Nuclear Decay Rate Calculator
Calculates the number of radioactive atoms remaining after a given time using the half-life decay law. Useful in nuclear medicine, radiological safety, and isotope dating applications.
About this calculator
Radioactive decay follows a well-established exponential law: the number of unstable atoms in a sample decreases by half during each half-life period. The governing equation is N(t) = N₀ × (0.5)^(t / t½), where N₀ is the initial number of atoms, t is the elapsed time, and t½ is the half-life of the isotope. This formula is equivalent to the continuous form N(t) = N₀ × e^(−λt), where the decay constant λ = ln(2) / t½. Activity A (decays per second, or Becquerels) equals λ × N(t), so it also declines exponentially. The half-life is a fixed nuclear property that ranges from fractions of a second for highly unstable nuclides to billions of years for long-lived isotopes like uranium-238. Understanding decay rates is essential for radiation safety calculations, medical dose planning, and radiometric dating.
How to use
Suppose you start with 1,000,000 atoms of iodine-131, which has a half-life of 8.02 days, and you want to know how many remain after 24 days. Step 1: Identify inputs — N₀ = 1,000,000 atoms, t½ = 8.02 days, t = 24 days. Step 2: Calculate the exponent: 24 / 8.02 ≈ 2.994. Step 3: Compute (0.5)^2.994 ≈ 0.1253. Step 4: Multiply: 1,000,000 × 0.1253 ≈ 125,300 atoms remaining. This means roughly 87.5% of the original iodine-131 has decayed after three half-lives, consistent with the rule that three half-lives reduce activity to about 12.5%.
Frequently asked questions
How does half-life determine how dangerous a radioactive isotope is over time?
A short half-life means an isotope decays quickly, releasing radiation at a high rate initially but becoming negligible within days or weeks. A long half-life indicates slow decay, lower immediate activity, but persistent contamination over centuries or millennia. The hazard posed by an isotope therefore depends on both its half-life and its radiation type and energy. For waste management, short-lived isotopes are held in interim storage until they decay to safe levels, while long-lived isotopes require deep geological repositories.
What is the difference between radioactive decay rate and activity?
Decay rate and activity are essentially the same quantity — the number of nuclear disintegrations occurring per unit time, measured in Becquerels (1 Bq = 1 decay per second) or Curies. Activity A = λ × N(t), where λ is the decay constant and N(t) is the current number of radioactive atoms. As atoms decay, N(t) decreases, so activity falls proportionally over time. Dose rate, by contrast, depends on activity but also on the energy and type of radiation emitted and the geometry of exposure.
Why do different radioactive isotopes have such vastly different half-lives?
Half-life is determined by the nuclear structure of the isotope — specifically, the balance of proton and neutron numbers and the nuclear binding energy. Nuclides far from the valley of nuclear stability decay rapidly because the energy released during the transition is large and the quantum tunneling probability is high. Isotopes close to stability decay slowly because the energy difference is small and the barrier to decay is large. This spans an enormous range, from microseconds for highly exotic nuclei to 4.47 billion years for uranium-238, reflecting the diversity of nuclear configurations.