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$\begingroup$ Just a remaining doubt: yes the entropy is a state function and as such its differential can be integrated exactly, over a reversible path. Now, why would the constant temperature of the bath imply a reversible path exists? Even the system is at constant temperature. $\endgroup$
– Smerdjakov
Commented Apr 17, 2014 at 20:52
$\begingroup$ We can introduce any amount of heat into the bath by lowering its temperature by an infinitesimal amount; this is reversible because raising the temperature by an infinitesimal amount will make the heat flow back the other way. The system is also at constant temperature, but may have other internal changes (like a chemical reaction going on), and without taking these into account we can’t calculate $\Delta S$ in the same way, because we can’t guarantee that the equivalent reversible path will end up in the right spot. $\endgroup$
– Ant
Commented Apr 17, 2014 at 22:55

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