QuestionThe radioactive isotope ${ }^{226} \mathrm{Ra}$ has a half-life of approximately 1599 years. Consider a lab that currently has 45 g of ${ }^{226} \mathrm{Ra}$. (a.) How much of the isotope remains after 1200 years? Round your answer to three decimal places.

Studdy Solution
Calculate the remaining amount after 1200 years:
$$N(1200) = 45 \left(\frac{1}{2}\right)^{\frac{1200}{1599}}$$
First, calculate the exponent:
$$\frac{1200}{1599} \approx 0.750469$$
Now calculate the power of $\frac{1}{2}$:
$$\left(\frac{1}{2}\right)^{0.750469} \approx 0.594$$
Finally, calculate the remaining amount:
$$N(1200) = 45 \times 0.594 \approx 26.730$$
The remaining amount of ${}^{226} \mathrm{Ra}$ after 1200 years is:
$$\boxed{26.730 \text{ grams}}$$