Efficiency of heat engines

I have been taught that the an engine running on the Carnot's cycle has the maximum energy among the maximum efficiency, however I never found a convincing proof for it. Wherever I have put this question up, everyone just said pit it up against any engine and it will come out with a higher efficiency.

I would like to know if there is any way to prove mathematically that the Carnot's cycle has the maximum efficiency among any cycle?

Also, I have the belief that the second law of thermodynamics is a consequence of the fact that the Carnot's cycle has the maximum efficiency among any cycle. Because the Carnot's cycle has come before the second law of thermodynamics, I suppose that the former points to the latter. Is it actually true that the second law of thermodynamics comes from Carnot's theorem, and is it possible to prove that a Carnot's cycle would have the maximum efficiency for a particular amount of heat given in?

• What do you mean by maximum energy. Nov 24, 2019 at 14:16
• Sorry it's efficiency Nov 24, 2019 at 15:58
• OK, then see my answer Nov 24, 2019 at 16:08
• It if I remember the definition of entropy comes from the fact that Carnot's cycle has the highest entropy. Nov 24, 2019 at 16:19
• The Carnot cycle is a reversible cycle in which zero entropy is generated. Not sure where you got the idea of it having the "highest entropy". Nov 24, 2019 at 16:34

Assuming you mean maximum efficiency when you say maximum energy, then the proof is by contradiction. Essentially, in order for a heat engine to be more efficient than a Carnot engine it would have to violate the second law of thermodynamics. Rather than for me to give you that proof, you can see it for yourself in the following link:

https://en.wikipedia.org/wiki/Carnot%27s_theorem_%28thermodynamics%29

Hope this helps.