Radioactivity and Its Uses
Simple Explanation
Radioactivity is the spontaneous emission of particles or energy from an unstable atomic nucleus. Each radioactive substance has a characteristic half-life — the time for exactly half of a sample to decay — letting the fraction remaining after any elapsed time be predicted.
Why Do We Need It?
Radioactivity has genuinely important uses — medical diagnosis and treatment, dating ancient artifacts, and smoke detectors — alongside the serious hazards it poses, making a solid quantitative understanding of decay essential.
See It
A curve starting at full amount and dropping by half at each successive half-life interval, approaching but never quite reaching zero
Formula
Radioactive Decay and Half-Life
fraction remaining = (1/2) raised to the power (t / T_half)
A radioactive sample decays so that its half-life, T_half, is the time for exactly half of it to decay — after n half-lives have passed, only (1/2) to the power n of the original sample remains.
- t
- — time elapsed, in the same units as the half-life
- T_half
- — the half-life of the radioactive substance — the time for half of it to decay
When to use it: Whenever the fraction (or amount) of a radioactive sample remaining after a given time needs to be found.
Worked Example
Find the fraction of a radioactive sample remaining
Iodine-131, used in medical treatment, has a half-life of 8 days. Find the fraction of a sample remaining after 24 days.
Why Does This Work?
Radioactive decay is a random process at the level of individual atoms, but for a large enough sample, the same fixed FRACTION decays in each successive half-life interval — this steady, repeating halving is exactly what produces the (1/2) to the power n pattern.
Real-Life Example
Carbon dating an ancient artifact
Archaeologists estimate the age of ancient organic remains by measuring how much radioactive carbon-14 remains in them.
Carbon-14 has a known half-life (about 5730 years) — measuring what fraction remains lets scientists work backward to calculate how many half-lives (and therefore how many years) have passed since the organism died.
Practice
Iodine-131 (half-life 8 days) starts as a sample. Find the percentage remaining after 32 days.
MediumCommon mistake
Assuming a sample decays completely after two half-lives — it actually never reaches exactly zero; after 2 half-lives, one quarter still remains, and some (an ever-smaller fraction) always remains mathematically.
Quick Review
- Half-life: the time for exactly half a sample to decay.
- After n half-lives, (1/2) to the power n of the sample remains.
- Used in medicine, dating, and smoke detectors.