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.

Answer 1

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))`