Chapter 20, Problem 43R

To determine

To Calculate: The number of half-lives elapsed when the amount of a sample’s radioactive isotope is reduced to $3.125\mathrm{\%}$ of the original amount on the sample.

Answer to Problem 43R

5 half-lives have elapsed when the amount of a sample’s isotope is reduced to 3.125 percent of original sample.

Explanation of Solution

Given info:

The given percentage is 3.125.

Formula used:

The number of half-lives is calculated by the formula given below:

  $\frac{\mathrm{N}}{{\mathrm{N}}_0}={\left(\frac{1}{2}\right)}^{\mathrm{n}}$

Where, $\mathrm{N}$ is final amount, ${\text{N}}_{\text{0}}$ is the initial amount of the sample, $\text{n}$ is the number of half-lives.

Calculation:

By using the given information in the formula,

  $\begin{array}{c}\frac{\mathrm{N}}{{\mathrm{N}}_0}=3.125\mathrm{\%}=0.03125\\ \frac{\mathrm{N}}{{\mathrm{N}}_0}=\frac{1}{32}\\ \left(\frac{1}{32}\right)={\left(\frac{1}{2}\right)}^{\mathrm{n}}\\ {\left(\frac{1}{2}\right)}^5={\left(\frac{1}{2}\right)}^{\mathrm{n}}\\ \mathrm{n}=5\end{array}$

Conclusion:

The number of half-lives is $5$ .