Half-Life Calculator
Choose what to solve for: the remaining quantity, the half-life, the elapsed time or the initial quantity. Enter the other three, and the calculator returns the answer with the half-lives elapsed and the percent remaining. Any time unit works if the half-life and the elapsed time use the same one. A result too small for four decimals prints in e-notation, such as 9.095e-11, instead of rounding to 0.
How to use the Half-Life Calculator
- Enter your numbers in the Half-Life Calculator above.
- Press the Calculate button to see the result.
- To try other numbers, change any value and press the button again.
The radioactive decay formula
N = N₀ × (1/2)^(t/T) · · · T = t ÷ log₂(N₀ ÷ N) · · · t = T × log₂(N₀ ÷ N) · · · N₀ = N × 2^(t/T)
N₀ is the initial quantity, N the remaining quantity, T the half-life and t the elapsed time. After one half-life, half remains; after two, one-quarter; after n half-lives, (1/2)^n. The same decay written with the decay constant λ = ln(2)/T is N = N₀ × e^(−λt).
Worked example
Carbon-14 has a half-life of 5,730 years. Start with 100 grams and let 11,460 years pass. That is 2 half-lives, so the calculator shows 25 remaining, 75 decayed and 25% remaining. With the default 10,000 years it shows 29.8292 remaining after 1.745 half-lives. To solve backward, take a sample that fell from 100 grams to 25 grams in 11,460 years. Solving for the half-life returns 5,730. With that half-life and the same two quantities, solving for the elapsed time returns 11,460. With 25 grams left after 11,460 years, solving for the initial quantity returns 100.
Frequently asked questions
Can it find the age of a sample or the half-life?
Yes. Set Solve for to Elapsed time or Half-life. The elapsed time is t = T × log₂(N₀ ÷ N) and the half-life is T = t ÷ log₂(N₀ ÷ N). A carbon-14 sample with 25% left has N₀ ÷ N = 4, and log₂(4) = 2, so it is 2 × 5,730 = 11,460 years old. Solving for the half-life also returns the decay constant.
What if the remaining quantity is larger than the initial quantity?
The calculator asks you to check them, because decay only shrinks a quantity. Equal quantities give an elapsed time of 0. They give no half-life, because a half-life needs some decay to measure.
What is the decay constant λ?
λ = ln(2) / T ≈ 0.693 / T. It's the fractional decay rate per unit time. Larger λ means faster decay.
Does this apply to anything other than radioactivity?
Yes, any exponential decay process works the same way, such as a drug clearing from the body or a capacitor discharging.
Why does decay never quite reach zero?
Each half-life halves what is left, so in this formula the amount keeps shrinking without ever reaching zero. Set the elapsed time to 10 times the half-life and the calculator shows 0.0977% remaining, which is (1/2)¹⁰. At 40 times it shows 9.095e-11%. Past about 1,000 half-lives the figure is too small to compute, and the calculator says below 1e-300.