Model radioactive decay over time with the exponential decay formula. Generate a decay curve showing how a radioactive sample diminishes over multiple half-lives.
Radioactive decay is the fundamental probabilistic process of nuclear physics: an unstable nucleus has a fixed probability per unit time of decaying, regardless of how old it is or what surrounds it. That probability is captured by the decay constant λ, and from λ comes the half-life t₁/₂ = ln(2)/λ — the time for half a sample to decay. Because decay is exponential, after one half-life 50% remains, after two 25%, after three 12.5%, after ten about 0.1%, and after twenty essentially none — but never zero exactly.
This calculator handles the standard problem: given an initial amount, a half-life, and an elapsed time, find what remains. It complements the related "half-life" calculator (which can also solve for half-life given amounts, or solve for elapsed time given amounts). Use this one when you have a known isotope and need to know what fraction is left at a future time — radioactive waste planning, medical isotope inventory, or carbon-14 dating predictions.
The decay law works for any radioactive isotope: short half-life (technetium-99m at 6 hours for medical imaging), long half-life (uranium-238 at 4.5 billion years for geological dating), and everything in between. It's also temperature-, pressure-, and chemistry-independent — uranium decays at exactly the same rate in a hot lava flow as in a cold mountain stream. The few exceptions (rare electron-capture decays influenced by electronic environment) are research curiosities, not practical concerns.
**Scenario:** Nuclear medicine ordered 10 GBq Tc-99m at 8 AM for a 2 PM patient scan. How much remains at scan time? **Calculation:** Tc-99m half-life = 6.0 hr. Elapsed = 6 hr = 1 half-life. N = 10 × 0.5 = 5 GBq. **Result:** 5 GBq at scan time — half the morning dose. Nuclear medicine departments time orders precisely because tracers are unusable within a few half-lives. By next day (24 hr, 4 half-lives), only 0.625 GBq remains — insufficient for most scans.
**Scenario:** Wooden artifact has 8% of the C-14 of a living tree. Date it. **Calculation:** Fraction remaining = 0.08. Half-lives elapsed = log(1/0.08)/log(2) = 1.10/0.301 = 3.64. Time = 3.64 × 5730 = 20,855 years old. **Result:** Approximately 21,000 years old — Last Glacial Maximum era (Pleistocene), possibly Solutrean culture site. C-14 dating is reliable to about 50,000 years; beyond that, the remaining C-14 is too low to measure accurately.
**Scenario:** Patient receives I-131 thyroid ablation dose of 1.5 GBq. When will residual activity drop below 30 MBq (release threshold)? **Calculation:** Need N/N₀ = 0.03/1.5 = 0.02. Half-lives = log(1/0.02)/log(2) = 5.64. I-131 half-life = 8.02 days. Time = 5.64 × 8.02 = 45.2 days. **Result:** ~45 days for the patient's residual activity to drop below 30 MBq. Hospitals typically discharge thyroid I-131 patients with restrictions for 2–3 weeks (limited contact with children, pregnant women). After 6 weeks, almost no activity remains. The biological half-life is shorter than physical half-life because urinary excretion accelerates clearance.
**Use radioactive decay calculations for:**
- **Medical isotope timing**: ordering, scan planning, patient release after treatment. - **Radioactive waste planning**: how long until disposal containers reach safe activity? - **Carbon-14 dating**: archaeological and geological age estimation (organic samples, < 50k years old). - **Uranium/thorium series dating**: older samples using U-238 (4.5 billion yr), K-40 (1.25 billion yr), etc. - **Radiation safety planning**: facility design, exposure modeling, evacuation timelines. - **Smoke detectors**: Am-241 source longevity (~450 yr half-life, essentially constant over device lifetime). - **Nuclear reactor operations**: fuel decay heat after shutdown, fission product timelines. - **Geochemistry**: K-Ar dating of volcanic rocks, lead isotope ratios for ore body genesis.
**Important isotopes by use case:**
**Medical imaging:**
- Tc-99m: 6 hr (most common, ~85% of nuclear medicine scans) - F-18: 110 min (PET scans with FDG) - I-131: 8.02 days (thyroid imaging and therapy) - Tl-201: 73 hr (cardiac perfusion)
**Industrial / safety:**
- Am-241: 432 yr (smoke detectors, industrial gauges) - Co-60: 5.27 yr (sterilization, radiotherapy) - Cs-137: 30.2 yr (sterilization, industrial gauging) - Sr-90: 28.8 yr (legacy fallout, betavoltaic batteries)
**Geological dating:**
- C-14: 5730 yr (organic, < 50k yr) - K-40 → Ar-40: 1.25 billion yr (volcanic rocks) - U-235 → Pb-207: 704 million yr (zircon, oldest rocks) - U-238 → Pb-206: 4.47 billion yr (oldest Earth materials) - Rb-87 → Sr-87: 49 billion yr (early Earth, lunar samples)
**Safety thresholds:**
- After 5 half-lives: ~3% remains, often considered "practically clear." - After 10 half-lives: ~0.1% remains — usually negligible. - After 20 half-lives: ~1 ppm — essentially gone.
**Branching decays:**
Some isotopes have multiple decay modes (e.g., K-40 decays by both β⁻ to Ca-40 AND electron capture to Ar-40). The total decay constant is the sum of branch constants: λ_total = λ_β + λ_EC. Half-life uses the total.
**Useful for environmental and forensic work:**
- **Cs-137 fallout dating**: Cs-137 from atmospheric weapons testing peaked 1955-1965. Soil or lake sediment cores show distinct Cs-137 signals that can date layers within ±1 year. - **Pu-239 in legacy contamination**: 24,110 yr half-life means weapons-era contamination persists effectively forever. - **Sr-90 in milk and bones**: shorter half-life means measurable in tissues; useful for monitoring environmental fallout history.
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Remaining
250.00
% Remaining
25.00%
Half-Lives
2.00
| Parameter | Value |
|---|---|
| Initial Amount (N0) | 1,000 |
| Half-Life | 5,730 years |
| Time Elapsed | 11,460 years |
| Half-Lives Elapsed | 2.0000 |
| Remaining Amount | 250.0000 |
| Decayed Amount | 750.0000 |
| Percent Remaining | 25.0000% |
| Decay Constant (lambda) | 1.209681e-4 / years |
| Activity (lambda x N) | 3.0242e-2 / years |
| Mean Lifetime (1/lambda) | 8266.6426 years |