Half-Life & Radioactive Decay — The Complete Physics Guide
Radioactive decay is a random process at the level of any individual nucleus, yet with a huge number of atoms it produces a remarkably precise, predictable pattern: the amount of a radioactive substance decreases exponentially over time, halving after a fixed interval called the half-life. This single number — unique to each isotope — lets scientists date ancient artefacts, calculate safe medical dosages, and predict how long nuclear waste remains hazardous.
Half-lives span an extraordinary range: from microseconds for the most unstable isotopes to billions of years for uranium-238. Despite this range, every radioactive decay follows the same mathematical form, making it one of the cleanest and most reliable relationships in all of physics.
What is Half-Life?
The half-life (t½) of a radioactive isotope is the time it takes for half of any sample of that isotope to decay. It does not matter how much you start with — after one half-life, exactly half remains; after two half-lives, a quarter remains; after three, an eighth, and so on. This is a direct consequence of decay being a random process with a constant probability per unit time for each nucleus, independent of the nucleus's age or history.
Crucially, half-life is a statistical property of a large population of nuclei, not a prediction for any single atom. You cannot say when one specific atom will decay — only the probability that it will decay within a given time. With a large enough sample (and even a milligram of material contains billions of billions of atoms), the statistical prediction becomes extremely accurate.
The Formula Explained
N₀ is the initial quantity present at t = 0 — this can be a number of atoms, a mass, or an activity (decays per second); the formula works with any consistent unit. N is the quantity remaining after time t. t½ is the half-life — the characteristic time constant for that specific isotope, found experimentally and tabulated for every known radioactive nuclide. The exponential form uses the decay constant λ (lambda), related to half-life by λ = ln(2)/t½ ≈ 0.693/t½ — λ represents the probability per unit time that any given nucleus decays.
Both forms of the equation are mathematically equivalent; the (½)^(t/t½) form is more intuitive for counting half-lives, while the e^(−λt) form is more convenient for calculus-based work (differentiating to find decay rate, or integrating to find total decays over an interval).
How to Use This Calculator
Select the mode that matches what you know. Use "N₀, t½ & t" when you know the starting amount, the isotope's half-life, and how much time has passed — this finds how much remains. Use "N₀, N & t½" when you've measured a remaining amount and know the half-life, and want to find how long decay has been occurring — this is the basis of radiometric dating. Use "N₀, N & t" when you've measured a remaining amount after a known time and want to determine the half-life itself — this is how half-lives are experimentally measured. Enter time in seconds (the calculator shows helpful conversions), and any consistent unit for N₀ and N — atoms, grams, or activity in becquerels all work identically since the formula is a ratio.
Worked Example 1 — Radiocarbon Dating
Problem: A wooden artefact contains 25% of the carbon-14 found in living wood. Carbon-14 has a half-life of 5,730 years. How old is the artefact?
25% = (½)^n → n = log(0.25)/log(0.5) = 2 half-lives
Age = 2 × 5,730 = 11,460 years
Worked Example 2 — Medical Isotope Dosing
Problem: A hospital receives a 200 MBq sample of technetium-99m (half-life 6.01 hours). What activity remains after 24 hours, and is it still usable for a scan requiring at least 10 MBq?
Half-lives elapsed = 24 / 6.01 = 3.99
N = 200 × (½)^3.99 = 12.6 MBq — still above the 10 MBq threshold, but only just; this is why Tc-99m must be used within hours of production.
Worked Example 3 — Determining an Unknown Half-Life
Problem: A 40 g sample of an unknown isotope decays to 5 g after 90 minutes. Find its half-life.
40 → 20 → 10 → 5 g is exactly 3 half-lives
t½ = 90 / 3 = 30 minutes
Worked Example 4 — Nuclear Waste Safety Timescales
Problem: Plutonium-239 has a half-life of 24,100 years. How long until only 0.1% of an initial sample remains?
0.001 = (½)^n → n = ln(0.001)/ln(0.5) = 9.97 half-lives
t = 9.97 × 24,100 ≈ 240,300 years — illustrating why plutonium-based nuclear waste requires such extraordinarily long-term storage solutions.
Common Mistakes
Treating decay as linear: a common misconception is that if half decays in one half-life, all of it decays in two half-lives. In reality, only three-quarters has decayed after two half-lives, and the sample never mathematically reaches exactly zero — it approaches zero asymptotically.
Mixing time units: t and t½ must be expressed in the same units before substituting — mixing years and seconds is the most common calculation error.
Confusing half-life with mean lifetime: the average (mean) lifetime of a nucleus, τ = 1/λ = t½/ln(2) ≈ 1.44 × t½, is a different (larger) quantity than the half-life, though the two are often conflated.
Forgetting decay is probabilistic, not deterministic: the formula predicts the expected amount remaining for a large sample — it does not predict when any individual atom decays, and for very small sample sizes the actual result can deviate noticeably from the formula due to statistical fluctuation.
Real-World Applications
Radiocarbon dating: archaeologists and geologists use carbon-14's 5,730-year half-life to date organic material up to roughly 50,000 years old, revolutionising our understanding of human history.
Nuclear medicine: diagnostic isotopes like technetium-99m (6 hours) and therapeutic isotopes like iodine-131 (8 days) are chosen specifically for half-lives short enough to minimise patient radiation exposure while long enough to complete the procedure.
Geological dating: uranium-lead dating (half-life 4.5 billion years) allows geologists to date the oldest rocks on Earth and even meteorites, establishing the age of the solar system itself at roughly 4.6 billion years.
Nuclear power and waste management: reactor operators track the decay of fission products to manage spent fuel storage, while regulators use half-life data to set safe long-term disposal timescales for radioactive waste.
Activity and the Becquerel
In practice, radioactive samples are rarely measured by counting individual atoms — instead, their activity (the rate of decay, in becquerels, Bq, where 1 Bq = 1 decay per second) is measured directly with a detector. Activity A relates to the number of atoms N by A = λN, and since N follows the same exponential decay law, activity also decays exponentially with the same half-life: A = A₀(½)^(t/t½). This is why this calculator works equally well whether you enter N₀ as a mass, an atom count, or an initial activity in becquerels.
The historical unit of activity, the curie (Ci), is still used in some contexts, particularly in the United States: 1 Ci = 3.7 × 10¹⁰ Bq, originally defined as the activity of exactly one gram of radium-226.
Types of Radioactive Decay
Not all radioactive decay is the same, though every type follows the same half-life mathematics. In alpha decay, a nucleus emits a helium nucleus (2 protons + 2 neutrons), reducing its mass number by 4 and atomic number by 2 — common in heavy elements like uranium and plutonium. In beta-minus decay, a neutron converts to a proton, emitting an electron and an antineutrino, increasing the atomic number by 1 while leaving the mass number unchanged. In beta-plus decay (positron emission), a proton converts to a neutron, decreasing the atomic number by 1 — this is how many medical imaging isotopes work, since the emitted positron annihilates with an electron to produce detectable gamma rays.
Gamma decay involves no change in the composition of the nucleus at all — it's simply the emission of high-energy photons as an excited nucleus (often one that just underwent alpha or beta decay) relaxes to a lower energy state, analogous to an electron emitting light as it drops to a lower atomic energy level. Each decay mode has its own characteristic half-life, and a single isotope may decay via more than one pathway with different probabilities (branching ratios) — but the overall exponential decay law N = N₀(½)^(t/t½) still describes the total remaining quantity regardless of which specific decay mode is involved.
Decay Chains and Secular Equilibrium
Many radioactive isotopes don't decay directly into a stable element — instead, they decay into another radioactive isotope, which decays into another, forming a decay chain that may involve a dozen or more steps before reaching stability. Uranium-238, for example, decays through a chain of 14 intermediate isotopes (including radium-226 and radon-222) before finally reaching stable lead-206, a process that takes billions of years overall despite some intermediate steps having half-lives of mere seconds.
When a long-lived parent isotope (like uranium-238) feeds a decay chain, the short-lived daughter isotopes reach a state called secular equilibrium, where each daughter's activity equals the parent's — effectively, the daughter decays exactly as fast as it's produced. This is why naturally occurring uranium ore always contains small but predictable amounts of radium and radon, continuously replenished by the slow decay of uranium, even though radium itself has a "short" half-life of only 1,600 years on a geological timescale.
Why Half-Life Determines Radiological Hazard
Counter-intuitively, isotopes with very short half-lives and very long half-lives are often less dangerous than isotopes with intermediate half-lives, measured in days to decades. An isotope with a half-life of seconds decays almost entirely before it can cause significant biological harm to be delivered slowly. An isotope with a half-life of billions of years, like uranium-238, has such a low decay rate (low activity per gram) that it emits relatively few particles per second. The isotopes of greatest concern — such as caesium-137 (30 years) and strontium-90 (29 years), both major components of nuclear fallout — sit in the "dangerous middle": long enough to persist in the environment and accumulate in the body for years, yet short enough to have high activity and deliver a significant radiation dose during that time.
This is why half-life data is central to nuclear safety planning, from setting evacuation zones after a reactor incident to determining how long contaminated land must be monitored. The general rule of thumb used in radiation protection is that after roughly ten half-lives, an isotope's activity has fallen to about 0.1% of its original value — often considered close enough to background levels to be considered safe, though the precise threshold depends on the isotope's initial activity and its biological effects.