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Chapter 21: Problem 20

One nuclide in each of these pairs is radioactive. Predict which is radioactive and which is stable: (a) \({ }_{20}^{40} \mathrm{Ca}\) and \({ }_{20}^{45} \mathrm{Ca},(\mathbf{b}){ }^{12} \mathrm{C}\) and \({ }^{14} \mathrm{C},(\mathrm{c})\) lead- 206 and thorium- \(230 .\) Explain your choice in each case.

Short Answer

Expert verified
In conclusion: - Pair (a): \({ }_{20}^{40} \mathrm{Ca}\) is stable and \({ }_{20}^{45} \mathrm{Ca}\) is radioactive, because of their even-even and even-odd configurations, respectively. - Pair (b): \({ }^{12} \mathrm{C}\) is stable and \({ }^{14} \mathrm{C}\) is radioactive, as \({ }^{12} \mathrm{C}\) has magic numbers while both are even-even configurations. - Pair (c): Lead-206 is stable and thorium-230 is radioactive, due to the presence of magic numbers in lead-206 in comparison to thorium-230, which only has even numbers of protons and neutrons.

Step by step solution

01

Understand Magic Numbers and Even Number Stability

Magic numbers refer to the specific number of protons or neutrons within a nucleus that leads to a stable configuration. For protons and neutrons, the magic numbers are 2, 8, 20, 28, 50, 82, and 126. In general, nuclides with an even number of protons and an even number of neutrons (even-even) are more stable than their odd counterparts (odd-even, even-odd, or odd-odd). This concept can be used to predict the stability of isotopes.
02

Analyze Pair (a)

For Pair (a), we have: \({ }_{20}^{40} \mathrm{Ca}\) : 20 protons (even) and 20 neutrons (even) \({ }_{20}^{45} \mathrm{Ca}\) : 20 protons (even) and 25 neutrons (odd) Using the notions stated in step 1, we can infer that \({ }_{20}^{40} \mathrm{Ca}\) is more stable due to its even-even configuration, while \({ }_{20}^{45} \mathrm{Ca}\) should be radioactive due to its even-odd configuration.
03

Analyze Pair (b)

For Pair (b), we have: \({ }^{12} \mathrm{C}\) : 6 protons (even) and 6 neutrons (even) \({ }^{14} \mathrm{C}\) : 6 protons (even) and 8 neutrons (even) Both isotopes have an even-even configuration. But \({ }^{12} \mathrm{C}\) possesses a magic number (6 protons and 6 neutrons) which makes it more stable than its counterpart \({ }^{14} \mathrm{C}\). Thus, \({ }^{12} \mathrm{C}\) is stable and \({ }^{14} \mathrm{C}\) is radioactive.
04

Analyze Pair (c)

For Pair (c), we have: Lead-206 (Pb-206): 82 protons (magic number) and 124 neutrons (near magic number) Thorium-230 (Th-230): 90 protons (even) and 140 neutrons (even) Lead-206 has a magic number of protons, and very close to a magic number of neutrons. In comparison, thorium-230 has even numbers of protons and neutrons but does not possess any magic numbers. Hence, Pb-206 (lead-206) is more stable and Th-230 (thorium-230) is radioactive. In conclusion: - Pair (a): \({ }_{20}^{40} \mathrm{Ca}\) is stable and \({ }_{20}^{45} \mathrm{Ca}\) is radioactive. - Pair (b): \({ }^{12} \mathrm{C}\) is stable and \({ }^{14} \mathrm{C}\) is radioactive. - Pair (c): Lead-206 is stable and thorium-230 is radioactive.

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Most popular questions from this chapter

The Sun radiates energy into space at the rate of \(3.9 \times 10^{26} \mathrm{~J} / \mathrm{s}\). (a) Calculate the rate of mass loss from the Sun in \(\mathrm{kg} / \mathrm{s}\). (b) How does this mass loss arise? (c) It is estimated that the Sun contains \(9 \times 10^{56}\) free protons. How many protons per second are consumed in nuclear reactions in the Sun?

A \(65-\mathrm{kg}\) person is accidentally exposed for \(240 \mathrm{~s}\) to a \(15-\mathrm{mCi}\) source of beta radiation coming from a sample of \({ }^{90}\) Sr. (a) What is the activity of the radiation source in disintegrations per second? In becquerels? (b) Each beta particle has an energy of \(8.75 \times 10^{-14} \mathrm{~J},\) and \(7.5 \%\) of the radiation is absorbed by the person. Assuming that the absorbed radiation is spread over the person's entire body, calculate the absorbed dose in rads and in grays. (c) If the \(\mathrm{RBE}\) of the beta particles is 1.0, what is the effective dose in mrem and in sieverts? (d) How does the magnitude of this dose of radiation compare with that of a mammogram ( \(300 \mathrm{mrem}\) )?

When two protons fuse in a star, the product is \({ }^{2} \mathrm{H}\) plus a positron (Equation 21.26 ). Why do you think the more obvious product of the reaction, \({ }^{2} \mathrm{He},\) is unstable?

A \(25.0-\mathrm{mL}\) sample of \(0.050 \mathrm{M}\) barium nitrate solution was mixed with \(25.0 \mathrm{~mL}\) of \(0.050 \mathrm{M}\) sodium sulfate solution labeled with radioactive sulfur-35. The activity of the initial sodium sulfate solution was \(1.22 \times 10^{6} \mathrm{~Bq} / \mathrm{mL}\). After the resultant precipitate was removed by filtration, the remaining filtrate was found to have an activity of \(250 \mathrm{~Bq} / \mathrm{mL}\). (a) Write a balanced chemical equation for the reaction that occurred. (b) Calculate the \(K_{s p}\) for the precipitate under the conditions of the experiment.

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