English: Entropy of classical ideal gas and quantum ideal gases (Fermi gas, Bose gas) as a function of temperature, for a fixed density of particles. This is for the case of non-relativistic (massive, slow) particles in three dimensions.

A few features can be seen:

• The classical gas follows the Sackur-Tetrode equation. As temperature increase, the Fermi and Bose gases approach the classical gas line, gradually.
• The Fermi gas is higher entropy, and Bose gas is lower entropy. (this is not always true! In 1D gas the order is flipped, and for 2D gas the Fermi and Bose are identical entropy for some reason, lying just above classical gas)
• The quantum gases don't go below 0 entropy, while Sackur-Tetrode does go negative, logarithmically.
• The Fermi gas has a linear dependence for low T, as expected since the thin Fermi 'shell' that contains all the uncertainty is shrinking linearly.
• The ★ marker indicates the upper temperature of Bose condensation. Moving below this temperature, the entropy approaches 0 rapidly as more and more particles are absorbed into the condensed phase. Interestingly, below the critical temperature the entropy is a constant per uncondensed particle, the constant being independent of temperature; the entropy drops because there are fewer uncondensed particles.

The figure has been scaled in a way that the particle degeneracy factor, density, mass, etc. are all factored out and irrelevant.

This was generated using the same source code as File:Quantum ideal gas pressure 3d.svg.