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
$A\to 2B+$ particles $+$ energy
$B\to 3C+$ particles $+$ energy
The decay constant for the species $A$ is ${\lambda }_1=1se{c}^{-1}$ and that for the species $B$ is ${\lambda }_2=100se{c}^{-1}.$ 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 $t_0=2ln(10)sec$ . 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 $\Rightarrow \frac{\mathrm{d}{\mathrm{N}}_{\mathrm{S}}}{\mathrm{dt}}=0$
$\Rightarrow 2{\lambda }_1{\mathrm{N}}_2={\lambda }_2{\mathrm{N}}_{{\mathrm{B}}_2}$
$\Rightarrow {\mathrm{N}}_{{\mathrm{B}}_{\max }}=\frac{2{\lambda }_1}{{\lambda }_2}{\mathrm{N}}_{\mathrm{A}}$
$\Rightarrow {\mathrm{N}}_{{\mathrm{B}}_{\max }}=\frac{2{\lambda }_1}{{\lambda }_2}{\mathrm{N}}_0{\mathrm{e}}^{-\lambda \mathrm{t}}=2$