Prove carnot engine is most efficient According to Carnot's theorem, Carnot engine is the most efficient engine. I wanted to know its proof. I looked up and I found a proof on Khan Academy in which he used a reverse carnot engine to prove that an engine more efficient than Carnot engine cannot exist. 
But my question is can there be an engine not having 2 adiabatic and 2 isothermal processes, and more efficient or equally efficient to Carnot Engine? If not, can anyone please give me an idea why having 2 isothermal and 2 adiabtic processes makes it the most efficient and why not any other series of processes?
 A: Yes, there can be engines NOT having 2 adiabatic and 2 isothermal processes and still be as efficient as the Carnot engine; however, no engine can be more efficient than the Carnot engine with both of them operating between the same 2 thermal reservoirs. Your query is addressed in Carnot's 2nd postulate which says: All reversible heat engines operating between the same 2 thermal reservoirs (i.e. fixed temperatures) have the same efficiency. A nice proof of this is given in https://www.youtube.com/watch?v=5q_MMdGINgQ&list=PLA17CE108925DC94F&index=7. The proof of this postulate starts at 00:09:30 in the video.
A: Posting it here as I exceeded the character count (Please let me know if there's a way around it!). I'll give you the answer to part a) and the same logic can be extended to part b). The difference between the reversible heat addition process in a Carnot engine and the heat addition in part a) of the problem is that in the Carnot engine case, the heat transfer was isothermal and reversible (internally and externally). However, in part a), as heat is added, the temperature of the system rises from T0 to nT0. Thus, the heat transfer is externally irreversible as the temperature of the system boundary and that of the heat source (assumed to be a constant temperature throughout!) have a finite difference during the heat addition process. Remember, that heat transfer across a finite temperature difference is irreversible and this is exactly what causes the efficiency obtained in part a) to be lesser than that proposed by Carnot. However, there are no internal irreversibilities in the system as we can see that there is a bold line joining (P1,V1) to (P2,V1) implying that the system has a well-defined state at every point of the heat addition process. If you find the concept of external and internal irreversbility confusing, I'd suggest you to watch the 4 lectures on 2nd law and its corollaries here: https://www.youtube.com/watch?v=lvy8h-yWhRQ&list=PLA17CE108925DC94F&index=6
