91Ó°ÊÓ

The relation between the activity R of a sample and the number of radioactive nuclei N in the sample is given by:

$R=\mathrm{λ}N$ ..............(i)

The relation between the half-life $T_{1/2}$and the decay constant $\mathrm{λ}$is given by

$\mathrm{λ}=\frac{In2}{T_{1/2}}$ ............(ii)

The half-life of ${}^{131}I$is role="math" localid="1664207830185" $T_{1/2}=8.04days$and its activity in a typical thyroid cancer treatment is$R=3.70×{10}^{9}Bq$.

We substitute our value for $T_{1/2}$into equation (ii), so we get the decay constant of ${}^{131}I$:

$$\begin{gathered} \mathrm{λ}=\frac{In2}{8.04d} \\ =0.086d^{-1} \\ =9.98×{10}^{-7}{s}^{-1} \end{gathered}$$

Now, we plug our values for $\mathrm{λ}$and R into equation (i) and evaluate for N, so we get the number of ${}^{131}I$atoms administered in a typical thyroid cancer treatment:

$$\begin{gathered} N=\frac{R}{\mathrm{λ}} \\ =\frac{3.70×{10}^{9}Bq}{9.98×{10}^{-7}{s}^{-1}} \\ =3.71×{10}^{15}atom \end{gathered}$$

Therefore, the right choice is (d) $3.71×{10}^{15}$.