The steps below show three of the steps in the radioactive decay chain for \(_{90}^{232}\) Th. The half-life of each isotope is shown below the symbol of the isotope. (a) Identify the type of radioactive decay for each of the steps (i), (ii), and (iii). (b) Which of the isotopes shown has the highest activity? (c) Which of the isotopes shown has the lowest activity? (d) The next step in the decay chain is an alpha emission. What is the next isotope in the chain? [Sections 21.2 and 21.4]

For each step in the decay chain, examine the change in atomic number and mass number to determine the type of decay. There are three types of radioactive decay we need to consider: alpha decay, beta decay, and gamma decay. (i) \(_{90}^{232}\text{Th} \rightarrow _{88}^{228}\text{Ra}\) In this step, the atomic number decreases by 2, and the mass number decreases by 4. This indicates an alpha decay. (ii) \(_{88}^{228}\text{Ra} \rightarrow _{89}^{228}\text{Ac}\) In this step, the atomic number increases by 1, and the mass number remains the same. This indicates a beta decay. (iii) \(_{89}^{228}\text{Ac} \rightarrow _{89}^{228}\text{Th}\) In this step, the atomic number remains the same and the mass number remains the same. This indicates a gamma decay.

To determine the isotope with the highest activity, we can compare their half-lives. The isotope with the shortest half-life will have the highest activity because it is decaying more quickly. Half-lives given: - Th: 14.0 billion years - Ra: 5.75 years - Ac: 6.25 hours Based on these half-lives, isotope Ac has the highest activity.

To determine the isotope with the lowest activity, we can again compare their half-lives. The isotope with the longest half-life will have the lowest activity because it is decaying less rapidly. Based on the half-lives given, isotope Th has the lowest activity.

According to the problem, the next step in the decay chain is an alpha emission. In an alpha emission, the atomic number decreases by 2 and the mass number decreases by 4. We will apply this to the current isotope (Th) to determine the next isotope. Current isotope: \(_{89}^{228}\text{Th}\) Applying alpha emission: - Atomic number: 89 - 2 = 87 - Mass number: 228 - 4 = 224 The next isotope in the decay chain is: \(_{87}^{224}\text{Fr}\), francium.