Physics · Particle & Nuclear Physics · Nuclear Physics
Radioactive Decay Calculator
Calculates the remaining activity, quantity, and decay constant of a radioactive isotope after a given time using the exponential decay law.
Calculator
Formula
N(t) is the number of undecayed nuclei (or remaining mass/activity fraction) at time t. N_0 is the initial quantity. \lambda is the decay constant (s\textsuperscript{-1}). t_{1/2} is the half-life. A(t) is the activity at time t, proportional to the number of remaining nuclei.
Source: Rutherford & Soddy (1902); standard nuclear physics formulation as presented in Krane, K.S., Introductory Nuclear Physics (1988).
How it works
Radioactive decay is a spontaneous, probabilistic process by which unstable atomic nuclei lose energy by emitting radiation. Because each nucleus decays independently with a fixed probability per unit time, large populations of nuclei follow a precise exponential decay law. This law is one of the most reliable and well-tested relationships in all of physics, forming the backbone of nuclear medicine dosing, carbon-14 dating, reactor fuel management, and radioactive waste storage planning.
The governing equation is N(t) = N₀ · e−λt, where N₀ is the initial number of undecayed nuclei (or equivalently, initial mass or initial activity in consistent units), λ is the decay constant measured in inverse seconds, and t is the elapsed time in seconds. The decay constant is directly related to the half-life t₁/₂ by the relation λ = ln(2) / t₁/₂ ≈ 0.693 / t₁/₂. The activity A(t) — the number of decay events per second — equals λ · N(t), so it also decreases exponentially with the same time constant. The number of half-lives elapsed is simply t / t₁/₂, which is a dimensionless ratio that immediately conveys how many times the sample has halved in size.
This calculator accepts half-life and elapsed time in any common unit (seconds, minutes, hours, days, or years) and converts internally to seconds for consistency. Outputs include the remaining quantity in the same units as the input, the percentage remaining, the percentage that has decayed, the decay constant in SI units (s⁻¹), and the number of half-lives elapsed. Practical applications span carbon dating of archaeological samples using carbon-14's 5,730-year half-life, calculating residual activity of nuclear medicine tracers such as technetium-99m (6-hour half-life), estimating repository requirements for nuclear waste isotopes like caesium-137 (30.2-year half-life), and radiation protection calculations.
Worked example
Example: Carbon-14 Dating
A piece of charcoal from an ancient campfire is found to contain 250 atoms (arbitrary units) of carbon-14, compared to a living tree's expected value of 1,000 atoms. Carbon-14 has a half-life of 5,730 years. How old is the sample?
First, we use the decay equation to find elapsed time. We know N(t)/N₀ = 250/1000 = 0.25. Setting 0.25 = e−λt and solving: −λt = ln(0.25) = −1.3863. With λ = ln(2) / 5730 = 0.6931 / 5730 = 1.2097 × 10−4 yr−1, we get t = 1.3863 / (1.2097 × 10−4) ≈ 11,460 years.
Checking with the calculator: enter N₀ = 1000, half-life = 5730 years, elapsed time = 11,460 years. The calculator returns N(t) = 250.0, percent remaining = 25.000%, percent decayed = 75.000%, decay constant = 3.835 × 10⁻¹² s⁻¹, and number of half-lives = 2.000. This confirms the sample is exactly two half-lives old — approximately 11,460 years.
Example: Medical Imaging — Technetium-99m
A nuclear medicine dose of 500 MBq of Tc-99m (half-life = 6.0 hours) is prepared at 7:00 AM. What activity remains at 1:00 PM (6 hours later)?
Enter N₀ = 500, half-life = 6 hours, elapsed time = 6 hours. The output is: remaining quantity = 250.00 MBq, percent remaining = 50.000%, number of half-lives = 1.000. Exactly one half-life has passed, so the activity has halved to 250 MBq — a straightforward but critical calculation for radiation dosimetry.
Limitations & notes
This calculator applies the standard first-order exponential decay model, which assumes a constant decay constant and a large enough sample that statistical fluctuations are negligible. For very small numbers of atoms (fewer than ~1,000 nuclei), individual decay events become statistically significant and the deterministic exponential model gives only an expected value rather than a precise prediction — a stochastic simulation would be more appropriate in that regime. The calculator does not account for secular equilibrium in decay chains (e.g., uranium-238 decaying through a long chain of daughters); if ingrowth of daughter products is important, a Bateman equation solver is required. It also assumes no addition or removal of the isotope from the system (closed system). Results for elapsed times vastly exceeding the half-life (more than ~50 half-lives) may approach floating-point underflow and should be interpreted carefully. Units must be consistent — the calculator converts all times to seconds internally, but the initial quantity is treated as dimensionless and must be supplied in consistent units by the user.
Frequently asked questions
What is the difference between decay constant and half-life?
The decay constant λ (lambda) is the probability per unit time that a single nucleus will decay, expressed in units of inverse time (e.g., s⁻¹). The half-life t₁/₂ is the time required for exactly half of a large sample to decay. They are related by λ = ln(2) / t₁/₂ ≈ 0.693 / t₁/₂, so a longer half-life corresponds to a smaller decay constant and a slower rate of decay.
Can I use this calculator for activity (Becquerels or Curies) instead of number of atoms?
Yes. Because activity A(t) = λ · N(t) and λ is constant, activity decays with exactly the same exponential law as the number of atoms. Simply enter your initial activity in Bq, Ci, or any consistent unit as N₀, and the output 'remaining quantity' will be the activity at time t in those same units.
How accurate is radiocarbon (carbon-14) dating?
Radiocarbon dating is accurate to within about ±30–40 years for samples up to roughly 10–15 half-lives old (approximately 50,000–60,000 years), beyond which the remaining C-14 becomes too small to measure reliably. Calibration against tree-ring records (dendrochronology) and other proxies is required because atmospheric C-14 levels have varied over time, meaning the initial concentration N₀ was not always the same as today's value.
Why does radioactive decay follow an exponential law?
Exponential decay arises because each nucleus decays independently with a fixed probability per unit time — a memoryless process. When the rate of change of a population is proportional to the population itself (dN/dt = −λN), the only solution is an exponential function. This probabilistic, memoryless property is a direct consequence of quantum mechanical tunneling and energy-level considerations within the nucleus.
What is meant by 'secular equilibrium' in a decay chain?
Secular equilibrium occurs in a radioactive decay chain when a long-lived parent isotope produces shorter-lived daughters at a constant rate, and the daughters' decay rates match their production rates. In this state, the activity of each daughter equals the activity of the parent. This calculator handles only single-step decay; for multi-step chains with significant daughter ingrowth, the Bateman equations must be used instead.
Last updated: 2025-01-15 · Formula verified against primary sources.