# Question #fca61

Dec 29, 2014

The answer is $62 \mathrm{da} y s$.

We know that an exponential decay can be expressed mathematically by

$A \left(t\right) = {A}_{0} \cdot {\left(\frac{1}{2}\right)}^{\frac{t}{t} _ \left(\text{1/2}\right)}$, where

$A \left(t\right)$ - the amount left after t years;
${A}_{0}$ - the initial quantity of the substance that will undergo decay;
${t}_{\text{1/2}}$ - the half-life of the decaying quantity.

$\text{Ra-223}$ has a molar mass of approximately $223 \frac{g}{m o l}$, which means that the sample's initial mass and the final mass will be

${A}_{0} = 0.240$ $m o l e s \cdot 223 \frac{g}{m o l} = 53.5 g$

$A \left(t\right) = 7.50 \cdot {10}^{- 3}$ $m o l e s \cdot 223 \frac{g}{m o l} = 1.67 g$

So,

$1.67 = 53.5 \cdot {\left(\frac{1}{2}\right)}^{\frac{t}{12.4}} \to 0.0312 = {\left(\frac{1}{2}\right)}^{\frac{t}{12.4}}$

$\frac{t}{12.4} = {\log}_{0.5} \left(0.0312\right) = 5.002$, which means that

$t = 5.002 \cdot 12.4 = 62$ days.