Chapter 25.2, Problem 12LC

Interpretation Introduction

Interpretation: To predict the sample of radioisotope after one half-life and two half-lives.

Concept Introduction: Half-life is the time needed for half the nuclei in a radioactive sample to decay into products. It is denoted by $t_{1/2}$ .

Answer to Problem 12LC

After one half-life, the sample of radioisotope that remains is 50 while after two half-lives, it is 25.

Explanation of Solution

The equation that is used to calculate the amount of isotope that remains in a radioactive sample can be given as follows:

  $A=A_{°}\times {\left(\frac{1}{2}\right)}^n$

Here, $A$ represents the remaining amount of isotope, $A_{°}$ represents the initial amount of isotope, and $n$ represents the number of half-lives.

If initially the amount of radioisotope is 100, then after one half-life, the remaining amount of radioisotope can be calculated as follows taking $n=1$:

  $\begin{array}{c}A=A_{°}\times {\left(\frac{1}{2}\right)}^n\\ =100\left(\frac{1}{2}\right)\\ =50\end{array}$

The amount of radioisotope after one half-life is 50.

If initially the amount of radioisotope is 100, then after two half-lives, the remaining amount of radioisotope can be calculated as follows taking $n=2$:

  $\begin{array}{c}A=A_{°}\times {\left(\frac{1}{2}\right)}^n\\ =100{\left(\frac{1}{2}\right)}^2\\ =\frac{100}{4}\\ =25\end{array}$

The amount of radioisotope after one half-life is 25.

Conclusion

After one half-life, the sample of radioisotope that remains is 50 while after two half-lives, it is 25.