Integrated Rate Laws
Relate concentration and time for common reaction orders
Zero Order
[A]_t = [A]_0 − k t
Linear in time; slope = −k when plotting [A] vs t
First Order
ln([A]_t) = ln([A]_0) − k t
Linear when plotting ln([A]) vs t; slope = −k
Half-life: t_1/2 = 0.693/k
Second Order
1/[A]_t = 1/[A]_0 + k t
Linear when plotting 1/[A] vs t; slope = k
Half-life: t_1/2 = 1/(k [A]_0)
Example (First Order)
Given: [A]_0 = 0.100 M, k = 0.050 min^-1, t = 20 min.
ln([A]_t) = ln(0.100) − 0.050 × 20 = −2.3026 − 1.0 = −3.3026
[A]_t = e^(−3.3026) ≈ 0.0367 M
Answer: [A]_t ≈ 0.0367 M
Common Mistakes
Wrong linear plot
Match the correct linearization to the reaction order.
Unit consistency
Use consistent units for k and time; check k dimensions for order.