Question

For a radioactive material, its activity $A$ and rate of change of its activity $R$ are defined as $A=\frac{-dN}{dt}$ and $R=\frac{-dA}{dt}$, where $N\left(t\right)$ is the number of nuclei at time $t$. Two radioactive sources $\text{P}$ (mean life $\tau$) and $\text{Q}$ (mean life $2\tau$) have the same activity at $t=0$. Their rates of change of activities at $t=2\tau$ are $R_P$ and $R_Q$, respectively, If $\frac{R_P}{R_Q}=\frac{n}{e},$ then the value of $n$ is ____.

Answer 1

According to the law of radioactivity,

$A_P=A_0{e}^{\frac{-t}{\tau }},A_Q=A_0{e}^{\frac{-t}{2\tau }}$

Given that, $R=\frac{-dA}{dt}$

$\therefore R_P=\frac{A_0}{\tau }{e}^{\frac{-t}{\tau }}\&R_Q=\frac{A_0}{2\tau }{e}^{\frac{-t}{2\tau }}$

$\text{at}t=2\tau$

$\frac{R_P}{R_Q}=\frac{\frac{A_0}{\tau }{e}^{-2}}{\frac{A_0}{2\tau }{e}^{-1}}$

$\Rightarrow \frac{R_P}{R_Q}=\frac{2}{e}$

Given that, $\frac{R_P}{R_Q}=\frac{n}{e}$

$\therefore n=2$

Hence, $2$ is the correct answer.