Question

1. the half - life of isotope x is 2 years. how many years would it take for a 4.0g sample of x to decay and have only 0.50g of it remain.

Explanation:

Step1: Determine 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 \(N = 0.50g\), \(N_0=4.0g\).
Substitute into the formula: \(0.50 = 4.0\times(\frac{1}{2})^n\).
Then \((\frac{1}{2})^n=\frac{0.50}{4.0}=\frac{1}{8}\).
Since \(\frac{1}{8}=(\frac{1}{2})^3\), so \(n = 3\).

Step2: Calculate the time

The half - life \(T = 2\) years.
The time \(t=n\times T\).
Substitute \(n = 3\) and \(T = 2\) into the formula: \(t=3\times2\) years.

Answer:

\(6\) years