Sackur-Tetrode equation

The Sackur-Tetrode equation gives the entropy of a 3D classical monatomic ideal gas. Given a gas of $N$ particles, it reads

\frac{S}{k_{B}N}=\ln\left[ \frac{V}{N}\left( \frac{4\pi m}{3h^{2}} \frac{U}{N} \right)^{3/2} \right]+ \frac{5}{2}=\ln\left( \frac{V}{N\lambda ^{3}} \right)+ \frac{5}{2}=-\ln(n\lambda ^{3})+ \frac{5}{2}

written in increasingly compact ways. $k_{B}$ is the Boltzmann constant, $V$ is the volume, $m$ is the mass of the particles, $h$ is a constant that is usually taken to be the Planck constant, $U$ is the internal energy, $\lambda$ is the de Broglie thermal wavelength and $n=N/V$ is the particle density.

The Sackur-Tetrode equation requires correct Boltzmann counting to be derived. Without it, the Gibbs paradox arises. Since this is a classical equation, it stops being valid when quantum phenomena are no longer negligible. This means that it is valid when

\frac{V}{N\lambda ^{3}}\gg 1\quad\text{or equivalently}\quad n\lambda ^{3}\ll 1

The second writing is particularly convenient as, since the interparticle distance is proportional to $n^{1/3}$ , it shows that the thermal wavelength must be much smaller than the interparticle distance for the system to be in a classical regime.