Question

In a certain hypothetical radioactive decay process, species $A$ decay into species $B$ and species $B$ decays into species $C$ , according to the reactions
$A \rightarrow 2B+$ particles $+$ energy
$B \rightarrow 3C+$ particles $+$ energy
The decay constant for the species $A$ is $\lambda _{1}=1sec^{- 1}$ and that for the species $B$ is $\lambda _{2}=100sec^{- 1}.$ Initially $10^{4}$ moles of the species of $A$ were present while there was none of $B$ and $C$ . It was found that species $\mathrm{B}$ reaches its maximum number at a time $t_{0}=2ln\left(\right.10\left.\right)sec$ . What will be the value of the maximum number of moles of $B$ ? $\left(e^{2 . 303} = 10\right)$

• A. 2
• B. 3
• C. 4
• D. 6

Answer 1

Answer: (A)

In a certain
$\frac{\mathrm{d} \mathrm{N}_{\mathrm{A}}}{\mathrm{dt}}=-\lambda_{1} \mathrm{N}_{\mathrm{A}} \frac{\mathrm{dN}_{\mathrm{S}}}{\mathrm{dt}}=2 \lambda_{\mathrm{T}} \mathrm{N}_{\mathrm{A}}-\lambda_{2} \mathrm{N}_{\mathrm{B}}$
$\mathrm{N}_{\mathrm{e}}=$ maximum $\Rightarrow \frac{\mathrm{d} \mathrm{N}_{\mathrm{S}}}{\mathrm{dt}}=0$
$\Rightarrow 2 \lambda_{1} \mathrm{N}_{2}=\lambda_{2} \mathrm{N}_{\mathrm{B}_{2}}$
$\Rightarrow \mathrm{N}_{\mathrm{B}_{\max }}=\frac{2 \lambda_{1}}{\lambda_{2}} \mathrm{N}_{\mathrm{A}}$
$\Rightarrow \mathrm{N}_{\mathrm{B}_{\mathrm{max}}}=\frac{2 \lambda_{1}}{\lambda_{2}} \mathrm{N}_{0} \mathrm{e}^{-\lambda \mathrm{t}}=2$