Half-Life: Measuring How Fast Radioisotopes Decay

High School⏱ 10 min readNuclear Chemistry

What half-life means, how to use N = N0(1/2)^(t/t1/2), and worked examples with carbon-14 dating.

Quick answer

The half-life (t1/2) of a radioisotope is the time required for half of the undecayed nuclei in a sample to decay. After each half-life, half of what remains decays, so the amount falls by halves: 1, 1/2, 1/4, 1/8, and so on.

$N = N_0\left(\tfrac{1}{2}\right)^{t/t_{1/2}}$

Example: Carbon-14 has a half-life of 5730 years. A 100 g sample becomes 50 g after 5730 years, 25 g after 11,460 years (two half-lives), and 12.5 g after 17,190 years (three half-lives).

What you will learn

Define half-life as the time for half of a sample's nuclei to decay
Use the half-life equation N = N0(1/2)^(t/t1/2) to calculate remaining amounts
Work out how many half-lives have passed from a given remaining fraction
Explain how carbon-14 dating estimates the age of once-living material
State why half-life is unaffected by temperature, pressure, or chemical state

What half-life means

The half-life of a radioisotope, written $t_{1/2}$ , is the time it takes for exactly half of the undecayed nuclei in a sample to decay. It is a fixed property of each isotope: carbon-14 always has a half-life of 5730 years, whether the sample is a gram or a ton, hot or cold, pure or in a compound.

The key idea is that the halving repeats. After one half-life, half remains. After a second half-life, half of that half decays, leaving one quarter. After three half-lives, one eighth remains. The sample never mathematically reaches zero; it just keeps halving toward ever smaller amounts.

The half-life equation

The pattern above is captured in one equation. If $N_0$ is the starting number of nuclei (or starting mass) and $t$ is the elapsed time, the amount remaining is:

$N = N_0\left(\tfrac{1}{2}\right)^{t/t_{1/2}}$

The exponent $t/t_{1/2}$ is simply the number of half-lives that have passed. If 2.5 half-lives have elapsed, the remaining fraction is $(1/2)^{2.5}$ , about 0.177. You can use mass, atom count, or activity in place of $N$ , because all three fall in the same proportion.

N = N_0\left(\tfrac{1}{2}\right)^{t/t_{1/2}}

Half-life equation: N = amount remaining, N0 = starting amount, t = elapsed time, t1/2 = half-life.

Counting half-lives

Always convert the problem to 'how many half-lives?' first. Divide the elapsed time by the half-life to get the exponent, then raise 1/2 to that power. Most half-life mistakes come from skipping this step.

Worked example: carbon-14

A wooden artifact contains 20 g of carbon-14 when the tree dies. The measured carbon-14 today is 5 g. How old is the artifact?

Remaining fraction: $5/20 = 1/4 = (1/2)^2$ , so 2 half-lives have passed.
Half-life of carbon-14: 5730 years.
Age: $2 \times 5730 = 11{,}460$ years.

Using the equation directly: $N/N_0 = 1/4$ , so $(1/2)^{t/5730} = (1/2)^2$ , giving $t/5730 = 2$ and $t = 11{,}460$ years. Either route gives the same answer; the fraction method is faster when the remaining amount is a clean power of one half.

Half-lives span an enormous range

Half-lives are among the most varied constants in science. Some isotopes, such as certain polonium isotopes, have half-lives of fractions of a second. Iodine-131, used in medicine, has a half-life of about 8 days — short enough to decay away in the body within weeks. Carbon-14's 5730 years makes it ideal for dating archaeological finds. Uranium-238, at about 4.5 billion years, decays so slowly that it has survived since the formation of Earth.

This variety is practical, not just trivia. Short half-lives suit medical tracers that must vanish quickly; long half-lives suit power sources and geological clocks.

How radiocarbon dating works

Living things constantly exchange carbon with the air, so while an organism is alive, the ratio of radioactive carbon-14 to stable carbon-12 in its tissues matches the atmosphere. When it dies, the exchange stops: carbon-14 keeps decaying, but no new carbon-14 arrives.

By measuring how much carbon-14 remains relative to carbon-12 and comparing it with the atmospheric ratio, scientists work out how many half-lives have passed since death. The method works back tens of thousands of years, covering all of recorded human history. It cannot date rocks or fossils millions of years old — after about ten half-lives, too little carbon-14 remains to measure reliably, so geologists use longer-lived isotopes like uranium-238 or potassium-40 instead.

What half-life does not depend on

It is worth stating plainly: nothing you do to a sample changes its half-life. Heating, cooling, crushing, dissolving, pressurizing, or bonding the atoms into different compounds all leave the decay rate untouched. Decay happens in the nucleus, and ordinary physical and chemical changes only affect electrons, which are too far removed to matter.

This independence is exactly what makes radioisotopes trustworthy clocks. A carbon-14 atom decays on the same schedule whether it sits in a living tree, a buried bone, or a laboratory vial.

Variables and units

Symbol	Name	Unit	Meaning
$N$	remaining quantity	grams, atoms, or becquerels	Amount of the parent radioisotope still undecayed after time t.
$N0$	initial quantity	grams, atoms, or becquerels	Starting amount of the parent radioisotope at time zero.
$t$	elapsed time	seconds, years (any time unit)	Time passed since the measurement of N0; must use the same unit as t1/2.
$t1/2$	half-life	seconds, years (any time unit)	Characteristic time for half of the nuclei to decay; constant for each isotope.

Common mistakes

✗ After two half-lives, all of the sample has decayed.

After two half-lives, one quarter remains: half decays in the first half-life, and half of the remainder decays in the second. The amount halves each time; it never reaches zero in a finite number of steps.

✗ Half-life means half the mass disappears.

The decayed nuclei do not vanish; they become daughter nuclei plus emitted particles. The total mass-energy is conserved — it is the amount of the original parent isotope that halves, not the total mass of the sample.

✗ You can speed up decay by heating a radioactive sample.

Decay rates are fixed by nuclear structure and are unaffected by temperature, pressure, or chemical state. No physical or chemical treatment changes a half-life.

Real-world applications

Radiocarbon dating of archaeological artifacts, ancient manuscripts, and fossils
Medical tracers such as iodine-131 and technetium-99m, chosen for convenient half-lives
Geological dating of rocks using long-lived isotopes like uranium-238 and potassium-40
Designing safe storage times for radioactive waste based on decay calculations

Frequently asked questions

How do I find the number of half-lives from a remaining fraction like 1/16?

Write the fraction as a power of 1/2. Since 1/16 = (1/2)^4, four half-lives have passed. Multiply 4 by the half-life to get the elapsed time. For fractions that are not clean powers of one half, take the base-2 logarithm: number of half-lives = log2(N0/N).

Can the half-life equation use mass instead of atom count?

Yes. Mass, number of atoms, and activity all decrease in the same proportion, because each decay removes one parent nucleus. Just be consistent: if N0 is in grams, N comes out in grams.

Why is carbon-14 dating limited to about 50,000 years?

After about nine or ten half-lives (roughly 50,000–60,000 years), less than a thousandth of the original carbon-14 remains, which is too little to measure accurately against background noise. Older samples need isotopes with longer half-lives.

Does a short half-life mean an isotope is more dangerous?

Not automatically. A short half-life means high activity per gram — many decays per second — but danger also depends on the type of radiation, the amount, and how exposure happens. A short half-life also means the material decays away quickly, which is why short-lived isotopes are preferred for medical use.

What is the difference between half-life and 'lifetime' of an isotope?

Half-life is the time for half the nuclei to decay. The mean lifetime (average life of a nucleus) is longer — about 1.44 times the half-life. In basic chemistry you will almost always work with half-life; mean lifetime appears mainly in physics contexts.