# Interpretation of entropy of an ideal gas (Sackur-Tetrode equation)

I understand how to obtain the expression for the entropy of a three dimensional ideal gas in the classical limit (Sackur-Tetrode equation):

$$S(V,N,T)=k_BN[\log(\frac{V}{N})+\frac{3}{2}\log(\frac{2\pi m k_BT}{h^2})+\frac{5}{2}]$$

But I fail in understanding a physical interpretation in terms of "disorder" or "loss of information" for the three summands in the above equation. For example, I guess that the term $$\log(\frac{V}{N})$$ has something to do with the number of ways of packing $$N$$ identical free particles in a volume $$V$$: The number of possible ways I can distribute particles should increase with the volume. Since entropy is additive, I would expect that the rest of terms in the sum also have a physical interpretation.

I don't even know if it is possible to interpret the equation as I wrote it or if I should reorganize terms. There is a brief discussion about this topic in Wikipedia, but is a summary rather than an explanation.

Question: Can the summands of the Sackur-Tetrode equation be interpreted in terms of "entropic sources"? if yes, could you offer a discussion about such interpretation and/or point me to a proper reference?

• In microcanonical ensemble, $S=k_B\ln \Omega$ where $\Omega$ is the number of microstates for the given total energy, which quantifies the degree of "disorder", or our complete ignorance of microstates. Evaluating this expression for ideal gas gives Sackur-Tetrode equation. The first term is indeed related to how particles can be distributed in space. The second term is about how kinetic energies are distributed among particles, when the total energy is fixed. Jan 4 at 21:48
• @MengCheng Please post answers as answers, not as comments. Jan 5 at 0:28
• @MengCheng Thank you! Could you give more insights about why the second term is related to the distribution of kinetic energy? Does the constant term have an interpretation? Are you aware of a reference that explains in more detail your statement about the interpretation?
– Javi
Jan 5 at 11:22