Define the radioactive decay constant.
Demonstrate that the decay constant $\lambda$ and the half-life $t_{1/2}$ of a radioactive isotope satisfy $$\lambda t_{1/2} = \ln 2.$$
A small amount of solution containing the radioactive isotope sodium-24 ($^{24}_{11}\text{Na}$) has an initial activity of $3.8 \times 10^{4}\,\text{Bq}$. Sodium-24, with half-life $15\,\text{hours}$, decays to produce a stable daughter isotope. The whole solution is then added to a container of water. After $36\,\text{hours}$, a water sample of volume $5.0\,\text{cm}^3$, taken from the container, is measured to have an activity of $1.2\,\text{Bq}$. Assuming that the radioactive isotope solution is evenly distributed throughout the container water, calculate the volume of water in the container.