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.
Key Formula
k = A × exp(−Ea / RT)
LaTeX: k = A e^{-E_a / (RT)}
| Symbol | Meaning | Unit |
|---|---|---|
| k | Rate constant | varies with order |
| A | Frequency factor (pre-exponential factor) | same as k |
| E_a | Activation energy | J/mol |
| R | Gas constant | 8.314 J/(mol·K) |
| T | Absolute temperature | K |
Worked Example
Problem
For a first-order decomposition reaction, k = 1.50 × 10⁻³ s⁻¹ at 300 K and the activation energy Eₐ = 75.0 kJ/mol. Calculate the rate constant at 400 K.
Solution
Step 1: Use the two-temperature Arrhenius expression: ln(k₂/k₁) = (Eₐ/R)(1/T₁ − 1/T₂) Step 2: Substitute values: Eₐ = 75,000 J/mol, R = 8.314 J/(mol·K) 1/T₁ = 1/300 = 3.333 × 10⁻³ K⁻¹ 1/T₂ = 1/400 = 2.500 × 10⁻³ K⁻¹ Step 3: ln(k₂/k₁) = (75,000/8.314)(3.333×10⁻³ − 2.500×10⁻³) = 9021 × 8.33×10⁻⁴ = 7.515 Step 4: k₂/k₁ = e^7.515 = 1826 k₂ = 1826 × 1.50 × 10⁻³ = 2.74 s⁻¹
Answer
k at 400 K ≈ 2.74 s⁻¹
Units of Rate Constant by Reaction Order
| Reaction Order | Rate Law Form | Units of k | Example Reaction |
|---|---|---|---|
| Zero | rate = k | mol·L⁻¹·s⁻¹ | Surface-catalyzed reactions |
| First | rate = k[A] | s⁻¹ | Radioactive decay, first-order decomposition |
| Second | rate = k[A]² | L·mol⁻¹·s⁻¹ | Dimerization, 2HI → H₂ + I₂ |
| Second (two) | rate = k[A][B] | L·mol⁻¹·s⁻¹ | SN2 reactions |
| Third | rate = k[A]³ | L²·mol⁻²·s⁻¹ | Termolecular gas-phase reactions |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
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.
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.
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).
From Latin "ratio" (reckoning) forming "rate," and Latin "constans" (standing firm, unchanging). The rate constant was formalized by Svante Arrhenius in 1889 when he showed that k varies exponentially with temperature, connecting microscopic molecular collision theory to macroscopic reaction speed.