Radioactive Decay Calculator
Calculate radioactive decay using three modes: amount remaining after elapsed time, time elapsed from initial and remaining amounts, or activity remaining (in Bq). Furthermore, half-life can be entered in seconds, minutes, hours, days, or years — covering everything from technetium-99m (6 hours) to uranium-238 (4.5 billion years).
How to use the Radioactive Decay Calculator
Mode 1: find remaining amount or fraction after a known elapsed time. Mode 2: find time elapsed from known initial and remaining amounts. Mode 3: find remaining activity in Becquerels. Furthermore, each mode shows half-life and time unit dropdowns for flexibility.
Type the half-life in the selected unit. Furthermore, common isotopes: C-14 (5,730 yr), I-131 (8.02 days), Tc-99m (6.0 hr), U-238 (4.47 × 10⁹ yr), Ra-226 (1,600 yr), Po-210 (138 days).
For mode 1: type elapsed time and initial amount (any unit — atoms, grams, Bq). Moreover, the result gives the remaining fraction and amount.
Enter N₀ (initial) and N (current remaining) alongside the half-life. Furthermore, the elapsed time in seconds and years is calculated — this is the basis of radiometric age dating.
The decay constant λ = ln(2)/t½ is shown for reference. Moreover, λ is the probability per second that any given nucleus will decay — it is an intrinsic nuclear property independent of chemical environment.
Variants, options and when to use each
| Isotope | Half-life | Application |
|---|---|---|
| Tc-99m | 6.01 hr | Nuclear medicine imaging |
| I-131 | 8.02 days | Thyroid therapy |
| C-14 | 5,730 yr | Archaeological dating |
| Ra-226 | 1,600 yr | Radium dial legacy contamination |
| U-238 | 4.47 × 10⁹ yr | Uranium-lead geochronology |
| Cs-137 | 30.2 yr | Nuclear fallout monitoring |
The formula explained
N₀ = initial amount (atoms, grams, Bq, or any consistent unit)
λ = decay constant (s⁻¹) = ln(2) / t½
t½ = half-life (time for half the nuclei to decay)
t = elapsed time (same units as t½ or converted to seconds)
Radioactive decay is a first-order process: the rate of decay is proportional to the number of radioactive nuclei present. Furthermore, this gives the exponential decay law N = N₀ × e^(−λt). The decay constant λ = ln(2)/t½ is the probability per unit time that a nucleus decays. Moreover, after one half-life, 50% remains; after 7 half-lives, less than 1% remains; after 10 half-lives, less than 0.1% remains.
Worked example — I-131 thyroid treatment dose after 16 days
I-131 has a half-life of 8.02 days. Furthermore, a patient receives 3.7 × 10⁹ Bq (3.7 GBq). What activity remains after 16 days?
| Step | Calculation | Result |
|---|---|---|
| Half-lives elapsed | 16 / 8.02 | 1.995 ≈ 2 |
| λ = ln(2)/t½ | 0.6931/8.02 days | 0.08642 day⁻¹ |
| A = A₀ × e^(−λt) | 3.7×10⁹ × e^(−0.08642×16) | 9.27 × 10⁸ Bq = 0.927 GBq |
| Fraction remaining | ~(0.5)² | 25.07% |
What is radioactive decay?
Radioactive decay is the spontaneous disintegration of unstable atomic nuclei, emitting radiation (alpha, beta, or gamma). Furthermore, the decay rate follows first-order kinetics — the number of decays per second (activity) is proportional to the number of radioactive nuclei. This gives the exponential decay law N = N₀e^(−λt), where λ is the isotope-specific decay constant.The half-life (t½) is the time for half the nuclei to decay — it is constant and characteristic of each isotope. Moreover, it is completely independent of chemical state, temperature, or pressure — unlike chemical reaction rates. Furthermore, radioactive half-lives span an extraordinary range: polonium-215 (0.0018 ms) to tellurium-128 (2.2 × 10²⁴ yr — longer than the age of the universe).
Activity is measured in Becquerels (Bq) — one disintegration per second. Additionally, the older unit curie (Ci) = 3.7 × 10¹⁰ Bq (defined as the activity of 1 gram of radium-226). Medical nuclear medicine typically uses GBq for therapeutic doses and MBq for diagnostic doses. Activity decreases with the same exponential law as the number of nuclei.
Who uses this calculator?
Nuclear medicine physicians use radioactive decay calculations to determine administered dose timing for diagnostic and therapeutic radiopharmaceuticals. Furthermore, archaeologists use C-14 decay calculations (radiocarbon dating) to determine the age of organic materials. Geologists use U-238, Rb-87, and K-40 decay systems for radiometric dating of rocks and minerals. Moreover, radiation protection officers calculate when radioactive waste or contaminated equipment has decayed to safe levels.
Historical context and related concepts
Henri Becquerel discovered radioactivity in 1896. Furthermore, Marie and Pierre Curie established that radioactivity is an atomic property independent of chemistry. Ernest Rutherford coined the term "half-life" and showed decay is exponential (1900–1902). Moreover, Rutherford and Soddy established the decay law N = N₀e^(−λt) in 1902, and Rutherford proposed using radioactive decay for geological age dating in 1904.
Why radioactive decay calculations are critical in nuclear medicine and safety
Nuclear medicine uses short-lived radioisotopes to minimise patient radiation dose while providing diagnostic information. Furthermore, Tc-99m (t½ = 6 hr) is used in 80% of nuclear medicine scans worldwide — it decays quickly enough that the patient clears the dose within 24–48 hours. The decay calculation determines the activity at the time of patient injection, given delivery time from the radiopharmacy. Moreover, hospital pharmacists routinely back-calculate activity to determine the preparation activity needed to deliver the required dose.Radioactive decay in nuclear waste management
High-level nuclear waste contains fission products with a wide range of half-lives. Furthermore, short-lived isotopes (Cs-137: 30 yr, Sr-90: 29 yr) dominate activity for the first few centuries. After ~300 years, activity falls to near natural background levels for many components. However, long-lived actinides (Pu-239: 24,100 yr, Am-241: 432 yr) require isolation for thousands of years. Moreover, the decay calculation guides the engineering lifetime and materials selection for deep geological repositories.
Frequently asked questions
Related tools
Half-Life Calculator
Calculate half-life from decay constant or vice versa. Furthermore, half-life and the decay constant are the two equivalent descriptions of decay rate.
→Radiocarbon Dating Calculator
Apply C-14 decay to determine archaeological ages. Moreover, radiocarbon dating is the time-elapsed mode of radioactive decay.
→Significant Figures Calculator
Express decay results to correct precision. Furthermore, radioactive decay measurements typically have 2–4 significant figures.
→Exponential Decay Calculator
General exponential decay for any first-order process. Moreover, radioactive decay is the canonical example of exponential decay.
→Scientific Notation Converter
Convert very large or small decay quantities. Additionally, activities (Bq) and decay constants (s⁻¹) span many orders of magnitude.
→Mole Calculator
Convert between atoms/nuclei and moles/grams. Furthermore, initial activity A₀ = λ × N = λ × (m/M_r) × Nₐ uses both decay and molar mass.
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