KINETICS · STORY 08

The concentration
countdown

Thermodynamics promised the downhill — it never said how long the walk takes. Season three straps a stopwatch to the reaction.

HOW TO READ

The stage on the right is driven by one decomposition: the beaker’s bubbles pace the instantaneous rate, the microscopic molecules vanish with the remaining fraction, and the symbolic panel draws the whole concentration-time curve with its half-life staircase. All three panels share one clock — rate is not a memorized formula, it is the slope of this curve.

One clock · three representations

t: 0 s
c/c₀: 1.000
v: 0.012 mol L⁻¹ s⁻¹

Macro · Beaker Bubble pace follows the computed instantaneous rate; the glassware is schematic.

Micro · Molecules Every 2 H₂O₂ consumed form 2 H₂O and 1 O₂; positions are schematic.

c(t) / c₀ t = 0 s · c/c₀ = 1.000

The peroxide bottle in the medicine cabinet, once opened, burbles away its oxygen:

\mathrm{2\,H_2O_2 \to 2\,H_2O + O_2}

Last season established this is thoroughly downhill (ΔG ≪ 0). Yet it does not go in one rush — the bubbles come one at a time.

Watch it for ten minutes. Will the bubbling speed up, hold steady, or slow down?

Place your bet.

Predict first: the bubbling will

Press play and watch the concentration fall.

The curve is steep at first, then gentle — because the rate follows the concentration: many molecules, many encounters, fast reaction; fewer molecules, and the rate ebbs with them. In symbols:

v = -\frac{\mathrm{d}c}{\mathrm{d}t} = k\,c

Rate is not one number; it is the slope of the curve at each point. The constant is k — the countdown's metronome, in s⁻¹, indifferent to concentration.

t = 0 s

A beautiful rhythm hides in this curve.

Falling from $c_0$ to $c_0/2$ takes some time; falling from $c_0/2$ to $c_0/4$ takes exactly the same time; halve again — the same again.

t_{1/2} = \frac{\ln 2}{k}

The half-life does not care how much is left — that is the fingerprint of a first-order reaction. Watch the staircase marks on the plot: evenly spaced drumbeats, all the way down (asymptotically) to zero.

But not every reaction keeps this beat. Switch the order and watch the fingerprint change:

Zero order: rate ignores concentration (saturated enzymes, surfaces) — successive half-lives halve each time, and the line crashes into zero
First order: half-life constant — radioactive decay; peroxide decomposition only within a stated concentration regime and at fixed catalyst concentration
Second order: rate ∝ c² — half-lives double each time, dragging a long tail

Measuring successive half-lives is dusting a reaction for fingerprints — the detective's way to read the order with no curve fitting at all.

Reaction order

Successive half-lives: 57.8 s → 57.8 s → 57.8 s

Write the fingerprint as mathematics. Integrating the first-order law:

c = c_0 e^{-kt} \qquad \ln c = \ln c_0 - kt

Take the logarithm and the curve straightens into a line with slope −k. The same move as straightening the vaporization curve in story two: find the right coordinates and the law stands up straight by itself.

The order m in $v = k c^m$ can only be read from experiment — never copied from stoichiometry. The 2 in $\mathrm{2\,H_2O_2}$ has nothing to do with "first order".

The same mathematics, with the stopwatch swapped for a calendar of millennia.

Carbon-14 is radioactive carbon made by cosmic rays; living things keep topping it up. At death the topping-up stops and the countdown begins — half-life 5730 years, rain or shine.

t = 5730 \cdot \log_2\frac{c_0}{c}

Ötzi the Iceman still carries about 53 % of his carbon-14: roughly 5300 years. The Dead Sea Scrolls were dated by this same exponential. A first-order stopwatch never asks how big the bottle is.

Carbon-14 remaining 53% 53% left → about 5248 years old

The honesty clause:

Peroxide's first-order behavior is a conditional experimental result, not something the equation reveals: fixed iodide can give pseudo-first-order behavior in H₂O₂, while an enzyme is approximately first order only in its low-substrate, unsaturated regime
k depends strongly on temperature (next story's protagonist) and on catalysts (the one after) — every curve on this page sits at one fixed temperature
Real reactions can change order: a saturated enzyme slides from first order toward zero; orders can even be fractional

A rate law is an empirical law read off measurements; the mechanism hides behind it — a deeper detective story.

The stopwatch is yours.

Three knobs — order, rate constant, starting concentration — recompute the curve and its half-life staircase live; the beaker and the molecule chamber answer in step.

Challenges: tune a first-order curve that drums exactly three half-life beats within 200 seconds; then work out why a second-order reaction never quite finishes dying.

Reaction order

Rate constant k0.012 s⁻¹ Starting concentration c₀1.0 mol L⁻¹ First half-life t½ = 57.8 s

THE STOPWATCH

The timekeeper’s handbook

Rate is slopev = −dc/dt: the tangent at every point, ebbing with concentration; k is the metronome that never changes.

Half-life fingerprintsEqual beats = first order, doubling = second, halving = zero — reading the order with no fitting at all.

Order is measuredThe m in v = kc^m comes only from experiment; stoichiometric coefficients get no vote.

The carbon-14 clockFirst-order decay with t½ = 5730 a: the same exponential runs from the medicine cabinet to the excavation site.

CONCEPT CHECK · PREDICT, THEN TEST

A reaction shows successive half-lives of 100 s, 200 s, 400 s. Its most likely order is:

Zero — the half-life is changing First — having a half-life at all means first order Second — half-lives double each round Impossible to say without temperature data

FREE PLAY · Leave the storyline with a question

The countdown bench

Order, rate constant, and starting concentration are all yours; the curve, the bubbles, and the molecules answer together.

Reaction order

Rate constant k0.012 s⁻¹ Starting concentration c₀1.0 mol L⁻¹ First half-life t½ = 57.8 s

First half-life t½ = 57.8 s

One clock · three representations

t: 80 s
c/c₀: 0.383
v: 0.0046 mol L⁻¹ s⁻¹

Macro · Beaker Bubble pace follows the computed instantaneous rate; the glassware is schematic.

Micro · Molecules Every 2 H₂O₂ consumed form 2 H₂O and 1 O₂; positions are schematic.

c(t) / c₀ t½ t = 80 s · c/c₀ = 0.383

The curve is computed from the integrated rate law; vertical lines drum the successive half-lives.

MODEL CARD

Model card · V0.1

System

H₂O₂ decomposition (2H₂O₂ → 2H₂O + O₂) is a classic classroom system. The first-order curve represents stated limiting conditions only: fixed iodide can make the law pseudo-first-order in H₂O₂; catalase is approximately first order at low, unsaturating substrate and tends toward zero order as substrate rises. The on-page k is a free knob, not one measured catalytic condition; the uncatalyzed room-temperature reaction is very slow.

Computation

Integrated rate laws for orders 0/1/2 evaluated analytically; successive half-lives advanced per-order (c₀/2k, ln2/k, 1/kc₀), with unit tests pinning the equal/halving/doubling fingerprints and curve-drumbeat alignment.

Carbon-14

Cambridge half-life 5730 a; age = 5730·log₂(c₀/c). Real archaeological dating additionally applies the IntCal atmospheric calibration curve; on-page ages are uncalibrated teaching values, off by up to centuries.

What cannot be inferred from this page

Rate laws are empirical: orders do not come from stoichiometry, and mechanisms (elementary steps, intermediates) are beyond this page; temperature lives in story nine, catalysis in story ten. Bubble pacing is proportional to the computed rate but is not a gas-volume measurement.

Rate is a slope, k is a metronome, half-life is a fingerprint. Thermodynamics drew the terrain; kinetics has started calling the time.

Next stop: how temperature winds this stopwatch — the mountain the molecules must cross.

Back to the story catalog GitHub ↗

The concentrationcountdown

The timekeeper’s handbook

A reaction shows successive half-lives of 100 s, 200 s, 400 s. Its most likely order is:

The countdown bench

Model card · V0.1

Next stop: how temperature winds this stopwatch — the mountain the molecules must cross.

The concentration
countdown