Question

how long does it take a 180 - gram of gold - 198 to decay to 22.5 grams if the half - life of gold - 198 is 2.68 days?

Explanation:

Step1: Calculate the number of half - lives

The formula for radioactive decay is \(N = N_0\times(\frac{1}{2})^n\), where \(N\) is the final amount, \(N_0\) is the initial amount, and \(n\) is the number of half - lives.
We have \(N = 22.5\) grams, \(N_0=180\) grams.
Substitute into the formula: \(22.5 = 180\times(\frac{1}{2})^n\)
\(\frac{22.5}{180}=(\frac{1}{2})^n\)
\(\frac{1}{8}=(\frac{1}{2})^n\)
Since \(\frac{1}{8}=(\frac{1}{2})^3\), so \(n = 3\)

Step2: Calculate the time

The half - life \(T = 2.68\) days.
The time \(t=n\times T\)
Substitute \(n = 3\) and \(T = 2.68\) into the formula: \(t=3\times2.68 = 8.04\) days

Answer:

8.04 days