In radioactive decay, an unstable nucleus transforms into a more stable one by emitting particles or energy. Think of it like a ticking clock that keeps on ticking until it stops – the ticking is the decay.
All radioactive decay follows the same mathematical rule:
\$x = x_0 e^{-\lambda t}\$
Where:
Because the rate of decay is proportional to the amount still present, the graph of \$x\$ vs. \$t\$ is a smooth, downward‑sloping curve that never quite reaches zero.
| Parameter | Definition | Formula |
|---|---|---|
| Decay Constant | Probability per unit time that a nucleus decays. | \$\lambda\$ |
| Half‑Life | Time taken for half the nuclei to decay. | \$T_{1/2} = \dfrac{\ln 2}{\lambda}\$ |
| Activity | Number of decays per unit time. | \$A = \lambda N\$ |
Imagine a steep hill that gradually flattens out. The hill’s slope represents the rate of decay. As time goes on, the hill becomes less steep because fewer nuclei remain to decay.
Key points to label on your sketch:
Use a smooth curve – no sharp corners – to show the continuous nature of the process.
Suppose we have \$N0 = 1.0 \times 10^{12}\$ atoms of a radioactive isotope with a half‑life of \$T{1/2} = 5\$ years.
\$\lambda = \dfrac{\ln 2}{T_{1/2}} = \dfrac{0.693}{5\,\text{yr}} = 0.1386\,\text{yr}^{-1}\$
\$N = N_0 e^{-\lambda t} = 1.0 \times 10^{12} e^{-0.1386 \times 10} \approx 2.5 \times 10^{11}\$
\$A = \lambda N \approx 0.1386 \times 2.5 \times 10^{11} \approx 3.5 \times 10^{10}\,\text{decays yr}^{-1}\$
Remember: Activity is often expressed in becquerels (decays per second). Convert if needed.