TL;DR:
The half-life is the time for half of the radioactive atoms in a sample to decay. It stays the same however much you start with.
After n half-lives, the fraction left is
(1/2)ⁿ. Starting with 80 g, three half-lives leave 10 g.Decay is exponential:
N = N₀ (1/2)^(t/T). Dating methods use it to turn a measured fraction into an age.
Imagine a jar of 80 g of a radioactive isotope, and a label that says its half-life is 5 days. After 5 days half of it is gone, but not all of it after 10. How does it work, and why does it never quite finish? The answer is a pattern of repeated halving.
Halving, again and again
Some atomic nuclei are unstable and sooner or later change into something else, giving off radiation. No one can say when a given nucleus will go. But for a large number of identical ones, the pattern is dependable. The half-life T is the time it takes for half of them to decay.
The key point is that the half-life does not depend on how many atoms you have. After one half-life half remain. After another, half of that half remain, which is a quarter. Each half-life cuts what is left in half again.
Why it is exponential
Each nucleus has a fixed chance of decaying in any given second, regardless of its age. So the number decaying per second is proportional to the number still there. More atoms means more decays, and fewer atoms means fewer. That rule gives an exponential curve that drops fast at first and then levels off, getting closer to zero without ever reaching it.
The formula is N = N₀ × (1/2)^(t/T), where N₀ is the starting amount, t the elapsed time and T the half-life. The exponent t/T just counts how many half-lives have gone by.
Worked example: 80 g after three half-lives
A sample starts with 80 g of a radioactive isotope with a half-life of 5 days. How much of the isotope is left after 15 days?
Count half-lives:
t/T = 15 / 5 = 3.Halve three times:
80 → 40 → 20 → 10.Formula check:
80 × (1/2)³ = 80 × 1/8 = 10 g.
So 10 g of the isotope is left. The other 70 g has not vanished: it has turned into other elements, which stay in the sample. Check by adding: 40 + 20 + 10 = 70 g decayed, and 70 + 10 = 80 g, the start.
What about a time that is not a whole number of half-lives? After 7.5 days, t/T = 1.5, so N = 80 × (1/2)^1.5 = 80 × 0.3536 ≈ 28.3 g. That sits between 40 g (one half-life) and 20 g (two), as it must.
How dating methods use it
Living things take in carbon, including a tiny, steady share of radioactive carbon-14. When the organism dies, it stops taking in more, and the carbon-14 in it slowly decays. The half-life of carbon-14 is about 5730 years.
Suppose a piece of wood has one quarter of the carbon-14 ratio of living wood. One quarter is two halvings, so about 2 × 5730 = 11,460 years have gone by. Scientists measure the remaining fraction, solve for the number of half-lives and multiply by T. The method suits organic material up to tens of thousands of years old. For much older rocks, isotopes with far longer half-lives are used instead.
Radiation safety: Radioactive sources are only for trained people with the right equipment. Never open a smoke detector, a lantern mantle or an old luminous dial to look inside, and never handle an unknown source. Distance, shielding and short exposure time are the basic ways to reduce a dose.
How to check your answer
Count first. Work out
t/Tbefore doing anything else, and keep time units the same.Direction. The amount left must be less than what you started with and more than zero.
Halving test. Whole half-lives: halve the amount that many times and compare with the formula.
Add back. Amount decayed plus amount left must equal the start.
Mistakes that cost marks
Thinking two half-lives destroy everything. Two half-lives leave a quarter.
Using total mass of the sample instead of the mass of the radioactive isotope, since the products stay.
Mixing days and years in
t/T.Believing a half-life tells you when a single atom will decay. It is statistical, so it only describes large numbers.
Thinking a bigger sample has a longer half-life. It does not.
Where you will use it next
The same halving pattern describes a hot drink cooling toward room temperature and a capacitor discharging. The energy side of the story links to heat vs temperature, and the oscillating behaviour in other systems comes up in springs and simple harmonic motion.
Frequently asked questions
Does a short half-life mean a dangerous isotope?
Not by itself. A short half-life means it decays quickly, so it is intense for a short while and then gone. What matters is the type of radiation, how much there is and how you are exposed.
Does the sample ever reach exactly zero?
Mathematically it only approaches zero. In practice, once only a few atoms are left, the last ones decay at random times.
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