# 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).

Chemistry (university) · Part 06: Chemical kinetics · Card 4 of 9 · Last updated 2026-10-19

## Analogy

Like a bath with a hole: the water drains faster when full, and slower as it empties.

## Example

**First order**

k = 0.0231 min⁻¹, [A]₀ = 0.80 M
After 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.

## Related concepts

- [Half-life](https://studyvisually.com/learn/science/chemistry-university/chemistry-university-part-6/chemistry-university-part-6-concept-5)

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