How can I calculate nuclear half life?

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Answer 1 · 2018-03-14T23:00:09

Consider that radioactive decay follow first order kinetics. Recall,

${ln [A]}_{t} = - k t + {ln [A]}_{0} (1)$ $ln[A]_"t" = -kt + ln[A]_0" " (1)$

Now, consider,

$\Rightarrow - ln (\frac{{[A]}_{t}}{{[A]}_{0}}) = k t$ $=> -ln(([A]_"t")/([A]_0)) = kt$

$\Rightarrow \frac{ln (\frac{{[A]}_{0}}{{[A]}_{t}})}{k} = t$ $=> ln(([A]_0)/([A]_t))/k = t$

$\Rightarrow \frac{ln (\frac{2}{1})}{k} = t$ $=> ln(2/1)/k = t$

$therefore t_(1/2) = ln(2)/k" "(2)$

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).