91Ó°ÊÓ

Half-life of the radioactive nuclide $T_{1/2}=30y$.

Consider the radioactive decay is due to the loss of the elementary particles from an unstable nucleus to convert them into a more stable one. The relation of the fraction of undecayed product after a given time and the time gives the fraction that is proportional to the exponential of the product of decay constant and time. Thus, using this relation, determine the required undecayed amount or the fraction.

Formula:

The activity of the undecayed sample after a given time is as follows:

$A=A_0{e}^{-\mathrm{Î\pm }t}$ …… (i)

Since, the time of decay is given as:

$$\begin{gathered} 60y=2\left(30y\right) \\ =2T_{\frac{1}{2}} \end{gathered}$$

The fraction left is given using equation (i) as:

$$\begin{gathered} 2^{-2}=\frac{1}{4} \\ =0.250 \end{gathered}$$

Hence, the fraction that remains undecayed is 0.250.

Since, the time of decay is given as:

$$\begin{gathered} 90y=3\left(30y\right) \\ =3T_{\frac{1}{2}} \end{gathered}$$

Consider the fraction left is given using equation (i) as:

$$\begin{gathered} 2^{-3}=\frac{1}{8} \\ =0.125 \end{gathered}$$

Hence, the fraction that remains undecayed is 0.125.