Maxwell's relations (thermodynamics): validity

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SUMMARY

The discussion centers on the validity of Maxwell's relations in thermodynamics, specifically under what conditions these relationships hold true. The participant highlights the necessity of assuming that internal energy (U), volume (V), and pressure (p) are functions of state. It is established that the first law of thermodynamics must be expressed as TdS - pdV, which is applicable primarily to simple gas systems. The mathematical principle of the equality of mixed partial derivatives is also noted as a foundational aspect of this topic.

PREREQUISITES
  • Understanding of thermodynamic functions of state
  • Familiarity with the first law of thermodynamics
  • Knowledge of Maxwell's relations
  • Basic principles of partial differentiation in calculus
NEXT STEPS
  • Study the derivation of Maxwell's relations in thermodynamics
  • Research the implications of the first law of thermodynamics for different systems
  • Explore the conditions under which functions of state are valid
  • Learn about the mathematical properties of partial derivatives and their applications in thermodynamics
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Students of thermodynamics, educators teaching thermodynamic principles, and researchers exploring the mathematical foundations of thermodynamic relationships.

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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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