The Ideal Gas, Phase-Space Volume, and the Sackur–Tetrode Entropy

The monatomic ideal gas is the system on which the microcanonical machinery first delivers a closed-form entropy. Its Hamiltonian is a sum of kinetic terms with no interactions, so the phase-space count factorizes into a configuration part and a momentum part, each of which can be evaluated exactly. The result is the Sackur–Tetrode equation, the first entropy formula in which Planck's constant appears as the size of the phase-space cell, and its numerical value agrees with calorimetry to the significant figures the constants allow.

The phase-space volume of the ideal gas

Take $N$ identical point particles of mass $m$ in a container of volume $V$, with no interactions. The Hamiltonian is purely kinetic,

so the energy constraint $H\le E$ restricts only the momenta, to the interior of a sphere ${\sum }_i{p}_i^2\le 2mE$ in the $3N$-dimensional momentum space. The coordinate integral is unconstrained over the box and contributes a factor $V$ per particle. The enclosed phase-space volume is therefore

where ${\Phi }_D(R)$ is the volume of a $D$-dimensional ball of radius $R$. The two factorized pieces are the configuration volume $V^N$ and the momentum-ball volume, and every ingredient of the ideal-gas entropy comes from evaluating the latter.

The volume of a high-dimensional ball

The volume of a ball of radius $R$ in $D$ dimensions is

with $\Gamma$ the gamma function, the continuous extension of the factorial through $\Gamma (n+1)=n!$. The familiar low-dimensional cases ${\Phi }_2=\pi R^2$ and ${\Phi }_3=\frac{4}{3}\pi R^3$ are the first two entries; the formula holds for any $D$. Setting $D=3N$ and $R=\sqrt{2mE}$,

so the enclosed phase-space volume becomes

Two features carry over into the entropy. The energy dependence is $E^{3N/2}$, confirming the exponent $\alpha N=3N/2$ used in the preceding lessons, so each of the $3N$ momentum degrees of freedom contributes a half-power of the energy. And the gamma function in the denominator, once expanded by Stirling's approximation, supplies the density-dependent term that makes the entropy extensive.

A geometric fact drives the physics: in high dimensions the ball's volume is dominated by a thin skin just inside its surface, because ${\Phi }_D\propto R^D$ makes $\mathbf{d}\Phi /\Phi =D\mathbf{d}R/R$ enormous. Almost all momentum configurations of fixed total energy have very nearly the maximal radius, which is why the enclosed volume $\Gamma$ and the shell count $\Omega =\mathbf{d}\Gamma /\mathbf{d}E\cdot \delta E$ share the same logarithm to leading order in $N$.

The Sackur–Tetrode entropy

The entropy is $S=k\ln \Gamma$, valid to leading order in $N$. Taking the logarithm and applying Stirling's approximation in the form $\ln N!\approx N\ln N-N$ and $\ln \Gamma (\frac{3N}{2}+1)\approx \frac{3N}{2}\ln \frac{3N}{2}-\frac{3N}{2}$,

Collecting the terms, the $N\ln N$ combines with $N\ln V$ into $N\ln (V/N)$, the two $\frac{3N}{2}$ pieces combine into $\frac{3N}{2}\ln (4\pi mE/3N{h}^2)$, and the loose constants add to $\frac{5N}{2}$:

This is the Sackur–Tetrode entropy in its energy form. Every argument of a logarithm is intensive — $V/N$ is the volume per particle and $E/N$ the energy per particle — so $S$ is proportional to $N$ at fixed intensities, as an extensive quantity must be.

The temperature follows from the microcanonical definition $1/T=(\partial S/\partial E{)}_{V,N}$. Only the middle term carries the energy, and its derivative is

The equipartition result $E=\frac{3}{2}NkT$ emerges as a derivative of the count, not as a separate assumption; each of the $3N$ quadratic momentum terms carries $\frac{1}{2}kT$. Substituting $E=\frac{3}{2}NkT$ into the entropy replaces the energy argument by temperature: $4\pi mE/3N{h}^2=2\pi mkT/h^2$, and

The thermal wavelength and quantum concentration

The combination inside the temperature logarithm has the dimensions of an inverse volume, and it defines a length scale.

Physically $\lambda$ is the de Broglie wavelength of a particle whose kinetic energy is of order $kT$; it is the size of a particle's quantum-mechanical wavepacket at temperature $T$. With this substitution the Sackur–Tetrode entropy takes its most compact form,

The argument $V/N{\lambda }^3$ compares the volume per particle $V/N=1/n$ to the thermal volume ${\lambda }^3$. Writing the quantum concentration$n_Q=1/{\lambda }^3=(2\pi mkT/h^2{)}^{3/2}$, the entropy reads $S=Nk[\ln (n_Q/n)+\frac{5}{2}]$. The classical treatment is self-consistent only where its argument is large and positive, $n\ll n_Q$: the particles are dilute compared to the quantum concentration, their wavepackets do not overlap, and their indistinguishability has not yet forced the quantum statistics that a later module develops. When $n$ approaches $n_Q$ the wavepackets overlap, the entropy formula would predict a negative or near-zero entropy, and the classical count breaks down.

The circles depict the particle wavepackets, each of size $\lambda$; on the left, where $\lambda \ll n^{-1/3}$, they are far apart and the classical count holds, while on the right they overlap once $n$ reaches $n_Q$ and the classical treatment fails.

Extensivity and the role of the $N!$

The $N!$ inserted in the phase-space measure is what makes the argument of the first logarithm the intensive density $V/N$ rather than the extensive volume $V$. Dropping the $1/N!$ and repeating the calculation removes the $-\ln N!$ term, and the entropy becomes

The offending term is $Nk\ln V$. Doubling the system at fixed density sends $N\to 2N$ and $V\to 2V$, and $Nk\ln V$ becomes $2Nk\ln (2V)=2Nk\ln V+2Nk\ln 2$, which exceeds twice the original by $2Nk\ln 2$. The labelled entropy is superextensive: the entropy per particle grows without bound with system size, which no thermodynamic entropy can do. The corrected Sackur–Tetrode entropy has $Nk\ln (V/N)$ instead, and doubling gives exactly $2Nk\ln (V/N)$, so $S$ scales linearly.

The physical failure the $N!$ repairs is the spurious entropy of mixing two samples of the same gas. Remove a partition between two equal volumes of identical gas at the same temperature and density; nothing observable happens, and the entropy must not change. The labelled count predicts an increase $2Nk\ln 2$ — the same it would predict for mixing two different gases — because it treats the interchange of identical atoms across the removed partition as new microstates. Dividing by $N!$ removes exactly this overcount, so the entropy of mixing vanishes for identical species and survives only for distinct ones. This is the Gibbs paradox, resolved here by indistinguishability and treated in full when the canonical ideal gas is built.

Summary

• The ideal-gas phase-space volume factorizes into the configuration part $V^N$ and the momentum-ball volume ${\Phi }_{3N}(\sqrt{2mE})={\pi }^{3N/2}(2mE{)}^{3N/2}/\Gamma (\frac{3N}{2}+1)$, giving $\Gamma (E)=V^N(2\pi mE/h^2{)}^{3N/2}/[N!\Gamma (\frac{3N}{2}+1)]$.
• Taking $S=k\ln \Gamma$ with Stirling's approximation yields the Sackur–Tetrode entropy $S=Nk[\ln (V/N{\lambda }^3)+\frac{5}{2}]$, and $1/T=\partial S/\partial E$ recovers $E=\frac{3}{2}NkT$.
• The thermal wavelength $\lambda =h/\sqrt{2\pi mkT}$ and quantum concentration $n_Q=1/{\lambda }^3$ set the classical regime $n\ll n_Q$; helium at $300\mathrm{K}$ and $1\mathrm{atm}$ has $\lambda /n^{-1/3}\approx 0.015$ and molar entropy $126\mathrm{J}{\mathrm{K}}^{-1}\mathrm{mo}{\mathrm{l}}^{-1}$, matching experiment.
• The $N!$ makes the entropy extensive by converting $\ln V$ into $\ln (V/N)$ and cancels the spurious entropy of mixing for identical gases, previewing the Gibbs paradox.