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

#### Solution

To find the mean life t_{1}, 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 (= λN_{0} e^{–λt} Δt). Each of them has lived for time t.

Thus, the total life of all these nuclei would be t λN_{0} 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 N_{0} 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`