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# CBSE Class 12-science Answered

how derive a equation for average halflife of a radioactive substance?
Asked by Abhishika John | 13 Feb, 2018, 04:20: PM
"Average half-life" is meaningless phrase because half-life is constant for given radioactive substance.
You can get the derivation for half-life from your physics text book or any physics book that has radioactivity as one of the chapter.

I assume that the student want 'mean-life' of radioactove substance and I am giving below the derivation of that

Radioactive decay is directly proportional to the amount of substance or number of atoms present at that  moment.

mathematically it means
where dN/dt is rate of disintegration, λ is decay constant and N is number of atoms present at that instant.
We can negelect the -ve sign in the above equation, it just indicates that number of atoms keep on decreasing.
Hence for dN atoms that lived between t and t+dt can be obtained from the above equation and it is given by

In the above equation, i have used the radioactice decay equation
where N0 is the number of atoms present when t = 0

Since these dN atoms have individual life time between t and t+dt and dt is very small, we get lifetime for dN atoms as

since life time varies from 0 to infinity , we get total life time of N0 atoms as
mean life time
above integration can be solved by considering
By solving the above integration we get maen life time

Answered by Thiyagarajan K | 13 Feb, 2018, 05:48: PM

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