Half-Life Calculator
Enter any three of initial quantity, remaining quantity, elapsed time, or half-life to solve for the fourth. Two 3D diagrams compare an early-decay and late-decay sample, and charts trace the exponential decay curve and compare common isotope half-lives.
Reviewed by the ToolNestr Editorial Team — July 2026
Enter any three values to solve for the fourth.
Early vs late decay
1. Early in the decay (many atoms remain)
A dense cluster of undecayed atoms (indigo) with only a few decayed atoms (grey) so far.
2. Several half-lives later (few atoms remain)
Most atoms have now decayed (grey) — only a small fraction of the original indigo atoms remain undecayed.
Half-life graphs
How it works
The core idea in one line: radioactive decay always removes the same fraction (one half) of whatever quantity currently remains during each half-life period — never a fixed absolute amount — which is exactly what makes the decay curve exponential rather than a straight line.
N = N₀ × (½)^(t / t½)
remaining quantity — N₀=initial, t=elapsed time, t½=half-life
N = N₀ × e^(−λt)
equivalent exponential form, λ = ln(2) / t½ (decay constant)
Each half-life period, by definition, halves whatever quantity remains: N = N₀ × (½)^(t/t½) captures this directly, where t/t½ counts how many half-life periods have elapsed (including fractional ones). The equivalent exponential form N = N₀ × e^(−λt) uses the decay constant λ = ln(2)/t½, which represents the same physics in the natural-log-based form more common in calculus-based derivations of decay kinetics.
Worked example 1 — Cobalt-60 decay over time
Given: A 10 g sample of Cobalt-60 (half-life 5.27 years) is stored for 15.81 years. How much remains?
After exactly 3 half-lives, only 12.5% of the original Cobalt-60 remains — this predictable decay pattern is why Cobalt-60 sources need periodic replacement in radiotherapy machines.
Worked example 2 — finding half-life from decay data
Given: A 50 g sample decays to 6.25 g after 12 years. Find the half-life.
Recognizing that 0.125 is exactly (½)³ makes this calculation straightforward — the general formula handles cases where the fraction isn't a clean power of ½.
Half-lives of common isotopes
Half-lives span an enormous range — from fractions of a second to billions of years — depending entirely on the isotope's nuclear stability.
| Isotope | Half-life | Common use |
|---|---|---|
| Technetium-99m | 6.01 hours | Medical imaging |
| Iodine-131 | 8.02 days | Thyroid treatment |
| Cobalt-60 ★ | 5.27 years | Radiotherapy, sterilization |
| Carbon-14 | 5,730 years | Archaeological dating |
| Uranium-238 | 4.47 × 10⁹ years | Geological dating |
★ Reference row (worked example 1). Each isotope's half-life dictates its practical use — short half-lives suit quick medical procedures, long ones suit dating ancient materials.
Where half-life actually matters
🕰️ Radiocarbon dating
Archaeologists estimate the age of once-living organic material by measuring how much Carbon-14 remains relative to stable Carbon-12, using its well-known 5,730-year half-life.
🏥 Nuclear medicine and radiotherapy
Medical physicists calculate how radioactive tracers and treatment sources decay in the body or in equipment, ensuring the correct dose reaches target tissue while minimizing exposure elsewhere.
☢️ Nuclear waste storage planning
Nuclear engineers calculate radioactive decay for spent fuel storage, determining how long waste must be safely contained before its radioactivity drops to acceptable levels.
🧬 Environmental contamination tracking
Environmental scientists use half-life data to predict how long radioactively contaminated areas will remain hazardous and to plan remediation timelines.
Common misconceptions
"After 2 half-lives, all of the substance has decayed."
After 2 half-lives, 25% remains, not 0% — decay is exponential, so the quantity approaches (but mathematically never quite reaches) zero, no matter how many half-lives pass.
"You can find remaining quantity by simple linear subtraction (e.g. half the original amount lost every year for a 1-year half-life)."
Decay is exponential, not linear — each half-life removes half of whatever remains at that point, not a fixed absolute amount. After 7 half-lives, less than 1% of the original material remains.
"A radioactive material's half-life depends on its temperature or chemical form."
Half-life is a fixed nuclear property of each isotope, completely unaffected by temperature, pressure, or the chemical compound the atom is part of — only the nucleus itself determines the decay rate.
"Half-life measures how long until a substance becomes completely safe."
Half-life only measures the time for half the material to decay — a substance can remain measurably radioactive for many half-lives beyond the first, so "safe" thresholds depend on the specific isotope and application, not a single half-life.
Formula sources & further reading
The formulas here are standard, traceable to:
- • OpenStax, Chemistry 2e — Chapter 21, "Nuclear Chemistry" (free, peer-reviewed). openstax.org
- • Brown, LeMay & Bursten, Chemistry: The Central Science — Chapter 21, Nuclear Chemistry.
- • Zumdahl & Zumdahl, Chemistry — Radioactive decay kinetics.
N = N₀×(½)^(t/t½), equivalently N = N₀×e^(−λt) with λ=ln(2)/t½. Reference half-lives are standard nuclear-data values. Results are rounded for display.
How to use this calculator
Enter any three values
Provide initial quantity, remaining quantity, elapsed time, or half-life — leave the unknown blank.
Read the fourth value
The missing value solves instantly, using consistent units throughout.
Reference common isotopes
Compare your result against known half-lives for medical, dating, or industrial isotopes.
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Frequently asked questions
What is half-life?
Half-life (t½) is the time required for a radioactive substance to decay to half of its original amount. After one half-life, 50% remains; after two, 25%; after three, 12.5%, and so on.
Is half-life always constant?
Yes — half-life is a fixed property of each radioactive isotope. It does not change with temperature, pressure, or chemical state. Carbon-14 has a half-life of 5,730 years regardless of its environment.
How is half-life used in carbon dating?
Carbon-14 has a half-life of 5,730 years. By measuring the remaining C-14 in an organic sample relative to the initial amount, scientists can determine roughly how long ago the organism died.
What is the decay constant?
The decay constant λ is related to half-life by λ = ln(2) / t½. It represents the probability of decay per unit time and appears in the exponential decay formula N = N₀ × e^(−λt).
Does half-life apply to non-radioactive processes?
Yes — the half-life concept applies to any process following exponential decay, including drug metabolism in the body, capacitor discharge, and the removal of pollutants from the environment.