Question

A radioactive substance has a half-life of $4$ moths. Three-Fourth of the substance will decay in

• A. $6$ months
• B. $8$ months
• C. $12$ months
• D. $16$ months

Answer 1

Answer: (B)

$8$ months

$\begin{array}{cc}\text{Given}\to T_{1/2}=4\text{months.}& \\ \text{Time to decay}{\left(\frac{3}{4}\right)}^{\text{th}}\text{times.}& \\ \text{i.e.}{\left(\frac{1}{4}\right)}^{\text{th}}\text{fraction left}& \\ \frac{N}{N_0}& =\frac{1}{4}\{\frac{N}{N_0}=\text{fraction of substance left}\end{array}$

$\begin{array}{c}\text{We know that}\\ \begin{array}{c}{N}_0={\left(\frac{1}{2}\right)}^n[n=n_0\text{of half cycle}]\\ =>\left(\frac{1}{4}\right)={\left(\frac{1}{2}\right)}^n\\ \Rightarrow {\left(\frac{1}{2}\right)}^2={\left(\frac{1}{2}\right)}^n\\ n=2\end{array}\end{array}$

$\begin{array}{cc}2& \text{half cycle require}\\ \text{Total time}& =2\times T_{1/2}\\ & =2\times 4=8\text{months}\end{array}$