Radioactive Decays

Exponential law describing remaining radioactive nuclei, activity, and half-life.

Conceptual context

Radioactive decay does not predict the exact instant at which one chosen nucleus will transform. It describes a stable statistical pattern for a large sample. The number of nuclei still undecayed, $N$, falls because each nucleus has the same probability per unit $t$ of decaying. That probability rate is represented by $\lambda$. The central question is therefore about a population: how much of the initial sample remains after some elapsed time, and how does that change when the decay constant is larger or smaller?

🟢 Essential level

At the start, $N_0$ tells how many radioactive nuclei are present. As $t$ passes, some nuclei decay and are no longer counted in $N$.

The key idea is that the sample does not lose the same number in every equal interval; it loses roughly the same fraction of what remains. This is why the curve falls quickly at first and then more slowly. $\lambda$ measures the internal tendency of that isotope to decay per unit time. A larger value means faster loss; a smaller value means slower change. $T_{1/2}$ translates that rate into a simple reading: after one half-life, half of the previous amount remains. $A$ counts the expected decays per second, so it falls with the surviving population statistically.

Picture a box full of popcorn kernels popping one by one at random, at a rate that depends on the kernel itself, not on how full the box is. At first many pop each second because so many are still unpopped; over time, even though each remaining kernel keeps the same chance of popping in the next instant, fewer kernels are left, so the total popping rate drops. That same logic, applied to nuclei instead of kernels, is all it takes to understand why the decay curve falls fast at first and then flattens without ever quite reaching zero.

🔵 Formal level

Basic formulas

The first-order model assumes a constant probability per unit $t$. Its central relation is:

Exponential decay law

This formula gives $N$ from $N_0$, $\lambda$, and $t$. The exponent must be negative because the remaining amount decreases. When solving for $\lambda$ or $t$, the ratio between $N$ and $N_0$ must lie between zero and one for a physical decay without replenishment.

The observable decay rate is $A$:

Radioactive activity

Real student questions

Why does everything not disappear after one $T_{1/2}$? Because $T_{1/2}$ acts on what remains, not once on the original amount. Each equal interval halves the previous value.

Why does $A$ decrease if $\lambda$ stays fixed? Because $A$ also depends on $N$. Each nucleus keeps the same decay probability, but fewer nuclei remain available.

Can one nucleus live longer than $\tau$? Yes. $\tau$ is a statistical mean, not an individual deadline.

Final synthesis

Radioactive decay describes how a nuclear population decreases through a constant probability per unit $t$. $\lambda$, $T_{1/2}$, $\tau$, and $A$ are complementary readings of the same exponential fall.