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Chapter 10: Problem 4

Why is the number of neutrons greater than the number of protons in stable nuclei that have an \(A\) greater than about 40? Why is this effect more pronounced for the heaviest nuclei?

Short Answer

Expert verified
In stable nuclei with a mass number \(A\) greater than 40, the number of neutrons is greater than the number of protons to balance the repulsive electrostatic force between protons and the attractive nuclear force between nucleons. Neutrons, being neutral, enhance the attractive nuclear force without contributing to electrostatic repulsion. As the number of protons in heavier nuclei increases, electrostatic repulsion also increases, requiring an even larger neutron-to-proton ratio to maintain nuclear stability. Thus, the effect becomes more pronounced for the heaviest nuclei.

Step by step solution

01

1. Introduction to Nuclear Forces and Electrostatic Forces

We can first consider two primary forces affecting protons and neutrons in the nucleus: the strong nuclear force, which is an attractive force that binds nucleons (protons and neutrons) together, and the electrostatic force (or Coulomb force), which is a repulsive force experienced between protons due to their positive charges.
02

2. Mass Number Greater Than 40

According to the question, the mass number \(A\) is greater than about 40. Mass number \(A = Z + N\), where \(Z\) is the number of protons and \(N\) is the number of neutrons. Stable nuclei with a mass number of around 40 or more show a neutron-to-proton ratio greater than 1. As the mass number increases, this ratio also increases.
03

3. Balancing Nuclear and Electrostatic Forces

For a nucleus to be stable, the attractive nuclear force must be stronger than the repulsive electrostatic force which acts mainly between protons. The strong nuclear force acts over short distances and is similar for protons and neutrons. The electrostatic force, however, acts over larger distances and is dependent on the number of protons. As we add more protons to the nucleus, the electrostatic repulsion between protons increases. This decreases the binding energy of the nucleus, making the nucleus less stable.
04

4. The Role of Neutrons

Neutrons play a crucial role in the stability of the nucleus. Since neutrons are neutral and do not contribute to the electrostatic repulsion, adding neutrons to the nucleus actually enhances the attractive nuclear force between nucleons. This, in turn, stabilizes the nucleus.
05

5. Heavy Nuclei and Neutron-To-Proton Ratio

As the number of protons in a nucleus increases (i.e., for heavier nuclei), the electrostatic repulsion between the protons also increases significantly. In order to compensate for this increased repulsion, more neutrons need to be added to the nucleus to provide enough attractive force to maintain the overall nuclear stability. Therefore, the neutron-to-proton ratio becomes even larger for heavier nuclei. In conclusion, nuclei with mass number \(A\) greater than about 40 have a higher number of neutrons than protons, and this effect becomes more pronounced for the heaviest nuclei to balance out the repulsive electrostatic forces and maintain nuclear stability.

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Most popular questions from this chapter

Two fusion reactions mentioned in the text are \(n+^{3} \mathrm{He} \rightarrow^{4} \mathrm{He}+\gamma\) and \(n+^{1} \mathrm{H} \rightarrow^{2} \mathrm{H}+\gamma\). Both reactions release energy, but the second also creates more fuel. Confirm that the energies produced in the reactions are 20.58 and \(2.22 \mathrm{MeV}\), respectively. Comment on which product nuclide is most tightly bound, \(^{4} \mathrm{He}\) or \(^{2} \mathrm{H}\).

Write a nuclear \(\beta^{-}\) decay reaction that produces the \(^{90} \mathrm{Y}\) nucleus. (Hint: The parent nuclide is a major waste product of reactors and has chemistry similar to calcium, so that it is concentrated in bones if ingested.)

The mass ( \(M\) ) and the radius ( \(r\) ) of a nucleus can be expressed in terms of the mass number, \(A\). (a) Show that the density of a nucleus is independent of \(A\). (b) Calculate the density of a gold (Au) nucleus. Compare your answer to that for iron (Fe).

The \(^{210}\) Po source used in a physics laboratory is labeled as having an activity of \(1.0 \mu \mathrm{Ci}\) on the date it was prepared. A student measures the radioactivity of this source with a Geiger counter and observes 1500 counts per minute. She notices that the source was prepared 120 days before her lab. What fraction of the decays is she observing with her apparatus?

If two nuclei are to fuse in a nuclear reaction, they must be moving fast enough so that the repulsive Coulomb force between them does not prevent them for getting within \(R \approx 10^{-14} \mathrm{m}\) of one another. At this distance or nearer, the attractive nuclear force can overcome the Coulomb force, and the nuclei are able to fuse. (a) Find a simple formula that can be used to estimate the minimum kinetic energy the nuclei must have if they are to fuse. To keep the calculation simple, assume the two nuclei are identical and moving toward one another with the same speed \(v\). (b) Use this minimum kinetic energy to estimate the minimum temperature a gas of the nuclei must have before a significant number of them will undergo fusion. Calculate this minimum temperature first for hydrogen and then for helium. (Hint: For fusion to occur, the minimum kinetic energy when the nuclei are far apart must be equal to the Coulomb potential energy when they are a distance \(R\) apart.)
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