Rate Law
The rate law (or rate equation) is a mathematical expression that relates the reaction rate to the concentrations of reactants raised to experimentally determined powers called reaction orders. It takes the general form rate = k[A]^m[B]^n, where k is the rate constant and m, n are the orders with respect to each reactant. Critically, the rate law cannot be deduced from the stoichiometric equation but must be determined experimentally, and it reflects the mechanism of the rate-determining step.
Key Formula
rate = k × [A]^m × [B]^n
LaTeX: r = k[A]^m[B]^n
| Symbol | Meaning | Unit |
|---|---|---|
| r | Reaction rate | mol/(L·s) |
| k | Rate constant | depends on overall order |
| [A], [B] | Molar concentrations of reactants | mol/L |
| m | Order with respect to A (experimentally determined) | dimensionless |
| n | Order with respect to B (experimentally determined) | dimensionless |
Worked Example
Problem
Given the following experimental data for the reaction A + B → C, determine the rate law and the rate constant k: Expt 1: [A]=0.10, [B]=0.10, rate=2.0×10⁻³ Expt 2: [A]=0.20, [B]=0.10, rate=4.0×10⁻³ Expt 3: [A]=0.10, [B]=0.30, rate=1.8×10⁻² (concentrations in mol/L, rates in mol/(L·s))
Solution
Step 1: Find order with respect to A (expts 1 and 2, [B] constant): rate₂/rate₁ = (4.0×10⁻³)/(2.0×10⁻³) = 2.0 ([A]₂/[A]₁)^m = (0.20/0.10)^m = 2^m 2^m = 2 → m = 1 (first order in A) Step 2: Find order with respect to B (expts 1 and 3, [A] constant): rate₃/rate₁ = (1.8×10⁻²)/(2.0×10⁻³) = 9.0 ([B]₃/[B]₁)^n = (0.30/0.10)^n = 3^n 3^n = 9 → n = 2 (second order in B) Step 3: Rate law: rate = k[A][B]² Step 4: Calculate k using Expt 1: k = rate / ([A][B]²) = (2.0×10⁻³) / (0.10 × (0.10)²) = (2.0×10⁻³) / (1.0×10⁻³) = 2.0 L²/(mol²·s)
Answer
Rate law: rate = 2.0[A][B]² L²/(mol²·s); k = 2.0 L²·mol⁻²·s⁻¹
Reaction Orders and Their Integrated Rate Laws
| Order | Rate Law | Integrated Form | Units of k |
|---|---|---|---|
| Zero | rate = k | [A] = [A]₀ − kt | mol/(L·s) |
| First | rate = k[A] | ln[A] = ln[A]₀ − kt | s⁻¹ |
| Second | rate = k[A]² | 1/[A] = 1/[A]₀ + kt | L/(mol·s) |
| Second (two reactants) | rate = k[A][B] | Complex integrated form | L/(mol·s) |
| Third | rate = k[A]³ | 1/[A]² = 1/[A]₀² + 2kt | L²/(mol²·s) |
Interactive Tools
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Related Terms
Rate Constant
The rate constant (k) is the proportionality constant in the rate law that relates the reaction rate to the concentrations of reactants; it is a characteristic value for a given reaction at a specific temperature. Unlike the reaction rate itself, k does not depend on concentrations but is strongly temperature-dependent, following the Arrhenius equation k = A·e^(−Eₐ/RT). The units of k vary with the overall reaction order, and a larger k indicates a faster inherent reaction speed.
Reaction Rate
The reaction rate is the change in concentration of a reactant or product per unit time in a chemical reaction, expressed in units of mol/(L·s) or mol·L⁻¹·s⁻¹. It quantifies how quickly reactants are consumed and products are formed, and is influenced by factors including concentration, temperature, surface area, catalysts, and the nature of the reactants. Understanding reaction rates is fundamental to chemical engineering (designing reactors), pharmacology (drug metabolism), and environmental chemistry (pollutant breakdown).
Activation Energy
Activation energy (Eₐ) is the minimum amount of energy that reacting molecules must possess for a collision to result in a chemical reaction — effectively the energy barrier that must be overcome to convert reactants into products. It determines how fast a reaction proceeds: reactions with low activation energies are generally fast (explosions, acid-base), while those with high activation energies are slow (rusting, digestion). The concept was introduced by Svante Arrhenius in 1889 and is central to the Arrhenius equation and transition state theory.
From Latin "ratio" (reckoning, reason) forming "rate," and Old French "loi" from Latin "lex" (law, rule). The rate law as a formal relationship between reaction speed and concentration was developed by Jacobus van't Hoff and Wilhelm Ostwald in the 1880s during the foundational era of chemical kinetics.