Half-Life Calculator — Decay Curve & AI Insights
This half-life calculator uses N(t) = N₀ × (½)^(t/t½) to find how much of a radioactive — or any exponentially decaying — substance remains after a given time, with a decay curve graph and plain-language AI insights.
Decay parameters
Enter the initial quantity, the substance's half-life, and the elapsed time.
What is half-life?
Half-life is the time required for half of a quantity of a radioactive (or otherwise exponentially decaying) substance to decay into a different form.
Every radioactive isotope has its own fixed half-life — independent of temperature, pressure, or chemical state. Some isotopes decay in microseconds; others take billions of years. The table below gives reference half-lives for isotopes commonly used in dating, medicine, and power generation.
| Isotope | Half-life (t½) | Common use |
|---|---|---|
| Carbon-14 (C-14) | 5,730 years | Radiocarbon dating of organic material |
| Uranium-238 (U-238) | 4.47 billion years | Geological / radiometric dating of rocks |
| Iodine-131 (I-131) | 8.02 days | Thyroid imaging and treatment |
| Cobalt-60 (Co-60) | 5.27 years | Radiotherapy, industrial radiography |
| Technetium-99m (Tc-99m) | 6.01 hours | Most common nuclear medicine imaging tracer |
| Polonium-214 (Po-214) | ~164 microseconds | One of the fastest-decaying natural isotopes |
Values are commonly cited reference figures rounded for readability; consult a nuclear data table (e.g. IAEA, NNDC) for precise values.
How the Half-Life Calculator Works — The Formula
Radioactive decay is a random quantum process, but large collections of atoms follow a precise statistical law. After every interval equal to one half-life, exactly half the remaining atoms have decayed:
N(t) = N₀ × (½)^(t ÷ t½)
Equivalent form:
N(t) = N₀ × e^(−λt) where λ = ln(2) ÷ t½
- N(t)
- Quantity remaining after time t
- N₀
- Initial quantity at t = 0
- t
- Elapsed time
- t½
- Half-life of the substance
- λ (lambda)
- Decay constant = ln(2) ÷ t½
The key insight is that after n half-lives, the fraction remaining is (½)^n regardless of the initial quantity. After 1 half-life: 50%. After 2: 25%. After 10: ~0.098%.
How to Use the Half-Life Calculator
- Enter the initial quantity (N₀) — this can be in any unit: grams, number of atoms, becquerels, or even a percentage.
- Enter the substance's half-life (t½) and choose its time unit from the dropdown.
- Enter the elapsed time (t) and choose its time unit — it doesn't need to match the half-life's unit.
- Click Calculate to see your results instantly.
- Scroll to the AI Insights section to understand what your result means.
How to Interpret Your Half-Life Calculator Results
What a Good Result Looks Like
There's no "good" or "bad" result in radioactive decay — it's a fixed physical process — but a result that makes sense usually has a recognizable relationship to the number of half-lives elapsed. For example, if your elapsed time equals exactly one half-life, you should see almost exactly 50% remaining; two half-lives should give 25%; three should give 12.5%. If the "Number of half-lives" output is a clean number like 1, 2, or 3 and the % remaining doesn't roughly match (½)ⁿ, double-check your unit selections.
Warning Signs in Your Results
Watch out for these signs that something may be off with your inputs: a "% remaining" value above 100% (impossible — check that the elapsed time isn't negative or that you haven't swapped N₀ and the result); a result that barely changes even though you entered a long elapsed time (likely a unit mismatch — e.g. half-life entered in years but elapsed time in seconds); or a "Number of half-lives" value that seems wildly different from what you expected given the isotope's known half-life.
How to Improve Your Result
To get the most accurate result: (1) verify your half-life value against a trusted nuclear data reference or the isotope table above; (2) keep the half-life and elapsed time in units you can sanity-check mentally (e.g. both in years for long-lived isotopes, both in hours for medical tracers); (3) for decay chains or mixtures of isotopes, remember this calculator models a single exponential decay — apply it separately to each isotope in the chain; (4) use the decay curve chart to visually confirm the marker lands where you expect relative to the curve's shape.
Half-Life Calculator Examples
What Is the Half-Life of Carbon-14? (Calculator Example)
Carbon-14 has a half-life of 5,730 years. Suppose an organic sample contains 25% of the C-14 it would have had when the organism died. How old is it?
- Initial quantity (N₀): 100
- Half-life (t½): 5,730 years
- Elapsed time (t): 11,460 years
Result: 25 remaining (25%), 2 half-lives elapsed. Since 25% = (½)², the sample is 2 × 5,730 = 11,460 years old. This is the basic principle behind radiocarbon dating.
What this means: knowing the remaining percentage of C-14 directly tells you the age of an organic sample, in units of half-lives. What to do next: try entering different "% remaining" values (by adjusting N₀ and elapsed time) to see how the age estimate changes.
How Much Substance Remains After 3 Half-Lives?
This example shows the (½)ⁿ pattern directly: after exactly 3 half-lives, 1/8 (12.5%) of any initial quantity remains, regardless of what the substance is.
- Initial quantity (N₀): 800 grams
- Half-life (t½): 10 days
- Elapsed time (t): 30 days (= 3 half-lives)
Result: 100 grams remaining (12.5%), decayed amount = 700 grams.
What this means: 800 → 400 → 200 → 100 grams across the three half-lives — each step halves the previous amount. What to do next: use the decay curve chart to see this step-down pattern visually, or try 7 half-lives to see how close the remaining amount gets to zero (~0.78%).
Half-Life of Iodine-131 Calculator Example
Iodine-131, used in thyroid imaging and treatment, has a half-life of 8.02 days. A hospital receives a dose of 50 MBq (megabecquerels) and needs to know how much activity remains after 16 days (storage/transport delay).
- Initial quantity (N₀): 50 MBq
- Half-life (t½): 8.02 days
- Elapsed time (t): 16 days
Result: ≈12.4 MBq remaining (≈24.8%), ≈1.995 half-lives elapsed.
What this means: after roughly two half-lives, only about a quarter of the original activity remains — important for dose planning in nuclear medicine. What to do next: try shorter elapsed times to see how quickly I-131 activity drops in the first day or two.
Frequently asked questions
What is half-life?
Half-life (t½) is the time it takes for exactly half of a quantity of a radioactive substance — or any exponentially decaying quantity — to decay or disappear. After one half-life, 50% remains. After two half-lives, 25% remains. After ten half-lives, less than 0.1% remains. Half-life is a constant property of each radioactive isotope, ranging from fractions of a second (polonium-214, ~164 μs) to billions of years (uranium-238, ~4.47 billion years).
How do I calculate the amount remaining after N half-lives?
Multiply the initial quantity by (½) raised to the number of half-lives elapsed: N(t) = N₀ × (½)ⁿ. After 1 half-life, 50% remains. After 2, 25%. After 3, 12.5%. After 7 half-lives, about 0.78% remains, and after 10, about 0.098% remains. This calculator does that math for you — enter N₀, the half-life, and the elapsed time, and it converts your elapsed time into a number of half-lives automatically.
What substances have the longest half-lives?
Some isotopes have half-lives far longer than the age of the universe — bismuth-209 has a half-life of about 2.01×10¹⁹ years, and tellurium-128 is around 2.2×10²⁴ years. Among commonly referenced isotopes, uranium-238 (≈4.47 billion years) and potassium-40 (≈1.25 billion years) are used for dating rocks and geological formations. At the other extreme, polonium-214 decays with a half-life of only ~164 microseconds. See the isotope reference table above for more examples.
What is the half-life formula?
The remaining quantity N after time t is given by: N(t) = N₀ × (½)^(t ÷ t½), where N₀ is the initial amount, t is elapsed time, and t½ is the half-life. An equivalent form using the decay constant λ = ln(2) ÷ t½ is N(t) = N₀ × e^(−λt). Both give identical results; the half-life form is easier to use directly.
What is the decay constant?
The decay constant (λ, lambda) is the probability per unit time that a single atom will decay. It is related to half-life by λ = ln(2) ÷ t½ ≈ 0.6931 ÷ t½. A shorter half-life means a larger decay constant — the substance decays more rapidly. Activity (decays per second) equals λ × N, where N is the current number of atoms.
What are real-world applications of half-life calculations?
Radiocarbon dating uses the 5,730-year half-life of carbon-14 to date organic materials up to ~50,000 years old. Nuclear medicine uses short-half-life isotopes (e.g. technetium-99m, t½ = 6 hours) to image organs. Nuclear power and waste management must account for the half-lives of fission products. Pharmacokinetics uses the same mathematics to model how drugs are eliminated from the body.
Does half-life apply to non-radioactive quantities?
Yes. Any exponentially decaying quantity obeys the same mathematics. In pharmacology, a drug's biological half-life is the time for its concentration in the bloodstream to halve. In finance, the same equations describe the decay of a debt with continuous compounding. In electronics, RC circuit discharge follows the same exponential decay pattern.
What are the limitations of the exponential decay model?
The N(t) = N₀ × (½)^(t/t½) formula assumes a single isotope decaying directly to a stable product — real samples are often more complex. Decay chains (where a daughter isotope is also radioactive) follow different curves until secular equilibrium is reached. Branching decay (an isotope that can decay via more than one pathway) splits the half-life between routes. For mixtures of isotopes or decay chains, this calculator is still useful for each individual step, but the overall sample activity may not follow a simple exponential curve.