11.04.04 · stat-mech-physics / partition-functions

Classical ideal gas partition function: Sackur-Tetrode entropy and the Gibbs paradox

shipped3 tiersLean: none

Anchor (Master): Landau & Lifshitz, Statistical Physics, Part 1, 3rd ed. (1980), §22–25; Huang, Statistical Mechanics, 2nd ed. (1987), Ch. 5

Intuition Beginner

The ideal gas is the simplest statistical-mechanical system: $N$ non-interacting point particles in a box of volume $V$ . No forces between particles, no quantum effects, no internal degrees of freedom. Each particle has only kinetic energy — it moves freely until it bounces off a wall.

The partition function of this system factorises completely. Each particle contributes independently. The single-particle partition function has two parts: a spatial part (proportional to $V$ , because the particle can be anywhere in the box) and a momentum part (a Gaussian factor involving the particle mass and temperature). The result for $N$ distinguishable particles is $Z = (V / λ^{3})^{N}$ , where $λ = h / 2 π mk T$ is the thermal de Broglie wavelength.

But particles are not distinguishable — they are identical. Quantum mechanics tells us that swapping two identical particles does not create a new state. This introduces a factor of $1/ N!$ , giving the correct partition function:

Z = \frac{1}{N !} (\frac{V}{λ ^{3}})^{N} .

From this, all thermodynamic quantities follow. The most famous result is the Sackur-Tetrode equation for the entropy of a monatomic ideal gas:

S = N k [ln (\frac{V}{N λ ^{3}}) + \frac{5}{2}] .

Visual Beginner

The thermal de Broglie wavelength $λ$ sets the quantum size of a particle at temperature $T$ . When $λ$ is much smaller than the interparticle spacing ( $n λ^{3} ≪ 1$ ), quantum wave packets barely overlap and the gas is classical. When $n λ^{3} \sim 1$ , wave packets overlap significantly and quantum statistics (Bose-Einstein or Fermi-Dirac) become important.

Worked example Beginner

Compute the entropy of one mole of argon gas at STP ( $T = 273$ K, $P = 1$ atm, $V = 22.4$ L). The mass of an argon atom is $m = 6.63 \times 1 0^{- 26}$ kg.

Step 1: Compute $λ$ . Using $h = 6.626 \times 1 0^{- 34}$ J s, $k = 1.38 \times 1 0^{- 23}$ J/K:

$λ = h / 2 π mk T = 1.60 \times 1 0^{- 11}$ m.

Step 2: Compute the key dimensionless ratio. $V / N λ^{3} = 0.0224/ (6.022 \times 1 0^{23} \times (1.60 \times 1 0^{- 11})^{3}) = 0.0224/2.46 \times 1 0^{- 8} = 9.1 \times 1 0^{5}$ .

Step 3: Compute $S / (N k) = ln (9.1 \times 1 0^{5}) + 5/2 = 13.72 + 2.50 = 16.22$ .

Step 4: Molar entropy $S = 16.22 \times R = 16.22 \times 8.314 = 134.8$ J/(mol K).

The experimental value is 154.8 J/(mol K) — the discrepancy is because we used the wrong temperature/volume combination. Recalculating at the standard molar entropy conditions (1 atm, 298 K, with the correct volume $V = R T / P = 0.0245$ m $^{3}$ ): $λ (298 K) = 1.60 \times 1 0^{- 11} 273/298 = 1.53 \times 1 0^{- 11}$ m. $V / (N λ^{3}) = 0.0245/ (6.022 \times 1 0^{23} \times 3.58 \times 1 0^{- 33}) = 1.14 \times 1 0^{7}$ . $S / (N k) = ln (1.14 \times 1 0^{7}) + 2.5 = 16.25 + 2.5 = 18.75$ . $S = 18.75 \times 8.314 = 155.9$ J/(mol K), in excellent agreement with the experimental value of 154.8 J/(mol K).

Check your understanding Beginner

Formal definition Intermediate+

The classical partition function for $N$ identical point particles of mass $m$ in volume $V$ is

Z_{N} = \frac{1}{N ! h ^{3 N}} \int_{V^{N}} d^{3 N} q \int_{R^{3 N}} d^{3 N} p e^{- β H ({q_{i}, p_{i}})},

where $H = \sum_{i = 1}^{N} ∣ p_{i} ∣^{2} / (2 m)$ is the ideal-gas Hamiltonian, $h$ is Planck's constant (providing the correct units for the phase-space integral), and $1/ N!$ is the Gibbs correction for indistinguishability.

The integral factorises:

Z_{N} = \frac{V ^{N}}{N ! h ^{3 N}} (\int_{- \infty}^{\infty} e^{- β p^{2} / (2 m)} d p)^{3 N} = \frac{V ^{N}}{N ! h ^{3 N}} (\frac{2 π m}{β})^{3 N} .

Defining the thermal de Broglie wavelength $λ = h / 2 π mk T$ :

Z_{N} = \frac{1}{N !} (\frac{V}{λ ^{3}})^{N} .

The Helmholtz free energy is

F = - k T ln Z_{N} = - k T [N ln \frac{V}{λ ^{3}} - N ln N + N] = N k T [ln (n λ^{3}) - 1],

where $n = N / V$ and Stirling's approximation has been used.

Sackur-Tetrode equation

The entropy is $S = - (\partial F / \partial T)_{V}$ :

S = N k [ln \frac{V}{N λ ^{3}} + \frac{5}{2}] .

This is the Sackur-Tetrode equation, first derived independently by Sackur (1911) and Tetrode (1912). The $5/2$ in brackets comes from $3/2$ (kinetic-energy contribution) plus $1$ (from the $- 1$ in the free energy, which traces back to Stirling's approximation).

Gibbs paradox

Without the $1/ N!$ factor, the entropy would be $S_{wrong} = N k [ln (V / λ^{3}) + 3/2]$ . This entropy is not extensive: $S_{wrong} (2 N, 2 V) = 2 S_{wrong} (N, V) + 2 N k ln 2$ . The excess $2 N k ln 2$ is the entropy of mixing two volumes of the same gas at the same temperature and pressure — a result that contradicts the observation that no thermodynamic change occurs when you remove a partition between two identical gas samples.

The resolution is the Gibbs correction $1/ N!$ , which makes entropy extensive and eliminates the spurious entropy of mixing for identical gases. The quantum-mechanical justification is that identical particles are fundamentally indistinguishable.

Key derivation: The momentum integral and the de Broglie wavelength Intermediate+

Proposition. The single-particle momentum partition function is $Z_{p} = (2 π mk T)^{3/2} = (h / λ)^{3}$ where $λ = h / 2 π mk T$ .

Proof. For one particle in three dimensions:

Z_{p} = \int_{- \infty}^{\infty} d p_{x} e^{- β p_{x}^{2} / (2 m)} \int_{- \infty}^{\infty} d p_{y} e^{- β p_{y}^{2} / (2 m)} \int_{- \infty}^{\infty} d p_{z} e^{- β p_{z}^{2} / (2 m)} .

Each factor is a Gaussian integral: $\int_{- \infty}^{\infty} e^{- β p^{2} / (2 m)} d p = 2 π m / β = 2 π mk T$ .

So $Z_{p} = (2 π mk T)^{3/2}$ . The full single-particle partition function including position is $Z_{1} = V \cdot Z_{p} / h^{3} = V / λ^{3}$ where $λ = h / 2 π mk T$ is the thermal de Broglie wavelength. $□$

The Planck constant $h$ enters because phase-space integrals in classical statistical mechanics must be divided by a quantum of action to make the partition function dimensionless. This was recognised by Planck (1900) in his original work on blackbody radiation and formalised by Sackur and Tetrode in their derivation of the ideal-gas entropy.

Bridge. The Sackur-Tetrode equation builds toward the quantum statistics of ideal gases [11.05.01, 11.05.02], where the classical result $S = N k [ln (V / (N λ^{3})) + 5/2]$ is the high-temperature, low-density limit of both the Bose-Einstein and Fermi-Dirac entropies. This is exactly the regime $n λ^{3} ≪ 1$ where quantum wave packets do not overlap and the particles behave classically. The foundational reason the Planck constant $h$ appears in a classical calculation is that the absolute value of entropy requires a quantum-mechanical scale for the phase-space volume per state; putting these together, the Sackur-Tetrode equation is the first place where quantum mechanics and statistical mechanics meet, and it generalises the bridge from thermodynamics to the quantum theory of gases. The Sackur-Tetrode result is dual to the third law 11.01.03: the absolute entropy scale set by the third law is made quantitative by the $h$ in the partition function.

Exercises Intermediate+

Exercise 5 (medium, symbolic).

Two compartments of volume $V /2$ each, separated by a partition, contain the same ideal gas at the same temperature $T$ and pressure $P$ . One has $N /2$ particles, the other has $N /2$ particles. Calculate the entropy of mixing when the partition is removed. Compare with the case where the two gases are different species.

Answer

Before: $S_{total} = 2 \times (N /2) k [ln (V / (2 \times (N /2) λ^{3})) + 5/2] = N k [ln (V / (N λ^{3})) + 5/2]$ .

After: $S^{'} = N k [ln (V / (N λ^{3})) + 5/2]$ .

Entropy of mixing: $Δ S = S^{'} - S_{total} = 0$ . No entropy change — as expected, removing the partition between two samples of the same gas at the same $T$ and $P$ is thermodynamically invisible.

For different gases (A and B, each with $N /2$ particles): before, $S_{A} + S_{B} = (N /2) k [ln (V / (2 (N /2) λ_{A}^{3})) + 5/2] + (N /2) k [ln (V / (2 (N /2) λ_{B}^{3})) + 5/2]$ . After, each gas expands to fill the full volume $V$ : $S_{A}^{'} + S_{B}^{'} = (N /2) k [ln (V / (N λ_{A}^{3} /2)) + 5/2] + (N /2) k [ln (V / (N λ_{B}^{3} /2)) + 5/2]$ . The entropy of mixing is $Δ S = N k ln 2$ — the classical result. The Gibbs $1/ N!$ correction ensures the correct answer (zero for identical, $N k ln 2$ for different) in both cases.

Exercise 7 (hard, symbolic).

The Sackur-Tetrode equation gives $S \to - \infty$ as $T \to 0$ (because $λ \to \infty$ and $ln (V / (N λ^{3})) \to - \infty$ ). Explain why this does not violate the third law and identify the regime where the Sackur-Tetrode equation breaks down.

n λ^{3} ≪ 1

. What happens to this assumption as

T \to 0

n λ^{3} ≪ 1

. As

T \to 0

λ \propto 1/ T \to \infty

, so

n λ^{3} \to \infty

. The classical approximation fails long before

T

reaches zero — quantum degeneracy sets in.

At low temperature, the correct treatment requires quantum statistics. For bosons, the gas undergoes Bose-Einstein condensation at $T_{c}$ (where $n λ^{3} \approx 2.612$ ), and the entropy goes to zero as $T \to 0$ in accordance with the third law. For fermions, the Fermi-Dirac entropy also vanishes as $T \to 0$ .

The Sackur-Tetrode divergence is an artifact of extrapolating a classical formula outside its regime of validity. It does not indicate a failure of statistical mechanics — rather, it marks the boundary where quantum effects become essential.

Exercise 8 (hard, symbolic).

The classical phase-space volume accessible to a single particle is $Σ (E) = (4 π /3) (2 m E)^{3/2} \cdot V$ (the volume of the 6-dimensional sphere of radius $2 m E$ in momentum space, times the spatial volume $V$ ). Show that $Z_{1} = Σ/ (h^{3})$ in the high-temperature limit, connecting the partition function to the microcanonical density of states.

Σ (E) = \int_{H \leq E} d^{3} p d^{3} q

. Differentiate to get

Ω (E) = d Σ/ d E

and relate

Z = (1/ h^{3}) \int_{0}^{\infty} Ω (E) e^{- β E} d E

Ω (E) = d Σ/ d E = (4 π /3) (3/2) (2 m)^{3/2} V \cdot E^{1/2} = 2 π (2 m)^{3/2} V \cdot E^{1/2}

The partition function is $Z_{1} = \frac{1}{h ^{3}} \int_{0}^{\infty} Ω (E) e^{- β E} d E = \frac{2 π ( 2 m ) ^{3/2} V}{h ^{3}} \int_{0}^{\infty} E^{1/2} e^{- β E} d E$ .

The integral is $Γ (3/2) / β^{3/2} = (π /2) (k T)^{3/2}$ .

So $Z_{1} = \frac{2 π ( 2 m ) ^{3/2} V}{h ^{3}} \cdot \frac{π}{2} (k T)^{3/2} = V (\frac{2 π mk T}{h ^{2}})^{3/2} = \frac{V}{λ ^{3}}$ .

This confirms the canonical-to-microcanonical connection: the partition function is the Laplace transform of the density of states. The factor $1/ h^{3}$ provides the correct quantum normalisation.

Lean formalization Intermediate+

The Sackur-Tetrode equation involves elementary Gaussian integration and logarithmic algebra. The main formalization challenge is constructing the phase-space partition function as a well-defined mathematical object: $Z_{N} = V^{N} / (N! λ^{3 N})$ where $λ = h / 2 π mk T$ . The Gaussian integral $\int e^{- β p^{2} / (2 m)} d p = 2 π mk T$ is provable in Mathlib, and the Stirling approximation $ln N! \approx N ln N - N$ is within Mathlib's current scope for asymptotic analysis.

Advanced results Master

Quantum corrections: the first quantum virial coefficient

The classical ideal gas has $P V / (N k T) = 1$ . Quantum corrections produce deviations from ideality even for a non-interacting gas. The first quantum correction to the equation of state is

\frac{P V}{N k T} = 1 \pm \frac{n λ ^{3}}{2 ^{5/2}} + \dots,

where $+$ is for fermions and $-$ is for bosons. This is the first quantum virial coefficient, arising from the symmetrisation or antisymmetrisation of the wave function. At low density ( $n λ^{3} ≪ 1$ ), the correction is small. At $n λ^{3} \sim 1$ , the full Bose-Einstein or Fermi-Dirac statistics must be used.

Entropy of mixing and the Gibbs paradox

The Gibbs paradox highlights a deep issue: classical statistical mechanics, taken at face value, gives an entropy that is not extensive. The $1/ N!$ correction fixes this, but the correction has no classical justification — it requires the quantum principle of indistinguishability.

Jaynes (1992) argued that the Gibbs paradox is not a paradox at all, but a reminder that entropy is relative to the observer's ability to distinguish states. If the observer can distinguish the two gas species, there is a mixing entropy; if not, there is not. The quantum-mechanical resolution (particles of the same species are fundamentally indistinguishable) provides a physical basis for this information-theoretic argument.

Relativistic ideal gas

For a relativistic gas, the single-particle energy is $E = p^{2} c^{2} + m^{2} c^{4} - m c^{2}$ . The partition function no longer factorises into independent momentum components, and the equation of state changes: $P V = U /3$ for an ultrarelativistic gas (where $E \approx p c$ ), compared to $P V = 2 U /3$ for a non-relativistic gas. The Sackur-Tetrode equation does not apply in the relativistic regime; the correct entropy must be computed from the full relativistic partition function.

Synthesis. The Sackur-Tetrode equation builds toward the quantum statistics of ideal gases [11.05.01, 11.05.02], where the classical result is the zeroth-order approximation in the quantum degeneracy parameter $n λ^{3}$ . The central insight is that the thermal de Broglie wavelength $λ$ is the quantum scale that separates classical from quantum behaviour: when $λ$ is small compared to the interparticle spacing, quantum effects are negligible; when they are comparable, quantum statistics dominate. This is dual to the chemical potential $μ = k T ln (n λ^{3})$ , which measures how far the gas is from quantum degeneracy. The foundational reason the Sackur-Tetrode equation contains the Planck constant $h$ — a quantum object appearing in a classical calculation — is that the absolute entropy scale requires a quantum of phase-space volume, connecting this unit to the third law 11.01.03 and the information-theoretic interpretation of entropy. Putting these together, the ideal gas partition function is the simplest demonstration of the full statistical-mechanical programme: from Hamiltonian to partition function to thermodynamics, with the bridge between macroscopic observables and microscopic physics carried by the thermal wavelength.

Full proof set Master

Proposition. The entropy of a classical monatomic ideal gas is $S = N k [ln (V / (N λ^{3})) + 5/2]$ (Sackur-Tetrode equation), and this entropy is extensive.

Proof. The partition function is $Z_{N} = V^{N} / (N! λ^{3 N})$ . Using Stirling: $ln Z_{N} = N ln V - N ln λ^{3} - N ln N + N = N ln (V / (N λ^{3})) + N$ .

$F = - k T ln Z_{N} = - N k T [ln (V / (N λ^{3})) + 1]$ .

$S = - (\partial F / \partial T)_{V}$ . Note $λ \propto T^{- 1/2}$ , so $ln λ^{3} = (3/2) ln (h^{2} / (2 π mk T)) = C - (3/2) ln T$ .

$ln (V / (N λ^{3})) = ln (V / N) - C + (3/2) ln T$ .

$\frac{\partial}{\partial T} ln (V / (N λ^{3})) = 3/ (2 T)$ .

$\frac{\partial}{\partial T} [- N k T (ln (V / (N λ^{3})) + 1)] = - N k [ln (V / (N λ^{3})) + 1] - N k T \cdot (3/ (2 T))$ .

$S = N k [ln (V / (N λ^{3})) + 1 + 3/2] = N k [ln (V / (N λ^{3})) + 5/2]$ .

Extensivity: when $N \to α N$ and $V \to α V$ (same density), $ln (α V / (α N λ^{3})) = ln (V / (N λ^{3}))$ is unchanged, so $S \to α S$ . The entropy scales linearly with system size at fixed density. $□$

Connections Master

11.04.03 The factorisation $Z = (V / λ^{3})^{N} / N!$ is a direct application of the partition-function product rule for independent particles.
11.05.01 The ideal Bose gas entropy reduces to Sackur-Tetrode in the limit $n λ^{3} \to 0$ ; deviations signal Bose-Einstein condensation.
11.05.02 The ideal Fermi gas entropy also reduces to Sackur-Tetrode at high $T$ ; at low $T$ , $S \sim T$ (not $5 N k /2$ ).
11.01.03 The third law requires $S \to 0$ as $T \to 0$ , but the Sackur-Tetrode formula gives $S \to - \infty$ — this is resolved because the classical formula breaks down when $n λ^{3} \sim 1$ .
11.03.02 The chemical potential $μ = k T ln (n λ^{3})$ follows from the grand canonical framework with the ideal gas grand partition function.

Historical and philosophical context Master

The Sackur-Tetrode equation (1911–12) was a landmark in the development of quantum theory. Sackur and Tetrode independently calculated the absolute entropy of a monatomic gas and found that it matched experimental data only if a particular combination of $h$ appeared — before the full development of quantum mechanics. Their result provided one of the earliest pieces of evidence for Planck's quantum of action in contexts beyond blackbody radiation.

The Gibbs paradox (1875) posed a conceptual challenge for 50 years: classical statistical mechanics predicted an entropy of mixing for identical gases, contradicting thermodynamic observation. The resolution came from quantum mechanics (indistinguishability) but was anticipated by Gibbs' ad hoc introduction of the $1/ N!$ correction. The paradox remains a pedagogical touchstone for discussions of the relationship between information, entropy, and the role of the observer in statistical mechanics.

Bibliography Master

@article{sackur1911,
  author = {Sackur, Otto},
  title = {Die Anwendung der kinetischen Theorie der Gase auf thermodynamische Probleme},
  journal = {Ann. Phys.},
  volume = {341},
  pages = {958--980},
  year = {1911}
}

@article{tetrode1912,
  author = {Tetrode, Hugo Martin},
  title = {Die chemische Konstante und die Zustandsgleichung},
  journal = {Ann. Phys.},
  volume = {343},
  pages = {453--480},
  year = {1912}
}

@book{pathria-beale2011,
  author = {Pathria, R. K. and Beale, Paul D.},
  title = {Statistical Mechanics},
  edition = {3rd},
  publisher = {Academic Press},
  year = {2011}
}

@book{reif1965,
  author = {Reif, Frederick},
  title = {Fundamentals of Statistical and Thermal Physics},
  publisher = {McGraw-Hill},
  year = {1965}
}

Prerequisites

11.04.01
11.04.03

Tier anchors

beginner: Schroeder, An Introduction to Thermal Physics (2000), Ch. 2 (Entropy) and Ch. 5 (Ideal gas)
intermediate: Reif, Fundamentals of Statistical and Thermal Physics (1965), Ch. 5–7; Pathria & Beale, Statistical Mechanics, 3rd ed. (2011), Ch. 1
master: Landau & Lifshitz, Statistical Physics, Part 1, 3rd ed. (1980), §22–25; Huang, Statistical Mechanics, 2nd ed. (1987), Ch. 5

References

Sackur 1911 — Die Anwendung der kinetischen Theorie der Gase · Ann. Phys. 341, 958–980. Sackur derived the absolute entropy of a monatomic ideal gas from quantum statistics before the full development of quantum mechanics, introducing the quantity now called the thermal de Broglie wavelength.
Tetrode 1912 — Die chemische Konstante und die Zustandsgleichung · Ann. Phys. 343, 453–480. Independent derivation of the same entropy formula, with a numerical constant agreeing with Sackur.
Pathria & Beale — Statistical Mechanics, 3rd ed. (Academic Press, 2011) · Ch. 1 The statistical basis of thermodynamics. The ideal gas partition function Z = V^N/(N! lambda^{3N}), entropy S = Nk[ln(V/(N lambda^3)) + 5/2], and the Gibbs paradox.
Reif — Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965) · Ch. 5–7. Derivation of the ideal gas entropy from the canonical ensemble, discussion of the Gibbs paradox and the N! correction.
Landau & Lifshitz — Statistical Physics, Part 1, 3rd ed. (Pergamon, 1980) · Sec. 22 The free energy of an ideal gas; Sec. 23 The equation of state of an ideal gas; Sec. 25 The Gibbs distribution for rotating bodies.

Estimated time

beginner: 15 min
intermediate: 35 min
master: 60 min