Derivation of sackur-tetrode equation

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Discussion Overview

The discussion centers around the derivation of the Sackur-Tetrode equation, exploring whether it can be derived using classical thermodynamics without resorting to the macrocanonical ensemble or quantum statistics. Participants examine the implications of indistinguishability and the role of classical versus quantum approaches in deriving the equation.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether the Sackur-Tetrode equation can be derived solely through classical thermodynamics, without using the macrocanonical ensemble.
  • Another participant asserts that the proper derivation of the Sackur-Tetrode formula requires quantum statistics, specifically Boltzmann statistics, and highlights the issue of indistinguishability of particles as a key factor.
  • A different viewpoint suggests that the equation can be derived using Stirling's approximation applied to the multiplicity formula, although the participant notes that their thermodynamics text does not provide a complete derivation.
  • One participant agrees with the previous claim but emphasizes the necessity of assuming particle indistinguishability, which introduces a factor of 1/N! that cannot be justified without invoking quantum theory.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of quantum mechanics for deriving the Sackur-Tetrode equation, indicating that multiple competing perspectives remain unresolved.

Contextual Notes

Participants highlight the dependence on the concept of indistinguishability and the implications of classical versus quantum statistical mechanics in the derivation process, suggesting limitations in the classical approach.

ralden
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how sackur-tetrode equation derive?, can it derive without the use of macrocanonical ensemble? only by classical thermodynamics? thank you.
 
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The Sackur-Tetrode formula can only be derived properly as the classical limit of quantum statistics, which is Boltzmann statistics for both fermions and bosons, taking into account the indistinguishability of particles, which is a specific quantum-mechanical phenomenon. Classical statistics leads to the Gibbs paradoxon and a wrong (non-extensive) expression for the entropy which is solved by the Sackur-Tetrode formula.
 
You can derive the Sackur-Tetrode equation by solving for the entropy of an ideal gas using Stirling's approximation applied to the multiplicity formula. My thermodynamics text does not go through the whole derivation, but that is how it says to derive it.
 
Yes, that you can do, but you have to assume the indistinguishability of particles, which leads to an additional factor [itex]1/N![/itex] compared to classical mechanics. This factor cannot justified without the indistinguishability argument that is generically quantum theoretical.
 

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