Radioactive Decay Calculator
Free Radioactive decay Calculator for nuclear chemistry. Enter variables to compute results with formulas and detailed steps.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
N(t) = N0 * e^(-lambda * t) | lambda = ln(2) / t_half
N(t) is the remaining quantity at time t, N0 is the initial quantity, lambda is the decay constant, and t_half is the half-life. The exponential decay law describes how the number of undecayed nuclei decreases over time.
Worked Examples
Example 1: Carbon-14 Decay
Problem:A sample contains 100 grams of Carbon-14 (half-life 5,730 years). How much remains after 17,190 years?
Solution:lambda = ln(2)/5730 = 0.000121 per year\nN(t) = 100 * e^(-0.000121 * 17190)\nN(t) = 100 * e^(-2.0794) = 100 * 0.125 = 12.5 g\nThis equals 3 half-lives: 100 -> 50 -> 25 -> 12.5
Result:12.5 grams remain (3 half-lives elapsed)
Example 2: Iodine-131 Medical Dose
Problem:A patient receives 200 mCi of Iodine-131 (half-life 8.02 days). How much activity remains after 24 days?
Solution:lambda = ln(2)/8.02 = 0.08643 per day\nN(t) = 200 * e^(-0.08643 * 24)\nN(t) = 200 * e^(-2.0743) = 200 * 0.1257 = 25.14 mCi\nAbout 2.99 half-lives elapsed
Result:25.14 mCi remain after 24 days
Frequently Asked Questions
What is radioactive decay?
Radioactive decay is the spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation in the form of alpha particles, beta particles, or gamma rays. The rate of decay is characterized by the half-life, which is the time required for half of the radioactive atoms in a sample to disintegrate. This process follows first-order kinetics, meaning the rate is proportional to the number of undecayed atoms present at any given time.
How is the decay constant related to half-life?
The decay constant (lambda) and half-life (t_half) are inversely related by the formula lambda = ln(2) / t_half, where ln(2) is approximately 0.6931. A larger decay constant means a shorter half-life and faster decay. The decay constant represents the probability per unit time that a given atom will decay. For example, Carbon-14 has a half-life of 5,730 years and a decay constant of about 1.21 times 10 to the negative 4 per year.
Can radioactive decay be sped up or slowed down?
Under normal physical and chemical conditions, radioactive decay rates cannot be altered. Unlike chemical reactions, nuclear decay is governed by the strong and weak nuclear forces, which are unaffected by temperature, pressure, or chemical bonding. However, in extreme conditions such as highly ionized atoms in stellar environments or under intense gravitational fields, very slight changes in certain types of decay (like electron capture) have been observed. For all practical purposes, the half-life is considered a fixed physical constant for each isotope.
How does radioactive decay determine safe storage time for nuclear medicine waste?
Hospitals using radioisotopes like technetium-99m (6-hour half-life) or iodine-131 (8-day half-life) for diagnostic scans and cancer treatment must hold contaminated waste in shielded storage until its activity decays to a level safe for regular disposal — typically 10 half-lives, per NRC and hospital radiation safety guidelines, at which point less than 0.1% of the original radioactivity remains. Radioactive Decay Calculator's decay-fraction output is exactly the number radiation safety officers use to determine minimum storage duration before waste can be released from a controlled area.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy