First-order integrated rate law
For a first-order reaction, ln[A] = ln[A]₀ − kt. A plot of ln[A] against time is a straight line with slope −k. Equivalent form: [A] = [A]₀e^(−kt).
Last updated
Analogy
Like a bath with a hole: the water drains faster when full, and slower as it empties.
Example
k = 0.0231 min⁻¹, [A]₀ = 0.80 MAfter 60 min:ln[A] = ln 0.80 − 0.0231 × 60[A] = 0.80 × e^(−1.386) = 0.20 M
In practice
Radioactive decay and many drug eliminations are first-order, so they follow this equation.
Practice
Flashcards
- Write the first-order integrated rate law.ln[A] = ln[A]₀ − kt.
Questions people ask
How do I tell the order from a graph?
Plot [A] against t, ln[A] against t, and 1/[A] against t. The one that gives a straight line shows the order. A straight [A] line means zero order, a straight ln[A] line means first order, and a straight 1/[A] line means second order.
What does the rate constant k tell me?
The rate constant measures how fast a reaction goes at a given temperature once the concentrations are accounted for. A larger k means a faster reaction. It changes with temperature, but not with concentration, so it is the number that makes the rate law work.