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 Quote by Studiot A heat bath is another name for a heat reservoir. It is not part of the system undergoing the process. http://en.wikipedia.org/wiki/Thermal_reservoir It has the property of remaining at constant temperature, however much heat it accepts or provides. However the system may or may not remain at constant temperature. Heat is transferred from/to the heat bath by the system process. If pressure remains constant no work is done so all the heat transferred increases/decreases the internal energy of the system. As an example, you melt some ice at constant pressure by adding latent heat from the surroundings (heat bath).
Thanks for the reply. That actually made a lot of sense. So my problem was differenating between whether it was talking about the heat bath or the system. However, if the system were at constant temperature and pressure, Gibbs energy would never change (from that relation in my OP), right?

If you wouldn't mind, would you clarify something for me again? There's another source for this topic which basically emulates a different proof;

Note: I believe he was meant to write $-ΔW < -ΔF$ and $-ΔW = PΔV$. In this case work done is the work done on to the system.

The theorem states that for a system kept at constant temperature and constant pressure, Gibbs never increases. True. But, by that thermodynamic relation in my OP, surely it will never decrease either? Its change is always zero. In this case there seems to be no reference to the system being connected to a constant pressure/temperature heat bath. Maybe that is automatically implied?

Also I've always learnt that $ΔW = -PΔV$ is only valid for reversible processes. P being the pressure of the system. He seems to make the equivalence without reference as to whether or not the process is reversible. The only way I can make sense of this is if by;

$-W = PΔV$, P is actually referring to the applied pressure, rather than the pressure of the system. The applied pressure P is always greater than system pressure due to frictional effects, and in the case of a reversible process $P_{applied} = P_{system}$?

Edit: The fact that it says that relation is only infinitesimal reversible processes also makes no sense to me. Infinitesimal? Okay. But the process depends on state variables and thus it works for all processes, not just reversible ones?