Radioactive Decay Calculator

When to Use Each Variable

• Solve for Remaining Quantity — when you know the initial quantity, decay constant, and elapsed time, e.g., determining how much of a medical isotope remains after shipping.
• Solve for Half-Life — when you know the decay constant and want to express the decay rate in the most commonly quoted form.
• Solve for Mean Lifetime — when you need the average survival time per atom, e.g., calculating expected detection rates in a particle physics experiment.
• Solve for Activity — when you know the decay constant and number of atoms, e.g., determining the disintegration rate of a radioactive source in becquerels.

Key Concepts

Radioactive decay is a random process governed by the exponential law N(t) = N₀e^(−λt). The decay constant λ is unique to each isotope and determines the half-life (t½ = ln2/λ) and mean lifetime (τ = 1/λ). Activity (A = λN) measures disintegrations per second in becquerels. After n half-lives, the fraction remaining is (½)ⁿ.

Applications

• Radiometric dating: determining the age of archaeological artifacts (carbon-14) and geological formations (uranium-lead, potassium-argon)
• Nuclear medicine: calculating dose decay rates for radiopharmaceuticals like technetium-99m used in diagnostic imaging
• Radiation safety: estimating how long radioactive waste must be stored before activity drops to safe levels
• Nuclear power: tracking fuel burnup and predicting fission product inventories in reactor cores

Common Mistakes

• Confusing half-life with mean lifetime — mean lifetime is always 1.443× longer than half-life; using them interchangeably gives wrong decay rates
• Applying the decay formula to a mixture of isotopes without separating them — each isotope has its own λ and must be decayed independently
• Assuming activity is constant — activity decreases exponentially along with the number of atoms; a freshly produced medical isotope is far more active than the same sample a few half-lives later