Radioactivity

Study of spontaneous nuclear decay and its measurable activity. Theory, formulas, worked examples and interactive graphs.

Conceptual context

Radioactivity appears when some atomic nuclei are unstable and transform spontaneously. It is not a chemical reaction and it is not caused by the material being consumed by air or light; the change comes from the nucleus itself. The central question is how $A$, the measurable rate of expected decays, changes with $t$. A sample with many radioactive nuclei can emit strongly at first and more weakly later because each decay removes one possible future decay. The $T_{1/2}$ gives a compact clock for this process: after each such interval, about half of the previous amount remains undecayed.

🟢 Essential level

A radioactive sample contains many unstable nuclei. Each nucleus has a small probability of decaying during a time interval, but we cannot predict which one will decay next. What we can predict with great precision, if the sample is large, is the collective behavior. At the beginning of the process, the sample presents a large quantity of unstable nuclei and its emission is very high. Over the course of $t$, the $N$ remaining undecayed decreases, which causes the $A$ (the rate of decays per second) to also decrease progressively.

The half-life $T_{1/2}$ summarizes the rate of this process: after an interval equal to one half-life, half of the nuclei remain, after two half-lives, a quarter remain, and so on. A larger decay constant $\lambda$ indicates that each nucleus is more prone to decay, so the remaining fraction $f$ of undecayed nuclei decreases more rapidly. It is a bit like a large urn full of balls where, every second, each ball has the same small chance of randomly "jumping" out: at first many balls jump because there are many inside, and over time fewer and fewer jump, simply because fewer balls remain in the urn.

🔵 Formal level

Basic formulas

The formal model describes a large nuclear population through macroscopic quantities. The remaining $N$ depends on $N_0$, the $\lambda$, and the elapsed $t$ measured from a chosen origin.

Exponential law of radioactive decay

$A$ is read as the expected number of decays per second, measured in becquerels. For a sample with available $N$, the connection between population and rate is direct: more undecayed nuclei give more expected decays in the same interval.

Activity of a radioactive sample

Since $N$ decreases with $t$, $A$ also decreases from $A_0$.

Activity as a function of time

Real student questions

Why does everything not disappear when one $T_{1/2}$ has passed? Because $T_{1/2}$ does not mark the end of every nucleus; it marks the statistical reduction of a large population.

Does $A$ decrease because nuclei get tired? No. It decreases because less $N$ remains available to decay.

Can I draw decay as a straight line? Only as a local approximation over very short intervals; the full physical model is curved.

Does high $A$ always mean higher risk? Not by itself. Risk also depends on radiation type, energy, distance, shielding, and exposure.

Final synthesis

Radioactivity is understood by reading together the remaining population, the decay rate, and the $T_{1/2}$ scale. The curve of $A$ decreases not by chemical wear, but by statistical reduction in the number of nuclei able to decay.