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[tex] dS \geq \frac{dQ}{T}[/tex]

Since dQ = 0 , it follows that if the process occurs reversibly dS = 0 and irreversibly dS > 0. But entropy is a state function , how could this possibly be ?

It then occured to me that the universe itself is an isolated system and its entropy is not a state function. Could we then generalize and say that the entropy of an isolated system is not a state function ?

I've found some very brief treatments on this in recent thermodynamic books , but none seems convincing to me. In one book , it is claimed that an irreversible process that corresponds to a reversible process cannot exist in an isolated system since there needs to be the 'uncompensated heat' ( N , according to Clausius ) that must be supplied to the system from the surroundings. Thus , the system is no longer isolated, and the clausius inequality holds no longer for such an irreversible path.

Makes sense ? ... any ideas ?