Half-Life Calculator

The Decay Equation

Radioactive decay is a random quantum process, but for large numbers of atoms the aggregate behaviour is precisely described by exponential decay. The two equivalent forms are:

Activity A (decays per second) equals λ × N(t). Activity is measured in becquerels (Bq, SI unit: 1 decay per second) or the older curie (Ci: 3.7 × 10¹° Bq, defined as the activity of 1 gram of radium-226).

Notable Half-Lives

Half-lives span an enormous range. Carbon-14 has a half-life of 5,730 years — long enough to date organic material up to about 50,000 years old, but short enough that measurable decay occurs on archaeological timescales. Iodine-131 decays in 8 days, making it useful as a short-lived medical isotope for thyroid treatment — it delivers a dose and then clears the body quickly. Caesium-137, a product of nuclear fission, has a 30-year half-life, which is why areas affected by the Chernobyl accident remain contaminated decades later. Uranium-235 has a half-life of 700 million years, and Uranium-238 approximately 4.5 billion years — comparable to the age of the Earth, which is why significant amounts still exist naturally.

Carbon Dating

Radiocarbon dating exploits the known atmospheric ratio of C-14 to C-12. Living organisms continuously exchange carbon with the atmosphere, maintaining this ratio. After death, C-14 decays without replacement. Measuring the remaining C-14 fraction and applying the decay equation gives the time since death. The technique is accurate to within a few decades for samples up to about 50,000 years old, beyond which the residual C-14 signal becomes too small to measure reliably.

Medical Isotope Applications

Nuclear medicine selects isotopes whose half-lives match their clinical purpose. Technetium-99m (half-life 6 hours) is the most widely used diagnostic isotope — it emits gamma rays detectable by a gamma camera and decays quickly enough to minimise patient radiation dose. Fluorine-18 (110 minutes) is used in PET scans. Longer-lived isotopes like Strontium-90 (29 years) are used in cancer radiotherapy devices where a sustained emission over years is needed.

Worked Examples

Example 1 — Carbon-14 dating a charcoal sample. An archaeological sample measures 25% of the C-14 activity of modern material. Using N/N₀ = (1/2)^(t/t½): 0.25 = 0.5^n, so n = 2 half-lives. t = 2 × 5 730 = 11 460 years old. Three half-lives (12.5%) would give 17 190 years; a doubling of age corresponds to squaring the remaining fraction.

Example 2 — Medical isotope decay to background. A hospital receives 10 GBq of Tc-99m at 8 AM for a gamma imaging procedure. Half-life 6.01 hours. By 8 PM (12 hours = 2 half-lives), activity = 10 × 0.25 = 2.5 GBq. This is why hospitals order Tc-99m fresh every morning from the on-site generator.

Example 3 — Waiting for radon to clear. After basement remediation, indoor radon-222 drops from 20 pCi/L to target 2 pCi/L. Rn-222 half-life is 3.82 days. Need factor of 10 reduction: ln(10)/ln(2) = 3.32 half-lives = 12.7 days (assuming no fresh radon ingress, which is rarely the case — ventilation remains the real fix).

Example 4 — Cesium-137 fallout persistence. Cs-137 half-life is 30.17 years. Starting from 1986 Chernobyl release, by 2026 (40 years), remaining fraction = 0.5^(40/30.17) = 39.9%. By 2086 (100 years from release), only 10% remains. Real-world decline is slightly faster due to weathering and root uptake.

Common Pitfalls

• Confusing half-life with mean lifetime. Mean life τ = t½ / ln(2) = 1.443 × t½. The mean is longer because a few atoms survive for many half-lives.
• Applying decay to small numbers. Exponential decay is a statistical average. For 10 atoms, random fluctuation dominates and the smooth curve is meaningless.
• Ignoring branching decay. Some isotopes decay into multiple products. K-40 becomes either Ca-40 (89%) or Ar-40 (11%). Total half-life is still 1.25 Gyr but each branch has its own partial decay constant.
• Using half-life for chemical reactions without checking order. True half-life applies cleanly only to first-order kinetics. Second-order reactions have concentration-dependent "half-life".
• Treating decay chains as simple. U-238 → ... → Pb-206 runs through 14 intermediates. Short-lived daughters reach secular equilibrium with the parent; the chain's effective behaviour is set by the longest-lived member.

Frequently Asked Questions

Does half-life depend on temperature or pressure? For nuclear decay, essentially no — the nucleus is shielded from chemical environment. Tiny shifts (<1%) exist for electron-capture decays under extreme pressure. For chemical kinetics, temperature matters enormously (Arrhenius law).

Why is the decay constant λ = ln(2) / t½? Setting N(t½) = N₀/2 in N(t) = N₀e⁻λᵗ: 1/2 = e⁻λᵗ₁₂, take ln: −ln(2) = −λt½, so λ = ln(2)/t½.

Can an isotope's half-life be changed? Barely. Be-7 in certain chemical environments shifts by ~0.1%. Free-neutron decay changes when the neutron is inside a nucleus. For practical purposes, half-lives are constants of nature.

How is half-life measured for very long-lived isotopes? Instead of watching atoms decay, count the daughter product. For U-238 (4.5 Gyr), measure the U/Pb ratio in a known-age zircon. Or count individual decays over years in a high-purity sample (one decay per second per 120 tonnes of U-238).

What does "half-life" mean for a drug? Biological half-life is the time for plasma concentration to drop by half — usually dominated by metabolism and excretion, not chemical decay. Drugs and radiopharmaceuticals both obey the same first-order math.

Related Calculators

For other exponential-growth and -decay models, try the Loan Calculator (compound interest). For the related chemistry, see the Ideal Gas Law Calculator and Molar Mass Calculator. For power-level math in decibels (logarithmic), the Decibels Calculator uses similar thinking. Browse the Science category for more.

Disclaimer

This calculator is provided for educational and informational purposes only. While we strive for accuracy, users should verify all calculations independently. We are not responsible for any errors, omissions, or damages arising from the use of this calculator.