Radioactive Decay — Question 7

Question 7

Constant production competes with radioactive loss

A process produces a radionuclide at constant rate P from t=0, starting with N=0. The nuclei decay with constant lambda. Production is stopped at $t_p=2T_{1/2}$.

1. Solve for N(t) while production continues and identify saturation activity.

2. Find activity immediately when production stops and one half-life later, in units of P.

Question 7 - Solution

Constant production competes with radioactive loss

(a) Method and calculation

$$\dot N=P-\lambda N\Rightarrow\boxed{N(t)=\frac P\lambda(1-e^{-\lambda t})}.$$$$\boxed{\mathcal A(t)=P(1-e^{-\lambda t}),\quad\mathcal A_{\mathrm{sat}}=P}.$$

(b) Interpretation and further calculation

$$\boxed{\mathcal A(t_p)=P(1-1/4)=3P/4}.$$$$\boxed{\mathcal A(t_p+T_{1/2})=(3P/4)(1/2)=3P/8}.$$

Physical check

At saturation, production and decay rates balance although individual nuclei continue to be replaced. After production stops, the ordinary decay law applies.