Question #43026

Consider a composite thermodynamic system consisting of $n$ moles of an ideal monoatomic gas and a macroscopic magnetic dipole of magnetic moment magnitude $\mu$. The system is confined in a perfectly insulated cylinder with a movable frictionless piston. The total internal energy of the system is $U = U_{\text{gas}} + U_{\text{mag}}$, where $U_{\text{gas}}$ is the thermal internal energy of the gas and $U_{\text{mag}} = -\vec{\mu} \cdot \vec{B}$ is the magnetic potential energy of the dipole in a uniform external magnetic field of magnitude $B$. Due to an internal reversible mechanism, the angle $\theta$ between $\vec{\mu}$ and $\vec{B}$ is constrained by the volume $V$ of the gas such that $\cos\theta = \left(\frac{V_0}{V}\right)^2$, where $V_0$ is the initial volume. The system expands quasi-statically, doing external work $P dV$. If the initial pressure of the gas is $P_0$ and the initial angle is $\theta = 0$, what is the gas pressure $P$ when the volume reaches $2V_0$?