Zero, First, and Second Order Kinetics

What "order" means

Reaction order is the exponent that concentration takes in the rate law. For a generic reaction aA + bB → products with rate law

      rate = k[A]^m [B]^n

the reaction is m-th order in A, n-th order in B, and (m+n)-th order overall. The exponents m and n are not the stoichiometric coefficients a and b. They reflect the slowest mechanistic step (the rate-determining step) — and the only way to know them is to measure them.

Three orders dominate elementary chemistry: zero (rate independent of concentration), first (rate proportional to one concentration), and second (rate depends on the square of one or the product of two). Mixed and fractional orders exist for chain reactions and surface kinetics, but the three cases below cover the textbook bulk.

Zero-order kinetics

The rate is constant: rate = k. Differentiating against time gives

      −d[A]/dt = k       (units of k: mol·L⁻¹·s⁻¹)

Integrating from [A]_0 at t = 0 to [A] at time t yields the integrated rate law:

      [A] = [A]_0 − k·t

So a plot of [A] versus t is a straight line with slope −k. The reaction reaches [A] = 0 in finite time, t = [A]_0 / k — at which point you must check whether the law still applies (it usually doesn't; some new mechanism takes over). Half-life:

      [A]_0 / 2 = [A]_0 − k·t_{1/2}    →    t_{1/2} = [A]_0 / (2k)

It depends on the starting concentration; each successive half-life is shorter than the last.

Where it shows up. Heterogeneous catalysis on a saturated surface (the rate is set by available active sites, not gas-phase pressure). Photochemistry in optically thick solutions (every photon entering the system gets absorbed regardless of dye concentration). Enzyme reactions at saturating substrate. Alcohol metabolism — the human liver clears ethanol at roughly 0.015 % blood alcohol per hour at any concentration above its K_M.

First-order kinetics

Rate is linear in one concentration: rate = k[A]. The differential equation

      −d[A]/dt = k[A]         (units of k: s⁻¹)

separates and integrates to

      ln[A] = ln[A]_0 − k·t
      [A] = [A]_0 · e^(−k·t)

The exponential decay shape is the trademark of first-order. A plot of ln[A] versus t is straight with slope −k. Half-life is concentration-independent:

      t_{1/2} = ln(2) / k ≈ 0.693 / k

Every half-life is the same length — the very property that makes radioactive decay and pharmacokinetics tractable.

Where it shows up. All radioactive decay (each nucleus decays independently with constant probability per unit time). Most unimolecular gas-phase decompositions in their high-pressure limit, including N₂O₅ → 2NO₂ + ½O₂ at 25 °C with k = 3.5 × 10⁻⁵ s⁻¹. Radioactive iodine-131 in thyroid uptake, with t_{1/2} = 8.02 days. Drug elimination — caffeine plasma half-life is roughly 5 hours in adults.

Worked example: caffeine clearance

A 200 mg dose of caffeine peaks in plasma at about [A]_0 = 5 mg/L. With first-order half-life t_{1/2} = 5 h:

      k = ln(2) / 5 = 0.139 h⁻¹
      [A](t) = 5 · e^(−0.139·t)
      [A](10 h) = 5 · e^(−1.39) = 5 × 0.249 = 1.25 mg/L  (one quarter — two half-lives)
      [A](24 h) = 5 · e^(−3.34) = 5 × 0.0356 = 0.18 mg/L (≈ 3.6 % left)

This is why a 4 pm coffee leaves a measurable plasma concentration at midnight: only 3 half-lives have passed.

Second-order kinetics

Two flavours. The simplest is second-order in one species, rate = k[A]²:

      −d[A]/dt = k[A]²         (units of k: L·mol⁻¹·s⁻¹)

Separating and integrating:

      1/[A] = 1/[A]_0 + k·t

So 1/[A] versus t is linear with slope +k. Half-life:

      t_{1/2} = 1 / (k·[A]_0)

Half-life grows as concentration drops, so reactions slow visibly as they proceed. The mixed second-order case rate = k[A][B] integrates differently — for [A]_0 = [B]_0 it reduces to the same form, otherwise the integrated law involves ln(([B]·[A]_0)/([A]·[B]_0)).

Where it shows up. Bimolecular gas-phase reactions: 2NO₂ → 2NO + O₂ obeys rate = k[NO₂]² with k = 0.775 L·mol⁻¹·s⁻¹ at 300 °C. Diels–Alder cycloadditions in solution. Saponification of ethyl acetate by NaOH (rate = k[ester][OH⁻]). Enzyme reactions at low substrate (k_cat/K_M is the second-order rate constant for the encounter-controlled limit).

Comparing the three orders side by side

	Zero-order	First-order	Second-order ([A]²)
Differential rate law	−d[A]/dt = k	−d[A]/dt = k[A]	−d[A]/dt = k[A]²
Integrated rate law	[A] = [A]_0 − k·t	ln[A] = ln[A]_0 − k·t	1/[A] = 1/[A]_0 + k·t
Linear plot	[A] vs t	ln[A] vs t	1/[A] vs t
Slope of linear plot	−k	−k	+k
Half-life	[A]_0/(2k)	ln(2)/k = 0.693/k	1/(k·[A]_0)
Half-life dependence on [A]_0	Proportional (shrinks as reaction proceeds)	Independent (constant)	Inversely proportional (grows as reaction proceeds)
Units of k	mol·L⁻¹·s⁻¹	s⁻¹	L·mol⁻¹·s⁻¹
Concentration-vs-time curve	Straight line	Exponential decay	Hyperbolic decay
Typical example	Surface-catalysed (saturated)	Radioactive decay, N₂O₅ → NO₂	2NO₂ → 2NO + O₂; ester hydrolysis

The three concentration profiles

  [A]
  │  zero-order: linear drop until depletion
  │ ●●●●●
  │      ●●●●●
  │           ●●●●●
  │                ●●●●●
  └────────────────────────────► t

  [A]
  │  first-order: exponential — same shape, just rescaled
  │ ●
  │  ●
  │   ●
  │    ●●
  │      ●●●●●
  │           ●●●●●●●●●●●●●●
  └────────────────────────────► t

  [A]
  │  second-order: starts steep, flattens fast, never reaches zero
  │ ●
  │  ●
  │    ●
  │       ●
  │           ●●●
  │                  ●●●●●●●●●●
  └────────────────────────────► t

Determining order experimentally

The integrated-rate-law method. Run the reaction, measure [A] at several times, and plot all three: [A] vs t, ln[A] vs t, 1/[A] vs t. Whichever is linear identifies the order; the slope gives k. This is the workhorse method for student labs and routine analysis.

The half-life method. Measure successive half-lives. Equal half-lives → first order. Doubling half-lives → second order. Halving half-lives → zero order. Useful when only a few well-spaced measurements are available.

The initial-rate method. Run several experiments with different starting concentrations, measuring only the early rate before [A] drops appreciably. Doubling [A]_0 doubles the rate for first order, quadruples it for second order, leaves it unchanged for zero order. The cleanest way to nail down both partial orders in a multi-reactant system.

Real-world rate constants

• N₂O₅ decomposition (gas, 45 °C). First-order, k = 4.8 × 10⁻⁴ s⁻¹, t_{1/2} ≈ 24 min. Cleanly first-order across four orders of magnitude in concentration — a textbook example.
• ²³⁵U fission decay (nuclear). First-order, t_{1/2} = 7.04 × 10⁸ years, k = ln(2)/t_{1/2} = 9.85 × 10⁻¹⁰ year⁻¹. Used to date zircon crystals up to 4.4 billion years old.
• 2NO₂ → 2NO + O₂ (gas, 300 °C). Second-order, k = 0.775 L·mol⁻¹·s⁻¹. Stoichiometry happens to match the rate law in this case — but only because the slow step is bimolecular collision of two NO₂ molecules.
• Hydrolysis of ethyl acetate by NaOH (saponification). Mixed second-order, rate = k[ester][OH⁻], with k = 0.11 L·mol⁻¹·s⁻¹ at 25 °C. Run in flooded base, the kinetics looks pseudo-first-order in ester.
• Ethanol metabolism (human liver). Zero-order at typical drinking concentrations because alcohol dehydrogenase is saturated. Clearance ≈ 0.015 % blood alcohol per hour, independent of how drunk you started.
• Photolysis of dilute Cl₂ by UV light. First-order in Cl₂ when light is far from being absorbed (most molecules see most photons); zero-order in optically thick samples (every photon is absorbed).

Pseudo-first-order — the lab trick

For a mixed second-order reaction A + B → P with rate = k[A][B], if you run the reaction with [B]_0 ≫ [A]_0, the concentration of B barely moves during the entire run. Effectively the rate becomes

      rate = k_obs · [A]    where k_obs = k · [B]_0

The kinetics now behave as first-order in A. Run several experiments at different [B]_0 values, extract k_obs from each ln[A]-vs-t plot, then plot k_obs against [B]_0 — the slope is the true second-order k. This "isolation method" is how nearly every solution-phase kinetic constant in the chemistry literature was actually measured.

Common mistakes

• Reading order off the balanced equation. Stoichiometry tells you nothing about rate law (except for elementary single-step reactions). Always determine order experimentally — even simple-looking equations can have square-root or fractional orders.
• Confusing instantaneous and average rates. Plot the slope of [A] vs t at a point, not the secant between two distant points. The integrated forms work directly with concentration values, not rates.
• Forgetting that k has units. Zero-order k is mol/(L·s); first-order is 1/s; second-order is L/(mol·s). If your reported k has the wrong units for the claimed order, the order is wrong.
• Extending the linear plot too far. Linearity often breaks down at high conversion — secondary reactions, depletion of catalyst, or product inhibition kick in. Stop the fit before the curve bends.
• Mixing temperatures. A rate constant at 25 °C and 35 °C are not the same number — k roughly doubles per 10 °C (Arrhenius). Always quote k with the temperature.
• Reporting "order = 1.5" without context. Fractional orders usually signal a chain mechanism. Don't fudge — investigate the mechanism behind the half-integer exponent.