$9.$ A heat engine adiabatically compresses a diatomic gas at $300$ K from a volume of $\backslash (4000\backslash$mathrm$\{$~cm$\}^\{3\}\backslash )$ to a volume of $\backslash (1000\backslash$mathrm$\{$~cm$\}^\{3\}\backslash )$ and a pressure of $400$ kPa $.$ The gas is then isothermally expanded back to the orginal volume. Finally, the gas pressure is isovolumetrically lowered to the original pressure.

$($a$)$ Draw the complete process on a $\backslash ($ P $\backslash )$ diagram. Make sure to correctly label the values on the axes.

$($b$)$ Determine the original pressure and temperature of the gas.

$($c$)$ Determine $\backslash (\backslash$Delta E$,$ W$\_\{$b y$\}\backslash ),$ and $\backslash ($ Q $\backslash )$ for each of the three processes. Tabulate your results in a table.

$($d$)$ How much work does this engine do per cycle? What is the thermal efficiency of the engine? Can an engine such as this one actually be built? Why or why not?