How radioactive half-life works
Radioactive decay is random for any single atom, but a large sample decays at a very predictable rate. In each half-life, half of the remaining radioactive nuclei decay. That means the amount drops by the same fraction every half-life, not by the same amount, which is called exponential decay.
The amount can be measured in grams, number of atoms, percent of the original, or activity (decays per second, in becquerels). Activity is proportional to the number of radioactive atoms, so it follows the same curve.
The formula
t = t½ × log(N ÷ N₀) ÷ log(1/2)
t½ = t × ln 2 ÷ ln(N₀ ÷ N)
N₀ = N ÷ (1/2)t / t½
λ = ln 2 ÷ t½ N = N₀ × e−λt
Example: 100 g of carbon-14 after 10,000 years
| Step | Result |
|---|---|
| Number of half-lives: 10,000 ÷ 5,730 | 1.745 |
| Fraction remaining: (1/2)1.745 | 0.2983 (29.83%) |
| Amount remaining: 100 g × 0.2983 | 29.83 g |
| Amount decayed: 100 g − 29.83 g | 70.17 g |
| Decay constant: 0.6931 ÷ 5,730 years | 1.21 × 10⁻⁴ per year |
Half-lives of common isotopes
| Isotope | Half-life | Where you meet it |
|---|---|---|
| Radon-222 | 3.82 days | Radioactive gas from uranium in rocks and soil; tested for in basements |
| Iodine-131 | 8.02 days | Thyroid treatment and imaging in nuclear medicine |
| Carbon-14 | 5,730 years | Radiocarbon dating of once-living material up to about 50,000 years old |
| Uranium-238 | 4.468 billion years | Dating the oldest rocks; about the age of the Earth |
Tips and common mistakes
- Match the time units. t and t½ must be in the same units before dividing. The calculator converts them for you.
- Don't subtract half each time. After two half-lives, 25% is left, not 0%. Each half-life halves what remains.
- Non-whole half-lives are fine. 1.745 half-lives is a perfectly valid exponent; use your calculator's xy key.
- Check the direction. The remaining amount is always less than the initial amount for decay.
- Quick check: after about 10 half-lives, roughly 0.1% (1/1,024) of the original remains.
Frequently asked questions
What is a half-life?
A half-life is the time it takes for half of a sample of a radioactive isotope to decay. After one half-life, 50% remains; after two, 25%; after three, 12.5%. The half-life is the same no matter how much material you start with.
How do I calculate the amount remaining?
Use N = N₀ × (1/2)^(t / t½). Divide the elapsed time by the half-life to get the number of half-lives, raise 1/2 to that power, and multiply by the starting amount. For 100 g of carbon-14 after 10,000 years: 10,000 ÷ 5,730 = 1.745 half-lives, and 100 × 0.5^1.745 = 29.83 g.
How do I solve for time?
Rearrange with logarithms: t = t½ × log(N/N₀) ÷ log(1/2), which is the same as t½ × log₂(N₀/N). For example, going from 100 g to 25 g is log₂(4) = 2 half-lives.
What is the decay constant?
The decay constant λ is the fraction of nuclei that decay per unit time. It is related to half-life by λ = ln 2 ÷ t½ ≈ 0.6931 ÷ t½. Using it, the decay law can also be written N = N₀ × e^(−λt), which gives exactly the same results.
Does a radioactive sample ever completely disappear?
Mathematically, the amount keeps halving and never reaches zero. In practice, after about 10 half-lives less than 0.1% remains, and eventually so few atoms are left that the formula only describes the probability that any are still there.
Can I use this for drug half-lives?
Yes. Many drugs are cleared from the body at a rate that follows the same first-order decay, so the same formula applies. Real dosing is more complicated, so follow medical guidance rather than this calculator for actual doses.
Rates and figures last checked October 2026.
This calculator gives estimates for planning. Check current rates and product labels before you buy, list or file.