Radioactive Decay by Half-Life Calculator
The Radioactive Decay by Half-Life Calculator determines what percentage of a radioactive substance remains after a given elapsed time, using the isotope's half-life. The core formula is N/N₀ = (1/2)^(t/t½), where t is elapsed time and t½ is the half-life. It is used in radiocarbon dating, nuclear medicine dosing, radiation safety planning, and nuclear waste management. For example, Carbon-14 (t½ = 5,730 years) decays to exactly 25% after 11,460 years (two half-lives). This calculator works for any isotope across timescales from milliseconds (e.g., Polonium-214, t½ = 164 µs) to billions of years (e.g., Uranium-238, t½ = 4.468 × 10⁹ years).
When to use this calculator
- Radiocarbon dating: A wood sample shows 12.5% of original C-14 (t½ = 5,730 yr) remaining → (½)³ means 3 half-lives → ~17,190 years old.
- Nuclear medicine: Iodine-131 (t½ = 8.02 days) is used to treat thyroid cancer; after 40 days (~5 half-lives) only ~3.1% of the administered dose remains active in the patient.
- Radiation safety / waste storage: Cesium-137 (t½ = 30.17 years) from nuclear accidents requires ~200 years (≈6.6 half-lives) to decay to less than 1% of its original activity.
- Smoke detector maintenance: Americium-241 (t½ = 432.2 years) in ionization smoke detectors loses only ~1.6% of its activity over a 10-year service life, confirming no replacement is needed.
- Nuclear power plant decommissioning: Cobalt-60 (t½ = 5.27 years) activated in reactor steel decays to <0.1% in ~53 years, informing safe dismantling timelines.
- Environmental contamination assessment: Strontium-90 (t½ = 28.8 years) released in nuclear tests requires ~96 years (≈3.3 half-lives) to drop below 10% of original concentration in soil.
Calculation example
- t½ = 5,730 years (C-14), t = 11,460 years
- N/N₀ = (½)² = 25%
How it works
3 min readHow It Is Calculated
The fundamental law of radioactive decay states that the number of undecayed nuclei decreases exponentially over time. The half-life form of the equation is:
N(t) / N₀ = (1/2)^(t / t½)
Where:
N(t) = quantity remaining at time t
N₀ = initial quantity (100%)
t = elapsed time (same units as t½)
t½ = half-life of the isotope
(1/2) = decay factor per half-life
Equivalent exponential form:
N(t) / N₀ = e^(−λt)
where λ = ln(2) / t½ ≈ 0.6931 / t½
To find elapsed time from remaining fraction:
t = t½ × log₂(N₀ / N(t))
t = t½ × [ln(N₀/N(t)) / ln(2)]Both forms are mathematically identical. The half-life form is more intuitive; the exponential form is preferred in physics calculations. The activity (decays per second, measured in Becquerels or Curies) follows the same equation: A(t) = A₀ × (½)^(t/t½).
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Reference Table — Common Isotopes and Their Half-Lives
| Isotope | Half-Life | Decay Type | Primary Use / Relevance |
|---|---|---|---|
| Polonium-214 (Po-214) | 164 microseconds | Alpha | Decay chain intermediate |
| Iodine-131 (I-131) | 8.02 days | Beta/Gamma | Thyroid cancer treatment |
| Technetium-99m (Tc-99m) | 6.01 hours | Gamma | Medical imaging (SPECT) |
| Cobalt-60 (Co-60) | 5.27 years | Beta/Gamma | Radiation therapy, food irradiation |
| Tritium (H-3) | 12.32 years | Beta | Nuclear weapons, luminous paint |
| Strontium-90 (Sr-90) | 28.8 years | Beta | Nuclear fallout contaminant |
| Cesium-137 (Cs-137) | 30.17 years | Beta/Gamma | Chernobyl/Fukushima fallout |
| Americium-241 (Am-241) | 432.2 years | Alpha | Smoke detectors |
| Carbon-14 (C-14) | 5,730 years | Beta | Radiocarbon dating |
| Plutonium-239 (Pu-239) | 24,110 years | Alpha | Nuclear weapons, reactor fuel |
| Uranium-235 (U-235) | 703.8 million years | Alpha | Nuclear fission fuel |
| Uranium-238 (U-238) | 4.468 billion years | Alpha | Age of Earth dating |
Half-life values from NIST Nuclear Data / National Nuclear Data Center (BNL).
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Typical Cases — Step-by-Step Examples
Example 1 — Radiocarbon Dating (C-14)
Example 2 — Medical Dose Decay (I-131)
Example 3 — Long-Term Waste Storage (Cs-137)
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Common Mistakes
1. Mixing time units: Using t½ in years but t in days without converting. Always express both in the same unit before calculating. Example: I-131 t½ = 8.02 days, so elapsed time must also be in days.
2. Confusing activity with mass: The formula gives the fraction of radioactive atoms (or activity) remaining — not the total mass. For long-lived isotopes like U-238, even after millions of years the mass barely changes, but activity follows the same exponential law.
3. Applying the formula to non-first-order processes: Radioactive decay is strictly first-order. Chemical reactions, biological half-lives of drugs (which can be zero- or mixed-order), and population decay may not follow the same exact equation. Never assume a biological or pharmacological "half-life" is identical in behavior to a nuclear half-life.
4. Forgetting secular equilibrium in decay chains: Uranium-238 decays through 14 daughter isotopes before reaching stable Pb-206. After a long period, daughter products build up and the total activity in a sample is much higher than U-238 alone would suggest. The simple (½)^(t/t½) formula applies to each individual nuclide, not the chain as a whole.
5. Rounding number of half-lives: Calculating n = t/t½ and rounding to a whole number introduces large errors. For example, t = 20 days with t½ = 8.02 days gives n = 2.494, not 2 or 3. Always use the exact ratio in the exponent.
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Related Calculators
Since no internal related slugs are defined for this calculator, explore other science tools on Hacé Cuentas for unit conversions, exponential growth/decay, and more scientific computations.
Frequently asked questions
What is a half-life in radioactive decay?
A half-life (t½) is the time required for exactly half of the radioactive atoms in a sample to decay into daughter nuclides. It is a fixed, isotope-specific constant unaffected by temperature, pressure, or chemical state. For example, Carbon-14 has t½ = 5,730 years, meaning a 1-gram sample of pure C-14 will contain only 0.5 g of C-14 after 5,730 years, regardless of environmental conditions.
After how many half-lives is a radioactive substance considered 'safe'?
A common practical threshold is 10 half-lives, after which only (½)¹⁰ ≈ 0.098% of the original activity remains — less than 0.1%. However, 'safe' depends on the isotope and initial activity. The NRC and EPA use specific dose-rate thresholds (e.g., <2 mrem/hr at 1 meter) rather than a fixed number of half-lives. For Cs-137 (t½ = 30.17 yr), 10 half-lives = ~302 years.
Can I use this calculator for the biological half-life of a drug?
The same mathematical formula (C(t) = C₀ × (½)^(t/t½)) applies to drugs that follow first-order pharmacokinetics, which is most common drugs. However, some drugs follow zero-order kinetics (constant elimination rate regardless of concentration, e.g., alcohol at typical doses), for which the half-life formula does NOT apply. Always confirm the pharmacokinetic model with clinical sources like the NIH DailyMed database before using this calculator for drug dosing.
Why does Carbon-14 dating only work up to ~50,000 years?
After about 8–9 half-lives of C-14 (8 × 5,730 ≈ 45,840 years), only ~0.4% of original C-14 remains. At this level, the signal becomes indistinguishable from background contamination and instrument noise. Modern Accelerator Mass Spectrometry (AMS) has pushed the practical limit to ~50,000–55,000 years. For older samples, geologists use isotopes with longer half-lives, such as Potassium-40 (t½ = 1.25 billion years) or Uranium-238 (t½ = 4.468 billion years).
What is the difference between the half-life formula and the decay constant λ?
They are two ways to express the same law. The decay constant λ = ln(2) / t½ ≈ 0.6931 / t½ represents the probability of decay per unit time. The exponential form is N(t) = N₀ × e^(−λt), while the half-life form is N(t) = N₀ × (½)^(t/t½). Both are mathematically identical. Physicists favor λ in differential equations; the half-life form is more intuitive for practical calculations since each half-life exactly halves the remaining quantity.
How accurate is the half-life of Carbon-14 used in radiocarbon dating?
The currently accepted value is t½ = 5,730 ± 40 years, established by Godwin (1962) and adopted internationally. Historically, Libby's original 1949 value was 5,568 years (the 'Libby half-life'), and many older radiocarbon dates in the literature still use this value. The difference is ~3%, so dates calculated using Libby's value should be multiplied by 1.029 to convert to the modern standard. NIST and the National Ocean Sciences AMS Facility confirm the 5,730-year value.
Does the half-life of an isotope ever change?
For practical purposes, no. Radioactive half-lives are determined by nuclear forces and are essentially constant under all normal conditions — changes in temperature, pressure, or chemical bonding affect half-lives by at most a few parts per thousand, only measurable in exotic laboratory settings. However, extreme environments like stellar interiors can alter decay rates via electron capture processes. For all Earth-based applications, treat half-lives as fixed constants as tabulated by NIST.
What units should I use for time in this calculator?
The half-life (t½) and elapsed time (t) must be in the same units — both in years, both in days, both in hours, etc. The calculator uses years by default, matching common isotopes like C-14 (5,730 yr) and Pu-239 (24,110 yr). For medical isotopes like Tc-99m (t½ = 6.01 hours), convert elapsed time to hours before entering values. The output percentage is dimensionless and independent of the unit chosen, as long as both inputs share the same unit.