ΔG° is plainly positive, yet the flask turns brown. A terrain map explains everything.
HOW TO READ
The stage on the right is driven by one extent of reaction: the macroscopic flask reports the NO₂ content through its color, the microscopic molecules show dimers splitting and re-pairing, and the symbolic panel draws the whole Gibbs-energy valley — the ball’s position is the flask’s state at this instant.
Current state: 25 °C, extent 0.19.
00
A sealed flask of colorless N₂O₄ gas. Look up the tables:
Positive. By the slogan "positive ΔG means non-spontaneous", nothing should happen in that flask.
Yet at room temperature the flask is visibly brown — the color of NO₂. About a fifth of the N₂O₄ has already split.
Is the slogan wrong? Place your bet.
01
Two forces wrestle inside the flask.
Energy sides with N₂O₄: splitting one dimer costs 57.2 kJ/mol. Enthalpy shouts "don't".
Counting sides with splitting: one molecule becomes two, the particle count grows from 1 to 1+ξ — more particles, more ways to mix. The W we counted last story shouts "do".
ΔG° referees only the match between pure reactants and pure products. But a reaction never has to go all the way — it can stop partway. And about partway, ΔG° says not a word.
02
So draw the partway. The axis is the extent : 0 is pure N₂O₄, 1 is fully split. For every ξ, compute the total Gibbs energy:
The first term is the standard straight line — tilted upward, since ΔG° is positive. The second is mixing entropy: every ln x of a mixture is negative, and it presses the middle of the curve down.
A line plus a sag makes a valley. Its floor sits at ξ ≈ 0.19 — not at 0, not at 1. The flask’s brown is the color of that floor.
03
Now untangle the two easily-confused siblings.
ΔrG (no °) is the valley’s slope at the current position:
Slope negative — the ball rolls right (forward reaction). Positive — it rolls left (reverse). Zero slope is the floor: equilibrium.
ΔrG° is merely one construction parameter of the valley: it sets which side the floor leans toward, never whether a floor exists. Mixing entropy guarantees one whenever two gases coexist — no reaction truly runs to completion.
04
The floor’s position has a name: the equilibrium constant.
K is no separate mystery — it is the floor’s coordinate in different units. ΔG° = 0 means K = 1, floor dead center; every 5.7 kJ/mol of ΔG° slides the floor one power of ten.
+4.7 kJ/mol converts to K ≈ 0.15: leaning toward N₂O₄, but nowhere near "doesn't happen". "Positive ΔG°" only ever means: the floor is in the left half.
05
Squeeze the valley: pressure.
In a 1-goes-to-2 reaction the particle count grows with extent; raising the pressure flattens the mixing advantage of the many-particle side, the right wall bulges, and the floor slides left — NO₂ is pressed back into dimers, the brown fades.
Drag the pressure slider and watch the floor follow . Le Chatelier’s principle is not an independent law — it is the geometry of a valley under load.
06
Now bake it: temperature.
Splitting the dimer is endothermic (ΔH° = +57.2 kJ/mol), and van ’t Hoff says K grows exponentially with temperature: the floor slides right.
In an ice bath the flask runs nearly colorless (0 °C: K ≈ 0.02); at 60 °C it browns deeply (K ≈ 1.3) — the classic classroom color demo is this same valley flexing continuously with temperature. Drag the slider and watch color and floor travel together.
07
Back to the slogan — now we can fix it precisely:
One sentence: ΔG° builds the valley, ΔrG reads the slope, the reaction rolls to the floor.
08
The valley is yours.
Two knobs — temperature and pressure — recompute the whole G(ξ) curve live; click anywhere on the valley to drop the ball and watch the slope shepherd it home; the flask and the molecules answer in step.
Challenge: find the temperature that centers the floor exactly at ξ = 0.5 — then squeeze it back below 0.3 with pressure alone.
Floor ξ_eq = 0.19 · K = 0.15 · ΔG° = 4.7 kJ/mol
THE SURVEYOR
CONCEPT CHECK · PREDICT, THEN TEST
FREE PLAY · Leave the storyline with a question
Temperature, pressure, and the ball are all yours; the valley recomputes live and the flask recolors in step.
Floor ξ_eq = 0.19 · K = 0.15 · ΔG° = 4.7 kJ/mol
Current state: 25 °C, extent 0.19.
MODEL CARD
N₂O₄(g) ⇌ 2 NO₂(g), standard formation data from CODATA/NIST-JANAF (298.15 K): ΔfG°(NO₂)=51.3, ΔfG°(N₂O₄)=97.9, ΔfH°(NO₂)=33.2, ΔfH°(N₂O₄)=9.16 kJ/mol, giving ΔrG°=+4.7 and ΔrH°=+57.2 kJ/mol. Unit tests pin K(298) ≈ 0.15 and the K = 1 temperature ≈ 325 K.
Ideal-gas mixing: G(ξ) evaluated analytically point by point; the floor has the closed form ξ_eq = √(K/(K+4p)), cross-checked against the bracketed zero of ΔrG to 1e-9. Temperature extrapolation is van ’t Hoff with constant ΔH°; the error within 80 °C is far below line width.
The flask browns linearly with the computed NO₂ mole fraction. Real NO₂ absorption (~400 nm) follows Beer’s law nonlinearly in path and concentration; the linear teaching map keeps the trend, not photometry.
The ideal-gas approximation degrades above ~5 bar, and the real vapor also holds minor species like N₂O₃ — neither is modeled. Rates and activation energies are entirely absent: the valley answers "where it stops", never "how fast it gets there". Non-ideality corrections and kinetics simulations are backend work (docs/后端待办.md).
ΔG° builds the valley, mixing entropy digs the floor, ΔrG reads the slope, Le Chatelier flexes the walls. A reaction has no finish line — only a floor.