Problem 35 - College Physics

Question

Which one of each of the following pairs of nuclei would you expect it to be easier to remove a neutron from: (a) \({ }^{16} \mathrm{O}\) or \({ }^{17} \mathrm{O} ;\) (b) \({ }_{20}^{40} \mathrm{Ca}\) or \({ }_{20}^{42} \mathrm{Ca}\) ; (c) \({ }^{10} \mathrm{~B}\) or \(\frac{11}{5} \mathrm{~B}\) (d) \({ }^{208} \mathrm{~Pb}\) or \({ }_{83}^{209}\) Bi? State your reasoning for your choice in each case.

Step-by-Step Solution

Answer

(a) 17O, (b) 42Ca, (c) 11B, (d) Neither.

Step 1: Analyze the nuclear composition

For each pair of nuclei presented, identify the number of protons (atomic number) and neutrons in each nucleus. This is crucial for understanding their stability and the ease with which a neutron might be removed.

Step 2: Compare neutron numbers

For each pair, determine the neutron numbers: \[(a)\] \(^{16}O\) has 8 neutrons vs \(^{17}O\) with 9 neutrons.\[(b)\] \(^{40}Ca\) has 20 neutrons vs \(^{42}Ca\) with 22 neutrons.\[(c)\] \(^{10}B\) has 5 neutrons vs \(^{11}B\) with 6 neutrons.\[(d)\] \(^{208}Pb\) has 126 neutrons vs \(^{209}Bi\) with 126 neutrons.

Step 3: Consider nuclear stability

The stability of nuclei often depends on the neutron-to-proton ratio. Neutron-rich isotopes tend to be less stable, making it easier to remove a neutron. More neutrons generally increase the likeliness of neutron removal.

Step 4: Application to each pair

Apply the stability observation to each pair:- \[(a)\] \(^{17}O\) is more neutron-rich (9 neutrons) than \(^{16}O\), making it generally easier to remove a neutron from \(^{17}O\).- \[(b)\] Similarly, \(^{42}Ca\) has more neutrons than \(^{40}Ca\), thus it is easier to remove a neutron from \(^{42}Ca\).- \[(c)\] \(^{11}B\) is more neutron-rich than \(^{10}B\), making it easier to remove a neutron from \(^{11}B\).- \[(d)\] In this case, both \(^{208}Pb\) and \(^{209}Bi\) have the same neutron number. Thus, additional context such as known energy levels or nuclear shell models might guide us, but technically neither is more favorable just by basic composition.

Step 5: Conclusion

Summarize the findings for clarity. For the given nuclei pairs, it is generally easier to remove a neutron from the isotope with the higher neutron number, assuming no other factors dominate.

Key Concepts

neutron removal

When discussing nuclear stability, the ease of neutron removal is an intriguing aspect. Neutrons play a significant role in an atom's nucleus, as they help bind protons together through the strong nuclear force. However, if a nucleus becomes too heavy with neutrons, it tends to be less stable. This instability can make neutron removal more achievable.

• For instance, between isotopes with differing neutron numbers, such as those compared in the exercise, the isotope with the extra neutron is typically less bound and thereby more prone to having a neutron removed.
• Removing a neutron can be spontaneous or require minimal external influence, particularly if the nucleus is neutron-rich and on the edge of a balance for stability.

This insight helps us understand why, in our exercise, isotopes like ^{17}O, ^{42}Ca, and ^{11}B are selected as having easier neutron removal compared to their counterparts with fewer neutrons.

neutron to proton ratio

The neutron-to-proton ratio is a crucial factor in determining an isotope's stability. A nucleus seeks a state where its energy is minimized, which often corresponds to the right balance between neutrons and protons. A stable ratio, particularly for lighter elements, tends to be close to 1:1.

• In heavier elements, more neutrons are needed to counteract the increasing repulsive force between the more significant number of protons.
• If the neutron-to-proton ratio is too high, the nucleus may become unstable, leading it to shed neutrons to regain stability.

This principle explains why neutron-rich isotopes like ^{17}O, ^{42}Ca, and other similar ones show a greater tendency towards neutron shedding as a pathway back to stable ratios.

isotopes

Isotopes are variants of a particular chemical element that have the same number of protons but differ in the number of neutrons. This difference often impacts the atomic mass and the stability of the isotope.

• For example, ^{16}O and ^{17}O are both isotopes of oxygen.
• Their isotopic nature influences properties such as how they behave in nuclear reactions or stability under certain conditions.

Isotopes with higher numbers of neutrons, like those mentioned in each nuclear pair from the exercise, tend to possess slightly different nuclear energies and stability characteristics, making the concept of isotopes critical in understanding nuclear behaviors and phenomena.

atomic structure

The atomic structure is foundational to understanding chemistry and physics. It consists of a dense central nucleus surrounded by a cloud of electrons. The nucleus itself has both neutrons and protons. The arrangement of these subatomic particles defines the element's properties and behavior.

• Protons determine the elemental identity of the atom. Their number is the atomic number.
• Neutrons contribute to the atomic mass and influence the stability of the nucleus.

In the context of our exercise, understanding atomic structure allows us to rationalize why extra neutrons, like those in ^{17}O or ^{42}Ca, affect the stability and make it easier for specific nucleons like neutrons to be removed.