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At a given instant , say t = 0, two radioactive substances A and B have equal activities. The ratio $\frac{R_{B}}{R_{A}}$ of their activities after time t itself decays with time t as e^-3t. If the half - life of A is $\ln 2$ , the half - life of B is:

Option 1)
$4\ln 2$
Option 2)
$2\ln2$
Option 3)
$\frac{\ln 2}{4}$
Option 4)
$\frac{\ln2}{2}$

Answers (1)

best_answer

Half Life Time -

$t_{1/2}=\frac{ln 2}{\lambda}=\frac{0.693}{\lambda}$

- wherein

Half life is time in which number of nuclei reduced to half of initial number of nuclei.

$t_{\frac{1}{2}}=\frac{\ln 2}{\lambda }$

$\lambda _{a}=1$

at t = 0 , $R _{A}=R _{B}$

$N_{A}e^{-\lambda _{A}T}=N _{B}e^{-\lambda _{B}T}$

at t = 0 , $N _{A}=N _{B}$

At t = t ,

$\frac{R_{A}}{R_{B}}=\frac{N_{o}e^{-\lambda _{A}t}}{N_{o}e^{-\lambda _{B}t}}$

$\frac{1}{e^{-3t}}=e^{-(\lambda _{A}-\lambda _{B})t}$

$\lambda _{A}-\lambda _{B}=-3$

$\lambda _{B}=4$

Option 1)

$4\ln 2$

Option 2)

$2\ln2$

Option 3)
$\frac{\ln 2}{4}$
Option 4)

$\frac{\ln2}{2}$

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