Question

A radioactive substance of half-life 69.3 days is kept in a container. The time in which 80% of the substance will disintegrate will be
A. 1.61 days
B. 16.1 days
C. 161 days
D. 1610 days

Answer 1

Hint: Throughout the course of the reactive disintegration the disintegration constant (\[\lambda \]) remains constant. Any radioactive decay follows the first-order kinetics. And is related to the half-life as given in the formula below. That \[\lambda \] then can be used to find the time and the relation to time is also given below.

Formulas Used:
\[{{t}_{\dfrac{1}{2}}}=\dfrac{\ln 2}{\lambda }\]
\[t=\dfrac{2.303}{\lambda }\log \dfrac{a}{a-x}\]

Complete step by step answer:
Through the course of any radioactive reaction, the disintegration constant (\[\lambda \]) remains constant. So let us start off by finding it out. In the question, it is given that the half-life of the substance is 69.3 days. And we also know that half-life and disintegration constant (\[\lambda \]) is related as
\[{{t}_{\dfrac{1}{2}}}=\dfrac{\ln 2}{\lambda }\]
Or, \[\lambda =\dfrac{\ln 2}{{{t}_{\dfrac{1}{2}}}}\]
We also know that \[\ln 2=0.693\], and it is given that \[{{t}_{\dfrac{1}{2}}}=69.3days\]
Plugging in all the values we get
\[\begin{align}
  & \lambda =\dfrac{\ln 2}{69.3} \\
 & \lambda =\dfrac{0.693}{69.3} \\
 & \lambda =0.01day{{s}^{-1}} \\
\end{align}\]
So the disintegration constant ($\lambda$) is \[0.01day{{s}^{-1}}\].

Now coming to the second part of the question we have to find out the time in which 80% of the substance will disintegrate.
We know that \[t=\dfrac{2.303}{\lambda }\log \dfrac{a}{a-x}\]
Where a is the original amount of substance let that be 100. x is the amount of substance left since 80% of the substance has disintegrated 20% remains. So x=20
Plugging in these values we get
\[\begin{align}
  & t=\dfrac{2.303}{\lambda }\log \dfrac{a}{a-x} \\
 & t=\dfrac{2.303}{0.01}\log \dfrac{100}{20} \\
 & t\approx 161days \\
\end{align}\]
So, the time in which 80% of the substance will disintegrate will be 161 days.
Option C is the correct one.

Note:
The relation between time and amount of radioactive substance remaining differs with the order of the radioactive reaction. But it should be remembered that radioactive decay is a first-order process and thus the time required for the amount of substance to get halved is constant. The relation between time and amount remaining is given by \[t=\dfrac{2.303}{\lambda }\log \dfrac{a}{a-x}\]