A rigid cylindrical container of fixed total volume \(2V_0\) is divided into two equal compartments, X and Y, by a light, frictionless, thermally insulating piston that can move freely. Initially, both compartments contain equal numbers of moles of an ideal gas at pressure \(P_0\), temperature \(T_0\), and volume \(V_0\). Compartment Y is held at constant temperature \(T_0\) by a thermal reservoir, while compartment X is slowly heated until the gas inside it reaches a uniform absolute temperature of \(2T_0\). What is the final volume of compartment X after the system reaches mechanical equilibrium?