Nuclear Decay & Half-Life Calculator

Radioactive Decay Guide

What should I know about Radioactive Decay Law?

N = N₀ × e^(−λt) = N₀ × (1/2)^(t/t½). Where N = quantity remaining, N₀ = initial quantity, λ = decay constant, t = time elapsed, t½ = half-life. The decay constant λ = ln(2)/t½ = 0.693/t½. After one half-life: 50% remains. After two: 25%. After three: 12.5%. After ten half-lives: 0.098% remains (essentially complete decay for practical purposes). The relationship between half-life and decay constant: t½ = 0.693/λ.

What do I need to know about Types of Radioactive Decay?

Alpha decay (α): emits ⁴₂He nucleus. Mass number decreases by 4, atomic number by 2. Penetrating power: low (stopped by paper or skin). Example: ²³⁸U → ²³⁴Th + ⁴He. Beta minus decay (β⁻): neutron → proton + electron + antineutrino. Mass number unchanged, atomic number increases by 1. Penetrating power: medium (stopped by aluminium). Example: ¹⁴C → ¹⁴N + e⁻. Gamma decay (γ): electromagnetic radiation released when nucleus transitions to lower energy state. No change in mass or atomic number. Penetrating power is highest for gamma radiation, which requires thick lead or concrete shielding, compared to alpha (stopped by paper) or beta (stopped by a few millimetres of aluminium).

What's the key thing to understand about Carbon-14 Dating?

Carbon-14 (t½ = 5,730 years) forms in the atmosphere from cosmic ray interactions with nitrogen. Living organisms maintain a constant ¹⁴C/¹²C ratio by exchanging carbon with the atmosphere. At death, ¹⁴C decays without replacement. Measuring the remaining fraction: t = −t½/ln(2) × ln(N/N₀). This method accurately dates organic material up to approximately 50,000 years (beyond that, too little ¹⁴C remains to measure accurately). Important assumption: constant atmospheric C-14 levels historically — an assumption that isn't perfectly true, which is why calibration curves derived from tree rings are used to correct raw radiocarbon dates.

What should I know about Medical and Industrial Applications?

Medical imaging: technetium-99m (t½ = 6 hours) is used in 80% of nuclear medicine procedures — short half-life minimises patient radiation dose while providing clear images. Cancer treatment: iodine-131 (t½ = 8 days) targets thyroid tissue. Cobalt-60 (t½ = 5.27 years) in external beam radiotherapy. Smoke detectors: americium-241 (t½ = 432 years) provides a stable alpha source. Food irradiation: gamma rays from cobalt-60 kill pathogens without radioactive contamination of the food. Nuclear waste: some fission products have very long half-lives, which is the central challenge in safely storing spent nuclear fuel for the thousands of years needed for radioactivity to decline to safe levels.

What should I know about Alpha, Beta, and Gamma Decay Differences?

Alpha decay emits a helium nucleus (2 protons, 2 neutrons), reducing the parent's mass number by 4 and atomic number by 2 — heavy, slow-moving, stopped by paper or a few cm of air, but highly ionising if inhaled or ingested. Beta decay emits an electron (or positron) as a neutron converts to a proton (or vice versa), changing atomic number by ±1 with no change in mass number — faster and more penetrating than alpha, stopped by a few mm of aluminium. Gamma decay emits high-energy photons with no change to mass number or atomic number at all (it's the nucleus releasing excess energy, often following an alpha or beta decay that leaves it in an excited state) — the most penetrating type, requiring thick lead or concrete shielding. Each decay type is governed by different conservation laws, but all three share the same exponential decay timing based on half-life.

What do I need to know about Writing Balanced Nuclear Equations?

A nuclear decay equation must balance both mass number (superscript, total nucleons) and atomic number (subscript, total protons) on both sides. For alpha decay of uranium-238: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He. Check: mass numbers 238=234+4 ✓; atomic numbers 92=90+2 ✓. For beta-minus decay of carbon-14: ¹⁴₆C → ¹⁴₇N + ⁰₋₁e (plus an antineutrino, often omitted in simplified equations). Check: mass numbers 14=14+0 ✓; atomic numbers 6=7+(−1) ✓. This balancing process is exactly like balancing a chemical equation but for nucleons and charge rather than atoms and electrons, and it's the standard method to identify what the daughter product must be from a given decay type, or to identify the decay type from a known parent and daughter.

What do I need to know about Radiometric Dating Beyond Carbon-14?

Carbon-14's 5,730-year half-life makes it useless beyond roughly 50,000 years, since too little remains to measure reliably. For older geological samples, scientists use isotopes with much longer half-lives instead: uranium-lead dating (uranium-238's 4.5-billion-year half-life) dates the oldest rocks and minerals, while potassium-argon dating (potassium-40's 1.25-billion-year half-life) is widely used for volcanic rock layers bracketing fossil sites. Choosing the right isotope for a given dating method always comes down to matching the half-life to the age range being investigated — too short and the signal has vanished, too long and not enough decay has happened to measure precisely.

What Are the Half-Lives of Commonly Referenced Radioisotopes?

Half-lives span an enormous range — from fractions of a second to billions of years — and the practical use of a given isotope (medical imaging, dating, power generation) is directly tied to where its half-life falls on that scale.

Worked example: a 10mg sample of Iodine-131 (half-life 8.02 days) decays to 5mg after one half-life (8.02 days), 2.5mg after two half-lives (16.04 days), and 1.25mg after three half-lives (24.06 days) — roughly 12.5% of the original amount remaining after about 24 days. This rapid decay is precisely why Iodine-131 is clinically useful for short-duration thyroid treatments: it delivers its therapeutic dose and then drops to a small fraction of its original activity within about a month, unlike a long-half-life isotope such as Cesium-137, which would remain hazardous for decades under the same starting quantity.

How Does Radioactive Decay Differ From a Chemical Reaction Going to Completion?

Radioactive decay never reaches zero in a finite, predictable time the way a chemical reaction running to completion does — it only ever halves the remaining quantity over each successive half-life, meaning a mathematically small but technically non-zero amount of the original isotope remains no matter how many half-lives pass. This is why half-life, rather than a 'total decay time,' is the standard way to describe how fast an isotope decays: after 10 half-lives, less than 0.1% of the original material remains, which is practically negligible for most purposes, but the decay process itself has no fixed endpoint the way a completed chemical reaction does. This distinction also explains why nuclear waste disposal planning uses multiples of a material's half-life (often 10 or more half-lives) to define a 'safe enough' storage timeframe, rather than waiting for the material to decay away entirely, since that point never technically arrives.

Worked Example: Activity Remaining After Multiple Half-Lives

A radioactive sample starts with an activity of 800 becquerels and has a half-life of 5 days. After 20 days have elapsed, how many half-lives is that? 20 / 5 = 4 half-lives. Using A = A_0 × (1/2)^n, the remaining activity is 800 × (1/2)^4 = 800 × 1/16 = 50 Bq. To find how long it takes to drop to a specific target, such as 100 Bq, solve 800 × (1/2)^n = 100, which gives (1/2)^n = 0.125 = 1/8, so n = 3 half-lives, meaning 3 × 5 = 15 days. This two-direction skill -- going from elapsed time to remaining activity, and from a target activity back to elapsed time -- is exactly what the nuclear decay calculator's input and output modes are built to handle, and it is the same arithmetic used to schedule medical isotope deliveries so they arrive with the needed activity still intact.

Decay Products and Decay Chains

A single radioactive decay rarely ends the story: the daughter nuclide produced by one decay is often itself radioactive, leading to a decay chain that continues through several more steps before reaching a stable isotope. Uranium-238, for instance, does not decay directly to lead; it passes through a chain of about 14 intermediate radioactive isotopes -- including radium-226 and radon-222 -- before finally reaching stable lead-206, a process that in total takes billions of years because the first step (U-238 to thorium-234) has by far the longest half-life in the chain and acts as the bottleneck. Radon-222, a step in this chain, is a gas and can seep out of rock and soil into buildings, which is why radon testing in basements is a standard home-safety recommendation: it is not the uranium itself but one of its unstable decay-chain descendants that poses the practical radiological risk to occupants.