Extremum of thermodynamic potentials: confusion

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SUMMARY

The discussion centers on the extremum of thermodynamic potentials, specifically the minimization of energy (U) under constant temperature and entropy conditions. The equation dU = TdS - pdV indicates that U remains constant when both volume (V) and entropy (S) are held constant. However, confusion arises regarding how U can be minimized without explicitly stating that volume must be constant. Steven Blundell's formulation in his book illustrates that under constant V and S, the availability (A) satisfies dA = dU + p_0dV - T_0dS ≤ 0, leading to the conclusion that U is minimized when these conditions are met.

PREREQUISITES
  • Understanding of thermodynamic potentials
  • Familiarity with the second law of thermodynamics
  • Knowledge of the concepts of entropy and temperature
  • Basic grasp of differential calculus in thermodynamics
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  • Study the derivation of thermodynamic potentials in Steven Blundell's "Thermal Physics"
  • Explore the implications of the second law of thermodynamics in various systems
  • Learn about the concept of availability and its applications in thermodynamics
  • Investigate the relationship between entropy, temperature, and volume in thermodynamic systems
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This discussion is beneficial for physics students, thermodynamics researchers, and professionals in engineering fields who seek a deeper understanding of thermodynamic potentials and their implications in energy minimization.

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An alternative formulation of the second law is that the energy of the system U is minimised if the temperature and entropy of the system are held constant.
However, dU= TdS -pdV
which means that U is presumably constant if the volume V and the entropy S are kept constant. How then can U change so that it is minimised?
 
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The formulation doesn't say anything about keeping volume constant.
 
Thanks for replying. I have seen two versions. One is in Steven Blundell's book where he derives the availability which satisfies
dA= dU + p_0dV -T_0dS \leq 0 where the subscripted variables are the reservoir ones. He then states that if V, S are constant then dA = dU \leq 0 so that U is minimised.

The other version uses a completely different approach but crucially no mention is made of V being constant as you say.
 

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