Chapter EH, Problem 20R

Solution

Interpretation : The amount of sample of the radium-226 left is to be calculated.

Concept Introduction : The chemical element radium has the symbol $\text{Ra}$ . The only radioactive member of the group of alkaline earth metals, radium is the hardest known alkaline earth metal. The formula to calculate the amounts of radioactive elements $\text{N}$

is given as:

  $\text{N}\text{=}{\text{N}}_{\text{o}}\left(\frac{\text{1}}{\text{2}}\right)\text{n}$

Where ${\text{N}}_{\text{o}}$ is the initial amount, $\text{n}$ is the number of half-lives passed.

The leftover sample of radium-226 is $15.625\text{g}$ .

Given information:

The initial amount of sample is $\text{500}\text{g}$ .

The elapsed time is $\text{1800 years}$ .

The half-life is $\text{1600 years}$ .

Explanation:

The ratio of elapsed time to the half-life is given as:

  $\text{n}\text{=}\frac{\text{t}}{\text{T}}$

Where $\text{t}$ is the elapsed time and $\text{T}$ is the half-life.

To calculate the ratio of elapsed time to the half-life, substitute the values in the above formula:

  $\begin{array}{c}\text{n}\text{=}\frac{\text{t}}{\text{T}}\\ \text{=}\frac{\text{1800 years}}{\text{1600 years}}\\ \text{n}\text{=}\text{5}\end{array}$

The formula to calculate the amounts of radioactive elements $\text{N}$

is given as:

  $\text{N}\text{=}{\text{N}}_{\text{o}}\left(\frac{\text{1}}{\text{2}}\right)\text{n}$

Where ${\text{N}}_{\text{o}}$ is the initial amount, $\text{n}$ is the number of half-lives passed.

To calculate the amount of radioactive element, substitute the values in the above formula:

  $\begin{array}{c}\text{N}\text{=}{\text{N}}_{\text{o}}{\left(\frac{\text{1}}{\text{2}}\right)}^{\text{n}}\\ \text{N}\text{=}\text{500}{\left(\frac{1}{2}\right)}^5\\ =500\left(\frac{1}{32}\right)\\ \text{N}=15.625\text{g}\end{array}$