Pressure, volume, and temperature — three faces of a gas — bound into one line by nothing but molecules bouncing off walls. Pressure is the momentum flux of those collisions: P = (1/3)(N/V)m⟨v²⟩. Feed in equipartition, ⟨v²⟩ = 3kT/m, and the microscopic picture collapses to the macroscopic law: PV = NkT. Down the center the state variables go in, the box computes, the equation of state comes out. The blue team derives and defends it; the red team tries to break it.
source É. Clapeyron, Mémoire sur la puissance motrice de la chaleur, Journal de l'École Polytechnique, Tome XIV (1834), pp. 153–190 — first statement of the combined law; kinetic derivation D. Bernoulli, Hydrodynamica (1738). No stable DOI (AMBER); fac-similé at gallica.bnf.fr · ark:/12148/bpt6k3414331n. Rendered, not quoted; natural units k = 1.
The law is not fitted; it falls out of collisions. A molecule of mass m hitting a wall reverses momentum 2m·vx; summing the flux over all molecules gives
P = (1/3)(N/V)·m·⟨v²⟩ (momentum flux, mass-independent once averaged).
Equipartition splits the kinetic energy across 3 translational modes — each carries (1/2)kT — so ⟨v²⟩ = 3kT/m. Substitute and the mass cancels: P = NkT/V. Live, the two routes must agree:
| route | pressure P |
|---|
P, V, T in a single equation is the shadow cast on the walls by the Maxwell–Boltzmann cloud of moving molecules. That sphere gives the distribution of speeds; average its ⟨v²⟩ and you get this sphere — equipartition made an equation of state.
The distribution is the premise; the ideal gas law is its integral. Each sphere is the next one's premise.
The blue team's live check: re-derive the pressure two ways and re-run Charles's linearity at the current state. If red tampers with the temperature power, the two routes split and linearity breaks — this badge is where it shows.
A gas in equilibrium is fixed by four numbers. Feed them to the box below; the pressure is not a fifth free number — it is forced.
| symbol | is | role |
|---|---|---|
| N | molecule count | more hits → more P |
| V | volume | bigger box → less P |
| T | temperature | hotter → faster → more P |
| m | molecular mass | cancels — P is mass-free |
That last row is the quiet miracle: at a given T, P does not care what the gas is. Heavy molecules move slower by exactly the amount that keeps momentum flux fixed. That universality is what you feed the panel.
Move any slider — pressure is computed live from momentum flux and cross-checked against the state relation, never looked up.
What the box produces, proven: PV = NkT — the two independent derivations (momentum flux and equipartition) agree to 1 part in 10⁹; the mean kinetic energy per molecule is exactly (3/2)kT; at fixed T pressure follows Boyle (P ∝ 1/V) and at fixed P volume follows Charles (V ∝ T). The current state's numbers are above; the invariant law is the output.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
And "T ∝ ⟨v²⟩" is classical only. At low T and high density the counting fails: electrons obey Fermi–Dirac, photons and helium-4 obey Bose–Einstein, and the equipartition ½kT per mode freezes out. The ideal gas is the first equation of state, not the last.
"Real gases obey PV = NkT." Cut. Only in the dilute limit. Near condensation Z = PV/NkT ≠ 1; van der Waals corrects for size and attraction — both computed away by the ideal assumption.
"Temperature measures how fast molecules move." Corrected. It measures mean kinetic energy (∝ ⟨v²⟩), not speed — and only the (3/2)kT translational share. Rotations and vibrations carry their own.
"Boyle discovered the gas law." Kept, corrected. Boyle (1662) gave P ∝ 1/V; Charles / Gay-Lussac the T part; Bernoulli (1738) the kinetic derivation; Clapeyron (1834) combined them into one line.
The red team's move: swap the temperature power, PV = NkT² instead of NkT. Doubling T no longer doubles P at fixed V, and Charles's straight line bends. The blue team's witness (window 7) is watching.
Raise the temperature to the wrong power and the momentum-flux pressure (honest, T¹) and the state relation (now T²) disagree — the witness recomputes, sees the split and the broken linearity, and turns red. Nothing is faked; the attack is real and it is caught.