\({ }^{235} \mathrm{U}\) decays to \({ }^{207} \mathrm{~Pb}\) via the decay series shown in Figure 30.15. The first decay in the chain, that of \({ }^{235} \mathrm{U}\), has a half-life of \(7.0 \times 10^{8}\) years. The subsequent decays are much more rapid, so we can take this as the half-life for the complete decay of \({ }^{235} \mathrm{U}\) to \({ }^{207} \mathrm{~Pb}\). Certain minerals exclude lead but not uranium from their crystal structure, so when the minerals form they have no lead, only uranium. As time goes on, the uranium decays to lead, so measuring the ratio of lead atoms to uranium atoms allows investigators to determine the ages of the minerals. If a sample of a mineral contains 3 atoms of \({ }^{207} \mathrm{~Pb}\) for every 1 atom of \({ }^{235} \mathrm{U}\), how many years ago was it formed?

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Mass number, A FIGURE 30.15 235U decay series. Alpha decay reduces A by 4 and Z by 2. Beta decay increases Z by I. 239 235 235U 231- 227 Some nuclei can undergo either & or B decay. 223- 219 215 211 207 207Pb is stable. 203 80 82 84 86 88 90 92 Atomic number, Z