Question
A radioactive substance has a half-life of 10 years. If we start with 100 g of this substance, how much will remain after 30 years?
Solution — Step by Step
The half-life () is the time required for exactly half of the radioactive nuclei in a sample to decay. After every half-life, the amount is reduced by half.
This is a fixed property of the substance — it doesn’t matter how much you start with. 100 g becomes 50 g in 10 years; 1 kg becomes 500 g in 10 years.
Total time = 30 years. Half-life = 10 years.
Number of half-lives =
Starting with 100 g:
| Time (years) | Half-lives elapsed | Amount remaining |
|---|---|---|
| 0 | 0 | 100 g |
| 10 | 1 | 50 g |
| 20 | 2 | 25 g |
| 30 | 3 | 12.5 g |
where = number of half-lives = 3.
Why This Works
Radioactive decay is a first-order process — the rate of decay is proportional to the amount of substance present. This mathematical property produces the exponential decay curve, and the half-life is constant regardless of how much material is present.
The general formula is:
where is the decay constant.
Alternative Method — Using Decay Constant
Same answer, as expected.
Common Mistake
Students sometimes think “50% decays per half-life, so 3 × 50% = 150% decays in 3 half-lives.” Percentages don’t add like that. Each half-life halves the remaining amount, not the original. After 3 half-lives, of the original remains — that’s 12.5%, not 50%.
A quick table to memorise: after half-lives, fraction remaining = . So: 1 → 1/2, 2 → 1/4, 3 → 1/8, 4 → 1/16, 5 → 1/32, 10 → 1/1024. After 10 half-lives, less than 0.1% of the original remains.