$(20$ points$)$ An insulated spring$-$loaded piston$/$cylinder device, shown in the figure, is connected to an air line flowing ait at $600$ kPa and $700$ K by a valve. Initially, the cylinder is empty and the spring force is zero. The valve is then opened until the cylinder pressure reaches $300$ kPa $.$ Air can be modeled as an ideal gas with constant specific heats. Take $P_0=100$kPa. Find the final temperature $T_2$ and the work done by air per unit mass, $w_{12}.$