Gibbs Paradox — Physica

Equivalent forms

\Delta S_{\text{mix}} = 2 N k_B \ln 2 \ \text{(equal mix of 2 distinct gases)}

S = N k_B\left[\ln\frac{V}{N\lambda^3} + \tfrac{5}{2}\right]

A bookkeeping puzzle whose only consistent resolution — the N! Gibbs factor — secretly demanded quantum indistinguishability decades early.

Symbol	Name	SI unit	Dimension	Range
$ΔS_mix$	Entropy of mixingoutput Entropy change when the partition is removed	J/K	M L² T⁻² Θ⁻¹	0 – 1e-20
$N$	Particle number Number of particles per compartment	dimensionless	1	10 – 60
$x_i$	Mole fraction Fraction of species i in the mixture	dimensionless	1	0 – 1

Unit systems

SI:: S in J/K
natural:: S in units of k_B
CGS:: S in erg/K

Where it holds

Classical ideal gases; the N! correction is exact in the dilute (non-degenerate) limit and is automatic in the correct quantum statistics.

Discovery

Josiah Willard Gibbs · 1875

Gibbs noted in the 1870s that the classical mixing entropy didn't vanish for identical gases; the fix — dividing phase space by N! — anticipated quantum indistinguishability by 50 years.

Try this

Why does removing a wall sometimes create entropy — and sometimes not?

Mix two different gases and entropy jumps. Mix a gas with an identical copy of itself and — nothing. The 'paradox' forced physics to admit that identical particles are truly indistinguishable.

Research status: stable

Common misconceptions

The paradox is not resolved by 'we can't tell the particles apart in practice' — it requires that identical particles are in principle indistinguishable, a quantum, not practical, statement.

Derivation

Without the 1/N! factor, the Boltzmann entropy of an ideal gas is not extensive, so combining two identical boxes appears to create entropy

\Delta S = 2N k_B \ln 2

even though nothing physical changed.

Dividing the phase-space volume by N!

(Gibbs' correction) restores extensivity and gives the Sackur-Tetrode formula; the spurious self-mixing term cancels.

For genuinely distinct gases the term survives as the real mixing entropy.

Limiting cases

Identical gases⟶

\Delta S_mix = 0

: removing the wall is thermodynamically a non-event

Two distinct gases, equal amounts⟶

\Delta S_mix = 2N k_B \ln 2

— maximal mixing entropy

Trace impurity

(x \to 0)

⟶ Per-particle mixing entropy diverges

as -\ln x

, but total stays finite

Sackur-Tetrode Equation→Boltzmann Entropy→Entropy Change→Second Law of Thermodynamics→