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This question already has an answer here:

I was recently looking at the entropy equation for an ideal gas, show below

$$S=NR\left\{\frac{5}{2}+\ln\left[\frac{V}{N}\left(\frac{2\pi mRT}{h^2}\right)^{3/2}\right]\right\}$$

And what troubles me is as $T\rightarrow0$ then entropy become negative, which violates the third law of thermodynamics, how can this be?

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marked as duplicate by Michael Seifert, John Rennie thermodynamics Mar 25 at 19:44

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The equation you're looking at (which is called the Sackur-Tetrode equation, by the way) is derived assuming that the gas follows Maxwell-Boltzmann statistics. Maxwell-Boltzmann statistics is a high-temperature approximation for either Fermi-Dirac statistics (for a fermion gas) or Bose-Einstein statistics (for a boson gas). As you lower the temperature to zero, the quantum statistics of the gas become important, so the Sackur-Tetrode equation is no longer valid.

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