Question

The radioactive isotope ${}^{\mathbf{32}}\mathbf{P}$decays by first-order kinetics and has a half-life of 14.3 days. How long does it take for 95.0% of a given sample of${}^{\mathbf{32}}\mathbf{P}$ to decay?

Short answer

The reaction is unimolecular.

Step-by-step solution

Describing the first-order rate constant

For half-life $t_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$,

localid="1663823449867" $\begin{array}{rcl}{\mathrm{t}}_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}& =& 0.693/\mathrm{k}\\ 14.3& =& 0.693/\mathrm{k}\\ \mathrm{k}& =& 0.04864\end{array}$

Describing the time taken for the decay

Using the First order concentration-time equation,

$\begin{array}{rcl}\mathrm{k}& =& \left[\mathrm{In}\left(\mathrm{a}/\mathrm{a}-\mathrm{x}\right)\right]/\mathrm{t}\\ 0.0693& =& \mathrm{In}\left(100/5\right)\\ \mathrm{t}& =& 61.82\mathrm{days}\end{array}$

Therefore, the time taken to decay 95% of a given sample is$61.82\mathrm{days}\cong 62\mathrm{days}$.