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A half-life is the amount of time required for half of a sample decay or degrade. It is a common misconception that half-life is only used in relation to radioactivity. Half-life is actually a general term that can be used to describe any decaying process. More importantly, a half-life does not even need to be exponential. Zeroth order half-lives exist, and describe processes that occur at a constant rate. With this being said, first order half-lives are most common, and certainly most pertinent to radioactivity.

Half-lives are determined experimentally by measuring the time it takes for half of the population to decay. Mathematically, $$N_{t}=\frac{1}{2}N_{\circ}=N_{\circ}(\frac{1}{2})^\frac{t}{t_{half}}$$ When the time, $t$, equals the half-life, half of the population decays.

A similar concept is the natural lifetime, $\tau$, which is the time it take for $\frac{1}{e}$ of the sample to remain. The natural lifetime is more common than the half life in many fields of chemistry and physics. It is related to the half life in the following manner.

$$N_{t}=\frac{1}{2}\cancel{N_{\circ}}=\cancel{N_{\circ}}e^{\frac{-t}{\tau}}$$ $$\frac{1}{2}=e^{\frac{-t}{\tau}}$$ $$ln(\frac{1}{2})=\frac{-t}{\tau}$$ $$\tau=\frac{-t}{ln(\frac{1}{2})}$$ $$\tau=\frac{t}{ln(2)}\approx 1.44t$$

Thus, we can see the natural lifetime of a substance is approximately 1.44 times as long as the half life.

Radioactive Half-Life & Activity

The condition defined above for the half-life (T0.5 = 0.693/lambda) is always true. If you know a substance's half-life, then you know its decay constant. The reverse is also true.

(Video) Radioactivity - Half Life - Physics

By: expertmathstutor

Radioactive nuclei are stable in their radioactive state. At some point, the nuclei will decay or become unstable, losing radioactivity. This decay cannot be tracked for individual nuclei, so instead the half-life is monitored.

The video describes this concept with a bag of 100 coins. All coins start out as "heads-up" or stable. After one half-life, 50 coins turn to "tails-up" or unstable. Another half-life occurs and now only 25 coins are head-up. This continues until all the coins are tails-up. This is how half-life works, where the amount of time it takes for the nuclei to become unstable is the half-life. It is noted that the original number of nuclei does not matter, the time remains the same for the same radioactive material.

Carbon-14 is looked at and has a known half-life of approximately 5,700 years. Half the nuclei decay every half-life. As the half-life occurs, the number of stable nuclei declines, as does the activity. If about 400 parent atoms (stable atoms) are present at time=0, or the initial time, it takes 34,200 years or six half-lives.

Iodine-123 has a much smaller half-life of 13 hours. Starting with same number of parent atoms using six half-lives to reach a decay of the atoms only takes 65 hours.