Integrated Rate Law Formula
Calculate concentration as a function of time for different reaction orders
Rate Law Forms
Zero Order (n = 0)
[A]t = [A]₀ - kt
Characteristics:
- Rate = k (constant)
- Linear [A] vs t plot
- t1/2 = [A]₀/(2k)
- Units of k: M/s
Examples:
- Surface reactions
- Enzyme saturation
- Photochemical reactions
First Order (n = 1)
ln[A]t = ln[A]₀ - kt
or: [A]t = [A]₀e-kt
Characteristics:
- Rate = k[A]
- Linear ln[A] vs t plot
- t1/2 = 0.693/k (constant!)
- Units of k: s⁻¹
Examples:
- Radioactive decay
- Many decompositions
- Most common order
Second Order (n = 2)
1/[A]t = 1/[A]₀ + kt
Characteristics:
- Rate = k[A]²
- Linear 1/[A] vs t plot
- t1/2 = 1/(k[A]₀)
- Units of k: M⁻¹s⁻¹
Examples:
- Gas phase reactions
- Dimerization
- 2A → products
Half-Life Formulas
Zero Order
t1/2 = [A]₀/(2k)
Depends on [A]₀
First Order
t1/2 = 0.693/k
Independent of [A]₀
Second Order
t1/2 = 1/(k[A]₀)
Depends on [A]₀
Worked Examples
Example 1: First Order Reaction
Problem: For a first order reaction with k = 0.0462 s⁻¹, if [A]₀ = 0.500 M, find [A] after 30.0 s.
Solution:
Use: ln[A]t = ln[A]₀ - kt
ln[A]t = ln(0.500) - (0.0462)(30.0)
ln[A]t = -0.693 - 1.386
ln[A]t = -2.079
[A]t = e-2.079
[A]t = 0.125 M
Check: t1/2 = 0.693/0.0462 = 15.0 s
After 2 half-lives (30 s): 0.500 → 0.250 → 0.125 M ✓
Example 2: Second Order Reaction
Problem: For 2NO₂ → 2NO + O₂, k = 0.543 M⁻¹s⁻¹. If [NO₂]₀ = 0.100 M, find [NO₂] after 10.0 s.
Solution:
Use: 1/[A]t = 1/[A]₀ + kt
1/[NO₂]t = 1/0.100 + (0.543)(10.0)
1/[NO₂]t = 10.0 + 5.43
1/[NO₂]t = 15.43 M⁻¹
[NO₂]t = 1/15.43
[NO₂]t = 0.0648 M
Example 3: Determine Reaction Order
Problem: Data for decomposition of A:
| Time (s) | [A] (M) | ln[A] | 1/[A] (M⁻¹) |
|---|---|---|---|
| 0 | 1.00 | 0.000 | 1.00 |
| 10 | 0.61 | -0.494 | 1.64 |
| 20 | 0.37 | -0.994 | 2.70 |
| 30 | 0.23 | -1.470 | 4.35 |
Solution:
Test for linearity:
• [A] vs t: Not linear (curved)
• ln[A] vs t: Linear! Slope = -0.0494 s⁻¹
• 1/[A] vs t: Not linear
First order, k = 0.0494 s⁻¹
Example 4: Calculate Half-Life
Problem: A zero order reaction has k = 0.020 M/s and [A]₀ = 1.50 M. Find t1/2.
Solution:
For zero order: t1/2 = [A]₀/(2k)
t1/2 = 1.50 / (2 × 0.020)
t1/2 = 1.50 / 0.040
t1/2 = 37.5 s
Note: For zero order, t1/2 decreases as [A] decreases!
Second t1/2 = 0.75/(2×0.020) = 18.75 s
Common Mistakes
Using Wrong Equation for Order
First order uses ln[A], second order uses 1/[A]. Don't mix them up!
Forgetting Natural Log
First order: ln[A], not log[A]. Use ln (base e), not log₁₀!
Assuming Constant Half-Life
Only first order has constant t1/2! Zero & second order t1/2 changes with concentration.
Graphical Method to Find Order
Plot [A] vs t, ln[A] vs t, and 1/[A] vs t. Whichever is linear tells you the order!