Question

There are two radioactive substances $A$ and $B$. Decay constant of $B$ is two times that of $A$. Initially, both have equal number of nuclei. After $n$ half lives of $A$, rate of disintegration of both are equal. The value of $n$ is

• A. $4$
• B. $2$
• C. $1$
• D. $5$

Answer 1

Answer: (C)

$Let{\lambda }_A=\lambda ,{\lambda }_B=2\lambda$

If No is the total number of atoms in A and B at $t=0$; then the initial rate of

disintegration of $A=\lambda No;$ and initial rate of disintegration of $B=2\lambda No$

$\begin{array}{c}As{\lambda }_B=2{\lambda }_A\\ T_B=\frac{1}{2}{T}_A\\ {\left(-\frac{dN}{dt}\right)}_A-\frac{\lambda No}{2}\\ {\left(-\frac{dN}{dt}\right)}_B=2\frac{2\lambda No}{4}=\frac{\lambda No}{2}\\ \therefore n=1,i.e.,onehalflifeofA\end{array}$

Option C.