One mole of an ideal paramagnetic gas obeys Curie's law, with a Curie constant \(C_{\mathrm{C}}\). Assume that the internal energy \(U\) is a function of \(T\) only, so that \(d U=C_{V, m} d T\), where \(C_{V, m}\) is a constant heat capacity. (a) Show that the equation of the family of adiabatic surfaces is $$ \frac{C_{V, m}}{n R} \ln T+\ln V=\frac{\mu_{0} m^{2}}{2 n R C_{\mathrm{C}}}+\ln A $$ where \(A\) is a constant for one surface. (b) Sketch one of these surfaces on a \(T V\) diagram.

The ideal gas law is a fundamental equation in thermodynamics that describes the relationship between the pressure (P), volume (V), and temperature (T) of an ideal gas. Mathematically, it is defined as: \[ PV = nRT \] Here, \(n\) is the number of moles of the gas, and \(R\) is the ideal gas constant. This law helps us understand how gases behave under different conditions of temperature and pressure. For instance, if the pressure is held constant, increasing the temperature will cause the volume to increase. Similarly, if the volume is kept constant, increasing the temperature will increase the pressure. It is essential for deriving other relations in thermodynamics, such as the ones involving adiabatic processes.

Curie's law describes the behavior of paramagnetic materials in response to applied magnetic fields and temperature. It states that the magnetization (M) of a paramagnetic material is directly proportional to the applied magnetic field (H) and inversely proportional to the absolute temperature (T):
\[ M = \frac{C}{T} H \]
Here, \(C\) is the Curie constant which depends on the material's nature. This principle helps in understanding how the internal energy of a paramagnetic gas is affected by temperature, which is crucial for deriving the equation of adiabatic surfaces in the provided exercise. Utilizing Curie’s law, we can identify how magnetic properties change, influencing the system's overall behavior during thermodynamic processes.

In an adiabatic process, there is no heat exchange with the surroundings. That means the system's internal energy changes solely due to the work done. For an ideal gas undergoing an adiabatic process, the change in internal energy (\(U\)) equals the work done on or by the system. The equation derived in the exercise, \ \frac{C_{V, m}}{n R} \text{ln} T + \text{ln} V = \frac{\text{\textmu}_{0} m^2}{2 n R C_{\text{C}}} + \text{\textln} A \ \, represents a family of surfaces where each surface corresponds to a specific adiabatic process.
By plotting one of these surfaces on a \(TV\) diagram, you can visualize how temperature and volume vary under adiabatic conditions. This equation encapsulates the combined effects of temperature, volume, and internal properties defined by Curie's law and the ideal gas constant.

The First Law of Thermodynamics, also known as the law of energy conservation, states that the total energy in an isolated system remains constant. It is formulated as:
\[ dU = dQ - dW \] ewline Here, \(dU\) represents the change in internal energy, \(dQ\) is the heat added to the system, and \(dW\) is the work done by the system. In the context of adiabatic processes, \(dQ = 0\), meaning no heat is exchanged. Thus, \(dU = -dW\).
For an ideal paramagnetic gas with internal energy as a function of temperature only, this relationship simplifies further. By integrating over relevant variables, like temperature and volume, and applying specific constraints, we derive key thermodynamic equations useful for practical applications, such as the adiabatic surface equation derived in the exercise.