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Half Life and Average Life

Half-life of ...

Question

Half-life of a radioactive substance A is two times the half-life of another radioactive substance B. Initially, the number of A and B are $N_{A}$ and $N_{B}$ , respectively. After three half-lives of A, number of nuclei of both are equal. Then, the ratio $N_{A} / N_{B}$ is

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Solution

The correct option is B $1 / 8$
$t_{1 / 2} = \frac{ln 2}{λ}$
So, if $λ_{A} = λ$ then $λ_{B} = 2 λ$
$t_{1 / 2 A} = \frac{ln 2}{λ}$
$N_{A} (t) = N_{A} e^{- λ t}$
$N_{B} (t) = N_{B} e^{- 2 λ t}$
Given after $3 t_{1 / 2 A}$ , both are equal
$N_{A} (t) = N_{B} (t)$
$N_{A} e^{- 3 λ \frac{ln 2}{λ}} = N_{B} e^{- 6 λ \frac{ln 2}{λ}}$
$\frac{N_{A}}{N_{B}} = \frac{1}{8}$

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