Problems related to Maxwell relations

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Homework Help Overview

The discussion revolves around the application of Maxwell relations in thermodynamics, specifically focusing on a problem involving the entropy of a system expressed in terms of various variables and parameters. The original poster seeks to establish mathematical constraints on the parameters α and β based on certain partial derivatives related to temperature, pressure, and chemical potential.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to understand the rationale behind Maxwell relations and expresses confusion regarding the transformation of variables in their equations. They explore the relationship between different thermodynamic potentials and their derivatives.

Discussion Status

Some participants are engaging with the original poster's questions, while one participant mentions that they will share ideas once a solution is released. There appears to be an ongoing exploration of the problem without a clear consensus or resolution at this stage.

Contextual Notes

The problem is noted as a bonus question, which may imply additional constraints or expectations regarding the solution. The original poster also indicates uncertainty about their previous attempts, suggesting a need for further clarification on the topic.

Mayan Fung
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Homework Statement


Given the entropy of a system :
$$ S = AU^αV^βN^{1-α-β} $$
The problem requires me to write
$$ (\frac{∂T}{∂U})_{V,N} > 0,  (\frac{∂P}{∂V})_{U,N} < 0, (\frac{∂μ}{∂N})_{U,V} > 0$$
to find the mathematical constraint of α and β

Homework Equations


dU = TdS - PdV + μdN

The Attempt at a Solution


Actually, I don't quite understand the rationale behind Maxwell relations.
I tried to write
dF = SdT - PdV + μdN
I can then get
$$ (\frac {∂T} {∂F})_{V,N}$$
but I just don't know how to transform the variable. Can anyone give me an idea of these kind of problems. Thanks!
 
This is actually a bonus question in my HW. It wrote something but I think they are wrong. The solution should be ready in the coming few days. I will share the ideas once it is released.
 
you can check it.http://odysriwo8.bkt.clouddn.com/wo.PNG
 
  • Like
Likes   Reactions: Mayan Fung
Thanks! I will try it!
 

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