Radioactive Decay and Half-Life

Radioactive decay is the ultimate example of randomness meeting predictability. Each individual radioactive atom has a completely unpredictable lifetime — it might decay in the next microsecond or in a million years, and there is absolutely no way to tell which. No measurement, no observation, no clever trick can predict when a specific atom will decay.

Yet if you have a large collection of identical atoms, the fraction that decays per unit time is rock-solid predictable. After one half-life, exactly half are gone. After two half-lives, a quarter remain. After ten half-lives, about one in a thousand. This perfect statistical regularity emerging from complete individual randomness is one of the most striking features of nature.

The half-life

The half-life $t_{1/2}$ is the time required for half the atoms in a sample to decay. After time $t$, the number of remaining atoms is:

$N(t) = N_{0}\left(\dfrac{1}{2}\right)^{t/t_{1/2}}$

Equivalently, using the natural exponential:

$N(t) = N_{0}\,e^{-\lambda t}$

where $\lambda = \dfrac{\ln 2}{t_{1/2}} \approx \dfrac{0.693}{t_{1/2}}$ is the decay constant (the probability per unit time that any given atom decays).

For our problem: $N_{0} = 1000$, $t_{1/2} = 5$ s, $\lambda = 0.693/5 = 0.1386$ s$^{-1}$.

Numerical evolution

• $t = 0$: $N = 1000$
• $t = 5$ s (1 half-life): $N = 500$
• $t = 10$ s (2 half-lives): $N = 250$
• $t = 15$ s: $N = 125$
• $t = 20$ s: $N = 62.5 \approx 63$
• $t = 25$ s (5 half-lives): $N \approx 31$
• $t = 50$ s (10 half-lives): $N \approx 1$

After 10 half-lives, only about 1 in 1000 atoms remains. After 20 half-lives, about 1 in a million. The decay is relentless.

The activity (decay rate)

The activity $A$ is the number of decays per second:

$A = \lambda N = \lambda N_{0}\,e^{-\lambda t}$

Activity is measured in becquerels (Bq): 1 Bq = 1 decay per second. For our sample at $t = 0$: $A_{0} = 0.1386 \times 1000 = 138.6$ Bq.

Activity also halves every half-life, following the same exponential curve.

Half-lives of real isotopes

The range is staggering:

• Polonium-214: $t_{1/2} = 164$ microseconds
• Radon-222: $t_{1/2} = 3.8$ days
• Iodine-131: $t_{1/2} = 8.02$ days (used in thyroid cancer treatment)
• Cobalt-60: $t_{1/2} = 5.27$ years (used in radiation therapy)
• Carbon-14: $t_{1/2} = 5730$ years (used in radiocarbon dating)
• Uranium-238: $t_{1/2} = 4.47$ billion years (age of the Earth)

Carbon-14 dating worked example

Living organisms absorb C-14 from the atmosphere. When they die, the C-14 decays with $t_{1/2} = 5730$ years. A bone fragment has 25% of the original C-14 level. How old is it?

$0.25 = (0.5)^{t/5730} \implies t/5730 = \dfrac{\ln 0.25}{\ln 0.5} = \dfrac{-1.386}{-0.693} = 2$

$t = 2 \times 5730 = 11{,}460$ years. Two half-lives have passed.

Common mistakes

• Thinking decay is linear. It's not — it's exponential. After one half-life, half remains; after two, a quarter (not zero).
• Confusing half-life with average lifetime. The mean lifetime $\tau = 1/\lambda = t_{1/2}/\ln 2 \approx 1.443 \times t_{1/2}$. They're related but not equal.
• Thinking "half-life" means "safe after two half-lives." After two half-lives, 25% remains — still potentially dangerous. Typically, 10 half-lives ($<0.1\%$) is a better "safe" threshold.

Try it in the visualization

Watch individual atoms (colored dots) randomly wink out as they decay. The remaining count follows the exponential curve. Speed up time to watch the pattern unfold quickly. Notice how the curve is smooth and predictable even though each individual decay is random. Toggle the activity bar to see the decay rate halving in lockstep with the population.