Radioactive Decay Chain for the special case of equal decay constants (parent-daughter)

Question

I have been trying to obtain an analytic solution for a daughter radionuclide's activity (or just the number of daughter atoms), as a function of $t\geq0$, resulting from the decay of a parent radionuclide specifically for the case where $\lambda_{parent}=\lambda_{daughter}$ (and no alternative decay modes so branching ratio=1), subject to initial conditions $\left.{N_{daughter}(t) }\right|_{\;t=0}^{\;}=0$ and $\left.{N_{parent}(t) }\right|_{\;t=0}^{\;}=C\neq0$. This is typically a pretty easy problem, I know, you just solve

$$\frac{dN_{parent}}{dt}=-\lambda_{parent}N_{parent}$$

to get parent solution (exponential decay with initial value of C and decay const $\lambda_{parent}$) then solve

$$\frac{dN_{daughter}}{dt}=\lambda_{parent}N_{parent}-\lambda_{daughter}N_{daughter}$$

to get daughter solution, which is (skipping to multiplying solution for $N_{daughter}$(t) by $\lambda_{daughter}$ to get daughter activity):

$$A_{daughter}=\lambda_{daughter}N_{daughter}(t)=\frac{C\lambda_{parent}\lambda_{daughter}}{\lambda_{daughter}-\lambda_{parent}}(e^{-\lambda_{parent}t}-e^{-\lambda_{daughter}t})+ [\left.{N_{daughter}(t) }\right|_{\;t=0}^{\;}]e^{-\lambda_{daughter}t}$$

$$=\frac{C\lambda_{parent}\lambda_{daughter}}{\lambda_{daughter}-\lambda_{parent}}(e^{-\lambda_{parent}t}-e^{-\lambda_{daughter}t})$$

which is obtained by applying initial cond's.

However, if you note what happens in the case $\lim_{\lambda_{daughter} \to \lambda_{parent}}[A_{daughter}(t)]$, you will see that $A_{daughter}(t)$ seems to go to an indeterminate form (0/0) if you apply direct substitution. In light of this, I wanted to try to apply L'Hopitals rule to this via:

Let

$$g(t,\lambda_{daughter})=C\lambda_{parent}\lambda_{daughter}(e^{-\lambda_{parent}t}-e^{-\lambda_{daughter}t})$$

and

$$h(\lambda_{daughter})=\lambda_{daughter}-\lambda_{parent}$$

then applying

$$\lim_{\lambda_{daughter} \to \lambda_{parent}}[\frac{g(t,\lambda_{daughter})}{h(\lambda_{daughter})}]=[\frac{\lim_{\lambda_{daughter} \to \lambda_{parent}}\frac{dg(t,\lambda_{daughter})}{d\lambda_{daughter}}}{\lim_{\lambda_{daughter} \to \lambda_{parent}}\frac{dh(\lambda_{daughter})}{d\lambda_{daughter}}}]$$

which gives me a solution that doesn't seem unreasonable

$$A_{daughter}(t)=\lambda_{daughter}[\left.{A_{parent}(t)}\right|_{\;t=0}]te^{-\lambda_{daughter}t}$$

but I'm not absolutely 100% sure that this approach is mathematically sound. Can anyone confirm or deny the validity of this approach, and if it is not valid could you help by suggesting an alternative (and valid) approach?

Answer 1

$$ \begin{align} \tag1N_p&=Ce^{-\lambda_pt}\\ \tag2\frac{\mathrm d\ }{\mathrm dt}\left(e^{\lambda_dt}N_d\right) &=\lambda_pCe^{(\lambda_d-\lambda_p)t}\\ \tag3A_d=\lambda_dN_d&=\frac{\lambda_d\lambda_pC}{\lambda_d-\lambda_p}\left(e^{-\lambda_pt}-e^{-\lambda_dt}\right) \end {align} $$ This last equation is in agreement with what you have.

However, it is extremely clear that Equation (2) has a totally different qualitative behaviour when $\lambda_d=\lambda_p=\lambda$; in that case, we would obtain that $$\tag4A_d=\lambda N_d=\lambda^2C\,t\,e^{-\lambda t}$$ which agrees with your L'Hôpital's approach (if you didn't have a typo here), but with manifest correctness. It is probably easier to see that this last equation is dimensionally correct by writing it as $\lambda C\cdot(\lambda t)e^{-\lambda t}$, where it is obvious that $\lambda t$ is dimensionless.