How much is left, and when. Watch a concentration fall — is it a straight line, an exponential, or a reciprocal creep? The answer is the reaction's order, and the tell is the half-life: first order keeps the same half-life forever (ln2/k, no matter the start), second order's half-life lengthens as the reactant thins, zeroth order's shrinks. Terms go in, the integrated law runs, the half-life comes out — and names the order. Blue builds it; red breaks it.
source J. H. van 't Hoff, Études de dynamique chimique (Amsterdam, 1884) — where reaction order and the integrated laws were first set out; the term "order" was formalized by Ostwald. archive.org/details/etudesdedynamiq00hoffgoog. Marked AMBER: rendering of the modern integrated forms, not a quote. Rendered, not quoted.
The differential rate d[A]/dt = −k[A]n integrates to a closed form for each order. Each has one signature: how its half-life depends on the starting amount.
| n | rate | integrated | t½ |
|---|---|---|---|
| 0 | k | [A]=[A]₀−kt | [A]₀/2k |
| 1 | k[A] | [A]=[A]₀e−kt | ln2/k |
| 2 | k[A]² | 1/[A]=1/[A]₀+kt | 1/(k[A]₀) |
First order's t½ is the odd one: no [A]₀ in it. Halve the start and it does not budge. That constant is the fingerprint the panel and witness are built to read.
This sphere is the time-integral of two neighbours. the-mass-action gives the differential rate d[A]/dt = −k[A]n; integrate over time and you get the curves here.
The single constant k is itself a function of temperature — the-arrhenius-equation's k = A·e−Ea/RT. So the half-life you measure is the mass-action law, integrated, with an Arrhenius k inside. Each sphere is the next one's premise.
The blue team's live check: measure the first-order half-life at four different starting concentrations and confirm they are identical — the signature of order 1. If red tampers, the four disagree and this badge turns red.
Three numbers fully specify a decay: the order n (0, 1, or 2), the rate constant k, and the starting concentration [A]₀. That is everything the integrated law needs.
| input | meaning | units |
|---|---|---|
| n | reaction order | — |
| k | rate constant | M1−n·s−1 |
| [A]₀ | start conc. | M |
k's units even encode the order: zeroth is M/s, first is 1/s, second is 1/(M·s). Feed these into the panel below.
| [A]₀ | t½ (measured) |
|---|
Change any control — the concentration curve, half-life, and order-verdict are computed from the integrated law on the spot, never looked up.
What the machine proves: the half-life names the order. First order's ln2/k is constant — halve [A]₀ and it is unchanged. Second order's 1/(k[A]₀) doubles when [A]₀ halves. Zeroth order's [A]₀/2k halves when [A]₀ halves. Three orders, three distinct fingerprints.
The blue team's witness (left) confirms the first-order constancy live; the red team (right) tries to break it.
Order is not the stoichiometric coefficient — it is empirical, and can be fractional or negative. "Second order overall" may hide a mechanism of several first-order steps. The clean curve is a fitted approximation, honest only over the window where it fits.
"Every reaction has a half-life like radioactivity." Cut. A constant, concentration-free half-life is the mark of first order only. Second and zeroth order half-lives depend on [A]₀ — both computed in the machine.
"The order equals the number of reactant molecules." Cut. Order is empirical, from the measured rate law — it can be 0, ½, or negative, and need not match stoichiometry.
"A first-order half-life gets longer as it runs out." Kept, corrected. That is second order (t½=1/k[A]₀ grows as [A]₀ falls). First order's half-life is fixed — the whole point.
The red team's move: rewrite the first-order half-life with the second-order formula 1/(k[A]₀), so it wrongly depends on concentration. The order-fingerprint collapses — and the witness (window 7) is watching.
Poison the first-order half-life and the four witness measurements stop agreeing — the constant-t½ signature breaks, the badge recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.