Sackur-Tetrode Equation

Equivalent forms

S = Nk_B\left[\ln\left(\frac{k_B T}{P}\,\left(\frac{2\pi m k_B T}{h^2}\right)^{3/2}\right) + \frac{5}{2}\right]

\frac{S}{Nk_B} = \ln\frac{V/N}{\lambda_{th}^3} + \frac{5}{2}, \quad \lambda_{th}=\frac{h}{\sqrt{2\pi m k_B T}}

The thermal de Broglie wavelength λ_th appears naturally: the classical regime is exactly when λ_th ≪ (V/N)^{1/3}.

Symbol	Name	SI unit	Dimension	Range
$S$	Entropyoutput Absolute entropy of N monatomic ideal gas particles	J/K	ML²T⁻²Θ⁻¹	0 – 10000
$T$	Temperature Absolute temperature	K	Θ	10 – 10000
$P$	Pressure Gas pressure	Pa	ML⁻¹T⁻²	100 – 1000000
$m$	Particle mass Mass of one gas atom (e.g., argon: 6.63e-26 kg)	kg	M	1e-27 – 1e-25

Unit systems

SI:: $h = 6.62607015e-34 J\cdot s$ , $k_B = 1.380649e-23\,\mathrm{J/K}$
natural:: $\hbar = k_B = 1$
CGS:: h in $erg\cdot s$

Where it holds

Monatomic ideal gas in the classical (non-degenerate) regime:

n\lambda _th^{3}

≪ 1. Breaks down near BEC or Fermi degeneracy.

Discovery

Otto Sackur & Hugo Tetrode · 1912

Sackur and Tetrode independently derived the absolute entropy of an ideal gas in 1912, two years before quantum mechanics was fully formulated — by introducing h as the 'elementary cell' in phase space.

Try this

What is the absolute entropy of a gas?

Classical thermodynamics can only measure entropy changes. The Sackur-Tetrode equation, derived entirely from quantum statistical mechanics, gives the absolute entropy of a monatomic ideal gas — including the quantum of phase space h.

Research status: stable

Common misconceptions

The 5/2 factor is exact for monatomic ideal gases; diatomic gases need rotational and vibrational contributions. The equation is not valid below the quantum degeneracy temperature

T_Q \approx \hbar ^{2}n^{2/3}/mk_B

.

Derivation

Compute ln Z for N indistinguishable particles:

\ln Z = N \ln (V/\lambda _th^{3}) - \ln N! \approx N[\ln (V/N\lambda _th^{3}) + 1]

(Stirling).

Free energy

F = -k_BT \ln Z

; entropy

S = -\partial F/\partial T

gives the Sackur-Tetrode form.

Limiting cases

\lambda _th \to (V/N)^{1/3}

(quantum regime)⟶

S \to 0

; Nernst theorem satisfied (third law)

High T or

low \rho

⟶ Classical limit; S/Nk_B grows logarithmically with T

Gibbs paradox check⟶ N! factor from indistinguishability prevents entropy increase on mixing identical gases

Boltzmann Entropy→Ideal Gas Law→Canonical Partition Function→