HW $-5($ii$)$

A closed piston$-$cylinder device contains an ideal gas $(PV=mRT)$

occupying a volume of $0\mathrm{.}05m^3$ and at an absolute temperature of $1200$ K

$($State $1)$ when the linear spring is touching the piston without exerting any

force. The gas is cooled until the piston hits the stops occupying a volume

of $0\mathrm{.}02m^3($State $2).$ Further cooling of the gas decreases the absolute

temperature to $250$ K $($State $3).$ The mass of the piston is $20$ kg $,$ its cross$-$

sectional area is $0\mathrm{.}03m^2,$ the spring constant is $1000\frac{N}{m},$ and atmospheric

pressure of $1$ bar acts on one side of the piston.

$($a$)$ Calculate the absolute pressure of the gas at States $1$ and $2,$ each in bar.

$($b$)$ Find the absolute pressure of the gas at State $3,$ in bar.

$($c$)$ Draw a $P-V$ diagram showing the two processes. Label states and identify process directions with arrows.

$($d$)$ Calculate the work for the gas during the process from State $1$ to State $3,$ in kJ$.$