Physics · Particle & Nuclear Physics · Nuclear Physics
Half-Life Calculator
Calculates the remaining quantity of a radioactive substance after a given time using its half-life, or determines the half-life from decay data.
Calculator
Formula
N(t) is the remaining quantity at time t; N₀ is the initial quantity (mass, atoms, or activity); t is the elapsed time; t₁/₂ is the half-life — the time required for exactly half of the substance to decay. The decay constant λ is related by λ = ln(2) / t₁/₂.
Source: Rutherford, E. & Soddy, F. (1902). The Cause and Nature of Radioactivity. Philosophical Magazine. Also: Krane, K.S. (1988). Introductory Nuclear Physics, Wiley.
How it works
Radioactive decay is a spontaneous, probabilistic process in which unstable atomic nuclei lose energy by emitting radiation. The half-life (t₁/₂) is the time required for exactly half of a given quantity of a radioactive isotope to decay into its daughter product. It is a constant for any given nuclide — independent of temperature, pressure, chemical state, or the initial amount present. This remarkable property makes it the cornerstone of nuclear physics and geochronology.
The governing equation is N(t) = N₀ × (1/2)^(t/t₁/₂), where N₀ is the initial quantity, t is the elapsed time, and t₁/₂ is the half-life. This is mathematically equivalent to N(t) = N₀ × e^(−λt), where the decay constant λ = ln(2) / t₁/₂ (approximately 0.693 / t₁/₂). The decay constant represents the probability per unit time that any individual nucleus will decay. After one half-life, 50% remains; after two half-lives, 25%; after ten half-lives, less than 0.1% of the original substance persists.
Applications span an enormous range of scales and fields. In radiocarbon dating (¹⁴C), the 5,730-year half-life allows archaeologists to date organic material up to roughly 50,000 years old. In nuclear medicine, short-lived isotopes like Technetium-99m (t₁/₂ = 6 hours) are used for diagnostic imaging precisely because they decay quickly, minimizing patient radiation dose. In nuclear power and waste management, understanding the half-lives of fission products — ranging from fractions of a second to thousands of years — is critical for safety planning. In pharmacology, a closely related concept governs the biological half-life of drugs in the body.
Worked example
Example: Radiocarbon Dating of an Archaeological Sample
Suppose a piece of ancient charcoal contains 25 grams of Carbon-14, whereas a modern sample of the same mass would contain 100 grams. The half-life of ¹⁴C is 5,730 years. How old is the sample?
Step 1 — Identify the fraction remaining: N(t)/N₀ = 25/100 = 0.25. This is exactly (1/2)², meaning two half-lives have elapsed.
Step 2 — Calculate elapsed time: t = 2 × 5,730 = 11,460 years.
Step 3 — Verify using the formula: N(t) = 100 × (0.5)^(11460/5730) = 100 × (0.5)² = 100 × 0.25 = 25 g ✓
Step 4 — Compute the decay constant: λ = ln(2) / 5730 years = 0.6931 / 5730 ≈ 1.21 × 10⁻⁴ yr⁻¹.
Step 5 — Percent remaining: 25/100 × 100% = 25% of the original ¹⁴C activity remains, confirming an age of approximately 11,460 years before present.
This same logic applies whether you are tracking the activity of a medical radioisotope in a patient, the power output of a radioisotope thermoelectric generator (RTG) on a spacecraft, or the safe storage timeline for nuclear waste.
Limitations & notes
The half-life formula assumes pure exponential decay with a single decay mode and a single nuclide. In reality, many isotopes have branching decay paths (e.g., both alpha and beta decay), and the formula must be applied to each branch separately. Daughter products that are themselves radioactive create decay chains (secular equilibrium), which require more complex Bateman equations to model accurately. The formula also assumes the half-life is constant — which holds for nuclear decay but not always for biological or environmental half-lives, which can depend on metabolic rate, temperature, and environmental conditions. For very small numbers of atoms (fewer than ~10⁴), the continuous exponential model breaks down and must be replaced by discrete stochastic models. Additionally, extremely short half-lives (femtoseconds) or very long ones (longer than the age of the universe) present measurement challenges and require specialized detection techniques. Always ensure that the units of elapsed time and half-life are consistent before performing calculations.
Frequently asked questions
What is a half-life in simple terms?
A half-life is the time it takes for exactly half of a radioactive substance to decay into a different element or isotope. For example, if you start with 80 grams of a substance with a 10-year half-life, you will have 40 grams after 10 years, 20 grams after 20 years, and so on. It is a fixed constant for each isotope, independent of how much you start with.
How do you calculate the remaining quantity after multiple half-lives?
Use the formula N(t) = N₀ × (1/2)^n, where n is the number of half-lives elapsed (n = t / t₁/₂). After 1 half-life, 50% remains; after 2, 25%; after 3, 12.5%; after 10 half-lives, only about 0.098% of the original material remains. The substance never fully reaches zero, but after roughly 10 half-lives it is considered negligible for most practical purposes.
What is the relationship between half-life and the decay constant?
The decay constant λ (lambda) and half-life t₁/₂ are inversely related by the equation λ = ln(2) / t₁/₂ ≈ 0.6931 / t₁/₂. The decay constant represents the instantaneous probability per unit time of a single nucleus decaying. A large decay constant means fast decay (short half-life), while a small decay constant means slow decay (long half-life).
Can the half-life formula be used for drug pharmacokinetics?
Yes — the same exponential decay model applies to the biological half-life of drugs, which describes how long it takes for the plasma concentration (or total body load) of a drug to reduce by half. However, biological half-lives depend on metabolism, kidney function, and body composition, unlike nuclear half-lives which are fixed physical constants. The mathematical form is identical, but the underlying mechanism is physiological rather than nuclear.
Which isotopes have the longest and shortest half-lives?
Tellurium-128 has one of the longest measured half-lives at approximately 2.2 × 10²⁴ years — far exceeding the age of the universe. At the other extreme, some excited nuclear states (isomers) have half-lives on the order of 10⁻²¹ seconds. Carbon-14 (5,730 years), Uranium-238 (4.47 billion years), and Iodine-131 (8 days) are commonly encountered isotopes across dating, geology, and medicine respectively.
Last updated: 2025-01-15 · Formula verified against primary sources.