Question

A radioactive sample contains two nucleoids $A$ and $B$ having decay constant $\lambda h{r}^{-1}$ and $2\lambda h{r}^{-1}$. Initially $25$ of the decay comes from $A$. The number of hours it will take before 75% of the total decay comes from $A$ is $n$ (take $\lambda =\ln 3$). Find $n$

Answer 1

Let the initial amount of A be $N_0$.

Since this is 25% of the total initial amount, initial amount of B is $3N_0$.

After time $t$,

$N_A=3N_B$

$⟹N_0{e}^{-\lambda t}=3\left[3N_0{e}^{-2\lambda t}\right]$

$⟹e^{-\lambda t}=9e^{-2\lambda t}$

$⟹e^{\lambda t}=9$

$⟹\lambda t=ln9$

Hence, $t=2h$

Hence, $n=2$