Half-Life Explained

Why does a 100 gram sample shrink to 12.5 grams after just three half-lives? Each half-life cuts the amount in half, no matter how much you start with. That one steady rule explains carbon dating, medical scans, and many other decay problems. This guide gives you the table, the formula, and the logic behind it.

Half-Life Quick Reference Table

Each half-life leaves exactly half of what was there before. Start with 100 percent, and the remaining share drops to 50, then 25, then 12.5 percent. The table below shows the first ten steps.

Notice that the amount never reaches zero on paper. After ten half-lives, less than one part in a thousand remains. Scientists often treat that level as practically gone for everyday purposes.

Four terms come up again and again in decay problems. Here is what each one means in plain words.

Half-life (T)

The fixed time it takes for half of the current amount to decay.

Initial amount (N0)

The starting quantity, measured in grams, atoms, or percent.

Remaining amount (N)

The quantity still present after the elapsed time has passed.

Decay constant

The fraction lost per unit of time, equal to 0.693 divided by the half-life.

Mean lifetime

The average life of one atom, about 1.44 times the half-life.

What Does Half-Life Actually Mean?

Half-life is the time it takes for half of a sample to decay or change. The EPA defines it as the time for half of the radioactive atoms present to decay. Each isotope has its own fixed half-life.

The key idea is that the rate is steady in proportion, not in amount. A sample loses half its atoms in one half-life, whether it holds a trillion atoms or a few thousand. That is why the curve bends and flattens instead of falling in a straight line.

Why the Rate Never Changes

Half-life stays constant because each atom has the same chance of decaying in a given time. The atoms do not age or wear out. A smaller sample simply has fewer atoms that can decay.

This is what makes decay exponential rather than linear. HyperPhysics writes the law as N = N0 x e^(-kt), where k is the decay constant. For carbon-14, k is about 0.000121 per year.

The constant also explains a common confusion. The average lifetime of one atom is longer than the half-life. For carbon-14, the mean lifetime is about 8,267 years, compared with its 5,730-year half-life.

How Do You Calculate the Amount Remaining?

Multiply the starting amount by one half raised to the number of half-lives passed. The formula is N = N0 x (1/2)^(t / T). Here t is the elapsed time and T is the half-life.

Try a clear example. A substance has a half-life of 5 years, and you start with 100 grams. After 15 years, three half-lives have passed, so 100 x (1/2)^3 leaves 12.5 grams.

The time does not need to be a whole number of half-lives. After 1.5 half-lives, (1/2)^1.5 equals about 0.354, so 35.4 percent remains. The power handles any fraction, which the table above cannot show.

The exponent does most of the work, so a quick review of powers helps. Our guide to exponents and scientific notation covers how fractional and negative powers behave. For fast answers with any three known values, the Half-Life Calculator solves for the missing one.

How Do You Solve for Time or Half-Life?

Count how many halvings separate the start and end amounts, then multiply by the half-life. In formula form, t = T x log2(N0 / N). Base-2 logarithms simply count the halvings for you.

Finding the Elapsed Time

Suppose 100 grams must fall to 10 grams, and the half-life is 5 years. The ratio is 10, and log2 of 10 is about 3.32. Multiply by 5 years, and the answer is about 16.6 years.

Finding the Half-Life Itself

Now reverse the question. A sample drops from 80 grams to 20 grams in 12 days. That is a ratio of 4, or exactly two halvings, so the half-life is 12 divided by 2, or 6 days.

Here is a handy check for any answer. The elapsed time divided by the half-life should match the number of halvings you counted. When the two numbers disagree, recheck the ratio before trusting the result.

Growth works the same way in reverse. A quantity that doubles has a doubling time, just as a shrinking one has a half-life. Read the mathematics of doubling time to see the growth side of this rule.

How Does Carbon Dating Use Half-Life?

Carbon dating compares the carbon-14 left in old organic material with the level found in living things. Carbon-14 has a half-life of about 5,730 years, so the remaining share reveals the age.

Living plants and animals keep taking in carbon, so their carbon-14 level stays steady. Once they die, intake stops and the clock starts. HyperPhysics notes that living matter shows about 15 decays per minute per gram of carbon.

The math follows the same formula. After 11,460 years, or two half-lives, 25 percent remains. After just 1,000 years, about 88.6 percent is still present. A piece of wood with 60 percent left is about 4,220 years old.

Newer accelerator methods count the atoms more directly. HyperPhysics notes that this approach can stretch the range to about 100,000 years in the best cases.

Which Isotopes Have Useful Half-Lives in Medicine?

Medical isotopes usually have short half-lives, from minutes to days. A short half-life lets a scan or treatment work, then fade quickly. The table uses half-life values listed by OpenStax.

The pattern is easy to read. Technetium-99m loses about seven eighths of its activity in one day. Cobalt-60 lasts for years, so equipment that uses it needs a steady source.

The same table logic helps with storage planning too. A source held for ten half-lives drops below 0.1 percent of its starting activity. For iodine-131, ten half-lives is about 80 days.

Short half-lives also explain why these isotopes are made close to where they are used. A fluorine-18 batch loses half its strength in under two hours, so long shipping times waste most of it.

These figures describe the isotope itself, not a dosing plan. Medical staff decide doses and timing for each patient.

Questions People Ask About Half-Life

Where These Numbers Come From

This article is general science and math education, not medical advice about radiation or treatment. Isotope values are rounded from the cited references. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 27, 2026.