Radioactive Decay Calculator
Reviewed by Hamza Gujjar, Algorithm Architect & Software Developer
Last updated: August 19, 2026A Process That Never Speeds Up or Slows Down
Radioactive decay is one of the few processes in nature that runs on a perfectly fixed clock, unaffected by temperature, pressure, or chemical environment. Every unstable isotope has a characteristic half-life — the time for half of any sample to decay — and that half-life never changes no matter how much material remains. This predictability is what makes radiometric dating and nuclear medicine dosing possible.
The Formula
N0 is the initial amount, N(t) is the amount remaining after time t, and half-life is the isotope's characteristic decay constant, expressed in whatever time unit t is measured in. Because the exponent is a ratio of elapsed time to half-life, the formula works regardless of whether half-life is measured in seconds, days, or years, as long as t uses the same unit.
Where This Clock Is Put to Use
- Radiometric dating — carbon-14 (half-life ~5,730 years) and other isotopes let geologists and archaeologists date organic material and rock formations.
- Nuclear medicine — diagnostic and therapeutic radioisotopes are selected partly by half-life, balancing enough activity to be useful against minimizing a patient's total radiation exposure.
- Nuclear waste management — storage timelines for spent fuel are planned around how long it takes various isotopes to decay to safe activity levels.
- Smoke detectors — ionization detectors use a small americium-241 source, relying on its long, stable half-life for a consistent decades-long service life.
The Universal Decay Curve
Because the formula is expressed in half-lives, this curve looks identical for every radioactive isotope — only the time axis stretches or compresses depending on the specific half-life:
| Half-lives elapsed | Percent remaining |
|---|---|
| 0 | 100% |
| 1 | 50% |
| 2 | 25% |
| 3 | 12.5% |
| 4 | 6.25% |
| 5 | 3.125% |
By 5 half-lives, less than 5% of the original material remains — a useful rule of thumb for estimating when an isotope's activity has become negligible.
How to Use This Calculator
- Enter the Initial Amount (any unit, such as grams or activity).
- Enter the Half-Life of the isotope (e.g., in years).
- Enter the Elapsed Time, using the same time unit as the half-life.
- Select Calculate to see the amount remaining, along with the percentage remaining and the worked formula.
Related Calculations
For the energy released in decay-related photon emission, see the Photon Energy Calculator, or explore other exponential-style relationships with the Frequency Calculator.
Nuclear Structure, Stability, and Radioactive Disintegration
Radioactive decay is the spontaneous quantum stochastic process by which an unstable atomic nucleus loses energy by emitting ionizing radiation. Nuclei with unfavorable neutron-to-proton ratios (lying outside the nuclear belt of stability) undergo nuclear transmutation to transform into more tightly bound, energetically stable daughter nuclides. Because each radioactive disintegration event is statistically independent, radioactive decay rate across macroscopic numbers of atoms follows exact exponential probability laws.
Where:
- A(t) (Activity): Rate of nuclear disintegrations per unit time, measured in Becquerels (1 Bq = 1 decay/second) or Curies (1 Ci = 3.7 × 1010 Bq).
- N(t) (Number of Radioactive Nuclei): Total remaining undecomposed radioactive parent nuclei.
- λ (Decay Constant): Characteristic disintegration probability per unit time (λ = ln 2 / t1/2).
- A0 (Initial Activity): Baseline radioactivity at time zero.
Primary Modes of Radioactive Nuclear Decay
Unstable nuclides release excess binding energy through three major classical radiation emissions:
- Alpha Decay (α): Heavy unstable nuclei (such as Uranium-238 or Radium-226) eject a Helium-4 nucleus (&sup4;2He2+), reducing atomic number Z by 2 and mass number A by 4. Alpha particles have high linear energy transfer (LET) but are stopped by a sheet of paper.
- Beta Decay (β- / β+): In neutron-rich nuclei, a neutron transforms into a proton, emitting a high-speed electron (β-) and an electron antineutrino. In proton-rich nuclei, positron emission (β+) occurs. Beta radiation is shielded by aluminum sheets.
- Gamma Emission (γ): High-energy nuclear electromagnetic photons emitted during isomeric de-excitation of daughter nuclei. Gamma rays possess high penetration power, requiring dense lead or thick concrete shielding.
Radiation Dose Units and Health Physics
| Measurement Quantity | SI Unit | Traditional Unit | Physical Definition |
|---|---|---|---|
| Source Radioactivity | Becquerel (Bq) | Curie (Ci) | 1 nuclear disintegration event per second |
| Absorbed Dose | Gray (Gy) | Rad (rad) | 1 Joule of ionizing energy absorbed per kg of tissue |
| Equivalent / Effective Dose | Sievert (Sv) | Rem (rem) | Biological damage dose adjusted for radiation weighting factor (wR) |
Step-by-Step Worked Calculation Example
Example: Industrial Radiography Cobalt-60 Source Decay
Problem: An industrial non-destructive gamma radiography camera utilizes a sealed Cobalt-60 (&sup6;°Co) source with a half-life of 5.271 years. When originally manufactured, the source had an activity of 3.70 × 1011 Bq (10.0 Curies). Calculate the remaining activity of the Cobalt-60 source after exactly 10.542 years (2 half-lives) in both Becquerels and Curies.
Step 1: Determine decay constant λ:
λ = 0.693147 / 5.271 years = 0.131502 year-1
Step 2: Calculate remaining activity using exponential decay:
A(t) = A0 × e(-λ × t) = (3.70 × 1011 Bq) × e(-0.131502 × 10.542)
A(t) = 3.70 × 1011 × e-1.3863 = 3.70 × 1011 × 0.2500 = 9.25 × 1010 Bq
Step 3: Convert Becquerels to Curies:
Activity in Ci = (9.25 × 1010 Bq) / (3.70 × 1010 Bq/Ci) = 2.50 Curies
Conclusion: After exactly two half-lives (10.542 years), the source activity decays to exactly 25.0% of its initial potency: 92.5 GBq (2.50 Ci).
Nuclear Waste Storage and Radioprotection Standards
- ALARA Safety Principle: In nuclear engineering and medical radiology, all radiation exposures must be kept "As Low As Reasonably Achievable" through three primary defense strategies: minimizing exposure Time, maximizing Distance from the source (Inverse Square Law: I ∝ 1/r²), and implementing appropriate Shielding.
- Secular Equilibrium: When a long-lived parent isotope (such as Uranium-238) decays into a short-lived daughter (such as Thorium-234), the daughter activity rises until it precisely matches parent activity, establishing secular equilibrium.
Radioisotope Thermoelectric Generators (RTGs) in Deep Space Missions
Deep space exploration spacecraft (such as NASA's Voyager 1, Cassini, New Horizons, and the Curiosity Mars rover) operate beyond the range of efficient solar panels. These probes rely on Radioisotope Thermoelectric Generators powered by Plutonium-238 (²³&sup8;Pu), an alpha-emitting isotope with an 87.7-year half-life and high thermal power density (0.57 W/g). Solid-state silicon-germanium thermocouples convert radioactive decay heat directly into hundreds of watts of continuous electrical power, enabling missions to operate for over five continuous decades in interstellar space.
Smoke Detectors and Americium-241 Ionization Chambers
Residential ionization smoke detectors contain a tiny sealed capsule of Americium-241 (activity ≈ 33 kBq or 0.9 μCi). Emitted alpha particles continuously ionize ambient nitrogen and oxygen molecules within a sensing chamber, creating a minute baseline electric current (picoamperes). When airborne combustion smoke particles enter the chamber, they neutralize ions, reducing chamber electrical conductivity and triggering the audible alarm circuit.