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A radioactivity decay is given by A unde...

A radioactivity decay is given by `A underset(t_(1//2)=8yrs)(rarr)B`
Only A is present at `t = 0`. Find the time at which if we are able to pick one atom out of the sample, then probability of getting B is `15` getting a.

Text Solution

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`{:(,,A,rarr,B),(at,t=0,N_(0),,0),(at,t=t,N,,N_(0)-N):}`
Probability of getting `A, P_(A) = (N)/(N_(0))`
Probility of getting `B, P_(B) = (N_(0) - N)/(N_(0)) rArr P_(B) = 15 P_(A) rArr (N_(0) - N)/(N_(0)) = 15 (N)/(N_(0)) rArr N_(0) = 16N rArr N = (N_(0))/(16)`
Remaining nuclei are `1/16` th of initial nuclei, hence required time `t = 4` half lives `= 32` years
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