According to Thermodynamics Everything should Pop into Existence?

[nonequilibrium]

(In the following discussion, when I use the word "always", I mean "as good as always" if you're willing to ignore exotic systems with negative temperature and such)

In the following discussion I will assume we're working in a heat bath with constant T and P:

So there are several ways to see the total Gibbs free energy of an object, defined G = U - TS + PV, is negative.

Two simple ways:
(*) Chemical potential is defined as \mu = -T \left( \frac{dS}{dN} \right)_{U,V} and thus is always negative. We also can prove G = \mu N.

(*) We know T \Delta S \geq Q = Q + P \Delta V - P \Delta V \geq Q + P \Delta V + W = \Delta U + P \Delta V so TS \geq U + PV or G \leq 0.

So now the problem is, when we 'make something' its G function goes from zero to something negative (as was just shown in two ways). This implies it should happen spontaneously, since in constant T and P the second law becomes "G goes to a minimum".

So does this say things should randomly pop into existence? Obviously there is a thinking error?

 

[Andy Resnick]

Like potential energy, the only meaning free energy has is in terms of *changes* (dG vs. G). Changes can be positive or negative. I'm not sure the (absolute) free energy has any meaning.

Sometimes (especially in biochemistry), you will enounter notation like \Delta \Delta G, which corresponds to changes to \Delta G.

 

[nonequilibrium]

Well, that's what I used, didn't I? First G = 0 and then G is negative, so the net change is negative (well the basic principle is that G goes down when something is created, check my math above)

 

[Mapes]

(*) Chemical potential is defined as \mu = -T \left( \frac{dS}{dN} \right)_{U,V} and thus is always negative. We also can prove G = \mu N.

Here you are assuming constant energy. How do you propose to change the amount of matter in a system without changing the total energy?

(*) We know T \Delta S \geq Q = Q + P \Delta V - P \Delta V \geq Q + P \Delta V + W = \Delta U + P \Delta V so TS \geq U + PV or G \leq 0.

How do you justify replacing \Delta S, \Delta V, and \Delta U with S, V, and U? I don't see how that's valid.

 

[nonequilibrium]

Okay, drop my first "derivation" then.

About the second: well, if I create the whole system, V_i = 0 and V_f = V, same for S and U, don't you agree?

EDIT: Btw thanks for the critique, I hope to discover my error before my exam in the morning, it's quite troubling I can't see where my reasoning goes astray

 

[Mapes]

As with the first derivation, I'm not seeing how this hypothesized system obeys energy conservation.

 

[nonequilibrium]

Everything enters as heat from the environment (that's the meaning of -TS in the definition of G, right? And in this case: TS > U + PV (as shown in the 2nd derivation)

But I've come to the conclusion "dS_tot > 0 <=> dG < 0" under constant P and T is only an equivalence if the system has a constant amount of particles! That's probably where I made my error. (G is still < 0, but now it just doesn't matter, really)