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When we speak about entropy, it's classically defined as

$\delta S = \frac {\delta Q}{T}$

Where, I emphasize that it's the definition for only a change in entropy and not entropy itself (That's how I understood it).

And now, while speaking of the Gibbs Free Energy in Thermodynamics, I understand it as a measure of the total energy that can be extracted from a thermodynamic entity. Now, the first term in its definition accounts for the inherent energy i.e. the $U$ term (which I suppose comes from the internal thermal motion of the entity's molecules). The second term accounts for the work done on the entity i.e. the $pV$ term (energy transferred to the entity) to give it a finite volume.

Now, my confusion is with the third term, i.e. $TS$.

I suppose this definition encompasses the entropy term in its classical view because it is related to Temperature and in the Boltzmann definition, I suppose that it isn't explicitly related to Temperature.

But classically, I think, we could define only a change in entropy and not entropy itself. So what does the $S$ factor mean here? Is it Boltzmann Entropy or Classical Entropy?

If this definition is classical, then shouldn't the term actually be something like $\int T.\delta S$ with some limits thrown in because I suppose Temperature is a function of Entropy.

Where am I going wrong with these fundamentals? Also, are the two definitions of Entropy equivalent? If so, give me some helpful intuition.

Kirusanth
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The Gibbs free energy is fundamentally defined by a Legendre transform of the enthalpy, $$G=H-TS$$ The reason for this definition is that under conditions of constant composition the total differential $dG$ expands nicely to, $$dG = VdP - SdT $$ This tells us that $G$ is an intrinsic function of $P$ and $T$, which differs from the other thermodynamic potentials like internal energy or enthalpy. It can be shown with a little extra effort that the change in Gibbs free energy is equivalent to the amount of non-$PV$ work that a system is capable of performing throughout a process. This is much closer to where you were starting from, but the fundamental definition of $G$ is a thermodynamic state function depending on $P$ and $T$ (and number of moles of each substance).

As an aside, entropy is one of the special thermodynamic quantities that actually can be explicitly defined unambiguously due to the 3rd Law. So there is no issue there.

Matt Hanson
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