Half-Life Calculator
Determine how much of a radioactive substance remains after a given time. Used in nuclear physics, carbon dating, and pharmacokinetics to model exponential decay.
About this calculator
Radioactive decay follows an exponential model where a substance loses half its quantity in each half-life period. The governing formula is: N(t) = N₀ × (0.5)^(t / t½), where N₀ is the initial amount, t is the elapsed time, and t½ is the half-life. Each time interval equal to t½ cuts the remaining quantity in half. This means after two half-lives, 25% remains; after three, 12.5%, and so on. The exponent t / t½ tells you how many half-life cycles have elapsed — it need not be a whole number. This same mathematical structure applies to drug clearance, carbon-14 dating of artifacts, and decay chains in nuclear reactors.
How to use
Suppose you start with 80 g of iodine-131, which has a half-life of 8 years, and want to know how much remains after 24 years. Plug into N(t) = 80 × (0.5)^(24 / 8) = 80 × (0.5)³ = 80 × 0.125 = 10 g. Enter 80 in Initial Amount, 24 in Time Elapsed, and 8 in Half-Life. The calculator instantly returns 10 g — exactly three half-lives of decay.
Frequently asked questions
What is the half-life formula and how does it work?
The half-life formula is N(t) = N₀ × (0.5)^(t / t½). N₀ is the starting quantity, t is elapsed time, and t½ is the half-life period. The exponent t / t½ counts how many half-lives have passed. After each full half-life, the remaining amount is exactly halved, creating a smooth exponential curve between whole-number intervals.
How do I use a half-life calculator for carbon-14 dating?
Carbon-14 has a half-life of approximately 5,730 years. By measuring the fraction of C-14 remaining in an organic sample and comparing it to the original atmospheric concentration, you can solve for elapsed time t = t½ × log₂(N₀ / N). Enter the known half-life and the measured remaining fraction into the calculator to find the sample's age. This technique reliably dates materials up to about 50,000 years old.
Why does radioactive decay follow an exponential rather than a linear pattern?
Each radioactive atom decays independently with a fixed probability per unit time, regardless of how many atoms are present. This memoryless property means the rate of decay is always proportional to the current amount — a hallmark of exponential processes. If decay were linear, the substance would eventually reach zero and then go negative, which is physically impossible. The exponential model ensures the quantity asymptotically approaches zero, never quite reaching it.