# How can I calculate nuclear half life?

Mar 14, 2018

$\ln {\left[A\right]}_{\text{t" = -kt + ln[A]_0" }} \left(1\right)$

Now, consider,

$\implies - \ln \left(\frac{{\left[A\right]}_{\text{t}}}{{\left[A\right]}_{0}}\right) = k t$

$\implies \ln \frac{\frac{{\left[A\right]}_{0}}{{\left[A\right]}_{t}}}{k} = t$

$\implies \ln \frac{\frac{2}{1}}{k} = t$

$\therefore {t}_{\frac{1}{2}} = \ln \frac{2}{k} \text{ } \left(2\right)$

Now, if we knew the rate constant for the specific decay, we could calculate the half life of that radioactive sample.

Or, practically, we could measure a sample at different intervals and calculate the rate constant in (1) and apply it to (2).