Half-life

It is generally very hard to say exactly when all of the nuclei in a sample will have decayed (since the decay law implies this technically takes infinite time), but it is straightforward to calculate the time needed for a given fraction of the sample to decay - and the most useful such fraction is one-half.

Half-life. The half-life $T_{1/2}$ of a radioactive sample is defined as the time required for the number of undecayed nuclei to fall to exactly half its initial value. Setting $N=N_0/2$ at $t=T_{1/2}$ in the decay law $N=N_0e^{-\lambda t}$ gives $\tfrac{1}{2}=e^{-\lambda T_{1/2}}$, i.e. $e^{\lambda T_{1/2}}=2$; taking logarithms of both sides,

Different radioactive species have wildly different half-lives - some as long as $10^{14}$ years, others as short as $10^{-14}$ s. After $n$ half-lives have elapsed (where $n=t/T_{1/2}$ need not be an integer), the number of undecayed nuclei remaining is

and correspondingly the activity after $n$ half-lives is $R=R_0(1/2)^n$ (8.41). It is a common misconception that a shorter half-life means a "safer" sample because it "won't last long" - in fact the opposite is true: a shorter half-life means a larger decay constant and hence a higher activity, making the sample more intensely radioactive (and more hazardous) while it lasts, even though it does not remain radioactive for as long. …