Half-life decay formula
The amount of a radioactive substance left after time t is N = N0 × (0.5)^(t ÷ half-life), where N0 is the starting amount. Each half-life cuts the remaining amount in half. The exponent t ÷ half-life is simply how many half-lives have passed. Keep t and the half-life in the same time unit. For decimal-fraction reasoning you can cross-check with the mole fraction calculator.
How to use this calculator
Enter the initial amount (any unit — grams, atoms, percent), the elapsed time, and the half-life in the same time unit as the elapsed time. The calculator raises 0.5 to the power of the number of half-lives and multiplies by N0. A zero half-life is rejected.
Because the units of t and the half-life cancel inside the exponent, only their ratio matters. Decay is also memoryless: the fraction remaining depends on elapsed time alone, not on how old the sample already was. Growth is the mirror image of decay, which is why the doubling time calculator uses the same algebra.
Worked example
Start with 100 g of an isotope whose half-life is 10 years. After 20 years, t ÷ half-life = 2 half-lives, so N = 100 × 0.5^2 = 25 g remaining. After 30 years it would be 12.5 g.
Notice that the amount never reaches exactly zero — each step removes half of what is left, so the curve approaches zero without touching it. The same maths describes drug clearance from the bloodstream and radiocarbon dating. Solving for an arbitrary elapsed time rather than a whole number of half-lives calls for the logarithm calculator.
Frequently asked questions
- What is a half-life?
- A half-life is the time it takes for half of a radioactive sample to decay. After each half-life, the remaining amount is halved again.
- What units should time be in?
- Any time unit works as long as the elapsed time and the half-life use the same unit, such as both in seconds, days or years.
- Can the elapsed time be a non-whole number of half-lives?
- Yes. The formula handles fractional half-lives, so an elapsed time of 15 years with a 10-year half-life gives 1.5 half-lives.
- Does the initial amount unit matter?
- No. The result comes out in the same unit as the initial amount, whether grams, atoms or a percentage.
- How do I find the half-life if I only know how much is left?
- Rearrange to half-life = t ÷ log₂(N0 ÷ N). If 100 g falls to 25 g in 20 years, N0 ÷ N = 4, log₂(4) = 2, so the half-life is 20 ÷ 2 = 10 years.