You can never say which nucleus will decay next, only the chance per second. Yet across billions of nuclei that randomness averages into one of the most reliable curves in physics: the exponential decay.
Decay is random and spontaneous: unpredictable for any one nucleus and unaffected by external conditions. The activity is A = λN, where the decay constant λ is the probability of decay per unit time. Because the rate is proportional to the number left, the count falls exponentially, N = N₀e⁻λᵗ, halving every half-life, with λt½ = ln2.
Each nucleus winks out at random, but watch the count: with a whole sample it halves in a fixed half-life and traces the exponential, whatever half-life you choose. Reset to run a fresh sample and see the randomness in the jagged line around the smooth law.
Four ideas, tightly linked.
Activity and count rate fall on the same exponential as N, because both are proportional to N. The half-life is a constant for a given isotope, independent of how much is left. After n half-lives the fraction remaining is (½)ⁿ, so three half-lives leave one eighth. Keep λ and t½ in consistent time units before substituting.
Four quick checks on radioactive decay. Each correct answer earns XP and lights this skill on your star map.
Radioactive decay is described as random and spontaneous because:
The activity of a radioactive source is given by:
The decay constant λ and the half-life t½ are related by:
After three half-lives, the fraction of the original nuclei still undecayed is:
To find a time, rearrange N = N₀e⁻λᵗ with natural logs: t = (1/λ) ln(N₀/N). Watch the difference between λ (per second) and t½ (seconds), and remember that a measured count rate is proportional to activity, not equal to it, because a detector only catches a fraction of the decays.
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