Problem $6$: $(20$ pts$)$ A piston cylinder contains $0\mathrm{.}1$ kg water at $500C$ and $1000$ kPa $.$ The piston has a stop at half the original volume. The water now cools to room temperature $25C.$

$($a$)$ Sketch the cooling process on a $P-v$ diagram.

$($b$)$ Find the final pressure and volume

$($c$)$ Find the work and heat transferred during the process.