Reaction Orders (0th, 1st & 2nd Order), Rate Laws & Half-Life t₁/₂

The kinetic order of a reaction reveals the true molecular mechanism hidden behind the overall stoichiometric equation.

Advanced Level · AP / University · Rate Laws & Half-Life

Mechanism

The reaction order n in the differential rate law v = -(d[A])/(dt) = k[A]^n dictates both the integrated concentration decay [A](t) and the scaling of the half-life t₁/2 with initial concentration [A]₀.

Equations and conservation laws

For zero-order kinetics (n=0) (e.g., surface-saturated catalysis), rate is constant (v = k), [A](t) = [A]₀ - kt, and t₁/2 = ([A]₀)/(2k) ∝ [A]₀. For first-order kinetics (n=1), decay is exponential [A](t) = [A]₀ e⁻kt and the half-life t₁/2 = ( ln 2)/(k) is strictly independent of [A]₀.

Measurements and applications

For second-order kinetics (n=2) (bimolecular collisions 2A → P), (1)/([A](t)) = (1)/([A]₀) + kt (linear in 1/[A] vs. t) and the half-life scales inversely with initial concentration: t₁/2 = (1)/(k[A]₀).

Experimental parameters and SI units

Reaction order n (0 = zero, 1 = first, 2 = second): 0 … 2, standard value 1
Initial reactant concentration [A]₀: 0.5 … 3 mol/L, standard value 2 mol/L
Rate constant k: 0.1 … 1 s⁻¹, standard value 0.35 s⁻¹
Reaction time t: 0 … 6 s, standard value 2 s

Physical model and application

Reaction order governs how half-life t₁/2 depends on initial reactant concentration [A]₀: in a 0th-order surface-saturated reaction, rate is constant and t₁/2 = ([A]₀)/(2k); in 1st-order decay, half-life t₁/2 = ( ln 2)/(k) is strictly concentration-independent; and in 2nd-order bimolecular kinetics, dilution slows pair collisions so t₁/2 = (1)/(k[A]₀) grows inversely with [A]₀.

v = -(d[A])/(dt) = k[A]^n n=0: t₁/2 = ([A]₀)/(2k), n=1: t₁/2 = ( ln 2)/(k), n=2: t₁/2 = (1)/(k[A]₀)

Question

How does the half-life t₁/2 depend on the initial reactant concentration [A]₀ for reactions of zero (n=0), first (n=1), and second (n=2) kinetic order?

Selected answer Option B

Integrating the differential rate law -(d[A])/(dt) = k[A]^n yields three distinct concentration profiles: 0th order (n=0, saturated heterogeneous catalysis): [A](t) = [A]₀ - kt, t₁/2 = ([A]₀)/(2k); 1st order (n=1, unimolecular decay): [A](t) = [A]₀ e⁻kt, t₁/2 = ( ln 2)/(k) ≈ (0.693)/(k) (independent of [A]₀); 2nd order (n=2, bimolecular collisions): (1)/([A](t)) = (1)/([A]₀) + kt, t₁/2 = (1)/(k[A]₀).

Formula

-(d[A])/(dt) = k[A]^n \Rightarrow [A] = [A]₀ - kt, t₁/2 = ([A]₀)/(2k) (n=0) [A] = [A]₀ e⁻kt, t₁/2 = ( ln 2)/(k) (n=1) (1)/([A]) = (1)/([A]₀) + kt, t₁/2 = (1)/(k[A]₀) (n=2)

Order n = 1: [A](2.0 s) = 0.993 M, t₁/2 = 1.98 s

Key indicators 0.99 mol/L

Reactant [A](t)

0.99 mol/L

Product [P](t)

1.01 mol/L

Reaction Rate v = k[A]ⁿ

0.35 mol/(L·s)

Fig. C-27.1 . Photochemical kinetic reactor with 3D concentration decay [A](t) and product growth [P](t) curves for 0th-, 1st-, and 2nd-order reactions

Reaction order n & integrated law n = 1: [A](t) = [A]₀·e^(−kt)

Reactant concentration [A](t) 0.993 mol/L (from [A]₀ = 2.00 M)

Product concentration [P](t) 1.007 mol/L

Instantaneous rate v(t) 0.348 mol/(L·s)

Half-life period t₁/₂ 1.98 s (independent of [A]₀!)

Reaction time t 2.0 s