Consider a process during which the pressure and volume are linearly

related, i$.$e$.$

P$=$k$\_(1)$v$+$k$\_(2)$

with k$\_(1)$ and k$\_(2)$ constant

$($a$)$ Show that if this process carries a fluid from an initial state defined by

$($P$\_(1),$v$\_(1))$ to a final state defined by $($P$\_(2),$v$\_(2))$ then the work performed is

equal to the following.

w$=(1)/(2)($P$\_(1)+$P$\_(2))($v$\_(1)-$v$\_(2))$

$($b$)$ If $1\mathrm{.}5$ kg of water undergoes this process beginning at $200$ kPa and $150$

$\backslash$deg C and ending at $600$ kPa and $350\backslash$deg C$,$ how much work was performed?