Maxwell's relations (thermodynamics): validity

AI Thread Summary
Maxwell's relations in thermodynamics are derived under specific conditions, primarily assuming that internal energy (U), volume (V), and pressure (p) are functions of state. The validity of these relations is contingent on the application of the first law of thermodynamics, which is typically applicable to simple gas systems without additional complexities. The discussion highlights the importance of recognizing that the order of partial differentiation for analytic functions remains consistent, reinforcing the reliability of these relations. The inquiry seeks confirmation on whether any systems exist where these assumptions might not hold. Overall, the conversation emphasizes the foundational principles that govern the applicability of Maxwell's relations in thermodynamics.
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Homework Statement


I came across a slightly unusual question today. It started out fine, just asking me to derive a maxwell relation but then asked under what conditions is this relationship valid.

Homework Equations



The Attempt at a Solution


In deriving the relation I need to assume U, V and p are functions of state, but I can't think of any system where this wouldn't be the case. Also the first law must take the form TdS-pdV which only applies for simple gas systems without any extra freedom. But is that is?

If someone could let me know if I'm missing something obvious here (or just confirm I'm right) I'd appreciate it.
Thanks
 
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Probably, it's the mathematical fact that the order of partial differentiation of an analytic function of two variables does not matter.

del/del x(del f/del y) = del/del y(del f/del x).
 
Thanks
 

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