Question

The half-life of a radioactive substance is 40 years. How long will it take to reduce to one fourth of its original amount and what is the value of decay constant?
A. 40 years, 0.9173/year
B. 90 years, 9.017/year
C. 80 years, 0.0173/year
D. None of these

Answer 1

Hint:The half-life of a radioactive substance can be described as the time needed for it to reduce into one-half of the initial amount. Half-life is the common way to characterize the rate of decay of radioactive substances, since they vary from each other.

Formula used :
The half-life of a radioactive substance is,
\[{T_{1/2}} = \dfrac{{0.6931}}{\lambda }\]
Where, \[{T_{1/2}}\] is the half-life and \[\lambda \] is the decay or disintegration constant.

Complete step by step solution:
Given, \[T_{1/2} = 40years\]
To find, \[{T_{1/4}} = ?\]
\[\lambda = ?\]
To find one-fourth of the substance, consider,
\[{N_0}\mathop \to \limits^{{T_{1/2}}} \dfrac{{{N_0}}}{2}\mathop \to \limits^{2{T_{1/2}}} \dfrac{{{N_0}}}{4} \\ \]
The half-life of a radioactive substance is,
\[{T_{1/2}} = \dfrac{{0.6931}}{\lambda } \\ \]
So, for the radioactive sample to get one-fourth of its initial amount, it needs two half-lives.
\[{T_{1/4}} = 2{T_{1/2}} \\ \]
\[\Rightarrow {T_{1/4}} = 2 \times {\rm{40}} \\ \]
\[\Rightarrow {T_{1/4}} = 80\,years \\ \]
To find the decay constant,
\[{T_{1/2}} = \dfrac{{0.6931}}{\lambda } \\ \]
\[\Rightarrow \lambda = \dfrac{{0.6931}}{{{T_{1/2}}}} \\ \]
\[\Rightarrow \lambda = \dfrac{{0.6931}}{{40}} \\ \]
\[\therefore \lambda = 0.0173/year\]
The time taken for the radioactive substance to get one-fourth of its initial amount is 80 years and the decay constant is 0.0173/year.

Hence, the correct option is C.

Note: Here, we need to find ,one-fourth of the initial amount of the radioactive substance, hence we calculated it as two half-lives, because during the first half-life the initial amount will be reduced to one-half \[\left( {1/2} \right)\] and when second half-life occurs, the amount of radioactive substance will be reduced by another \[\left( {1/2} \right)\], i,e., \[\left( {\dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{4}} \right)\] which gives us the result.