Chapter 19.1, Problem 6RQ

Interpretation Introduction

Interpretation: The fraction of ${}^{224}\text{Ra}$ sample that remains after one month is to be calculated.

Concept introduction: Half-life of a radioactive sample is the time taken by it to reduce its concentration to half. The formula used to calculate the amount of sample left after half-lives is

  $\text{Half-life}=\frac{1}{2^n}$

Answer to Problem 6RQ

Fraction of ${}^{224}\text{Ra}$ sample that remains after one month is $=\frac{1}{321.79}$ .

Explanation of Solution

Half-life of ${}^{224}\text{Ra}=3.6\text{ days}$

Formula used to calculate the amount of sample left after half-lives is

  $\text{Half-life}=\frac{1}{2^n}$

  $\begin{array}{c}\text{Number of half lives in one month = }\frac{\text{30 days}}{3.6\text{ day}}\\ \text{ = 8}\text{.33}\end{array}$

The amount of ${}^{224}\text{Ra}$ sample left after $\text{ 8}\text{.33}$ half-lives is

  $\begin{array}{l}=\frac{1}{{\left(2\right)}^{8.33}}\\ =\frac{1}{321.79}\end{array}$

Conclusion

After $\text{ 8}\text{.33}$ half-lives, fraction of ${}^{224}Ra$ sample that remains after one month is $=\frac{1}{321.79}$ .