A more comprehensive version of this tool is now available: Radioactive Isotope Half-Life Computer - with dual-mode isotope search, tri-variable auto-solve, amount unit toggles, and a live SVG decay curve with half-life markers.

Half-Life Radioactive Decay Visualizer

Select an isotope preset or enter custom values. Fill any three fields to instantly solve the fourth, visualize the exponential decay curve, and generate a breakdown schedule.

Decay Calculator
Initial Amount (N0) Empty
N0
Remaining Amount (Nt) Empty
Nt
Half-Life (t1/2) Empty
t1/2
Time Elapsed (t) Empty
t
Fill any three fields above to calculate the fourth.
Key Terms Explained
Half-Life The time required for exactly half of the atoms in a radioactive sample to decay. It is a fixed constant for each isotope, regardless of sample size.
Exponential Decay A process where a quantity decreases by a consistent percentage over equal time intervals, producing a curved graph that approaches but never reaches zero.
Parent Isotope The original, unstable radioactive atom before it undergoes decay. For example, Uranium-238 is the parent in the uranium decay series.
Daughter Isotope The new atom produced when a parent isotope decays. The daughter may be stable or itself radioactive, creating a decay chain.
Radiometric Dating A technique for determining the age of materials by measuring the known decay rate of radioactive isotopes and the ratio of parent to daughter atoms present.
Isotope Atoms of the same element that have the same number of protons but different numbers of neutrons, giving them different atomic masses and stability.
Decay Constant The probability per unit time that an individual atom will decay, symbolized as lambda. It is related to half-life by the formula: lambda = ln(2) divided by t1/2.
Radioactive Decay The spontaneous transformation of an unstable atomic nucleus into a more stable configuration, releasing energy in the form of alpha particles, beta particles, or gamma rays.
Logarithm The inverse of exponentiation. The natural logarithm (ln) is used in half-life formulas to solve for time when the amounts and half-life are known.
Activity The number of decay events per second in a radioactive sample, measured in Becquerels (Bq). Activity decreases as the sample decays, following the same exponential curve as the amount.

The Complete Guide to Radioactive Half-Life and Decay

Half-life is one of the most elegant concepts in nuclear physics: no matter how large or small your sample, exactly half of it will be gone after one half-life. This tool lets you explore that mathematics visually - from the scale of bacteria-killing Iodine-131 (half-life of just 8 days) to the incomprehensible patience of Uranium-238 (half-life longer than the age of Earth).

How to Use This Tool

Enter values in any three of the four fields: Initial Amount (N0), Remaining Amount (Nt), Half-Life (t1/2), and Time Elapsed (t). Use the unit dropdowns to match your real-world problem - the calculator automatically converts all time values to the same unit before computing, so you can enter a half-life in Years and a time elapsed in Days without errors. The fourth variable solves instantly, the decay curve draws in real-time, and the schedule table shows you exactly how much remains after each of the first five half-lives.

The Core Decay Formula

Nt = N0 x (0.5)^(t / t1/2)

Solve for N0: N0 = Nt / (0.5)^(t / t1/2)
Solve for t: t = t1/2 x ln(Nt/N0) / ln(0.5)
Solve for t1/2: t1/2 = t x ln(Nt/N0) / ln(0.5)

The formula says: take the initial amount, multiply it by 0.5 raised to the power of (time elapsed divided by half-life). Each time t equals one full half-life, the exponent equals 1, and you multiply by 0.5 exactly once - cutting the amount in half.

Half-Life Rules of Thumb

These benchmarks are used constantly in nuclear medicine, radiation safety, and geology. Memorizing them gives you instant intuition about any decay problem:

1
half-life
50.00% remains
2
half-lives
25.00% remains
3
half-lives
12.50% remains
5
half-lives
3.125% remains
7
half-lives
Less than 1% remains
10
half-lives
Less than 0.1% remains

The 7-half-life and 10-half-life rules are particularly useful in nuclear medicine and radiation safety. A radioactive tracer used in a medical scan becomes essentially inert after 7 half-lives (less than 1% of original activity). A nuclear waste storage plan must account for at least 10 half-lives before a site is considered safe.

Real-World Applications

Carbon-14 dating works because the isotope is constantly replenished in living organisms through atmospheric CO2. Death stops replenishment, and the clock starts. At 5,730 years per half-life, an artifact with half the normal C-14 ratio is approximately 5,730 years old. This method reliably dates organic material up to about 50,000 years.

Iodine-131's 8.02-day half-life makes it both medically useful and manageable from a safety perspective. It is used to treat thyroid cancer precisely because the thyroid absorbs iodine - the isotope delivers a targeted radiation dose. After 10 half-lives (about 80 days), less than 0.1% of the original dose remains.

Uranium-238, with its 4.468-billion-year half-life, is used to date the oldest rocks on Earth. When molten rock solidifies, it traps Uranium-238 but excludes its daughter product Lead-206. The ratio of U-238 to Pb-206 in a rock sample directly reveals its age - a method that has confirmed Earth is approximately 4.54 billion years old.

Frequently Asked Questions

The decay formula is an exponential function. Exponential functions approach zero asymptotically - they get infinitely close but the math never produces an exact zero. Each half-life cuts the remaining amount in half, so you always have half of something remaining, no matter how small. In reality, atoms are discrete: you eventually reach one atom and it either decays or it does not. But as a mathematical model for large populations of atoms, the curve never truly touches zero. This is why in practice we use the 7-half-life and 10-half-life rules of thumb to define "effectively gone" for safety and medical purposes.
Living organisms continuously absorb Carbon-14 from the atmosphere, maintaining a known ratio of C-14 to stable C-12. When an organism dies, it stops absorbing C-14, and the existing C-14 begins decaying with a half-life of 5,730 years. Scientists measure the current C-14 to C-12 ratio in a sample and compare it to the known starting ratio. By rearranging the decay formula to solve for time (t = t1/2 x ln(Nt/N0) / ln(0.5)), they calculate how many years ago the organism died. The technique is reliable for objects up to about 50,000 years old, after which too little C-14 remains to measure accurately.
A parent isotope is the unstable radioactive atom that undergoes decay. The daughter isotope is the new atom produced after the parent decays. For example, when Uranium-238 decays, its immediate daughter product is Thorium-234. The daughter may itself be radioactive, leading to a decay chain that ends at a stable isotope (in U-238's case, Lead-206, after 14 intermediate steps). In radiometric dating, scientists measure the ratio of parent to daughter isotopes in a rock or mineral to determine its age, based on the known half-life of the parent.
No. Most elements have both stable and unstable (radioactive) isotopes. Carbon has two stable isotopes - Carbon-12 and Carbon-13 - and one common radioactive isotope, Carbon-14. Hydrogen has stable Protium and Deuterium, and radioactive Tritium. Generally, isotopes with too many or too few neutrons relative to their protons are unstable and will decay. The band of stability on a nuclide chart shows the neutron-to-proton ratios that produce stable nuclei. Elements with atomic number above 83 (Bismuth) have no stable isotopes at all - all are radioactive.
No - under the conditions found on Earth or in most laboratory settings, temperature and pressure have no measurable effect on radioactive half-lives. Decay rates are governed by nuclear forces, which are far stronger than the chemical or physical forces affected by heat and pressure. This constancy is precisely what makes radiometric dating reliable: the decay clock ticks at the same rate whether a rock is buried deep underground or sitting on the surface. Only extreme conditions found inside stars - or under very specific electron-capture scenarios in laboratory settings - can cause measurable shifts in decay rates.