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Lessons · Kinetics

When the reaction never slows down: zero-order decomposition

Species accounting machine-checked — the integrated rate law + half-lifek data-sourced (openstax-chemistry-2e)2 modeling assumptions (disclosed)

On a hot tungsten wire, ammonia breaks down: 2 NH₃ → N₂ + 3 H₂. The surface is saturated with NH₃, so speeding it up by adding more does nothing — the reaction is zero order: rate=k\text{rate} = k, a constant, with k=1.3×10−6 M s−1k = 1.3\times10^{-6}\ \mathrm{M\,s^{-1}}. The amount left falls in a straight line, [NH3]=[NH3]0−kt[\mathrm{NH_3}] = [\mathrm{NH_3}]_0 - kt, so unlike an exponential decay it reaches exactly zero at a finite time t=[NH3]0/kt = [\mathrm{NH_3}]_0/k. Its half-life is t1/2=[NH3]0/(2k)t_{1/2} = [\mathrm{NH_3}]_0/(2k) — proportional to how much is present, so as the ammonia runs low the half-life shrinks: the last half disappears faster than the first. The species ledger again, extent marching in time — at a steady pace that never slackens.

2 NH3→N2+3 H22\,\mathrm{NH_{3}} \rightarrow \mathrm{N_{2}} + 3\,\mathrm{H_{2}}
rate=k\text{rate} = k·k = 1.3×10⁻⁶ M s⁻¹zero order · on a hot tungsten (W) surface (source states no temperature)
Integrated rate lawOne constant fixes the whole future — [NH3]=[NH3]0−kt[\mathrm{NH_{3}}] = [\mathrm{NH_{3}}]_0 - kt.

Starting at 0.01 M, the concentration falls in a straight line — the rate = k is constant, independent of how much is left — until the reactant runs out completely at 2.14 h.

The decay curveEvery point is the integrated rate law — the ledger marched forward in time.
0.000.002500.005000.007500.01000.01.12.1hours[NH₃] (M)½ left¼ left⅛ left
to halving #10.005 Mstep took 1.07 h · 50% gone
to halving #20.0025 Mstep took 0.534 h · 75% gone
to halving #30.00125 Mstep took 0.267 h · 87.5% gone
first half-life t½1.07 h= 3850 s
t½ relationt1/2=[NH3]02kt_{1/2} = \dfrac{[\mathrm{NH_{3}}]_0}{2k}t½ ∝ [A]₀ — shrinks as [A] falls ✓
the half-life shrinks1.07 → 0.534 → 0.267 heach half takes ~½ as long

The zero-order tellBecause t1/2=[NH3]02kt_{1/2} = \dfrac{[\mathrm{NH_{3}}]_0}{2k} is proportional to [NH₃]₀, each half-life is shorter than the last (1.07 → 0.534 → 0.267 h — roughly halving): the rate never changes (rate = k, the surface is saturated), so a fixed amount disappears every hour and the reactant runs out entirely at 2.14 h. The machine checks each successive halving takes about half as long — and that [NH₃] hits exactly zero, not an asymptote.

VerificationProven at build time — not asserted.
  • ✓ The reaction conserves every element [reaction balanced]
  • ✓ Every curve point matches the order-0 integrated law — re-derived independently [integrated law]
  • ✓ The first t½ = [A]₀/2k (order 0) [half-life relation]
  • ✓ Successive half-lives halve — the order-0 fingerprint [half-life progression]
✗ Common misconception: “As the ammonia gets used up there is less of it, so the reaction must slow down and only taper off toward zero, never quite reaching it.”

Not for a zero-order reaction. The rate = k is constant — it does not depend on [NH₃] at all (the tungsten surface is saturated, so adding more changes nothing). A fixed amount disappears each hour, so far from slowing, the reactant runs out completely at 2.14 h. Because the drop is steady, each successive half-life is shorter: 1.07 → 0.534 → 0.267 h — the machine checks the halving and the finite finish.

Modeling assumptions — author-asserted, disclosed not discharged
  • model The reaction is zero order in NH₃ (rate=k\text{rate} = k) — an experimentally determined rate law that holds while the tungsten surface is saturated, not read off the balanced equation. The order and rate constant kk are the sourced data; the integrated law and half-life follow exactly.
  • model The zero-order regime holds throughout the run modeled here — fixed temperature, the surface kept saturated — so the straight-line decay applies until the ammonia is nearly gone (at very low [NH₃] the surface is no longer saturated and the true order changes).

Concepts in this lesson

Linked into the Chemical Atlas where an entry exists; the rest fill in as the Atlas grows.

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