A radioactive source has decayed to one tenth of one percent of its initial activity in one hundred days. What is its half-life period?

Solution:

Let A₀ be the initial activity.

From Question:

$A = \frac{1}{{10}} \times 1\% {\rm{ }}of{\rm{ }}{A_0} = \frac{1}{{10}} \times \frac{1}{{100}} \times {A_0} = \frac{{{A_0}}}{{1000}}$

We have,

$\begin{array}{l}A = {A_0}{e^{ - \lambda  \times 100}}\\or,\frac{{{A_0}}}{{1000}} = {A_0}{e^{ - \lambda  \times 100}}\\or,{\rm{ }}{e^{ - \lambda  \times 100}} = 1000\\or,\lambda  \times 100 = \ln 1000\\or,{\rm{ }}\lambda  = 0.069\\Half\;{\rm{ Life  =  }}{T_{\frac{1}{2}}} = \frac{{0.693}}{\lambda } = \frac{{0.693}}{{0.69}} = 10days\end{array}$