Chapter 24.2, Problem 11PP

Interpretation Introduction

Interpretation:

The amount of sample B and C is to be calculated when sample A has 16.2 g remaining.

Concept introduction:

The time needed for one half of nuclei of radioisotope to decay into its products is the half life. The formula to calculate remaining amount of radioactive element is,

$N=N_0{\left(\frac{1}{2}\right)}^n$

Where $N$ is remaining amount, $N_0$ is the initial amount and n is the number of half lives that have passed.

The n can be replaced with the equivalent quantity $t/T$ where t is the elapsed time and T is duration of half life

Answer to Problem 11PP

The amount of B is 32.3 g and amount of C is 29.2 g.

Explanation of Solution

Sample A will have 16.2 grams reaming after two half-lives, or 10.54 years. For sample B, amount remaining,

$\begin{array}{c}\text{Amount}\text{remaining}\text{=}\left(\text{Initial}\text{amount}\right){\left(\frac{1}{2}\right)}^{\frac{t}{T}}\\ =\left(58.4\text{g}\right){\left(\frac{1}{2}\right)}^{\frac{10.54y}{12.32y}}\\ =\left(58.4\text{g}\right){\left(\frac{1}{2}\right)}^{0.856}\\ =32.3\text{g}\end{array}$

For sample C, amount remaining,

$\begin{array}{c}\text{Amount}\text{remaining}\text{=}\left(\text{Initial}\text{amount}\right){\left(\frac{1}{2}\right)}^{\frac{t}{T}}\\ =\left(37.6\text{g}\right){\left(\frac{1}{2}\right)}^{\frac{10.54y}{28.79y}}\\ =\left(58.4\text{g}\right){\left(\frac{1}{2}\right)}^{0.366}\\ =29.2\text{g}\end{array}$

Conclusion

The measurement of radioactive decay is done in half lives.