Chapter 25, Problem 13STP

Interpretation Introduction

Interpretation:

Estimating the percentage of remaining radioisotope.

Concept introduction:

Radioisotopes can be defined as atoms that contain an unstable combination of neutrons and protons, or excess energy in their nucleus.

Answer to Problem 13STP

  $\text{7}\text{.8\%}$

Explanation of Solution

Step 1:

The given graph represents the decay curve for radioactive isotope where on the x-axis number of half-lives is given, and on the y-axis radioisotope remaining in % is given.

We have to find out the percentage of remaining radioisotope after $\text{3}{\text{.75 t}}_{\text{1}/\text{2}}$

If we use $\text{100\%}$ of radioisotope initially, after one half-life we will have $\frac{1}{2}$ of that percentage $\left(50\%\right)$ . After the second half-life, we will have $\frac{1}{2}$ of the remaining percentage and that is $25\%$ , while after the third half-life we will have $12.5\%$ of the remaining isotope.

Step 2:

After $0.75$ half-lives, we will have an additional $75\%$ from the half nuclei (6.25%) that will decay which is $\text{0}\text{.75×6}\text{.25=4}\text{.7\%}$ more.

Additional nuclei that will decay are represented by this value, so we now have $\text{12}\text{.5\%-4}\text{.7\%=7}\text{.8\%}$ .

In conclusion, after $\text{3}{\text{.75t}}_{\frac{\text{1}}{\text{2}}}$ we will have $\text{7}\text{.8\%}$ of remaining radioisotope.

This value can be confirmed if we simply $\text{3}{\text{.75t}}_{\frac{\text{1}}{\text{2}}}$ on the x-axis and find the correspondent value on the y-axis which is around $\text{7}\text{.5\%}$