Half Life Calculator · Practical guide
Half-Life Formula Explained with an Example
One exponential formula powers radioactive dating, drug dosing, and physics homework — here is how it works.
At a glance
The Formula
The half-life formula is Remaining = Initial × (0.5)^(Elapsed ÷ Half-Life). The exponent — elapsed time divided by half-life — counts how many half-lives have passed, and each one halves whatever is left. This is exactly the equation the calculator above evaluates.
Interactive tool
Half Life Calculator
Enter your own values, then compare the result with the examples below.
Result
Step-by-Step Example
Start with 100 grams of a substance whose half-life is 5 years, and let 15 years pass:
- Half-lives passed: 15 ÷ 5 = 3
- Decay factor: (0.5)³ = 0.125
- Remaining: 100 × 0.125 = 12.5 grams
- Decayed: 100 − 12.5 = 87.5 grams (87.5% of the original)
Fractional Half-Lives Work Too
Time rarely lines up in whole half-lives. With the same 5-year half-life, 8 years gives an exponent of 8 ÷ 5 = 1.6, so the remaining fraction is (0.5)^1.6 ≈ 0.330 — about 33.0% left, or 33.0 g from the original 100 g. The calculator handles these fractional exponents automatically and converts mixed units (hours vs. days, for example) before computing.
Where the Formula Shows Up
- Archaeology: carbon-14 dating of organic remains.
- Medicine: how fast a drug clears the bloodstream.
- Physics: decay of radioactive isotopes in labs and reactors.
Frequently Asked Questions
What is the decay constant in the half-life formula?
The decay constant λ equals ln(2) ÷ half-life, about 0.693 divided by the half-life. It converts the formula into the equivalent exponential form N = N₀e^(−λt).
Does the starting amount change the half-life?
No. Half-life is a property of the substance, not the sample size. A gram and a kilogram of the same isotope both halve over the same time span.
Can the elapsed time be longer than the half-life?
Yes — that just means multiple half-lives have passed. After 3 half-lives one eighth (12.5%) remains; after 10, less than a thousandth.
Why does half-life decay never reach zero?
Because each half-life removes half of what is left, not half of the original. The remainder shrinks toward zero forever, which is why trace radioactivity persists for ages.