Question

Which of the following nuclides would you expect to be radioactive: \({ }_{28}^{62} \mathrm{Ni},{ }_{29}^{58} \mathrm{Cu},{ }_{47}^{108} \mathrm{Ag},\) tungsten- \(184,\) polonium- \(206 ?\) Justify your choices.

Short answer

In conclusion, we expect the following nuclides to be radioactive: \( _{29}^{58}\mathrm{Cu}, _{47}^{108}\mathrm{Ag},\) and \( _{84}^{206}\mathrm{Po}\). This is based on the analysis of magic numbers of protons and neutrons and checking the evenness of protons and neutrons in each nuclide.

Step-by-step solution

Check for magic numbers of protons and neutrons

First, compare the number of protons and neutrons for each nuclide to the magic numbers list. Here are the comparisons: - \( _{28}^{62}\mathrm{Ni} \): 28 protons (magic number), 34 neutrons (not magic number) - \( _{29}^{58}\mathrm{Cu} \): 29 protons (not magic number), 29 neutrons (not magic number) - \( _{47}^{108}\mathrm{Ag} \): 47 protons (not magic number), 61 neutrons (not magic number) - \( _{74}^{184}\mathrm{W} \): 74 protons (not magic number), 110 neutrons (not magic number) - \( _{84}^{206}\mathrm{Po} \): 84 protons (not magic number), 122 neutrons (not magic number)

Check the evenness of protons and neutrons

Now, we'll examine if the nuclei contain even numbers of protons and neutrons: - \( _{28}^{62}\mathrm{Ni} \): 28 protons (even), 34 neutrons (even) - \( _{29}^{58}\mathrm{Cu} \): 29 protons (odd), 29 neutrons (odd) - \( _{47}^{108}\mathrm{Ag} \): 47 protons (odd), 61 neutrons (odd) - \( _{74}^{184}\mathrm{W} \): 74 protons (even), 110 neutrons (even) - \( _{84}^{206}\mathrm{Po} \): 84 protons (even), 122 neutrons (even)

Determine radioactivity

Based on our analysis, we can make the following predictions about the radioactivity of the given nuclides: - \( _{28}^{62}\mathrm{Ni} \): Non-radioactive (28 protons is a magic number, and both protons and neutrons are even) - \( _{29}^{58}\mathrm{Cu} \): Radioactive (no magic numbers, and both protons and neutrons are odd) - \( _{47}^{108}\mathrm{Ag} \): Radioactive (no magic numbers, and both protons and neutrons are odd) - \( _{74}^{184}\mathrm{W} \): Non-radioactive (no magic numbers, but both protons and neutrons are even) - \( _{84}^{206}\mathrm{Po} \): Radioactive (no magic numbers, but both protons and neutrons are even; however, Po is a well-known radioactive element) In conclusion, we expect the following nuclides to be radioactive: \( _{29}^{58}\mathrm{Cu}, _{47}^{108}\mathrm{Ag},\) and \( _{84}^{206}\mathrm{Po}\).

Key concepts

Magic Numbers in Nuclear Chemistry

In nuclear chemistry, the term 'magic number' refers to the number of nucleons (either protons or neutrons) such that they are arranged into complete shells within the nucleus. These magic numbers for protons and neutrons are 2, 8, 20, 28, 50, 82, and 126. When a nuclide has a magic number of protons or neutrons, it is generally more stable than those that do not.

This concept is analogous to the noble gases in chemical terms, where a full valence shell of electrons results in enhanced stability. Nuclides with magic numbers of protons and neutrons exhibit a lower level of energy and increased binding energy, making them less likely to undergo radioactive decay.

For instance, \( _{28}^{62}\mathrm{Ni} \) has 28 protons, which is a magic number, contributing to its stability. On the contrary, none of the other nuclides mentioned – like \( _{29}^{58}\mathrm{Cu} \) or \( _{47}^{108}\mathrm{Ag} \) – have proton counts that correspond to magic numbers, likely making them less stable and more prone to being radioactive.

Nuclear Stability

The stability of a nucleus is determined by various factors, including the ratio of neutrons to protons and the presence of magic numbers. A stable nucleus has a balanced ratio that allows the strong nuclear force to maintain the integrity of the nucleus against the repulsive force of the protons. Usually, a ratio close to 1:1 is favorable for lighter elements, but heavier elements require more neutrons than protons to remain stable.

Another factor is the evenness of the neutron and proton counts. Nuclei with even numbers of both protons and neutrons are generally more stable than those with odd numbers. This is because nucleons tend to form pairs, and unpaired nucleons can lead to instability and radioactivity. For example, \( _{74}^{184}\mathrm{W} \) which has even numbers of protons and neutrons is more likely to be stable, though it doesn't feature magic numbers.

In our exercise, \( _{28}^{62}\mathrm{Ni} \) is likely stable due to its even numbers of protons and neutrons in addition to having a magic number of protons. On the other hand, nuclides like \( _{29}^{58}\mathrm{Cu} \) and \( _{47}^{108}\mathrm{Ag} \) are expected to be radioactive due to having odd numbers of both protons and neutrons, falling outside of the favorability for stability.

Radioactivity

Radioactivity is the process by which unstable atomic nuclei lose energy by emitting radiation. This occurs in the spontaneous process of radioactive decay, where unstable isotopes transform into more stable forms by emitting alpha, beta, or gamma radiation. The drive for a nucleus to decay is the pursuit of increased stability.

Identifying whether a nuclide is likely to be radioactive involves considering whether it is stable, as we've discussed with magic numbers and nuclear stability. It's not just about odd or even numbers of nucleons, but also the overall energy state of the nucleus and whether it is more energetically favorable for it to transform into something else.

In cases like \( _{84}^{206}\mathrm{Po} \) from our exercise, despite both proton and neutron counts being even, polonium is a well-known radioactive element, often decaying via alpha emission due to its position far from the band of stability. This instance shows that while general trends exist in predicting radioactivity, there are always exceptions, and real-world observation is crucial. Historical and experimental evidence for the radioactivity of certain elements, like polonium, play a key role in confirming the theoretical predictions of nuclear behavior.