One mole of air, initially at $300C$ and $9$ bar undergoes a series of mechanically reversible processes. The process starts with a decrease in pressure adiabatically to $2$ bar, followed by expansion to four times of its initial volume at constant pressure. The change continues with an increase of pressure to $3$ bar at constant volume. It finally returns to its initial state through the adiabatic process followed by isochoric process. Assume air is an ideal gas and the heat capacities are independent of temperature: heat capacity at constant volume, $C_V=(\frac{5}{2})R$ and heat capacity at constant pressure, $C_P=(\frac{7}{2})R.$

a$.$ Sketch the processes in a single pressure$-$volume $(P-V)$ diagram.

$[5$ marks$]$

b$.$ Determine the heat transferred, $Q(J)$ and work required, $W(J)$ for each process.

$[15$ marks$]$

c$.$ If the process is irreversible with an efficiency of $85\%,$ determine the heat transferred, $Q(J)$ and work required, $W(J)$ for each process.

$[6$ marks$]$