Half-life: the clock of first-order reactions
Learn what half-life means in chemical kinetics, why it is constant only for first-order reactions, and how to use t½ = 0.693/k.
Quick answer
Half-life is the time required for a reactant's concentration (or a sample's amount) to fall to half its initial value. For first-order reactions it is constant: t½ = 0.693/k.
Unit: s (or min, h, yr — same as 1/k)
Example: Dinitrogen pentoxide decomposes by a first-order reaction with k = 6.93 × 10⁻³ min⁻¹, so its half-life is 0.693/0.00693 = 100 min — after 100 min, half the N₂O₅ remains.
What you will learn
- Define half-life and relate it to the first-order integrated rate law.
- Use t½ = 0.693/k to convert between rate constants and half-lives.
- Explain why half-life is constant only for first-order reactions.
- Apply half-life reasoning to dating, medicine, and decay problems.
What half-life means
The half-life (t½) of a reaction is the time it takes for the reactant concentration to drop to exactly half its starting value. It is the most intuitive way to talk about speed: "this drug's half-life is 4 hours" instantly tells a doctor how dosing works.
The famous formula
comes straight from the first-order integrated law. Set [A]t = ½[A]₀ in [A]t = [A]₀e^(−kt): the [A]₀ cancels, leaving e^(−kt½) = ½, and taking logs gives kt½ = ln 2 = 0.693. Because the starting concentration canceled out, the half-life does not depend on how much you began with.
Why constancy is a first-order privilege
The constant-half-life property is special to first order. For other orders the half-life changes as the reaction proceeds:
- First order: — constant forever
- Second order: — each half-life takes longer than the last
- Zeroth order: — half-lives get shorter
This gives a neat experimental test: if you measure the half-life at two different starting concentrations and it comes out the same, the reaction is first order.
A worked half-life example
A classic worked case: dinitrogen pentoxide decomposes by a first-order reaction with k = 6.93 × 10⁻³ min⁻¹. Its half-life is
Starting from 0.80 M: after 100 min, 0.40 M remains; after 200 min, 0.20 M; after 300 min, 0.10 M. Each 100-minute block halves whatever is left — the signature pattern of first-order kinetics.
Half-life beyond the beaker
Radioactive decay is always first order, which is why every isotope has a characteristic half-life: carbon-14 (5730 years) for archaeology, iodine-131 (8 days) for thyroid treatment, uranium-238 (4.5 billion years) for dating rocks.
In medicine, half-life reasoning balances effectiveness against safety: technetium-99m's 6-hour half-life gives doctors time to image a patient while ensuring the radioactivity fades quickly afterward.
Variables and units
| Symbol | Name | Unit | Meaning |
|---|---|---|---|
| Half-life | s, min, h, yr | Time for concentration to fall to half its initial value | |
| Rate constant | s⁻¹ (time⁻¹) | First-order rate constant |
Common mistakes
Each half-life removes half of what remains, not half of the original. After one half-life 50% remains, after two 25%, after three 12.5% — the amount approaches zero but mathematically never reaches it.
That is true only for zeroth and second order. For first-order processes — including all radioactive decay — the half-life is a property of the substance (via k) and is completely independent of the starting amount.
Any first-order process has a half-life: chemical decompositions, drug elimination, and even the dimming of some phosphorescent materials. Nuclear decay is just the most famous example.
Real-world applications
Radiocarbon dating: carbon-14's 5730-year half-life dates archaeological finds.
Nuclear medicine: technetium-99m's 6-hour half-life is short enough to be safe but long enough to image.
Pharmacology: a drug's half-life sets how often you take it.
Environmental science: half-lives of pesticides predict how long residues persist in soil.
Frequently asked questions
Is half-life always the same no matter how much you start with?
For first order, yes: t½ = 0.693/k depends only on k, not on concentration. For zeroth order, t½ = [A]₀/2k, and for second order, t½ = 1/(k[A]₀) — both shrink as the starting concentration drops. Only first-order half-lives are truly constant.
How much is left after several half-lives?
After n half-lives, the fraction remaining is (1/2)ⁿ. Five half-lives leave about 3% (1/32); ten leave about 0.1% (1/1024). This is why 'ten half-lives' is a rule of thumb for essentially complete decay.
What is the difference between chemical and nuclear half-life?
Both describe the same first-order mathematics: each nucleus (or molecule) has a fixed probability of reacting per unit time. Chemical half-life usually refers to molecules reacting; nuclear half-life refers to nuclei decaying. The formulas are identical.
What does it mean that half-life 'is not constant' for other orders?
It means the time for the amount to fall to half is not a fixed number — it changes as the reaction proceeds. For second order, each successive half-life takes twice as long as the previous one because the rate depends on concentration squared.