Question

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

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

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.

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.

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.

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.

Key concepts

Magic Numbers

Magic numbers are special numbers in nuclear physics that represent configurations of protons or neutrons that create exceptionally stable nuclei. These numbers are 2, 8, 20, 28, 50, 82, and 126. They are called "magic" because nuclei with these numbers of protons or neutrons are more stable compared to their neighbors with slightly different numbers. This is similar to the concept of filled shells in atomic physics.

• Magic numbers are derived from nuclear shell models that predict energy levels for protons and neutrons.
• Nuclei with both proton and neutron numbers that are magic are doubly magic, leading to extraordinary stability.

An example of a magic number effect is seen in lead-206, which has a magic number of 82 protons. This contributes to its stability, despite having a neutron number that is not magic but still favorable.

Nuclear Stability

Nuclear stability refers to how likely a nucleus will hold together without undergoing radioactive decay. Several factors can influence nuclear stability.

• One of the key factors is the ratio of neutrons to protons. A balanced ratio makes a nucleus more stable. Most stable isotopes have a nearly equal number of neutrons and protons.
• Presence of magic numbers also plays a crucial role in determining stability. Nuclei with magic numbers of protons, neutrons, or both are far more stable.

An example of nuclear stability consideration is the comparison between carbon-12 and carbon-14. Carbon-12 is very stable due to its even numbers of both protons and neutrons, and being close to a magic number, unlike carbon-14, which is unstable due to the presence of extra neutrons.

Even-Even Configuration

An even-even configuration occurs when a nuclide has an even number of both protons and neutrons. This type of configuration contributes significantly to nuclear stability.

• Even-even nuclei are generally more stable because having pairs of nucleons lowers the energy, making the nucleus more tightly bound.
• Approximately 60% of stable nuclei have an even number of protons and an even number of neutrons, reaffirming the stability of this configuration.

The calcium-40 nuclide, for instance, has 20 protons and 20 neutrons, both even numbers, which leads to its greater stability compared to the isotope calcium-45, which has an odd number of neutrons.

Nuclide Analysis

Nuclide analysis is a method used to predict the stability of isotopes by examining their number of protons and neutrons.

• Through nuclide analysis, unstable isotopes can be identified based on their odd or non-magic proton or neutron numbers.
• This analysis is crucial for understanding why certain isotopes like thorium-230 are radioactive while others like lead-206 are not.

The process involves examining whether an isotope's proton or neutron numbers match any magic numbers or follow the even-even rule. This helps predict whether the isotope is radioactive or stable, aiding in fields like nuclear medicine and dating techniques.