Not quite sure of your symbols here. E is not normally used for enthalpy. Assuming you are using F for what is normally called the Helmoltz free energy (or work function) and H for enthalpy, equation 38 should read.
G = H - TS = F + PV.
since there is some doubt as to the meaning of E I shall avoid it and use U for internal energy.
So for a change at constant temperature along a reversible path
dU = TdS + dwrev
At constant temp TdS = d(TS) so we can rewrite the above as
d(U-TS) = dwrev
(at constant T)
Let F = A-TS then dF = dwrev
(at const T)
Now the expression wrev
refers to all possible kinds of work.
If only pressure/volume work is envisaged
At const volume dV = 0 so dF = 0 so F is a minimum, at equilibrium.
So the condition for equilibrium at constant temp and volume is that the Helmholtz free energy is a minimum and dF = 0.
Now to develop a similar discussion for constant pressure Note that
H = U + PV
dH = dU + PdV + VdP
Substitute from the above
dH = TdS + dwrev
+ PdV + VdP
or at constant T and P
d(H-TS) = PdV + dwrev
This new function is called the Gibbs free energy and is G = H - TS = F+PV
So dG = PdV + dwrev
Again restricting this to PV work dwrev
So at const T and P dG = 0, Thus G is a minimum, at equilibrium.
Now applying all that to my melting ice example.
dG= 0 = d(H-TS) = dH - d(TS) = dH - TdS
dS = dH / T
The entropy of melting (fusion) can be calculated by dividing the enthalpy (=latent heat) by the melting point.
You might like to look at this thread and in particular post#11 (Which no longer contains a typo)
does this help?