/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 35 Which one of each of the followi... [FREE SOLUTION] | 91影视

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Chapter 29: Problem 35

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.

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.
04

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.
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Neutron activation analysis was performed on small pieces of hair that had been taken from the exiled Napoleon after he died on the island of St. Helena in 1821\. This procedure involves exposing the samples to a source of neutrons. Some (stable) arsenic nuclei, if present in the sample, will absorb a neutron. In Napoleon's case the samples did contain abnormally high levels of arsenic, which supported the theory that his death was not due to natural causes. (a) These results came from studying beta emissions of the resulting \({ }^{76} \mathrm{As},\) nucleus. Write the nuclear equation for the neutron absorption and use it to determine the arsenic isotope initially present in the hair. (b) Write the nuclear equation for the subsequent beta decay of \({ }^{76} \mathrm{As}\). Use it to determine the nucleus after this decay.

Write the nuclear equations for (a) the alpha decay of \(237 \mathrm{~Np},\) (b) the \(\beta^{-}\) decay of \(32 \mathrm{P}\), (c) the \(\beta^{+}\) decay of \({ }^{56} \mathrm{Co}\) (d) electron capture in \({ }^{56} \mathrm{Co},\) and \((\mathrm{e})\) the \(\gamma\) decay of \({ }^{42} \mathrm{~K}\) from an excited nuclear state to the ground state (not excited).

A sample of \({ }^{215} \mathrm{Bi}\), which beta decays \(\left(t_{1 / 2}=2.4 \mathrm{~min}\right)\) initially contains one-hundreth of Avogadro's number of nuclei. (a) What is the sample's mass? (b) Write down the beta decay equation and predict the product nucleus. (c) How many bismuth nuclei are present after \(10 \mathrm{~min} ?\) (d) After \(1.0 \mathrm{~h} ?\) (d) What are the activities, in curies and becquerels, at these times?

The half-life of a radioactive isotope is known to be exactly \(1 \mathrm{~h}\). (a) What fraction of a sample would be left after exactly \(3 \mathrm{~h}:(1)\) one-third, (2) one-eighth, or (3) oneninth? (b) What fraction of a sample would be left after exactly 1 day?

The radioactive source in most smoke detectors is \(241 \mathrm{Am},\) which has a half-life of 432 years. In a typical detector, only about \(0.100 \mathrm{mg}\) of this material is needed. (a) Write down its alpha decay equation and predict the product nucleus. (b) What is the initial source activity? (c) What would be the source's activity after 40 years in operation? (d) How many \({ }^{241} \mathrm{Am},\) nuclei would have decaved in this 40 -vear period?
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