# Radioactive decay half-life calculator: remaining atoms, activity, and isotope examples
Use this radioactive decay calculator to estimate how much of an unstable isotope remains after a chosen amount of time. It is designed for the most common search intent behind half-life questions: finding the formula, applying it to real isotopes, comparing remaining parent nuclei with decayed nuclei, and understanding why activity decreases as a sample ages.The tool combines two complementary models. The numerical results use the standard exponential decay equation, while the atom field simulates individual nuclei with stochastic thresholds. That makes it useful both as a quick half-life calculator and as a visual explanation of why radioactive decay is predictable in bulk but random for any single atom.# Radioactive decay formula used by the calculator
The calculator uses N(t) = N0 x (1/2)^(t / T1/2). In this equation, N0 is the starting number of parent nuclei, N(t) is the expected number remaining after time t, and T1/2 is the isotope half-life. The exponent t / T1/2 counts how many half-lives have passed.For example, if a sample starts with 1,000 parent nuclei and two half-lives pass, the expected remaining amount is 1,000 x (1/2)^2 = 250 nuclei. The decayed amount is the difference between the original and remaining sample, so 750 nuclei have decayed. The same calculation works whether the half-life is measured in seconds, hours, days, years, or billions of years.| Elapsed time | Formula factor | Parent nuclei remaining | Relative activity |
|---|---|---|---|
| 0 half-lives | (1/2)^0 | 100% | 100% |
| 1 half-life | (1/2)^1 | 50% | 50% |
| 2 half-lives | (1/2)^2 | 25% | 25% |
| 3 half-lives | (1/2)^3 | 12.5% | 12.5% |
| 5 half-lives | (1/2)^5 | 3.125% | 3.125% |
| 10 half-lives | (1/2)^10 | 0.098% | 0.098% |
# How to calculate remaining activity after a half-life
For a single parent isotope, activity is proportional to the number of undecayed nuclei. If 25% of the parent isotope remains, the activity from that isotope is also about 25% of the starting activity. This is why the calculator reports relative activity alongside remaining and decayed atoms: for one isotope, they follow the same exponential factor.This proportional relationship is especially important in nuclear medicine and radiation safety. A Technetium-99m tracer loses activity over hours, so imaging schedules are planned around its short half-life. Iodine-131 remains important over days, which affects therapy timing, contamination monitoring, and instructions for limiting exposure after treatment.# Examples: Carbon-14, Iodine-131, Technetium-99m, Uranium-238, and Radon-222
| Isotope | Approximate half-life | Common search use | What the result tells you |
|---|---|---|---|
| Carbon-14 | 5,730 years | Radiocarbon dating | How much parent Carbon-14 activity remains in once-living material. |
| Iodine-131 | 8.02 days | Medical therapy and nuclear incidents | How quickly activity falls over days after release or treatment. |
| Technetium-99m | 6.01 hours | Diagnostic imaging | Why useful medical activity fades within a clinical workday. |
| Uranium-238 | 4.47 billion years | Geologic dating | Why very long-lived isotopes remain measurable over Earth history. |
| Radon-222 | 3.82 days | Indoor radiation and decay chains | How a gas exposure source changes over days. |
# How to read the stochastic atom simulation
The animated atom field is intentionally stochastic. The equation gives the expected value for a large sample, but individual nuclei decay randomly. With a small sample, one run after one half-life might leave slightly more or fewer than 50% of the atoms. With a larger sample, the visual result tends to sit closer to the theoretical curve because random fluctuations average out.This distinction matters for learning. Half-life does not mean every atom waits for a timer and then half of them disappear at once. Each unstable nucleus has a constant probability of decay per unit time. The smooth curve appears only when many independent random events are counted together.# Half-life calculator use cases
- Homework and classroom physics: calculate remaining parent nuclei after a given number of half-lives and connect the formula to a visual model.
- Chemistry and nuclear science: compare isotope stability, decay speed, and relative activity across very different time scales.
- Radiocarbon dating intuition: understand why older samples contain less Carbon-14 and why dating becomes harder as activity approaches background levels.
- Medical isotope planning: see why short half-lives are useful for imaging and why activity changes quickly after administration.
- Radiation safety education: estimate how relative activity falls over time for a single isotope without confusing half-life with immediate disappearance.