Rate Law
Understanding the relationship between reaction rates and reactant concentrations
General Rate Law Formula
For reaction: aA + bB → products
Variables:
- Rate = Reaction rate (M/s)
- k = Rate constant
- [A], [B] = Concentrations (M)
- m, n = Reaction orders
Key Points:
- • Rate law must be determined experimentally
- • Orders (m, n) are NOT equal to coefficients (a, b)
- • Overall order = m + n
- • k depends on temperature
Understanding Reaction Orders
Zero Order (m = 0):
- • Rate = k
- • Rate independent of [A]
- • Linear decrease in concentration
- • Common in enzyme reactions at saturation
First Order (m = 1):
- • Rate = k[A]
- • Rate proportional to [A]
- • Exponential decrease
- • Half-life is constant
Second Order (m = 2):
- • Rate = k[A]²
- • Rate proportional to [A]²
- • 1/[A] vs time is linear
- • Half-life depends on concentration
Determining Rate Law: Step-by-Step Example
Problem:
For the reaction 2A + B → C, determine the rate law from the following data:
| Experiment | [A] (M) | [B] (M) | Initial Rate (M/s) |
|---|---|---|---|
| 1 | 0.10 | 0.10 | 2.0 × 10⁻⁴ |
| 2 | 0.20 | 0.10 | 8.0 × 10⁻⁴ |
| 3 | 0.10 | 0.20 | 4.0 × 10⁻⁴ |
Step 1: Find order with respect to A
Compare Experiments 1 and 2 (B constant):
Rate₂/Rate₁ = (8.0 × 10⁻⁴)/(2.0 × 10⁻⁴) = 4
[A]₂/[A]₁ = 0.20/0.10 = 2
4 = 2^m → m = 2 (second order in A)
Step 2: Find order with respect to B
Compare Experiments 1 and 3 (A constant):
Rate₃/Rate₁ = (4.0 × 10⁻⁴)/(2.0 × 10⁻⁴) = 2
[B]₃/[B]₁ = 0.20/0.10 = 2
2 = 2^n → n = 1 (first order in B)
Step 3: Write the rate law
Rate = k[A]²[B]¹
Overall order = 2 + 1 = 3
Step 4: Calculate rate constant
Using Experiment 1:
2.0 × 10⁻⁴ = k(0.10)²(0.10)
k = (2.0 × 10⁻⁴)/(1.0 × 10⁻³) = 0.20 M⁻²s⁻¹
Answer:
Rate = 0.20[A]²[B] M⁻²s⁻¹
Integrated Rate Laws
| Order | Differential Rate Law | Integrated Rate Law | Linear Plot | Half-Life |
|---|---|---|---|---|
| 0 | Rate = k | [A] = [A]₀ - kt | [A] vs t | t₁/₂ = [A]₀/(2k) |
| 1 | Rate = k[A] | ln[A] = ln[A]₀ - kt | ln[A] vs t | t₁/₂ = 0.693/k |
| 2 | Rate = k[A]² | 1/[A] = 1/[A]₀ + kt | 1/[A] vs t | t₁/₂ = 1/(k[A]₀) |
Common Mistakes to Avoid
Stoichiometric Coefficients
Reaction orders ≠ stoichiometric coefficients. Orders must be determined experimentally.
Rate vs. Rate Constant
Rate changes with concentration; rate constant k only changes with temperature.
Units of Rate Constant
Units depend on overall order: M^(1-n)s^(-1) where n = overall order.
Initial Rates Method
Change only one concentration at a time when comparing experiments.
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Frequently Asked Questions
Why can't I use stoichiometric coefficients as orders?
The rate law describes the mechanism, not just the overall stoichiometry. Elementary steps may differ from the overall reaction.
How do I determine the rate law experimentally?
Use the initial rates method: measure initial rates while varying one reactant concentration at a time, keeping others constant.
Can reaction orders be fractions or negative?
Yes! Fractional orders suggest complex mechanisms, while negative orders indicate inhibition by that species.
How does temperature affect the rate law?
Temperature doesn't change the orders but dramatically affects the rate constant k, typically doubling every 10°C increase.