Calculate remaining quantity ($N_t$), initial quantity ($N_0$), elapsed decay time ($t$), or half-life ($t_0.5$) for radioactive isotopes and pharmacological compounds with step-by-step mathematical derivations.
• Module 1: Enter any 3 of the 4 decay values ($N_t, N_0, t, t_0.5$) and leave the target field blank to calculate the unknown variable.
• Module 2: Enter any 1 parameter (Half-Life $t_0.5$, Mean Lifetime $\tau$, or Decay Constant $\lambda$) to convert and compute the other two.
Please provide any three of the following to calculate the fourth value.
Please provide any one of the following to get the other two.
An in-depth technical resource for students, researchers, nuclear engineers, and medical professionals.
In physics and chemistry, the half-life (t½) of a substance is the time required for a quantity to reduce to exactly half of its initial value. The term is most commonly used in nuclear physics to describe the radioactive decay of unstable atomic nuclei, but it applies equally to chemical reaction kinetics, first-order biological elimination of pharmaceutical drugs, and ecological pollutant degradation.
Because radioactive decay is a stochastic (random quantum) process at the single-atom level, it is impossible to predict when a specific individual nucleus will decay. However, for macroscopic samples containing billions of atoms, the decay rate is strictly governed by an exponential decrease law:
After 1 half-life, 50% remains; after 2 half-lives, 25% remains; after 3 half-lives, 12.5% remains; after 4 half-lives, 6.25% remains; and after 5 half-lives, only 3.125% remains (over 96.8% has decayed).
| Isotope | Symbol | Half-Life (t½) | Decay Mode | Primary Application |
|---|---|---|---|---|
| Technetium-99m | ⁹⁹ᵐTc | 6.01 hours | Isomeric Transition (γ) | Diagnostic Nuclear Imaging (SPECT) |
| Iodine-131 | ¹³¹I | 8.02 days | Beta (β⁻), Gamma (γ) | Thyroid Disease & Cancer Therapy |
| Radon-222 | ²²²Rn | 3.82 days | Alpha (α) | Environmental Health & Geology |
| Fluorine-18 | ¹⁸F | 109.8 minutes | Positron (β⁺) | PET Medical Scans (FDG Tracers) |
| Cobalt-60 | ⁶⁰Co | 5.27 years | Beta (β⁻), Gamma (γ) | Industrial Radiography & Cancer Radiotherapy |
| Tritium | ³H | 12.32 years | Beta (β⁻) | Luminous Exit Signs & Fusion Research |
| Cesium-137 | ¹³⁷Cs | 30.17 years | Beta (β⁻), Gamma (γ) | Industrial Gauging & Environmental Tracers |
| Carbon-14 | ¹⁴C | 5,730 years | Beta (β⁻) | Archaeological Radiocarbon Dating |
| Plutonium-239 | ²³⁹Pu | 24,110 years | Alpha (α) | Nuclear Power & Spacecraft RTGs |
| Uranium-235 | ²³⁵U | 703.8 million years | Alpha (α) | Nuclear Fission Reactors & Naval Propulsion |
| Potassium-40 | ⁴⁰K | 1.25 billion years | Beta (β⁻), EC | Geological Rock Dating (K-Ar Dating) |
| Uranium-238 | ²³⁸U | 4.468 billion years | Alpha (α) | Age of the Earth & Radiometric Dating |
Archaeologists and geologists measure the ratio of unstable Carbon-14 (t½ = 5,730 years) to stable Carbon-12 in organic matter to accurately date wooden tools, bone fragments, and fossils up to 50,000 years old.
In medicine, a drug's elimination half-life determines how frequently patients must take medication to maintain a steady therapeutic blood concentration without toxicity. It generally takes 4 to 5 half-lives to fully clear a drug from the body.
Hospitals utilize short-lived isotopes such as Technetium-99m (t½ = 6.01 hours) and Fluorine-18 (t½ = 109.8 minutes) for medical imaging so patients receive high-clarity diagnostic scans with minimal long-term radiation exposure.
The mean lifetime (τ) represents the average time an individual radioactive atom survives before decaying. Because of exponential kinetics, mean lifetime is longer than the half-life by a factor of 1 / ln(2) ≈ 1.4427. Therefore, τ = t_½ / 0.693147.
Theoretically, exponential decay approaches zero asymptotically and never reaches absolute zero. However, practically, after 5 half-lives, only 3.125% of the original substance remains (96.875% decayed). After 10 half-lives, less than 0.1% remains, which is universally considered clinically or radiologically cleared.
No. Nuclear half-lives are determined by quantum nuclear binding forces inside the atomic nucleus, which are millions of times stronger than chemical bonds or ordinary thermodynamic temperatures and pressures. Half-life remains invariant across almost all physical and chemical environments.
You can use any unit of time (seconds, minutes, hours, days, years, or millennia) as long as both Elapsed Time (t) and Half-Life (t1/2) are expressed in the exact same unit. The resulting calculated time will naturally be in that same unit.
Yes. If the remaining quantity (Nt) exceeds the initial quantity (N0), the system evaluates the process as exponential growth (doubling time) rather than decay, and calculates the corresponding growth parameter or negative elapsed decay time.