Problem #$2(10$ pts$)$

Located near the bottom of an insulated cylinder is a rigid plate of negligible mass. Between the rigid plate and the bottom of the cylinder is $0\mathrm{.}5kg(m_s)$ of air at $250$kPa$(P_a)$ and $0\mathrm{.}287k\frac{J}{k}g-K.$ Between the rigid plate and a semi$-$constrained piston $0\mathrm{.}1kg(m_w)$ exists superheated water vapor at $400$ Celsius $(T_{w1})$ and $150$kPa$(P_w).$ Consider these to be the initial conditions $($state $1)$ of this multi$-$component system, before the energy conducts between the two substances. The system boundary includes the air, the water and the rigid plate.

The rigid plate is conductive allowing heat to pass between the air and the water vapor, but the piston is well$-$insulated. The water vapor is manually compressed until the volume of water vapor $(V_w)$ is halved and the pressure is $300$kPa$(P_w),$ at which point the air and water vapor are in temperature equilibrium. Determine the following:

a$.$ The final temperature of the water vapor, $T_2,$ in Celsius

b$.$ The boundary work for the water vapor

c$.$ The final pressure of the air, $P{a}_2,$ in kPa

d$.$ The change in entropy of the air, $S_{a2}-S_{a1},$ in $k\frac{J}{K}$

e$.$ The change in entropy of the water, $S_{w2}-S_{w1},$ in $k\frac{J}{K}$

f$.$ The change in entropy of the combined system, $S_2-S_1$ in $k\frac{J}{K},$ neglecting the mass of the rigid plate

g $.$ The entropy produced, $,$ of the combined system during the process

h$.$ Draw a T$-$s and P$-$v diagram for the water

i$.$ Draw a T$-$s and P$-$v diagram for the air