Nuclear Decay Calculator
Determine how much of a radioactive isotope remains after a given time using its half-life. Used in radiometric dating, medical isotope dosing, and nuclear waste management.
About this calculator
Radioactive decay follows an exponential law: each nucleus has a constant probability of decaying per unit time. The governing formula is N(t) = N₀ × (0.5)^(t / t½), where N₀ is the initial quantity of atoms, t is the elapsed time, and t½ is the half-life of the isotope. After one half-life, exactly half the original atoms remain; after two half-lives, one quarter remain, and so on. This relationship is universal across all radioactive isotopes regardless of chemical environment. The formula is equivalent to N(t) = N₀ × e^(−λt), where the decay constant λ = ln(2) / t½. Scientists use it to date archaeological artifacts with carbon-14 (t½ ≈ 5,730 years), predict hospital isotope inventory for Tc-99m (t½ ≈ 6 hours), and model the long-term hazard of nuclear waste.
How to use
Suppose you start with 1,000,000 atoms of Iodine-131, which has a half-life of 8.02 days, and want to know how many remain after 24.06 days (3 half-lives). Enter: Initial Quantity = 1,000,000 atoms, Half-Life = 8.02 years (or days — keep units consistent), Time Elapsed = 24.06. The calculator evaluates: N = 1,000,000 × (0.5)^(24.06 / 8.02) = 1,000,000 × (0.5)^3 = 1,000,000 × 0.125 = 125,000 atoms. After 24.06 days, 125,000 atoms of I-131 remain — exactly 12.5% of the original sample.
Frequently asked questions
What is the difference between half-life and decay constant in nuclear decay?
The half-life (t½) is the time it takes for exactly half of a radioactive sample to decay and is the more intuitive quantity. The decay constant (λ) expresses the probability per unit time that a single nucleus will decay, and the two are related by λ = ln(2) / t½ ≈ 0.693 / t½. Both describe the same underlying exponential decay process — choosing which to use is a matter of convention. Nuclear engineers often prefer the decay constant for differential equations, while geologists and medical physicists typically cite the half-life.
How do I use the nuclear decay calculator for carbon-14 radiocarbon dating?
Carbon-14 has a well-established half-life of 5,730 years. Enter the initial quantity as 100% (or a measured initial C-14/C-12 ratio), enter 5,730 as the half-life in years, then enter the measured remaining fraction as the current quantity and solve for elapsed time. For example, if a sample retains 25% of its original C-14, the elapsed time is 2 × 5,730 = 11,460 years. The calculator automates this algebra, making it straightforward for archaeologists and geologists to estimate sample age.
Why does radioactive decay follow an exponential curve rather than a linear decline?
Each nucleus decays independently and randomly, with a fixed probability of decaying per unit time regardless of how many other nuclei have already decayed. This memoryless, constant-probability process is the hallmark of an exponential decay. Mathematically, the rate of decay at any moment is proportional to the number of atoms still present: dN/dt = −λN, whose solution is the exponential N(t) = N₀e^(−λt). A linear decline would imply each atom 'knows' how many others remain, which has no physical basis.