Nuclei $A$ and $B$ convert into a stable nucleus $C$ . Nucleus $A$ is converted into $C$ by emitting $2 α$ -particles and $3 β$ - particles. Nucleus $B$ is converted into $C$ by emitting one $α$ -particle and $5 β$ -particles. At time $t = 0$ , nuclei of $A$ and $4 N_{0}$ and nuclei of $B$ are $N_{0}$ . Initially, number of nuclei of $C$ are zoer. Half-life of $A$ (into conversion of $C$ ) is $1 m i n$ and that of $B$ is $2 m i n$ . Find the time (in minutes) at which rate of disintegration of $A$ and $B$ are equal.

Question

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Answer 1

We have to find the time at which
$λ_{A} N u_{A} = λ_{B} N u_{B}$
$(\frac{ln 2}{T _{A}}) (4 N_{0} e^{- λ_{A} t}) = (\frac{ln 2}{T _{B}}) (N_{0} e^{- λ_{B} t})$
$\Rightarrow e^{(λ_{A} - λ_{B}) t} = 8$
$\Rightarrow (λ_{A} - λ_{B}) t = ln 8 = 3 (ln 2)$
$(\frac{ln 2}{1} - \frac{ln 2}{2}) t = 3 ln (2)$
$\Rightarrow t = 6 m i n$

Answer 2

We have to find the time at which

λ_{A} N u_{A} = λ_{B} N u_{B}

(\frac{ln 2}{T _{A}}) (4 N_{0} e^{- λ_{A} t}) = (\frac{ln 2}{T _{B}}) (N_{0} e^{- λ_{B} t})

\Rightarrow e^{(λ_{A} - λ_{B}) t} = 8

\Rightarrow (λ_{A} - λ_{B}) t = ln 8 = 3 (ln 2)

(\frac{ln 2}{1} - \frac{ln 2}{2}) t = 3 ln (2)

\Rightarrow t = 6 m i n