1) Gathering the data
The half-life of Radium 223: 11.4 days
Initial Mass of Rd-223 : 120g
The final mass of Rd-223: 100
2) Let's use one of our possible formulas for exponential decay to calculate how long until Radium-223 gets from 120g to 100grams. And applying properties of the Logarithm
[tex]\begin{gathered} N=N_0(\frac{1}{2})^{\frac{t}{t\text{ 1/2}}} \\ 100=120\text{ (}\frac{1}{2})^{\frac{t}{11.4}} \\ \frac{100}{120}=\frac{120}{120}\text{(}\frac{1}{2})^{\frac{t}{11.4}} \\ \frac{5}{6}=(\frac{1}{2})^{\frac{t}{11.4}} \\ \log _{10}\frac{5}{6}=\log _{10}(\frac{1}{2})^{\frac{t}{11.4}} \\ -0.0792=\frac{t}{11.4}\log _{10}\frac{1}{2} \\ -0.0792=\frac{t}{11.4}(-0.301) \\ -0.0792=-\frac{0.301t}{11.4} \\ -0.90288=-0.301t \\ t=2.9996\approx\text{ 3 days} \end{gathered}[/tex]3) So it will take approximately 3 days to the Radium-223 isotope reduce to 100 grams.
