Question

Nuclei A and B convert into a stable nucleus C. Nucleus A is converted into C by emitting 2 $\alpha -$ particles and $3\beta -$ particles. Nucleus B is converted into C by emitting one $\alpha -particle$ and 5 $\beta -particles$. At time $t=0$, nuclei of A are $4N_0$ and nuclei of B are $N_0$. Initially, number of nuclei of C are zero. Half-life of A (into conversion of C) is $1$ min and that of B is $2$ min. Find the time (in minutes) at which rate of disintegration of A and B are equal.

Answer 1

$A\to C$
$B\to C$
First order decay:
$\frac{dA}{dt}=-{\lambda }_{A}A$
$\frac{dB}{dt}=-{\lambda }_{B}B$
$A=4N_o{e}^{-{\lambda }_{A}t}$
$B=N_o{e}^{-{\lambda }_{B}t}$

Rates of disintegration equal:
$\frac{dA}{dt}=\frac{dB}{dt}$
$4{\lambda }_A{e}^{-{\lambda }_{A}t}={\lambda }_B{e}^{-{\lambda }_{B}t}$
$4\frac{ln2}{1}{e}^{-\frac{ln2}{1}t}=\frac{ln2}{2}{e}^{-\frac{ln2}{2}t}$
$t=6min$