We consider a piston$-$cylinder system attached to a spring with a linear constant $\backslash ($ k $\backslash ).$ Inside the cylinder we have $1$ kg of water, initially at $\backslash (120^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ with quality $\backslash ($ x $\backslash )$ of $\backslash (20\backslash \%\backslash ).$ We then heat the system until pressure rises to $1$ MPa and temperature is $\backslash (250^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ We assume that the process is slow enough to be quasi$-$static.

$1.$ What is the initial pressure? What is the initial and final state of water? What is the initial and final volume? What is the initial and final internal energy?

$2.$ What are forces applied on the piston? Considering that such forces satisfy equilibrium in this quasi$-$static process, show that pressure $\backslash ($ P $\backslash )$ inside the cylinder increases linearly with its volume $\backslash ($ V $\backslash )($ie$,$ show that we can write $\backslash ($ P$=$a$+$b V $\backslash )$ where $\backslash ($ a $\backslash )$ and $\backslash ($ b $\backslash )$ are some constants$).$

$3.$ Using $1$ and $2,$ draw the path followed in this process in a P$-$V diagram. $($Show regions for different phases.$)$

$4.$ Show that the boundary work on the gas is $\backslash ($ W$=-\backslash$frac$\{$P$\_\{$I$\}+$P$\_\{$F$\}\}\{2\}\backslash$left$($V$\_\{$F$\}-$V$\_\{$I$\}\backslash$right$)\backslash ),$ where subscripts $\backslash ($ I $\backslash )$ and $\backslash ($ F $\backslash )$ refer to initial and final states. $($Hint: This does not require a complicated computation.$)\backslash ($ W $\backslash )$ therefore does not explicitly depend on many details such as initial position of spring, mass of the piston, spring constant...

$5.$ Compute the value of $\backslash ($ W $\backslash ).$ Using the first law, deduce the amount of heat $\backslash ($ Q $\backslash )$ injected into the system.