Radioactive Decay — Question 6

Question 6

Competing branches add decay constants rather than half-lives

A parent can decay through two independent channels. Their partial half-lives, defined by $T_i=\ln2/\lambda_i$, are $T_1=6.00\,\mathrm h$ and $T_2=12.0\,\mathrm h$.

1. Find the total half-life and branching probabilities.

2. From $N_0=9000$ parents with no initial daughters, find the expected number of channel-1 decays after 8.00 h.

Question 6 - Solution

Competing branches add decay constants rather than half-lives

(a) Method and calculation

$$\lambda=\lambda_1+\lambda_2\Rightarrow\boxed{T^{-1}=T_1^{-1}+T_2^{-1},\quad T=4.00\,\mathrm h}.$$$$\boxed{b_1=\lambda_1/\lambda=2/3,\quad b_2=1/3}.$$

(b) Interpretation and further calculation

$$N_{\mathrm{dec}}=N_0(1-2^{-8/4})=9000(3/4)=6750.$$$$\boxed{N_1=b_1N_{\mathrm{dec}}=4500}.$$

Physical check

A partial half-life is the lifetime the parent would have if that channel acted alone. The actual lifetime is shorter when both channels are available.