Q.A radioactive isotope has a half-life of T years. How long will it take the activity to reduce to a) 3.125%, b) 1% of its original value?

Half-Life: The Heartbeat of Radioactive Decay

Imagine you have a giant jar of popcorn kernels, and every minute, exactly half of the kernels left in the jar pop. You start with 1000 kernels. After one minute, 500 are left. After two minutes, 250. After three minutes, 125. After four minutes, about 62. And so on.

That constant "halving time" — the fixed interval it takes for half of whatever remains to disappear — is the core idea of half-life.

Radioactive decay works the same way, except the "popping" is a nucleus spontaneously transforming into a different nucleus by emitting radiation. The key insight: each nucleus has the same fixed probability of decaying per second, regardless of how old it is or how many other nuclei are around. This is a purely random, memoryless process.

The Precise Statement

The half-life, denoted $T_{1/2}$, is the time required for exactly half of the radioactive nuclei in a sample to decay.

If you start with $N_0$ nuclei, after one half-life you have $\frac{N_0}{2}$ left. After two half-lives, you have $\frac{N_0}{4}$ left. After three, $\frac{N_0}{8}$, and so on.

Mathematically, the number of nuclei remaining after time $t$ follows an exponential decay law:

where $\lambda$ is the decay constant — the probability per unit time that a given nucleus will decay. The half-life is the value of $t$ that makes $N(t) = N_0/2$:

Cancelling $N_0$ and taking natural logs:

The number $0.693$ is just $\ln 2$ to three decimal places. This formula is the exact bridge between the decay constant (a microscopic probability) and the half-life (a macroscopic, measurable time).

Why "Independent of Initial Amount"?

This is the most counterintuitive part for beginners. Suppose you have two samples of the same isotope: one with 1 million atoms and one with 10 atoms. The half-life is identical for both. …