Abstract: There is heat exchange between two real gases at the same temperature. According to the first law of thermodynamics, the temperature changes of V1 and V2 in adiabatic cycles are not equal to 0. When the temperature change of the cycle is less than 0, thermal work conversion is achieved. Set a single heat source and restore the initial temperature T0 of the system. This is the second type of perpetual motion machine.
Please refer to the attached diagram for details
Please make a drawing for your system. Right now I cannot understand what you mean.
I am afraid that you are deeply wrong, as you overestimate your communication capability. Probably everything is clear to you personally but this is not enough. You have to make it clear to others as well. But after all, this is your choice. If you do not want to discuss your question it is up to you. In general, before you write equations, you should clearly specify your thermodynamic system. You have not done it so far. A good drawing in this respect helps a lot.
If you cannot help me to understand your question, I cannot help you with your question.
As for your picture, it is a result of confusion. The second law says that the maximum thermal engine efficiency is independent from a working fluid. As a result, you can take any fluid.
Well, the simplest is to employ the ideal gas, but feel free to take any fluid. If you do it correctly, the result will be the same.
You are just repeating the content of the book, and I am also aware of this knowledge. But you need a more general perspective to view the second law of thermodynamics as a whole. My paper on dual gas perpetual motion machine is understandable to Chinese students and there is no requirement for additional content.
Indeed, there were and there are people who did / do not like the second law. Yet, they were unable to make a perpetural motion of the second kind. There is nice book about such efforts:
Capek, Vladislav, and Daniel P. Sheehan. Challenges to the second law of thermodynamics. Dordrecht: Springer, 2005.
"In this volume we will attempt to remain clear on this point; that is, while the second law might be potentially violable, it has not been violated in practice. This being the case, it is our position that the second law should be considered absolute unless experiment demonstrates otherwise."
I have answered this question in your new post:
https://www.researchgate.net/post/The_second_law_of_thermodynamics_switches_formulas_surreptitiously_in_mathematical_calculations
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