Seeking formal derivation for common thermo equation

[saybrook1]

Hi guys,

I was hoping that someone might be able to help me out with a formal derivation of this common thermodynamic equation regarding the change in entropy during an isothermal change of state.

The first equation is what I would like to derive, and the second is where the book tells me to derive it from once we acknowledge that there is no energy change in an isothermal process. I've tried a few different ways but haven't had success yet. I'm thinking it has to do with the first law and then somehow relating heat to entropy.

Anyway, thanks for any help. Even a link would rock!

Best regards

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[Charles Link]

Hello again. The best book I know of for Statistical Physics/Thermodynamics is F.Reif's book. He derives the case $F=-kT ln Z$ and gives an extensive treatment of $Z$ for $N$ atoms using the Maxwell-Boltzmann Statistics with the $N!$ Boltzmann factor in the denominator. $Z=\zeta^N/N!$ where $\zeta$ is the partition function for a single atom (of the gas). The derivation of $\zeta$ is somewhat lengthy but not difficult. Once you get Z and F, the minus partial of F w.r.t. T at constant V I think is the entropy $S$.

 

[saybrook1]

Hello my friend! I've just returned from getting some late night tacos; I appreciate you responding to my post again. So, the text that I'm going through is Pathria and I also have the Schroeder undergrad text "Thermal Physics". The problem is that I think Pathria implies that you can derive eqn (1) without use of the partition function. This is at the end of chapter 1 of his book and the partition function has not been introduced in any form yet. I'm just banging my head on a way to derive eqn (1) from eqn (2) under an isothermal change of state(fixed N,T). I will however attempt your method. Thanks again!

 

[saybrook1]

Wooo I think I figured it out. I think you need to take eqn (2) and just take the difference like so: $$S_2(N,V_2,E) - S_1(N,V_1,E)$$
I'll report back and let you know how it goes!

 

[saybrook1]

Alright, so here it is. I actually found this solution while looking through the publicly available lectures notes of Alejandro L. Garcia of San Jose State University.
If anyone has anything to add, please do. A question that I still have about this, is that it doesn't seem like we needed to invoke an isothermal condition, so is this a general entropy change equation for adiabatic processes?