In the statement below, a physics student describes a thermodynamic cycle through which a sample of an ideal monatomic gas is taken.

$"$At the start of the cycle, the gas has a volume of $1{10}^{-3}{m}^3$ and a pressure of $1{10}^{5}Pa.$ The gas is contained inside a closed cylinder that has a movable piston on one end. Call the initial state of the gas state $A.$ Process $AB$ takes the gas at constant pressure from state $A$ to state $B,$ where $B$ has three times the volume of state $A.$ The gas is then compressed isothermally until it is back to its initial volume, reaching state $C.$ Finally, a single process $CA$ returns the gas to its initial state $A."$

$($a$)$ i$.$ On the axes below, plot states $A,B,$ and $C,$ and sketch a graph of pressure $P$ as a function of volume $V$ for cycle ABCA. Label states $A,B,$ and $C,$ and label the axes with numerical values.

ii$.$ Let the temperature for state $A$ be $T_0.$ In terms of $T_0,$ determine the temperature for state $B.$

iii. Compare the temperature at state $C$ to the temperature at state $A.$ Explain how your comparison is consistent with your graph above.

$($b$)$ i$.$ Briefly describe an experimental procedure that would take the gas from state $C$ to state $A.$

ii$.$ Relate the temperature of the gas to the microscopic processes involved in the energy transfer as the gas goes from state $C$ to state A for the procedure you described in part $($b$)($i$).$

$($c$)$ i$.$ Use the student's verbal description or your graph in part $($a$)($i$)$ to calculate a numerical value for the work done on the gas in process $AB.$

ii$.$ The magnitude of the change in internal energy for process $AB$ is $U=300J.$ Calculate a numerical value fo the energy added to or removed from the gas by heating in process $AB,$