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Question
Define the term 'decay constant' of a radioactive sample. The rate of disintegration of a given radioactive nucleus is 10000 disintegrations/s and 5,000 disintegrations/s after 20 hr. and 30 hr. respectively from start. Calculate the half-life and the initial number of nuclei at t= 0.
Solution
The decay constant is the fraction of the number of atoms that decay in one second.
It is denoted by A.
Let N0 be the initial number of nuclei,
Let λ be the decay constant
Let t1/2 be the half-life
The instantaneous activity of radioactive material is given by `A = A_0e^(-lambdat)`
Where A0 is activity at t = 0
Therefore, after 20 hours is 10,000 disintegrations per second
`10,000 = A_0e^(-lambda(20 xx 3600))` ...(1)
Activity after 30 hours is 5,000 disintegrations per second
`5000 = A_0e^(-lambda(30 xx 3600))` ..(2)
On dividing (1) by (2),
`2 = e^(lambda xx 3600)`
⇒ `lambda = ln 2/36000 = 1.92 xx 10^-5`
And half life is,
`ln 2/(1.92 xx 10^-5) = 36,000"s" = 10` hours
Since,
`(dN)/(dt) = lambdaN`
`1000 = (1.92 xx 10^-5) xx N_1`
⇒ `N_1 = (10,000)/(1.92 xx 10^-5) = 5.208 xx 10^8`
Therefore, the half life is 10 hours, thus the initial number of nuclei is `N_0 = 10.416 xx 10^8`
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