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A radioactive sample can decay by two different processes. The half-life for the first process is T_1and that for the second process is T_2Find the effective half - life T of the radioactive sample. |
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Answer» Solution :Let N be the total number of ATOMS of the radioactive sample initially. Let `(dN_1)/(dt)` and `(dN_2)/(dt)` be the initial rates of DISINTEGRATIONS of the radioactive sample by the two processes respectively. Then `(dN_1)/(dt)=lambda_1N` and `(dN_2)/(dt)=lambda_2N` where `lambda_1` and `lambda_2` are the decay constants for the first and second processes respectively. The initial rate of disintegrations of the radioactive sample by both the processes `=(dN_1)/(dt)+(dN_2)/(dt)=lambda_1N+lambda_2N=(lambda_1+lambda_2)N` If `lambda`is the effective decay CONSTANT of the radioactive sample, its initial rate of disintegration. `(dN)/(dt)=lambdaN`, But `(dN)/(dt)=(dN_1)/(dt)+(dN_2)/(dt)` `lambdaN=(lambda_1+lambda_2)N, lambda=lambda_1+lambda_2` `0.693/T_1+0.693/T_2=0.693/T , 1/T=1/T_1+1/T_2 ,T=(T_1T_2)/(T_1+T_2)` |
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