Integrated Rate Laws: Half-Life by Order

Integrated Rate Laws and Half-Life by Reaction Order

By StudyStore · Published 2026-11-05 · 1 min read · 244 words

Integrated rate laws give concentration as a function of time. The order is read from which plot of the data is a straight line.

The three orders

Order	Integrated law	Straight-line plot	Slope	Half-life
Zero	[A] = [A]₀ - kt	[A] vs t	-k	[A]₀ / 2k
First	ln[A] = ln[A]₀ - kt	ln[A] vs t	-k	0.693 / k
Second	1/[A] = 1/[A]₀ + kt	1/[A] vs t	+k	1 / (k[A]₀)

Units of k

Zero order: M/s. First order: 1/s. Second order: 1/(M s).

Worked example

A first order decay has k = 0.0231 per minute. Half-life = 0.693 / 0.0231 = 30 minutes. After 90 minutes (3 half-lives) one eighth remains.

Key point on half-life

Only first order has a half-life that does not depend on starting concentration. Zero order half-lives shrink as the reactant is used up; second order half-lives grow.

Common traps

Using 0.693 / k for every order. It only works for first order.
Forgetting the sign. The slope is -k for zero and first order, +k for second.
Mixing up this with initial rates. Initial rates compare experiments; integrated laws follow one run over time.

One-minute recap

Linear plot picks the order: [A], ln[A], 1/[A].
k units: M/s, 1/s, 1/(M s).
Constant half-life means first order.

Check yourself: 5-question mini-quiz

Try each question first, then click it to reveal the answer.

1. Which plot is linear for a first order reaction?

ln[A] versus time.

2. What is the half-life of a first order reaction with k = 0.0693 per second?

10 seconds.

3. Which order has a half-life that increases as the reaction proceeds?

Second order.

4. What are the units of k for a zero order reaction?

M/s.

5. What fraction remains after 3 first order half-lives?

One eighth.

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Written by StudyStore from standard scientific facts. Study aid only. Always check values against the reference your exam provides.