$2)(55$ points$)$ A heat engine uses air $($diatomic$)$ as the working fluid in a three$-$step cycle. First, a volume of $\backslash (2\backslash$mathrm$\{$~L$\}\backslash$left$(2\backslash$times $10^\{-3\}\backslash$mathrm$\{$~m$\}^\{3\}\backslash$right$)\backslash )$ at $300$ K and $1$ atm $($state $1)$ is heated under constant volume to a temperature of $1000$ K $($state $2).$ Second, the gas is expanded adiabatically to a pressure of $1$ atm $($state $3)$ and volume $\backslash (4\mathrm{.}726\backslash$mathrm$\{$~L$\}\backslash$left$(4\mathrm{.}726\backslash$times $10^\{-3\}\backslash$mathrm$\{$~m$\}^\{3\}\backslash$right$)\backslash ).$ Third, the gas is allowed to cool under constant pressure back to the original state $($state $1).$

$($a$)$ Draw the three$-$step cycle on a P$-$V diagram and name each leg in the process. Label states $1,2,$ and $3.$

$($b$)$ Calculate the number of moles of the gas.

$($c$)$ Calculate the pressure at state $2.$

$($d$)$ Calculate the temperature at state $3.$

$($e$)$ Calculate the change in internal energy of the gas during each of the process steps $1-2,2-3,$ and $3-1$ and for the entire cycle.

$($f$)$ Calculate the heat added to the gas during each of the process steps $1-2,2-3,$ and $3-1$ and for the entire cycle.

$($g$)$ Calculate the work done by the gas during each of the process steps $1-2,2-3,$ and $3-1$ and for the entire cycle.

$($h$)$ Using the previous heat and work calculations, draw the schematic heat and work energy transfers diagram for the engine and show the amounts for $\backslash ($ Q$\_\{$H$\},$ Q$\_\{$L$\}\backslash ),$ and $\backslash ($ W $\backslash )$ for one cycle. Calculate the heat engine efficiency. Does the engine obey the first law of thermodynamics?

$($i$)$ Calculate the change in entropy of the gas during each of the process steps $1-2,2-3,3-1$ and for the entire cycle. Does the heat engine obey the second law of thermodynamics?