Pull the partition and two gases interpenetrate; disorder jumps by ΔS = −nR Σ xi ln xi, positive for every distinct pair and maximal at an even split. Make the two gases identical and the same removed partition changes nothing at all — ΔS = 0. That discontinuous cliff is the Gibbs paradox. Down the center, data flows: the composition goes in, the engine computes, the entropy comes out. The blue team builds and defends it; the red team tries to break it.
source J. W. Gibbs, On the Equilibrium of Heterogeneous Substances, Trans. Connecticut Acad. Arts & Sci., vol. III (1875–1878) — no single stable primary link; reprint at wikisource.org/…/Gibbs,_Volume_1. Rendered, not quoted.
For an ideal-gas mixture the entropy rise on removing the partitions is configurational only — it counts arrangements, nothing else:
ΔS = − n R Σi xi ln xi
xi are the mole fractions (Σ xi = 1), n the total moles, R the gas constant. Each x ln x is negative, so −Σ is positive. It is symmetric — it knows only the fractions, never which gases.
Current split, term by term:
| component | xi | −xi ln xi |
|---|
Mixing is spontaneous and irreversible: the gases never un-mix on their own. That one-way arrow is exactly what the-clausius-inequality states globally — ∮dQ/T ≤ 0, entropy of an isolated system only climbs.
This sphere supplies the configurational entropy that sits underneath that inequality: ΔS = −nR Σ xi ln xi for distinct gases, zero for identical ones. Each sphere is the next one's premise.
The blue team's live check: recompute the 50/50 distinct case (must equal nR ln2) and the identical-gas case (must equal 0). If red tampers, the identical case stops being zero and this badge turns red.
Two gas samples sit either side of a partition in a rigid, insulated box, at the same T and P. You supply three things: the mole fraction split x : (1−x), the total moles n, and whether the two gases are distinct or identical.
Then the partition is pulled. For an ideal gas each species free-expands into the whole volume with no energy change (dU = 0, dQ = 0 to the box) — the entire effect is the entropy of arrangement. That, and only that, is what the panel computes.
Distinct gases: each interpenetrates new space, disorder rises.
ΔS/nR vs. split — dome peaks at ln2 (x = ½)
Move any control — ΔS is computed from −nR Σ x ln x on the spot, never looked up.
What the machine produces, proven: for distinct gases ΔS = −nR Σ xi ln xi > 0, maximal at the even split where it equals nR ln2 ≈ 5.763 J/K per mole; a 30/70 split gives −R(0.3 ln0.3 + 0.7 ln0.7) ≈ 5.080 J/K. For identical gases every split gives ΔS = 0 — the Gibbs paradox.
The blue team's witness (left) confirms these numbers live; the red team (right) tries to make them wrong.
And the discontinuity is genuinely disturbing: ΔS stays at nR ln2 for gases as alike as you please, then drops to 0 the instant they are the same. Classical mechanics gives no natural place for that cliff — its resolution rests on indistinguishability (the N! counting; made fundamental by quantum statistics). AMBER
"Mixing entropy depends on how different the two gases are." Cut. For ideal gases it depends only on the mole fractions — identical value for a near-twin pair or wildly different ones. The dependence on difference is a step, not a slope.
"Removing a partition always raises entropy." Cut. Between two samples of the same gas the states are indistinguishable; ΔS = 0. Nothing observable changed.
"The paradox proves classical thermodynamics is wrong." Kept, corrected. It is not wrong — it forces correct counting of indistinguishable states. Gibbs saw the resolution; quantum mechanics later made it inevitable. AMBER
The red team's move: apply the mixing formula to identical gases too — report nR ln2 for pulling a partition between the same gas, so entropy seems to rise from nothing. The blue team's witness (window 7) is watching.
Force the formula onto identical gases and the Gibbs-paradox check fails: the witness recomputes the identical case, finds it is no longer zero, and turns red. Nothing is faked; the attack is real and it is caught.