$3.(40$ points$)1\mathrm{.}2$ kg of air $\backslash (\backslash$left$(\backslash$mathrm$\{$R$\}=0\mathrm{.}287\backslash$mathrm$\{$~kJ$\}/\backslash$mathrm$\{$kg$\}^\{*\}\backslash$mathrm$\{$~K$\},\backslash$mathrm$\{$C$\}\_\{\backslash$mathrm$\{$P$\}\}=1\mathrm{.}2\backslash$mathrm$\{$~kJ$\}/\backslash$mathrm$\{$kg$\}^\{*\}\backslash$mathrm$\{$~K$\}\backslash$right$)\backslash )$ is initially contained in a piston$-$cylinder at $300$ K and $100$ kPa $($state $1).$ The air undergoes the following processes:

$1-2$: Isothermal compression to one$-$fourth of the initial volume

$2-3$: Isochoric heat addition to a maximum temperature of $1000$ K

$3-4$: Isothermal expansion to the initial pressure

$4-1$: Isobaric compression to the initial state

Use specific heats for air as an ideal gas at room temperature conditions.

$($a$)(6$ points$)$ Sketch the cycle on a P$-$V diagram.

$($b$)(12$ points$)$ Calculate the temperature, pressure, and volume at each state in the cycle. HINT: You do not need to go through the cycle states in numerical order $-$ carefully determine which values are known first from the given information, and relate between states using the known processes.

$($c$)(12$ points$)$ Calculate the change in internal energy, work, and heat transfers for each process in the cycle.

$($d$)(5$ points$)$ Calculate the cycle thermal efficiency.

$($e$)(5$ points$)$ Calculate the second law efficiency of the cycle, assuming that heat is received at the maximum temperature in the cycle and rejected to the minimum temperature in the cycle.