Radioactive Decay Calculator

Understanding Radioactive Decay

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable configuration, accompanied by the emission of particles or electromagnetic radiation. The rate of decay follows an exponential law: N(t) = N₀ × (½)^t/t½, where N₀ is the initial quantity, t is the elapsed time, and t½ is the half-life. This mathematical relationship means the decay rate is always proportional to the amount remaining, producing the characteristic exponential curve.

This calculator lets you solve for the remaining amount, the time elapsed, or the half-life, given the other two values. It also computes the decay constant (λ = ln(2)/t½), the activity (A = λN), the number of half-lives elapsed, and the percentage remaining. Units are flexible — use years, days, seconds, or any consistent time unit for both half-life and elapsed time.

Half-Life and Decay Constant

The half-life is the most intuitive measure of decay rate: it is the time for exactly half the atoms to decay. After n half-lives, the fraction remaining is (1/2)ⁿ. The decay constant λ gives the probability per unit time that any individual atom will decay. Activity A = λN measures the total number of decays per second in the sample, expressed in becquerels (Bq) or curies (Ci). As atoms decay and N decreases, activity decreases proportionally.

Types of Radioactive Decay

The three main types are alpha decay (emission of a helium-4 nucleus), beta decay (conversion of a neutron to a proton with electron emission, or vice versa), and gamma decay (emission of high-energy photons). Each type changes the nucleus differently: alpha decay reduces atomic number by 2 and mass number by 4, beta-minus decay increases atomic number by 1, and gamma decay changes neither. Many isotopes undergo decay chains, passing through several intermediate isotopes before reaching stability.

Applications

Carbon-14 dating uses the 5,730-year half-life of C-14 to date organic materials up to about 50,000 years old. Nuclear medicine uses short-lived isotopes like technetium-99m (6-hour half-life) for diagnostic imaging. Nuclear power harnesses the energy from fission of uranium-235 and plutonium-239. Geological dating uses isotopes like potassium-40 (1.25 billion year half-life) and uranium-238 (4.47 billion year half-life) to date rocks and determine the age of the Earth.

Frequently Asked Questions

What is radioactive decay?

The spontaneous transformation of unstable nuclei, emitting radiation. Governed by N(t) = N₀(½)^t/t½, where the rate is proportional to the remaining amount.

What is a half-life?

The time for half the atoms to decay. After n half-lives, (1/2)ⁿ of the original amount remains. Half-lives range from microseconds to billions of years.

What is the decay constant?

λ = ln(2)/t½ is the probability per unit time of an atom decaying. Activity A = λN gives total decays per second.

What is carbon-14 dating?

A method using C-14's 5,730-year half-life to date organic materials. The remaining C-14 fraction reveals how many half-lives have passed since the organism died.

Can radioactive decay be changed?

Under normal conditions, no. Decay rates are governed by nuclear forces unaffected by temperature, pressure, or chemistry. Only extreme conditions can produce tiny changes.