Radioactive Decay &
Half-Life Calculator
Solve for any variable in the radioactive decay equation — remaining amount, initial mass, elapsed time, or half-life — with step-by-step working.
Calculating…
Choose a solve mode, enter your known values, and press Calculate to see the result with step-by-step working.
What Is Radioactive Decay?
Radioactive decay is the spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. Unlike chemical reactions, decay is entirely random at the individual atom level — but for large populations of atoms the rate follows a precise exponential law governed by a single constant: the half-life.
The half-life (t½) is the time it takes for exactly half of a radioactive sample to decay. After one half-life, 50% of the original material remains. After two half-lives, 25% remains. After ten half-lives, less than 0.1% survives. Half-lives range from fractions of a second for highly unstable isotopes to billions of years for the most stable ones.
This decay law applies to any quantity proportional to the number of radioactive nuclei: mass, activity (disintegrations per second), number of atoms, or radioactive concentration. The calculator above handles all four common solve directions: finding the remaining amount, the original amount, the elapsed time, or the half-life itself.
The Half-Life Formula Explained
The fundamental radioactive decay law expresses how the quantity N of a substance changes over time t:
| Symbol | Meaning | Solve Direction |
|---|---|---|
| N(t) | Remaining quantity after time t | Mode 1 — Find N(t) |
| N₀ | Initial quantity at t = 0 | Mode 2 — Find N₀ |
| t½ | Half-life — time for 50% decay | Mode 4 — Find t½ |
| t | Elapsed time since t = 0 | Mode 3 — Find t |
| λ | Decay constant (λ = ln 2 / t½) | Derived — shown in results |
| k | Number of half-lives elapsed (k = t / t½) | Derived — shown in results |
How to Calculate Remaining Amount After Decay
The most common radioactive decay problem is finding how much material remains after a given time. The 3-4-5 step process is the same every time:
Exponential Decay Curve — Amount vs. Time
Each half-life cuts the remaining amount in half. The curve never reaches zero.
Worked example — Carbon-14 (t½ = 5,730 years), starting with 100 g, after 11,460 years:
- k = t / t½ = 11,460 / 5,730 = 2.000 half-lives
- Decay factor = (½)^k = (0.5)² = 0.25
- N(t) = 100 × 0.25 = 25 g remaining
Common Radioactive Isotopes and Their Half-Lives
Half-lives span an enormous range — from minutes for medical radiotracers to billions of years for geological isotopes. Use this reference table to verify your inputs or explore how different isotopes behave.
| Isotope | Symbol | Half-Life |
|---|---|---|
| Carbon-14 | ¹⁴C | 5,730 years |
| Uranium-238 | ²³⁸U | 4.47 × 10⁹ years |
| Iodine-131 | ¹³¹I | 8.02 days |
| Caesium-137 | ¹³⁷Cs | 30.17 years |
| Tritium (H-3) | ³H | 12.32 years |
| Radon-222 | ²²²Rn | 3.82 days |
| Technetium-99m | ⁹⁹ᵐTc | 6.01 hours |
| Polonium-210 | ²¹⁰Po | 138.4 days |
| Fluorine-18 | ¹⁸F | 109.8 minutes |
Applications of Half-Life Calculations
Radiometric Dating
Carbon-14 dating determines the age of organic material up to ~50,000 years old. Geologists use uranium-238 (t½ = 4.47 × 10⁹ yr) for dating ancient rocks and meteorites.
Nuclear Medicine
Technetium-99m (t½ = 6 h) is the world's most used medical radioisotope for SPECT imaging. Iodine-131 (t½ = 8 days) treats thyroid cancer. Short half-lives minimise patient radiation exposure.
Home Safety
Radon-222 (t½ = 3.82 days) seeps from soil into buildings. Half-life calculations help health physicists assess exposure risk and decay rates during remediation work.
Nuclear Power & Waste
Nuclear engineers calculate when spent fuel is safe to handle based on its decay. Caesium-137 (t½ = 30 yr) drives long-term waste storage decisions for power plant decommissioning.
Half-Life Calculator FAQ
What is the half-life formula?
The standard half-life formula is N(t) = N₀ × (½)^(t / t½), where N₀ is the initial amount, N(t) is the amount after time t, and t½ is the half-life. An equivalent form using the natural exponential is N(t) = N₀ × e^(−λt), where the decay constant λ = ln(2) / t½.
How do I calculate the remaining amount after radioactive decay?
To find remaining amount: (1) divide elapsed time by the half-life to get k (number of half-lives); (2) raise 0.5 to the power k; (3) multiply the initial amount by this factor. Example: 100 g with t½ = 10 years after 30 years → k = 3, factor = 0.125, N(t) = 12.5 g.
How long does it take for a radioactive substance to fully decay?
Theoretically, never — exponential decay approaches zero asymptotically and never exactly reaches it. In practice, after about 10 half-lives roughly 99.9% has decayed and the remaining amount is negligible. Medical protocols typically consider an isotope "gone" after 5–7 half-lives when activity drops below detectable thresholds.
What is the decay constant λ and how is it related to half-life?
The decay constant λ (lambda) is the probability per unit time that any given nucleus will decay. It is related to the half-life by λ = ln(2) / t½ ≈ 0.6931 / t½. A shorter half-life means a larger λ — the substance decays faster. The mean lifetime (average time an atom survives) is τ = 1/λ = t½ / ln(2).
Can I use this calculator for carbon-14 dating?
Yes. Select "Find Time" mode, enter 100 as N₀ (representing 100% original ¹⁴C), enter the measured percentage as N(t), set the half-life to 5730, and select years as the time unit. The result is the estimated age of the sample. Real-world radiocarbon dating involves calibration curves for atmospheric ¹⁴C variation, but the decay calculation itself is what this tool provides.
Why can't the remaining amount exceed the initial amount?
Radioactive decay only removes nuclei — it cannot create new ones of the same isotope. If N(t) > N₀, either the measurement is wrong, the substance is being replenished by another source (a decay chain), or the two values refer to different points in time in the wrong order. The calculator rejects this case to prevent physically meaningless results.
What is the difference between half-life and mean lifetime?
The half-life is the time after which exactly 50% of atoms have decayed (the median survival time). The mean lifetime τ is the average time an atom survives before decaying, and equals t½ / ln(2) ≈ 1.4427 × t½. The mean lifetime is always longer than the half-life because atoms that survive longer pull up the average.
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