Problem $1.$ A piston$-$cylinder assembly fitted with stops contains $0\mathrm{.}209$ kg of water, initially at $1\mathrm{.}0$MPa,$581C.$ The water undergoes two processes in series:

Process $1-2$: Constant$-$pressure cooling until the piston face rests against the stops. The volume occupied by the water is then one$-$half its initial volume.

Process $2-3$: With the piston face resting against the stops, the water cools to $28C.$

$($a$)$ Sketch the two processes on a p$-$v diagram.

$($b$)$ Neglecting kinetic and potential energy effects, evaluate for each process the work and heat transfer, each in kJ$.$

Problem $2.$ A reversible power cycle receives energy $Q_H$ by heat transfer from a hot reservoir at $T_H$ and rejects energy $Q_C$ by heat transfer to a cold reservoir at $T_C.$

$($a$)$ If $T_H=1165K$ and $T_C=438K,$ what is the thermal efficiency?

$($b$)$ If $T_H=530C,T_C=30C,$ and $W_{\text{c}\text{y}\text{c}\text{l}\text{e}}=1081kJ,$ what are $Q_H$ and $Q_C,$ each in kJ $?$

$($c$)$ If $=64\%$ and $T_C=37F,$ what is $T_H,$ in $F?$

$($d$)$ If $=49\%$ and $T_H=744C,$ what is $T_C,$ in $C?$