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Define the term ‘decay constant’ of a radioactive sample. The rate of disintegration of a given radioactive nucleus is 10000 disintegrations/s and 5000 disintegrations/s after 20 h and 30 h respectively from start. Calculate the half-life and initial number of nuclei at t = 0. |
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Answer» Solution :DECAY CONSTANT of a radioactive ELEMENT is the reciprocal of the time during which the number of nuclides left undecayed is `1/e` times the original number. As per question at time `t_(1) =20h`, decay rate `R_(1) = 10000s^(-1)` and at time `t_(2) = 30h`, decay rate `R_(2) =5000s^(-1)` Thus during the time `t_(2)- t_(1) = (30 - 20) h= 10h`, the decay rate falls to one HALF i.e., `R_(2)/R_(1) = 5000/10000= 1/2` Thus by definition of half-life period it is clear that `T_(1/2) =10h` If the initial decay rate by `R_(0)` at time t = 0, then `t_(1) = 20 h=2T_(1/2)` and hence `R_(1)=(1/2)^(2)=R_(0)/4 IMPLIES R_(0)=4R_(1)=4xx10000=40000s^(-1)` `since R_(0)=-lambdaN_(0) implies N_(0)=R_(0)/(lambda)=(R_(0)xxT_(1/2))/0.693=((40000s^(-1))xx(10h))/0.693=(40000xx10xx60xx60)/0.693=2.08xx10^(10)` |
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