Question

1. a 50.0 g sample of ¹⁴n decays to 12.5 g in 14.4 seconds. what is its half - life?

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.
Given \(N_0 = 50.0\space g\), \(N=12.5\space g\).
Substitute into the formula: \(12.5 = 50.0\times(\frac{1}{2})^n\)
\(\frac{12.5}{50.0}=(\frac{1}{2})^n\)
\(\frac{1}{4}=(\frac{1}{2})^n\)
Since \(\frac{1}{4}=(\frac{1}{2})^2\), so \(n = 2\)

Step2: Calculate the half - life

We know that the total time \(t = 14.4\space s\) and the number of half - lives \(n = 2\).
The formula for half - life \(t_{1/2}=\frac{t}{n}\)
Substitute \(t = 14.4\space s\) and \(n = 2\) into the formula: \(t_{1/2}=\frac{14.4}{2}\)

Answer:

\(7.2\space s\)