Q.On your birthday, you measure the activity of a sample of $^{210}\text{Bi}$ which has a half-life of 5.01 days. The initial activity that you measure is $1\,\mu\text{Ci}$.

Activity of a Radioactive Sample

Imagine you have a jar of fireflies. Each firefly glows at a random moment, and once it glows, it stops forever. You don't care about any single firefly — you care about how many flashes you see per second. That rate of flashing is the activity of your jar.

A radioactive sample is exactly that jar. Inside are unstable nuclei. Each nucleus will "flash" (decay) at some unpredictable instant, and after that it becomes a stable atom. The activity $A$ is simply the number of decays happening each second.

The precise definition

If you have $N$ unstable nuclei at a given moment, and each has a constant probability $\lambda$ of decaying per second (the decay constant), then the number of decays per second is:

$\lambda$ is a fixed property of the isotope — it tells you how "eager" each nucleus is to decay. A large $\lambda$ means each nucleus is very likely to decay soon, so the activity is high even with few nuclei.

Why activity itself decays

As nuclei decay, $N$ drops. Since $A$ is proportional to $N$, the activity must drop too. The mathematics is straightforward: the rate at which $N$ decreases is exactly the activity (each decay removes one nucleus), so

This differential equation gives the familiar exponential decay:

Multiply both sides by $\lambda$ and you get the activity:

The activity itself decays exponentially with the same decay constant $\lambda$.

Units

The SI unit is the becquerel (Bq): 1 Bq = 1 decay per second. That is a tiny number. Historically, the curie (Ci) was used: 1 Ci = $3.7 \times 10^{10}$ Bq, roughly the activity of one gram of radium-226. …