Sackur–Tetrode entropy (monatomic ideal gas)

Entropy of a dilute classical monatomic ideal gas including the quantum/indistinguishability constant: canonical and microcanonical forms.

Sackur–Tetrode entropy (monatomic ideal gas)

This formula gives the entropy of a dilute monatomic ideal gas (classical regime), and fixes the additive constant by incorporating the Gibbs $1/N!$ factor in the classical phase-space count.

Let $\lambda_T$ be the thermal wavelength

\lambda_T=\frac{h}{\sqrt{2\pi m k_B T}}.

Statement (canonical form)

For a monatomic ideal gas of $N$ particles of mass $m$ in volume $V$ at temperature $T$ (with $n\lambda_T^3\ll 1$ for number density $n=N/V$ ),

S(T,V,N)=N k_B\left[\ln\!\left(\frac{V}{N\lambda_T^3}\right)+\frac{5}{2}\right].

This is consistent with the classical ideal-gas partition function computation .

Statement (microcanonical form)

In terms of internal energy $U$ (see internal energy ),

S(U,V,N)=N k_B\left[\ln\!\left(\frac{V}{N}\left(\frac{4\pi m U}{3N h^2}\right)^{3/2}\right)+\frac{5}{2}\right].

Context / derivation sketch

Start from the canonical partition function for the ideal-gas Hamiltonian and include the Gibbs factor $1/N!$ for indistinguishable particles:
$Z_N=\frac{1}{N!}\left(\frac{V}{\lambda_T^3}\right)^N.$
Use the canonical identity
$S = k_B\big(\log Z_N + \beta U\big), \qquad \beta=\frac{1}{k_B T},$
together with $U=(3/2)Nk_B T$ .
The $1/N!$ term is what resolves the classical Gibbs paradox and produces the $\ln(V/N)$ dependence rather than $\ln V$ .

Scope and limitations

The condition $n\lambda_T^3\ll 1$ is the classical/dilute requirement; when it fails, quantum statistics become essential (see Bose–Einstein condensation and degenerate Fermi gas ).
Interactions modify the entropy beyond the ideal-gas form (compare van der Waals gas ).