$(1)$ Shown below is a PV$-$Digram for a refrigerator which follows a cycle called a Brayton cycle. A Brayton cycle consists of two isobaric processes and two adiabatic processes. Note: The numbers on this chart are rounded. Any number that should be an integer can be rounded to the nearest number.

$($a$)$ How many degrees of freedom does this gas have?

$($b$)$ Is this gas monatomic, diatomic, or something else?

$($c$)$ How much heat is transferred into and out of the system during one full cycle? Please report the heat transferred into the system and out of the system separately with the appropriate label $Q_C$ and $Q_H.$

$($d$)$ What is the total work done on the gas by the surroundings for this refrigerator?

$($e$)$ What is the change in thermal energy for the gas in the refrigerator for one for cycle?