Impossible refrigerator (thermodynamics and PV diagrams)

[Avatrin]

That's pretty much correct. Then take for instance the junction between legs D and C. In leg D, as you calculated, Q < 0, so heat is going from the working substance to the hot reservoir, so T_ws > T_h. Then in leg C Q > 0, so T_ws < T_c. What happens then at point 4 on the diagram?

Well, if we graph T_ws, there will either be an impossible discontinuity, or T_h<T_ws<T_c, which is a contradiction.

 

[DrClaude]

Well, if we graph T_ws, there will either be an impossible discontinuity,

But T_ws is a state variable, so it has a defined value at each point in the cycle.

or T_h<T_ws<T_c, which is a contradiction.

And that's the solution.

To summarize, any cycle can be made into a heat engine, as we only need to use a hot reservoir with T_h > max(T_ws) and a cold reservoir with T_c < min(T_ws). The problem with a refrigerator is that it has to cycle between T_ws < T_c and T_ws > T_h, which is not possible for all cycles.