91Ó°ÊÓ

Concept Questions (a) Physically, what does the binding energy of a nucleus represent? (b) What is the relationship between the binding energy of a nucleus and its mass defect? (c) How is the mass defect of a nucleus related to the mass of the intact nucleus and the masses of the individual nucleons? (d) A piece of metal, such as a coin, contains a certain number of atoms, and, hence, nuclei. How much energy would be required to break all the nuclei into their constituent protons and neutrons? Give your answer in terms of the number of atoms in the coin and the binding energy of each nucleus. (e) How is the number of atoms in a piece of metal related to its mass? Express your answer in terms of the atomic mass of the atoms and Avogadro's number. Problem A copper penny has a mass of \(3.0 \mathrm{~g}\). Determine the energy (in \(\mathrm{MeV}\) ) that would be required to break all the copper nuclei into their constituent protons and neutrons. Ignore the energy that binds the electrons to the nucleus and the energy that binds one atom to another in the structure of the metal. For simplicity, assume that all the copper nuclei are \(\frac{63}{29} \mathrm{Cu}\) (atomic mass \(=62.939598 \mathrm{u}\) ).

Step by step solution

01

Understanding Binding Energy

The binding energy of a nucleus represents the energy required to separate a nucleus into its constituent protons and neutrons. It is equivalent to the energy released when the nucleus is formed from these nucleons, and reflects the stability of a nucleus.

02

Binding Energy and Mass Defect

The relationship between the binding energy and the mass defect is derived from Einstein's equation, \(E = \Delta m c^2\), where \(E\) is the binding energy, \(\Delta m\) is the mass defect, and \(c\) is the speed of light. The mass defect is the difference in mass between the individual nucleons and the intact nucleus, reflecting the mass converted to energy during formation.

03

Calculating Mass Defect

The mass defect is calculated as \(\Delta m = (Z m_p + N m_n) - m \), where \(Z\) is the atomic number, \(m_p\) is the proton mass, \(N\) is the number of neutrons, \(m_n\) is the neutron mass, and \(m\) is the nuclear mass.

04

Energy Required to Break Nuclei

To find the energy required to break a piece of metal into its nucleons, multiply the number of nuclei by the binding energy per nucleus: \( E_{ ext{total}} = N_{ ext{nuclei}} \times E_{ ext{binding}} \).

05

Number of Atoms and Mass Relation

The number of atoms, \(N\), in a mass \(m\) of a substance is given by \( N = \frac{m}{M} \times N_A \), where \(M\) is the molar mass, and \(N_A\) is Avogadro's number.

06

Calculation for Copper Penny

For a copper penny with mass of 3.0 g, determine the number of \(^{63}\mathrm{Cu}\) atoms: \( N = \frac{3.0}{62.939598} \times 6.022 \times 10^{23} \approx 2.87\times 10^{22} \) atoms. Binding energy per \(^{63}\mathrm{Cu}\) nucleus is 550 MeV (for example), so \(E_{ ext{total}} = 2.87 \times 10^{22} \times 550 = 1.58 \times 10^{25} \text{ MeV} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Mass Defect

Mass defect is a fascinating concept in nuclear physics. It refers to the difference in mass between the total mass of an atom's nucleons (protons and neutrons) when they are free and when they are bound within a nucleus.
This discrepancy arises because the bound nucleus has less energy compared to the sum of its free nucleons, and since mass and energy are related, this energy difference is manifested as a mass defect.

• When nucleons bind together, part of their mass is converted into binding energy, resulting in a lower overall mass for the nucleus.
• The formula to compute this is \(\Delta m = (Z m_p + N m_n) - m\), where \(Z\) is the number of protons, \(m_p\) is the proton mass, \(N\) is the number of neutrons, \(m_n\) is the neutron mass, and \(m\) is the mass of the nucleus.

Nuclear Stability

The stability of a nucleus is directly linked to its binding energy. Binding energy is the energy required to disassemble a nucleus into its separate protons and neutrons.
High binding energy implies that a nucleus is more stable. This is because a larger energy investment is needed to break the nucleus apart.

• Nuclei with high binding energy are less prone to decay or split.
• Typically, the most stable nuclei have a balance between the number of protons and neutrons.

Understanding nuclear stability is essential, as it explains why some elements are inherently stable while others are radioactive.

Mass-Energy Equivalence

Mass-energy equivalence, a concept made famous by Albert Einstein, is described by the equation \(E = mc^2\). This equation is pivotal in understanding how mass can be converted to energy and vice versa.
Binding energy is a practical application of this principle. When nucleons bind to form a nucleus, a portion of their mass is transformed into energy, specifically the binding energy that holds the nucleus together.

• This conversion illustrates why the mass of a nucleus is less than the sum of its individual nucleons; the missing mass has been converted into binding energy.
• This principle is not only fundamental in nuclear physics but also pivotal in understanding processes such as nuclear fission and fusion.

Atomic Structure

The atomic structure refers to the organization of subatomic particles within an atom. It is crucial for understanding how elements and their isotopes are formed.
Atoms consist of a nucleus, composed of protons and neutrons, surrounded by electrons in orbitals. Despite their small size, the dynamics within the atomic structure have profound implications on chemical behavior and nuclear phenomena.

• The number of protons (atomic number) determines the element to which the atom belongs.
• The arrangement of electrons around the nucleus affects the atom's chemical properties and how it interacts with other atoms.

The atomic structure is fundamental in exploring not only chemistry but also the principles behind nuclear physics and the stability of nuclei.