Question

1. chromium - 48 has a short half - life of 21.6 hours. how long will it take 360 g of chromium - 48 to decay to 11.25 g?

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 know that \(N = 11.25\space g\), \(N_0=360\space g\).
Substitute into the formula: \(11.25 = 360\times(\frac{1}{2})^n\).
Then \((\frac{1}{2})^n=\frac{11.25}{360}=\frac{1}{32}\).
Since \(\frac{1}{32}=(\frac{1}{2})^5\), so \(n = 5\).

Step2: Calculate the time

The half - life \(T = 21.6\space h\).
The total time \(t=n\times T\).
Substitute \(n = 5\) and \(T = 21.6\space h\) into the formula: \(t=5\times21.6\space h\).
\(t = 108\space h\).

Answer:

\(108\space h\)