Radioactive Decay — Question 3

Question 3

Activity and sample mass are connected through the decay constant

A pure radionuclide has molar mass $M=100\,\mathrm{g/mol}$ and half-life 10.0 days. Its initial mass is $1.00\,\mu\mathrm g$. Use $N_A=6.022\times10^{23}\,\mathrm{mol^{-1}}$ and 86400 s per day.

1. Find the initial number of nuclei and activity in becquerels.

2. Find its activity and remaining mass after 30.0 days.

Question 3 - Solution

Activity and sample mass are connected through the decay constant

(a) Method and calculation

$$\boxed{N_0=(m/M)N_A=(10^{-6}/100)(6.022\times10^{23})=6.022\times10^{15}}.$$$$\boxed{\mathcal A_0=\lambda N_0=\frac{\ln2}{864000}(6.022\times10^{15})=4.831\times10^9\,\mathrm{Bq}}.$$

(b) Interpretation and further calculation

$$30.0\,\mathrm d=3T_{1/2}\Rightarrow\boxed{\mathcal A=\mathcal A_0/8=6.039\times10^8\,\mathrm{Bq}}.$$$$\boxed{m=m_0/8=.125\,\mu\mathrm g}.$$

Physical check

Activity measures decays per second, not energy released per decay. The mass specified is the remaining parent mass, excluding accumulated daughters.