A radionuclide $A_1$ with decay constant $\lambda_1$ transforms into a radionuclide $A_2$ with decay

Question

A radionuclide $A_1$ with decay constant $\lambda_1$ transforms into a radionuclide $A_2$ with decay constant $\lambda_2$. Assuming that at the initial moment the preparation contained only the radionuclide $A_1$, after what time interval will the activity of radionuclide $A_2$ reach its maximum value?

Professor PC Thomas Classes · Answer

The number of nuclei of $A_2$ as a function of time is given by:
$N_2(t) = \frac{\lambda_1 N_{10}}{\lambda_2 - \lambda_1} (e^{-\lambda_1 t} - e^{-\lambda_2 t})$
Activity is proportional to $N_2$. To find the maximum, set $dN_2/dt = 0$:
$-\lambda_1 e^{-\lambda_1 t} - (-\lambda_2 e^{-\lambda_2 t}) = 0 \implies \lambda_1 e^{-\lambda_1 t} = \lambda_2 e^{-\lambda_2 t}$
$\ln(\lambda_1) - \lambda_1 t = \ln(\lambda_2) - \lambda_2 t \implies (\lambda_2 - \lambda_1)t = \ln(\lambda_1 / \lambda_2)$
$t = \frac{1}{\lambda_2 - \lambda_1} \ln\left(\frac{\lambda_1}{\lambda_2}\right)$.