The different textbooks for P.Chem excel at different things. Many people like Atkins the best because it is the most thorough when it comes to the math derivations and most people who study P.Chem are better at math than they are at English. Levine's book on the other hand excels because it is the most clearly explained book in English. He tells you what every single term means in all the equations he introduces. To someone like me that struggles with the derivations that approach is the best. It depends on who you are.
I sort of avoided the derivation of dG = -RTlnKeq because to do it for real would take too long, but so long as you plan to read up on it in a formal book I can give you a sneak peak of where it will come from. This isn't really a derivation because I am introducing things that I am not explaining but the book will cover the rest for me I think, and with that disclaimer:
dG = dH - TdS
Now to measure the difference in material equilibrium only keep pressure and temperature in the system constant for both the initial and final state. This is easy to do in real life: 1) Pressure is constant due to the atmosphere 2) You can keep the temperature constant by having the reaction start at room temp and cool down to room temp before measuring the final step. This makes the dH term equal to zero so that:
dG = 0 - TdS = -TdS
Now introducing the boltzman statistical definition of entropy which is klnS or RlnS (the mole version which is the one we want):
dG = - TR(LnS2 - lnS1) = -RTln(S2/S1)
Now, you can show (which I won't do...every P.Chem book shows this derivation) that S2/S1 can be related to the Volume of two gases expanding in the system as V2/V1.
dG = -RTln(V2/V1)
Now from the ideal gas law you can see that when T and P are constant that V is directly proportional to moles.
V = nRT/P = Cn (where C is a constant)
You can also arrange this in terms of molarity (which is what we want to ultimately do since we are using Keq):
n/V = RT/P = [M]
Making the substitutions into dG:
dG = -RTln(n2/n1) = -RTln([n2/V2]/[n1/V1]) = -RTln([M2]/[M1])
Finally if we define Keq as:
Keq = [M2]/[M1]
dG = - RTln(Keq)
A sloppy derivation in a few ways, but at least you can see that there is no magic involved.