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  • $\begingroup$ I am kind of following what you are saying, but I can't make sense of how it would fix what is in the quoted text. Is there anything in the quote that is incorrect, or there just needs to be a more-detailed explanation? Also, what if when $T>T_R$? $\endgroup$
    – BioPhysicist
    Commented Apr 16 at 19:06
  • $\begingroup$ It is not that it is incorrect, just it is obscure. It is obscure because it makes you conflate, for lack of better term, the Gibbs equation that holds for two infinitesimally close equilibrium states with an irreversible process that is not described by such equation. If $T>T_R$ then the entropy flow ("heat" flow) is in the opposite direction and the result is the same because then $dS<0$, the system is rejecting thermal energy and the environment absorbs it, and thus $dA=(T-T_R)dS\le 0$ or $dF\le 0$ still holds. $\endgroup$
    – hyportnex
    Commented Apr 16 at 21:11
  • $\begingroup$ I also dislike this so-called "proof" because it is really backwards from how it should be taught, I believe. That poetically speaking everything after a while turns to shit is really what the meaning of the 2nd law is. And it being the most intuitive concept of all natural phenomena it should be the at the very start of thermodynamics not something to be derived from less obvious considerations. Instead this is what Schroeder does for a very special no-work only thermal energy transport case. My preference would be to start with the axiom $dA\le 0$ or something similar and then get the rest. $\endgroup$
    – hyportnex
    Commented Apr 16 at 21:26