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Question
Consider the situation of the previous problem. Suppose the production of the radioactive isotope starts at t = 0. Find the number of active nuclei at time t.
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Solution
Let the number of atoms present at t = 0 be N0.
Let N be the number of radio-active isotopes present at time t.
Then,
N = N0e−λt
Here, `lambda` = Disintegration constant
∴ Number of radioactive isotopes decayed = `N_0 - N = N_0 - N_0e^(-lambdat)`
= `N_0(1-e^(-lambdat))` ...(1)
Rate of decay (R) is given by
`R = lambdaN_0` ...(2)
Substituting the value of `N_0` from equation (2) to equation (1), we get
`N = N_0(1-e^(-lambdat))`
= `R/lambda (1 - e^(-lambdat))`
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