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

Question

Dr. Abubacker · Accepted Answer

(c) Let $\lambda_{A} = \lambda	herefore\lambda_{B} = 2\lambda$
If $N_{0}$ is total number of atoms in $A$ and $B$ at $t = 0$, then initial rate of disintegration of $A = \lambda N_{0}$, and initial rate of disintegration of $B = 2\lambda\ N_{0}$
As $\lambda_{B} = 2\lambda_{A}$
$	herefore T_{B} = \frac{1}{2}T_{A}$
$ie,\ $half-life of $B$ is half the half-life of $A$.
After one half-life of $A$
$\left( - \frac{dN}{dt} ight)_{A} = \frac{\lambda N_{0}}{2}$
Equivalently, after two half lives of $B$
$\left( - \frac{dN}{dt} ight)_{B} = \frac{2\lambda N_{0}}{4} = \frac{\lambda N_{0}}{2}$
Clearly, $\left( - \frac{dN}{dt} ight)_{A} = - \left( \frac{dN}{dt} ight)_{B}$
after $n = 1\ ie,\ $one half-life of $A$

Q. 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

Explanation

Reviewed by Dr. Abubacker

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