Radioactive Decay Simulator

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Radioactive decay is the spontaneous process by which unstable atomic nuclei lose energy by emitting radiation. This process transforms the atom into a different element or a more stable isotope. The decay follows exponential mathematics described by N(t) = N₀ × e^(-λt), where N is the number of atoms, t is time, and λ is the decay constant.

The deterministic model uses the mathematical formula N(t) = N₀ × e^(-λt) to calculate the exact expected number of remaining atoms at any time. The stochastic (Monte Carlo) model simulates random decay events for each atom, reflecting the probabilistic nature of real radioactive decay. With large numbers of atoms, both models converge to similar results, but the stochastic model shows natural statistical variations.

The decay constant (λ) and half-life (T½) are inversely related by the formula λ = ln(2)/T½ ≈ 0.693/T½. The decay constant represents the probability per unit time that an atom will decay, while the half-life is the time required for half of the atoms to decay. A larger decay constant means shorter half-life and faster decay.

Each radioactive atom has the same probability of decaying per unit time, regardless of how long it has existed or how many other atoms have already decayed. This memoryless property is fundamental to radioactive decay and leads to the exponential decay law. It's analogous to flipping a coin – past results don't affect future probabilities.

Radioactive decay has numerous applications: Carbon-14 dating determines the age of archaeological artifacts (up to ~50,000 years), Uranium-238 dating measures geological time scales (billions of years), medical isotopes like Iodine-131 treat thyroid conditions, Technetium-99m is used in medical imaging, and Plutonium-239 powers spacecraft and nuclear reactors.

The stochastic model is actually most important for small numbers of atoms because individual decay events significantly affect the total count. With millions of atoms, statistical averaging makes the deterministic model highly accurate. But with tens or hundreds of atoms, the random nature of decay causes noticeable fluctuations that only the stochastic model can represent.

After each half-life, half of the remaining atoms decay. After 1 half-life: 50% remain, after 2: 25%, after 3: 12.5%, after 4: 6.25%, and so on. After 10 half-lives, less than 0.1% remains. Theoretically, complete decay takes infinite time, but practically, after 7-10 half-lives, the remaining activity becomes negligible.

While this simulator focuses on single-isotope decay for clarity, you can run multiple simulations with different isotopes and compare their decay curves. In decay chains (like Uranium-238 → Thorium-234 → ...), daughter products have their own decay rates, creating complex multi-stage decay patterns that require specialized chain decay calculators.

Unlike chemical reactions, radioactive decay rates are essentially unaffected by temperature, pressure, chemical bonds, or electric/magnetic fields. This is because decay occurs in the nucleus, which is largely isolated from external conditions. This stability makes radioactive decay perfect for absolute dating methods.

The exponential decay curve (appearing as a straight line on a logarithmic scale) reflects the constant decay probability. The curve's shape is universal for all radioactive isotopes – only the time scale (half-life) changes. This predictable pattern enables precise dating, radiation safety calculations, and medical dosing.