Problem #$4(10$ points$)$

Steam at constant pipeline conditions of $300$ kPa and $250C$ passes slowly through a valve into

a well$-$insulated, frictionless, and initially evacuated piston$-$cylinder assembly. A pressure of $100$

kPa in the cylinder is needed to suppcti the piston due to its own weight and the ambient

atmosphere. The volume under the piston is $5m^3$ with $0\mathrm{.}5$ kg of water inside the cylinder at the

end of the process.

Find: $($a$)$ the boundary work done during this process, $($b$)$ the final temperature of the water in

the cylinder, and $($c$)$ the phase of water in the cylinder at the final state $($please calculate quality

if it is a saturated mixture$).$