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Question
How is the mean life of a given radioactive nucleus related to the decay constant?
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Solution
To find the mean life t1, we need to use the equation of radioactive law.
The number of nuclei which decay in the time interval t to t + Δt is R(t)Δt (= λN0 e–λt Δt). Each of them has lived for time t.
Thus, the total life of all these nuclei would be t λN0 e–λt Δt. It is clear that some nuclei may live for a short time, while others may live longer. Therefore, to obtain the mean life, we have to integrate the above expression over all times from 0 to ∞ and divide it by the total number N0 of nuclei at t = 0.
Therefore, we get
`t=(lambdaN_0int_0^oote^(-lambdat)dt)/N_0=lambdaint_0^oote^(-lambdat)dt`
Solving by integration-by-parts we get
`t=lambdaxx1/lambda^2=1/lambda`
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