Question

A radioactive nuclide has a half-life of 30.0y. What fraction of an initially pure sample of this nuclide will remain undecayed at the end of (a) 60.0 yand (b) 90.0y?

Short answer

1. The fraction of the sample that will remain undecayed after ${}_{60y}$is 0.250.
2. The fraction of the sample that will remain undecayed after 90y is 0.125.

Step-by-step solution

Write the given data

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

Determine the concept of decay rate

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:

$A=A_0{e}^{-\alpha t}$ …… (i)

a) Calculate the fraction that will remain undecayed after 60y

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.

b) Calculate the fraction that will remain undecayed after 90y

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.