Thermodynamics Maxwell relation

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SUMMARY

The discussion focuses on the Maxwell relation in thermodynamics, specifically the equation \(\left( \frac{\partial S}{\partial V} \right)_{E, N} = \frac{P}{T}\). This relation is derived from the fundamental thermodynamic equation \(\mathrm dE = T \, \mathrm dS - p \, \mathrm dV + \mu \, \mathrm dN\) under the conditions that energy (E) and particle number (N) remain constant. The participants explore the implications of this relation and question the possibility of deriving it without invoking the first law of thermodynamics.

PREREQUISITES
  • Understanding of thermodynamic principles, specifically the first law of thermodynamics.
  • Familiarity with the concepts of entropy (S), pressure (P), and temperature (T).
  • Knowledge of partial derivatives in the context of thermodynamic equations.
  • Basic grasp of the relationship between energy (E), volume (V), and particle number (N) in thermodynamic systems.
NEXT STEPS
  • Study the derivation of Maxwell relations in thermodynamics.
  • Explore the implications of the first law of thermodynamics on state functions.
  • Learn about the role of entropy in thermodynamic processes.
  • Investigate alternative derivations of thermodynamic relations without relying on the first law.
USEFUL FOR

This discussion is beneficial for students and professionals in physics and engineering, particularly those specializing in thermodynamics, as well as researchers exploring advanced thermodynamic relations and their applications.

LagrangeEuler
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When can be used
[tex](\frac{\partial S}{\partial V})=\frac{P}{T}[/tex]?
 
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The fundamental relation is, if I recall correctly,
##\mathrm dE = T \, \mathrm dS - p \, \mathrm dV + \mu \, \mathrm dN##

So setting dE = 0, you obtain P/T = dS/dV when dN = 0, so if the number of particles and the energy in the system don't change.
For convenience you can put this in the subscript:
##\left( \frac{\partial S}{\partial V} \right)_{E, N} = \frac{P}{T},##
as a reminder that E and N should be kept constant.
 
And could you get this relation without using first law of thermodynamics?
 

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