Problem $3.$ A power cycle operates between reservoirs at $1000$ K and $500$ K $.$ The cycle receives energy via heat transfer at the high temperature at a rate of $1500$ kW and discharges energy via heat transfer to the cold reservoir.

a$.$ The cycle develops power at a rate of $1000$ kW $.$ Use the Clausius inequality to determine whether this cycle operates reversibly, irreversibly, or is impossible.

b$.$ Do the same as in part a$,$ but now assume the cycle power generation rate is $750$ kW $.$

c$.$ Repeat one more time, but now the power generation rate is $0$ kW $.$

d$.$ Comparing your answers to parts a$,$ b$,$ and c$,$ are you surprised? Why or why not?