How is the mean life of a given radioactive nucleus related to the decay constant?
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`
