Radioactive Activity (A = λN)

Every nucleus of a given isotope has the same fixed probability $\lambda$ of decaying in the next second, and that probability does not depend on how long the nucleus has already existed, on what its neighbours are doing, or on anything you can do to it with heat or pressure or chemistry. A nucleus does not age. Given that, the arithmetic is immediate: $N$ nuclei each with probability $\lambda$ per second produce, on average, $\lambda N$ decays per second. The equation is a probability multiplied by a count, and its content lies entirely in the memorylessness that justifies the multiplication.

Activity is measured in becquerels, one decay per second, and the decay constant relates to the more familiar half-life by $\lambda = \ln 2 / T_{1/2}$. Take caesium-137: $T_{1/2} = 30.08$ years $= 9.49 \times 10^{8}$ s, so $\lambda = 7.30 \times 10^{-10}$ per second. One gram contains $(1/137) \times 6.022 \times 10^{23} = 4.40 \times 10^{21}$ nuclei, giving $A = 3.2 \times 10^{12}$ Bq — 3.2 TBq per gram, which matches the handbook value. The older unit, the curie, is $3.7 \times 10^{10}$ Bq, chosen because that is roughly the activity of a gram of radium-226.

The relation is the derivative of the exponential decay law. If $N(t) = N_0 e^{-\lambda t}$, then $-dN/dt = \lambda N$, which is this page. Rutherford and Soddy established both in 1902–03 at McGill, and the consequence that matters is that activity and population fall together at the same rate, so their ratio stays fixed. That is what makes radiometric dating possible: count the clicks and you have counted the atoms, without ever separating them. A gram of carbon from a living organism gives about 0.23 decays per second, corresponding to roughly one carbon-14 atom in $10^{12}$. Willard Libby's insight, which won the 1960 chemistry Nobel, was that this ratio is fixed while the organism is exchanging carbon with the atmosphere and starts falling the moment it stops — every halving of the rate marking another 5730 years.

Four places this goes wrong. The decay constant is not the half-life. $\lambda = 0.693/T_{1/2}$, and substituting a half-life where $\lambda$ belongs makes the answer wrong by a factor of about 0.69 — small enough to look believable, which is what makes it dangerous. Note too that $\lambda$ here is a rate in reciprocal seconds and has nothing to do with the wavelength that shares the symbol elsewhere in this shard. Second, activity is not dose. Becquerels count disintegrations and say nothing about the energy released, the type of radiation, or whether the source is outside you or inside; grays and sieverts are the dose units and you cannot convert without knowing the nuclide and the geometry. A banana is roughly 15 Bq of potassium-40 and is not a hazard. Third, both $A$ and $N$ are falling exponentially, so what this page returns is an instantaneous rate at one moment, not a total over any period. And fourth, decay chains: a long-lived parent in secular equilibrium with its daughters emits several particles per parent decay, so a detector counting everything will read considerably more than $\lambda N$ computed for the parent alone. Radium-226 with its radon progeny is the standard case.