Question

1. if the half - life of uranium - 235 is 7.04 x 10^8 years, and 12.5 g of uranium - 235 remains after 2.82 x 10^9 years, how much of the radioactive isotope was in the original sample?

Explanation:

Step1: Calculate the number of half - lives

The formula for the number of half - lives \(n=\frac{t}{T}\), where \(t = 2.82\times10^{9}\) years (time elapsed) and \(T=7.04\times 10^{8}\) years (half - life).
\(n=\frac{2.82\times 10^{9}}{7.04\times 10^{8}}\approx4\)

Step2: Use the radioactive decay formula

The radioactive decay formula is \(N = N_{0}(\frac{1}{2})^{n}\), where \(N = 12.5\) g (remaining amount), \(n = 4\) (number of half - lives), and \(N_{0}\) (initial amount) is what we want to find.
We can rewrite the formula as \(N_{0}=N\times2^{n}\)
Substitute \(N = 12.5\) g and \(n = 4\) into the formula:
\(N_{0}=12.5\times2^{4}\)
\(N_{0}=12.5\times16\)

Answer:

\(200\) g