The rate of an elementary reaction is proportional to the product of the concentrations of its reactants. For A + B ⇌ C the forward rate is kf·[A]·[B] and the reverse is kr·[C]; integrate the two ODEs and the system relaxes until forward equals reverse — fixing the quotient Q = [C]/([A][B]) at exactly Keq = kf/kr. Terms go in, the engine integrates, the equilibrium comes out. The blue team builds it; the red team tries to break it.
source Guldberg & Waage, Studies concerning affinity (1864), read to the Christiania Academy — Norwegian original, no stable primary DOI (facsimile 1964; Eng. tr. J. Chem. Educ. 63, 1044, 1986). AMBER. Rendered, not quoted.
Two coupled ODEs, one for each species; the reverse of the forward:
d[A]/dt = d[B]/dt = kr[C] − kf[A][B]
d[C]/dt = kf[A][B] − kr[C]
The forward term is second order — first order in each reactant. Two combinations never change: [A]+[C] and [B]+[C] (atoms are conserved). Live, for the current run:
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Guldberg & Waage, 1864: rate is set by concentration. Everything downstream is this one idea specialised.
Hold C as an enzyme–substrate complex and add a catalytic step and you get the-michaelis-menten. Take the ratio at steady state and you get the-equilibrium-constant Keq=kf/kr. Reinterpret kf[A][B] as a per-time propensity for discrete molecules and you get the-gillespie. Each sphere is this one's next reading.
The blue team's live re-check: doubling [A] must double the forward rate, atoms must stay conserved, and the integrated quotient must land on kf/kr. If red tampers, this badge is where it shows.
Four numbers feed the engine: the rate constants kf, kr and the starting amounts [A]₀, [B]₀ (with [C]₀ = 0). Nothing else is needed — the whole trajectory and the equilibrium are consequences of these four, computed live below.
kf carries the forward affinity; kr the reverse. Their ratio alone fixes where the reaction lands, no matter the path taken to get there.
RK4 integration, dt=0.02, to t=30. The forward rate is recomputed from concentrations at every step — never looked up.
Blue = [A], cyan = [B], green = [C] relaxing to equilibrium; the dashed line is the equilibrium [C] the ratio kf/kr demands.
What the machine proves, live: the integrated quotient Q = [C]/([A][B]) lands on Keq = kf/kr to 1e−6; forward rate equals reverse rate there; Q climbs (or falls) monotonically toward Keq from either side; and [A]+[C], [B]+[C] hold constant to 1e−12 the whole way.
The blue team's witness (left) re-checks these live; the red team (right) tries to make them wrong.
A real overall reaction is a mechanism of hidden elementary steps, so its measured rate law is empirical and need not match the balanced equation at all. The law is exact for the step, not the summary.
"Reaction order equals the stoichiometric coefficients." Cut. True only for an elementary step. Overall order is measured, not read off the equation.
"Mass action gives the equilibrium constant's value." Cut. Kinetics gives Keq only as the ratio kf/kr; its numerical value is fixed by thermodynamics (ΔG) — the two must agree, and that agreement is the constraint.
"Guldberg & Waage wrote rate = k·[A][B]." Kept, corrected. Their 1864 term was active mass; strict validity in non-ideal solutions needs activities, not raw concentrations.
The red team's move: make the forward rate a constant (zeroth order) — independent of concentration. Then doubling [A] does nothing, and the equilibrium ratio no longer equals kf/kr. The blue team's witness (window 7) is watching.
Cut the concentration out of the forward rate and mass-action proportionality dies — the witness recomputes, finds doubling no longer doubles, and turns red. Nothing is faked; the attack is real and it is caught.