$7.(20$ points$)$ Air initially at $300$ K and $200$ kPa $,$ in a piston cylinder assembly undergoes the following cycle that consists of three processes.

$\backslash (1\backslash$rightarrow $2\backslash )$ : Adiabatic expansion to a pressure of $100$ kPa

$\backslash (2\backslash$rightarrow $3\backslash )$ : Constant volume heating to $500$ K

$\backslash (3\backslash$rightarrow $1\backslash )$ : Constant pressure compression to the initial conditions.

a$.$ Fix each state by determining the pressure, temperature, specific volume and specific internal energy.

b$.$ Determine the specific work and specific heat transfer for each process.

c$.$ Find the net specific work and the net specific heat transfer for the cycle and explain if your answer makes sense.