Decay Law and Half-Life

Describes how a radioactive sample decreases and how its half-life is interpreted.

Conceptual context

A radioactive sample does not fade like a lamp with a fixed drain. Each nucleus has its own chance of decaying, and that chance does not remember how long the nucleus has waited. The individual event is unpredictable, but a large population forms a clean statistical pattern. The central question is how $N$ changes as $t$ passes and how $T12$ turns that change into a readable $t$ scale. This leaf links remaining amount, measured $A$, remaining fraction, and dating: the clock is not inside one nucleus, but in the collective statistics of many nuclei.

🟢 Essential level

The decay law describes a sample in which $N$ decreases because some nuclei transform spontaneously.

You cannot predict which specific nucleus will decay next, but you can predict the collective trend when the sample contains many nuclei. $T12$ is the $t$ compass: after each interval of $T12$, half of the radioactive nuclei present at the start of that interval remain. The decrease therefore does not subtract the same amount each $t$; it keeps reducing the same proportion, like a statistical clock for the population. If $\lambda$ is large, the sample changes quickly and $T12$ is short. If $\lambda$ is small, the sample persists for a longer $t$. $A$ follows the same fall because it depends on how many nuclei are still available to decay.

Picture a full theater where, minute by minute, a fixed small percentage of the people still seated get up and leave — never the same absolute number. At first many people leave per minute because the hall is full; over time, even though the percentage leaving per minute never changes, fewer people leave each minute because fewer remain seated. That is exactly the logic behind $T12$ and the fall of $N$.

🔵 Formal level

Basic formulas

The formal model measures $t$ from the start of observation and relates $N$ to $N0$ through an exponential decrease. The core relation is:

Exponential decay law

Here $\lambda$ sets the probabilistic speed of the process. It is not an absolute loss per second, but a probability per unit $t$ applied to each nucleus that has not yet decayed. $A$ uses the same $t$ structure:

Activity as a function of time

The instantaneous connection between $A$ and $N$ is:

Relation between activity and remaining nuclei

Real student questions

Does the sample disappear after many $T12$ intervals? Not exactly in the continuous model: it approaches zero. In real matter, the number of nuclei is discrete and may eventually reach none.

Does $T12$ depend on sample size? No. It depends on the isotope through $\lambda$, not on $N0$.

Why is the fall not linear? Because each interval removes a fraction of what remains, not a fixed amount.

Does high $A$ always mean a large mass? Not necessarily. There may be few nuclei with large $\lambda$, or many nuclei with small $\lambda$.

Why can it date samples? Because $f$ stores information about elapsed $t$ if the initial amount is known or estimated.

Final synthesis

The decay law turns microscopic randomness into macroscopic regularity. $T12$ does not say when one nucleus decays; it describes how a whole population shrinks. Reading $N$, $A$, and $f$ links $t$, rate, and physical persistence.