity, and the constants \(c\) and \(D\). It provides insight into the equilibrium properties of the system at constant volume and temperature.

The Gibbs free energy, symbolized by \(G\), is a crucial concept in thermodynamics because it tells us about the maximum amount of work that can be performed by a thermodynamic process at constant pressure and temperature. It is defined as:
\[ G = H - TS \]
Using the values \(H = cT\) and \(S = c\ln{T} + \frac{DH}{T}\), we calculate:
\[ G = cT - T\left( c \ln{T} + \frac{DH}{T} \right) \]
Simplifying gives:
\[ G = cT - Tc \ln{T} - DH = cT (1 - \ln{T}) - DH \]
This formula indicates that Gibbs free energy depends on the system's temperature, magnetic field intensity, and constants \(c\) and \(D\). It helps to predict the direction of chemical processes and reactions in the system.