Chapter 25.2, Problem 10SP

Interpretation Introduction

Interpretation: To determine if all atoms of thorium would decay in 48 days.

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 10SP

Thus, in this case, the half-life of thorium-234 is 24 days, which means that every 24 days, half of the atoms of thorium will decay.

Explanation of Solution

The half-life is 24.1 days.

The time is 48 days.

Thus, the number of half-lives can be calculated as follows:

  $\begin{array}{c}n=\frac{t}{t_{1/2}}\\ =\frac{\text{48 }\cancel{\text{days}}}{\text{24}\text{.1 }\cancel{\text{days}}}\\ \simeq 2\end{array}$

Hence, the total number of half-lives are 2.

The equation that is used to calculate the mass of radioisotope that remains in a sample can be given as follows:

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

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

So, in this case,

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

Conclusion

Thus, in this case, the half-life of thorium-234 is 24.1 days, which means that every 24.1 days, half of the atoms of thorium will decay.