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At time $t=0$, a material is composed of two radioactive atoms ${A}$ and ${B}$, where ${N}_{{A}}(0)=2 {N}_{{B}}(0)$ The decay constant of both kind of radioactive atoms is $\lambda$. However, A disintegrates to ${B}$ and ${B}$ disintegrates to ${C}$. Which of the following figures represents the evolution of ${N}_{{B}}({t}) / {N}_{{B}}(0)$ with respect to time $t$ ?
${N}_{{A}}(0)={No} . \text { of } {A} \text { atoms at } {t}=0$
${N}_{{B}}(0)={No} . \text { of } {B} \text { atoms at } {t}=0$
Solution
${A} \rightarrow {B}, {B} \rightarrow {C}$
$\frac{{d} {N}_{{B}}}{{dt}}=\lambda {N}_{{A}}-\lambda {N}_{{B}}$
$\frac{{d} {N}_{{B}}}{{dt}}=2 \lambda {N}_{{B}_{0}} {e}^{-\lambda t}-\lambda {N}_{{B}}$
${e}^{-\lambda t}\left(\frac{{d} {N}_{{B}}}{{dt}}+\lambda {N}_{{B}}\right)=2 \lambda {N}_{{B}_{0}} {e}^{-\lambda {t}} \times {e}^{\lambda {t}}$
$\frac{{d}}{{dt}}\left({N}_{{B}} {e}^{\lambda t}\right)=2 \lambda {N}_{{B}_{0}}$, on integrating
${N}_{{B}} {e}^{\lambda t}=2 \lambda {tN}_{{B}_{0}}+{N}_{{B}_{0}}$
${N}_{{B}}={N}_{{B}_{0}}[1+2 \lambda {t}] {e}^{-\lambda {t}}$
$\frac{{d} {N}_{{B}}}{{dt}}=0$ at $-\lambda[1+2 \lambda {t}) {e}^{-\lambda {t}}+2 \lambda {e}^{-\lambda {t}}=0$
${N}_{{B}_{{max}}}$ at ${t}=\frac{1}{2 \lambda}$
