I'm trying to resolve some of my conceptual sticking points with the Carnot cycle. For one thing, I'm reading Reiss's Methods of Thermodynamics, and he offers a proof that the Carnot cycle obtains maximum efficiency when conducted reversibly. How can the cycle be conducted reversibly? During the two isothermal expansions, heat is exchanged and temperature is nonzero. Therefore there is is a finite nonzero change in entropy. I was under the impression that processes for which ΔS≠0 are irreversible. Is this correct? Secondly, since an adiabatic process has no heat exchange by definition, what is the difference between an adiabatic and a reversible process (or is there none)? Thirdly, there are two isothermal legs of the Carnot cycle, one wherein the engine performs work on the environment, and one for the opposite. Hence dS must be positive in one direction, and negative in the other. Since dS [itex]\geq[/itex] 0 by the second law, am I right in thinking there must be some associated dS of the environment which balances things here? Otherwise, I do not see how one can have a process for which dS < 0.