Question

A stable nuclei C is formed from two radioactive nuclei A and B with decay constant of `lambda_1` and `lambda_2` respectively. Initially, the number of nuclei of A is `N_0` and that of B is zero. Nuclei B are produced at a constant rate of P. Find the number of the nuclei of C after time t.

Answer 1

Correct Answer - A::B::C::D
We have for B `(dN_B)/(dt)=P-lambda_2N_B`
`implies int_0^(N_B)(dN_B)/(P-lambda_2N_B)=int_0^tdt`
`implies1n((P-lambda_2N_B)/(P))=-lambda_2t`
`implies N_B=P(1-e^(-lambda_2t))/(lambda_2)`
The number of nuclei of A after time t is
`N_A=N_0e^(-lambda_1t)`
Thus, `(dN_c)/(dt)=lambda_1N_A+lambda_2N_B`
`(dN_c)/(dt)=lambda_1N_0e^(-lambda_1t)+P(1-e^(-lambda_2t))`
`implies N_c=N_0(1-e^(-lambda_1t))+P(t+((e^(-lambda_2t))-1)/(lambda_2))`