Maxwell relation with 3 variables?

[anony]

Hello,
I have the thermodynamic potential dG = -SdT + VdP - MdB and I find that

(dG/dB)_(T,P) = -M and (dG/dT)_(P,B) = -S, where I have used _(letters) to denote constants and that these are partial differentials. I want to prove the Maxwell relation that

(dS/dB)_(T,P) = (dM/dT)_(B,P) *

so I do

(d/dB (dG/dT)_(P,B))_(T,P) = (d/dT (dG/dB)_(T,P))_(P,B)

But then bring these partials together, I no longer no what it supposed to be kept constant, or how I get to *

Sorry, about the messiness, and cheers for any help. I didn't put this in the homework section because its a pretty general question rather that an actual problem.

EDIT: Nevermind, I'm being an idiot. I got it now.

 

[JANm]

' Have found my thermodynamics book and see that I understand What S,T,V and P are.
What are M and B here?

 

[astrokreng]

Dear reader,

Thank you for your question. The Maxwell relation is a fundamental concept in thermodynamics that relates the partial derivatives of thermodynamic potentials to each other. In your case, you have three variables: temperature (T), pressure (P), and magnetic field (B). The Maxwell relation with three variables states that the partial derivative of entropy (S) with respect to magnetic field (B), at constant temperature (T) and pressure (P), is equal to the partial derivative of magnetization (M) with respect to temperature (T), at constant magnetic field (B) and pressure (P). This can be written as:

(dS/dB)_(T,P) = (dM/dT)_(B,P)

To prove this relation, you can use the thermodynamic potential dG that you have provided. By taking partial derivatives of dG with respect to T and B, you can show that the left side of the equation is equal to the right side. The key is to remember that when taking partial derivatives, all other variables are held constant. In this case, T and P are held constant for the derivative with respect to B, and P and B are held constant for the derivative with respect to T.

I hope this helps clarify the Maxwell relation with three variables. It is an important concept to understand in thermodynamics and can be applied to many different systems and equations. Keep up the good work in your studies!