Chapter 11, Problem 11.21UKC

Magnesium-28 decays by β emission to give aluminum-28. If yellow spheres represent ${}_{12}^{28}\text{Mg}$ atoms and blue spheres represent ${}_{13}^{28}\text{Al}$ atoms, how many half-lives have passed in the following sample?

Interpretation Introduction

Interpretation:

The number of half-lives when $\text{Mg-28}$ decays by $\text{β}$-emission to give $\text{Al-28}$ has to be calculated.

Concept Introduction:

Half-life:

The time required to reduce a radioactive isotope to half of its initial value.

Explanation of Solution

Figure 1

From the above figure,

We know that, yellow spheres are representing ${}_{\text{12}}^{\text{28}}\text{Mg}$ and blue spheres are ${}_{\text{13}}^{\text{28}}\text{Al}$

The total number of yellow spheres present in first box is $\text{16}$

The number of yellow spheres present in second box after beta decay is 4

$\underset{\text{100\%}}{\text{16}}\underset{\text{β}}{\overset{\text{1}\text{half-life}}{\to }}\underset{\text{50\%}}{\text{8}}\underset{\text{β}}{\overset{\text{2 half-life}}{\to }}\underset{\text{25\%}}{\text{4}}$

So, the number of half-lives needed is $\text{2}$

Conclusion

The number of half-lives when $\text{Mg-28}$ decays by $\text{β}$-emission to give $\text{Al-28}$ was calculated as two half-lives.