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# Half-life

## Half-life

The half-life $(t_{1/2})$ is the time taken for half of the original number of radioactive nuclei in a material to decay.

Particular radioactive isotopes have specific half-lives (e.g. the half-life of $^{14}C$). These are independent of temperature and pressure.

The count rate and activity halve after every half-life (due to the proportionality relations with the number of nuclei).

A radioactive isotope with a high half-life (e.g. $\ce{^{235}U}$) emits radiation for a long period of time and hence remains dangerous to the surrounding environment for a long period of time.

Like nuclear decay itself, the half-life of a radioactive material is probabilistic. If one examines many samples of the same radioactive material, on average, approximately half of each sample will have decayed during the half-life.

## Half-life formulae

The half-life can be derived from the formula for the number of radioactive nuclei $N(t)$ by:\begin{align*}N(t_{1/2})=N_{0}e^{-\lambda t_{1/2}}&=\frac{N_{0}}{2}\\\Rightarrow e^{-\lambda t_{1/2}}&=\frac{1}{2}\end{align*}Taking the natural logarithm of both sides, we have:\begin{align*}&\ln{(e^{-\lambda t_{1/2}})}=\ln{(\frac{1}{2})}\\\Rightarrow&-\lambda t_{1/2}=-\ln{(2)}\\\Rightarrow& t_{1/2}=\frac{\ln{(2)}}{\lambda}=\frac{0.693}{\lambda}\end{align*}$N_{0}=$number of nuclei at time $t=0$, $N_{1/2}=$number of nuclei at time $t_{1/2}$ and $\lambda=$decay constant.