Define the term 'decay constant' of a radioactive sample. The of disintergration of a given radioactive nucleus is 10000 disintegrations/s and 5,000 disintegration/s after 20 hr. and 30 hr. respectively from start. Calculate the half life and initial number of nuclei at t = 0.

Explanation

The decay constant of a radioactive element is the reciprocal of the time in which the number of its nuclei reduces to 1/e of its original number. We haveR = `lambda N`
R(20 hrs) = 1000 = `lambdaN_(20)`
`R(30 hrs) = 5000 = lambdaN_(30)`
`N_(20)/N_(30)=2`
This means that the number of nuclei, of the given radioactive nucleus, gets halved in a time of (30 - 20)
hours = 10 hours
Half-life = 10 hours.
This means that in 20 hours (= 2 half-lives), the original number of nuclei must have gone down by a factor of 4. Hence rate of decay at t = 0
`lambdaN_(0)=4lambdaN_(20)`
`R_(0) = 4 xx 10,000=40,000` disintegration per second
`N_(0)=40000/lambda = 40000/(ln_(e)2) xx 10 xx 60 xx 60`
`therefore lambda = (ln_(e)2)/("Half-life")`
`N_(0) = (144 xx 10^(7))/0.693=2.08 xx 10^(9)` nuclei.