Half-Life & Radioactive Decay Calculator

Half-Life & Radioactive Decay Guide

What Half-Life Means?

Half-life is the time for half of a radioactive substance to decay. After one half-life: 50% remains. After two half-lives: 25% remains. After three: 12.5%. After n half-lives: (1/2)^n remains. The decay is exponential, not linear. Formula: N(t) = N₀ × (1/2)^(t/t½). Where: N(t) = quantity at time t. N₀ = initial quantity. t½ = half-life. Alternative form: N(t) = N₀ × e^(-λt). Where λ (decay constant) = ln(2) / t½ = 0.693 / t½. The two forms are equivalent. Why exponential: each atom has a fixed, constant probability of decaying in any given time interval, regardless of how long it has already existed — unlike, say, a light bulb which becomes more likely to fail the older it gets.

What should I know about Carbon Dating?

Carbon-14 (radiocarbon) dating uses the known half-life of C-14 (5,730 years) to date organic materials. How it works: living organisms constantly exchange carbon with the atmosphere, maintaining a constant ratio of C-14 to stable C-12. When an organism dies, it stops taking in carbon. The C-14 decays (half-life 5,730 years). C-12 remains stable. By measuring the remaining C-14 ratio, you calculate how long ago the organism died. Calculation: age = t½ × log₂(N₀/N) = 5,730 × log₂(100/percent remaining), giving an age estimate directly from the ratio of C-14 measured in the sample compared to a living organism.

What should I know about Medical and Industrial Uses?

Medical isotopes (chosen for specific half-lives): Technetium-99m: 6 hours. Most-used medical imaging isotope. Short half-life means rapid clearance from body. Iodine-131: 8 days. Thyroid treatment and imaging. Fluorine-18: 110 minutes. PET scans. Made on-site (decays too fast to transport far). Cobalt-60: 5.27 years. Cancer radiotherapy, sterilisation. Why half-life matters medically: short half-life: less radiation dose to patient (decays quickly), but must be used soon after production. Longer half-lives give a more stable, longer-lasting source but deliver radiation to the patient or environment over a longer period, so the choice always balances practicality against dose.

What do I need to know about Decay Constant and Mean Lifetime?

Three related quantities: half-life (t½): time for half to decay. Most intuitive. Decay constant (λ): probability of decay per unit time. λ = ln(2)/t½ = 0.693/t½. Used in the exponential decay equation. Mean lifetime (τ): average lifetime of an atom. τ = 1/λ = t½/ln(2) = 1.443 × t½. Always longer than half-life. Relationships: τ = t½ / 0.693 = 1.443 × t½. λ = 1/τ = 0.693/t½. Activity (decay rate): A = λN. Measured in Becquerels (Bq = 1 decay/second) or Curies (Ci = 3.7×10¹⁰ Bq). Activity also decreases exponentially over time, following the same half-life as the number of remaining atoms, since activity is directly proportional to how many radioactive atoms are still present.

What do I need to know about Effective (Biological) Half-Life?

In nuclear medicine, a radioisotope inside the body disappears by two independent routes at once: physical radioactive decay and biological elimination (metabolism, excretion). The two combine as 1/T_effective = 1/T_physical + 1/T_biological, and because the two rates add as reciprocals, the effective half-life is always shorter than either one alone. This is why a radiotracer with a long physical half-life can still clear from a patient relatively quickly if the body eliminates it efficiently — a key safety consideration when choosing isotopes for diagnostic imaging.

What should I know about Common Half-Lives in Practice?

Half-lives span an enormous range depending on the isotope: Technetium-99m (medical imaging) decays with a half-life of about 6 hours, making it ideal for same-day scans with minimal lingering radiation dose. Iodine-131 (thyroid treatment) has a half-life of about 8 days. Carbon-14 (archaeological dating) has a half-life of 5,730 years, useful for dating organic material up to roughly 50,000 years old. Uranium-238 (geological dating) has a half-life of about 4.5 billion years — comparable to the age of the Earth itself — which is exactly why it's used to date the oldest rocks.

What should I know about Doubling Time: Half-Life's Mirror Image?

Half-life describes exponential decay, where a quantity repeatedly loses half its value over a fixed interval. The same mathematics applies in reverse to exponential growth through the concept of doubling time — the fixed interval over which a growing quantity repeatedly doubles. Both are governed by the same type of equation, N(t) = N₀ × 2^(±t/T), with a minus sign for decay (half-life T) and a plus sign for growth (doubling time T). Population growth, compound interest, and the early spread of an epidemic all follow doubling-time behaviour, while radioactive decay, drug elimination, and capacitor discharge follow half-life behaviour; mathematically they are the same curve reflected about the time axis. This connection also explains the 'rule of 70' used in finance: for a quantity growing at r% per period, the doubling time is approximately 70/r periods, derived from the same natural-logarithm relationship (T = ln2/k) that links half-life to the decay constant λ via T½ = ln2/λ. Recognising that decay and growth share one mathematical framework makes it easy to move between radioactive half-lives, financial doubling times, and biological population models using the same core formula.

Worked Example: Remaining Activity After Three Half-Lives

Starting with 800 mg of a radioactive isotope, the amount remaining after each half-life is found by halving the previous amount, since by definition exactly half decays away in one half-life period. After 1 half-life: 800 ÷ 2 = 400 mg. After 2 half-lives: 400 ÷ 2 = 200 mg. After 3 half-lives: 200 ÷ 2 = 100 mg. This matches the general formula N(t) = N₀ × (1/2)^(t/T½): with t equal to exactly 3 half-life periods, N = 800 × (1/2)³ = 800 × (1/8) = 100 mg. If the isotope's half-life is, say, 5 days, then 100 mg remains after 15 days (3 × 5), and the amount never reaches exactly zero — it keeps halving indefinitely, which is why radioactive decay is described as asymptotic rather than having a hard 'finish' date.

Half-Lives of Commonly Referenced Isotopes

The huge range in this table — from hours to billions of years — explains why half-life is chosen specifically for the job: a medical tracer needs a short half-life so it clears the body quickly, while dating ancient rock needs an isotope with a half-life on a geological timescale.

Half-Life Is Independent of How Much Material You Start With

A key, often counter-intuitive property of radioactive decay is that the half-life itself never changes regardless of the starting quantity: 10 grams takes the same half-life period to fall to 5 grams as 10 kilograms takes to fall to 5 kilograms. This is because decay is a probabilistic process acting on individual atoms (each atom has a fixed chance of decaying in a given time window), not a process that 'runs out of steam' as the quantity shrinks. It's also why half-life measurements are considered one of the most reliable constants in physics, unaffected by temperature, pressure, or chemical state, unlike most other physical and chemical rates.

Why Seven Half-Lives Is a Common Rule of Thumb for 'Safe' Levels

After 7 half-lives, roughly 1/128 (about 0.8%) of the original radioactive material remains, which is why many radiation safety guidelines use 7-10 half-lives as a rough benchmark for when activity has dropped to a level considered negligible for general handling, though the specific threshold always depends on the isotope's starting activity and the applicable safety regulations. After 10 half-lives, less than 0.1% remains, which is close enough to background levels for most practical purposes, even though mathematically the quantity never reaches exactly zero.

Half-Life in Nuclear Medicine Dosing Schedules

Nuclear medicine deliberately chooses radioactive tracers with short half-lives to minimise a patient's radiation exposure while still allowing enough time to complete an imaging scan. Technetium-99m, used in the majority of nuclear medicine scans worldwide, has a half-life of only about 6 hours — long enough to prepare, inject, and image a patient within a single clinic visit, but short enough that the radioactivity has largely decayed away within a day or two, well before most patients would otherwise need another scan. This short half-life creates a genuine logistics challenge: because Technetium-99m decays so quickly, it can't be stockpiled or shipped long distances economically, so hospitals typically generate it on-site from a longer-lived parent isotope (Molybdenum-99, with a roughly 66-hour half-life) delivered periodically from specialist production facilities. The choice of isotope half-life in medical imaging is therefore a careful balance between diagnostic usefulness, patient radiation safety, and the practical realities of producing and distributing a radioactive material that is, by design, disappearing the whole time it's being used.