Hi all, I've been having some difficulty understanding the derivation of the Fundamental Theorem of Thermodynamics, [itex]dU=T \ dS-P \ dV[/itex]. The derivation, which can be found at Wikipedia (http://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation) first starts with the universal First Law of Thermodynamics, [itex]dU=dQ-dW[/itex] and then supposes that the situation is reversible, allowing the substitution of [itex]T \ dS[/itex] and [itex] P \ dV [/itex] for [itex]dQ[/itex] and [itex]dU[/itex] respectively. I stress again these individual substitutions are only valid for the reversible case. However, at this point, both Understanding Thermodynamics by Van Ness and Wikipedia do something I don't quite understand. They say that, even through we derived the equation for the reversible case, because this equation only contains functions of state (or 'properties of the system' in Van Ross' terms) this equation must apply for all processes, both reversible and irreversible. I do not understand this at all. These substitutions were only valid for reversible processes and, as far as I can see, from the moment you make these substitutions and implicitly assume that the process is reversible, the equation can only be applied (with certainty) to the reversible case, irrespective of its final form. As a 'counter-example' to this logic, consider the equation [itex]dS=0[/itex]. This is applicable for all adiabatic, reversible processes - any reversible processes that take place within an 'insulated' environment. Now why can't we apply this same logic and say that, because this equation only contains 'functions of state' (just [itex]S[/itex]) then, logically, all processes, both reversible and irreversible, which occur in this insulated environment must obey this 'law'. Because of course this final conclusion would be false. I would appreciate it immensely if anyone could lead me in the right direction, thanks!