⚗️ Reaction Order Detective — initial rates, half-lives, Arrhenius

Rate law

d[A]/dt = −k[A]ⁿ. Integrated: n=0 → [A]=[A]₀−kt; n=1 → [A]=[A]₀e^−kt; n=2 → 1/[A]=1/[A]₀+kt. The plot that comes out straight (a different one for each order) identifies n.

Half-life

n=0: t½=[A]₀/2k; n=1: t½=ln2/k (independent of [A]₀); n=2: t½=1/(k[A]₀). So t½(2[A]₀)/t½([A]₀) = 2¹⁻ⁿ.

Initial rates

Rate₀=k[A]₀ⁿ, so the slope of log(rate) vs log[A]₀ is n.

Arrhenius

k=k₂₉₈·exp[−(E_a/R)(1/T−1/298.15)], R=8.314 J/(mol·K); the slope of ln k vs 1/T is −E_a/R.

Molecule view

N₀=300 molecules; each time step the number of reactions is drawn from a Poisson distribution with mean (rate × dt), so the jagged trace scatters around the smooth curve (fewer molecules = more noise). Molecule motion is decorative (it speeds up with √T) and does not itself determine the rate.

Simplifications and honesty

Single-step irreversible reaction, constant volume, perfectly mixed, no reverse reaction. Zero order is realised in real life when a catalyst surface is saturated; here it is simply defined. Measurement noise in the experiments (3% on rates, 2% on k) is added on purpose to mimic a lab and is not a chemical effect. Rate constants are illustrative, not data for a named reaction.

Independent check

The exact curve is the closed form above; a separate RK4 integration of the differential equation is compared against it live (relative error shown in the readouts).