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13.9 - EXERCISES

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Binding Energy Calculations

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Teacher
Teacher

Today we'll learn how to calculate binding energy using the mass of the nitrogen nucleus. Who remembers what binding energy means?

Student 1
Student 1

Isn't it the energy required to separate the nucleons in a nucleus?

Teacher
Teacher

Exactly! Now, let's calculate the binding energy for 14N, given that its mass is 14.00307 u. Remember, to find the binding energy, we first need the mass defect.

Student 2
Student 2

How do we find the mass defect?

Teacher
Teacher

Good question! We take the sum of the masses of protons and neutrons in the nucleus and subtract it from the actual mass of the nucleus. Let's calculate together.

Student 3
Student 3

Are we going to use the formula DE = DMc^2 for energy?

Teacher
Teacher

Exactly! Remember, we will convert the mass defect into energy using that relation. Let's summarize: Binding energy is key for understanding nuclear stability.

Nuclear Reaction Dynamics

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Teacher
Teacher

Next, we’ll explore the Q-value of nuclear reactions. What does Q represent in a reaction?

Student 4
Student 4

Isn't it the difference in total mass-energy before and after the reaction?

Teacher
Teacher

Yes! It informs us whether a reaction is exothermic or endothermic. Let's check a reaction: What happens in the fusion of deuterons?

Student 1
Student 1

Isn’t energy released when they fuse?

Teacher
Teacher

Correct! As deuterons combine, you get more tightly bound products, releasing energy. Great! Now, let’s summarize Q-value calculations.

Nuclear Radii and Density

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Teacher
Teacher

Now, let's discuss nuclear radii. Who can write the formula for nuclear radius?

Student 2
Student 2

It's R = R0 A^(1/3), right?

Teacher
Teacher

Well done! This tells us that the radius scales with the cube root of the mass number. Now, what does that mean for density?

Student 3
Student 3

Does it mean density is constant across different nuclei?

Teacher
Teacher

Exactly! Regardless of mass, nuclear matter density remains consistent. Remember this: Denser nuclei are more stable.

Applications of Nuclear Properties

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Teacher
Teacher

Let’s now consider the practical applications of our nuclear concepts in energy generation. Can someone explain energy release during nuclear fission?

Student 4
Student 4

Fission of heavy nuclei releases a lot of energy because they split into lighter nuclei which are more stable!

Teacher
Teacher

Exactly! The released energy can be harnessed. How does this compare to fusion?

Student 1
Student 1

Fusion combines light nuclei to form a heavier nucleus, also releasing energy, especially in stars!

Teacher
Teacher

Fantastic! Energy release from fission and fusion showcases the significance of understanding binding energy.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section provides exercises to reinforce the concepts of binding energy, nuclear reactions, and nuclear properties.

Standard

The exercises in this section challenge students to apply their understanding of nuclear concepts, including binding energy calculations, nuclear reaction dynamics, and the properties of specific isotopes and elements. The problems range in difficulty and promote critical thinking about nuclear physics.

Detailed

Exercises Overview

This section presents a series of exercises that are focused on nuclear physics, specifically revolving around binding energy, nuclear reactions, and related calculations. The exercises aim to test and enhance the understanding of key concepts introduced in the chapter about nuclei and help students apply theoretical knowledge to practical scenarios. Students will undertake calculations involving mass defects, determine the energy associated with separating nucleons, and explore properties of isotopes through problem-solving. Such exercises are crucial in consolidating learning and ensuring that students can practically apply their knowledge in biological and chemical contexts.

Key Areas Covered:

  • Binding Energy Calculations: Students will calculate the binding energy of different isotopes using their mass data.
  • Energy Release in Reactions: Understanding how to quantify energy released in nuclear fission or fusion processes.
  • Nuclear Radius Ratios: Students will explore properties of nuclei in terms of their physical dimensions and concepts like density and volume.
  • Applications of Nuclear Concepts: Practical applications of nuclear physics principles in real-world scenarios.

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Audio Book

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Exercise 1: Binding Energy of Nitrogen Nucleus

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13.1 Obtain the binding energy (in MeV) of a nitrogen nucleus 14N, given m 14N = 14.00307 u.

Detailed Explanation

In this exercise, we are tasked with calculating the binding energy of the nitrogen nucleus (14N). The binding energy can be calculated using the mass of the nucleus and the mass of its individual protons and neutrons. This calculation typically involves determining the difference between the mass of the separate nucleons (8 protons and 6 neutrons for 14N) and its actual mass (14.00307 u). The binding energy is derived from the mass defect using the equation: \( E = \Delta m c^2 \). Here, c is the speed of light. First, we find the mass defect (\(\Delta m\)), which is the difference between the expected mass of the separate nucleons and the actual mass of the nucleus. Then, we multiply the mass defect by \(c^2\) to get the energy in MeV.

Examples & Analogies

Think of a tightly wrapped gift box (nucleus) versus the individual items packed inside (nucleons). The binding energy is like the energy spent to keep those gifts tightly bound in the box. If the box is too big (more mass), it will take more energy to keep everything in. If you unwrap it (calculate the mass defect), you can find out just how much energy was used to keep it sealed.

Exercise 2: Binding Energy of Iron and Bismuth Nuclei

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13.2 Obtain the binding energy of the nuclei 56Fe and 209Bi in units of MeV from the following data: m (56Fe) = 55.934939 u and m (209Bi) = 208.980388 u.

Detailed Explanation

This exercise focuses on determining the binding energies of the iron (56Fe) and bismuth (209Bi) nuclei. Similar to the previous exercise, we find the total mass of the protons and neutrons expected for each nucleus. For 56Fe, which has 26 protons and 30 neutrons, and 209Bi, which has 83 protons and 126 neutrons, we calculate the expected mass based on the individual nucleon masses. The binding energy is calculated by finding the mass defect using their actual masses and applying the mass-energy equivalence formula to convert this defect into energy measured in MeV.

Examples & Analogies

Imagine a car (the nucleus) and the fuel it consumes (binding energy). The bigger the car (the more nucleons), the more fuel it needs to operate efficiently (more binding energy). By calculating how much fuel would be consumed in a trip (binding energy), we gain insight into how 'heavy' the car feels when it's on the road.

Exercise 3: Energy from a Coin

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13.3 A given coin has a mass of 3.0 g. Calculate the nuclear energy that would be required to separate all the neutrons and protons from each other. For simplicity, assume that the coin is entirely made of 63Cu atoms (of mass 62.92960 u).

Detailed Explanation

In this exercise, we want to find the total nuclear energy needed to separate all nucleons in a coin. We first determine how many 63Cu atoms are in the 3.0 g coin using its molar mass. Next, we multiply the number of atoms by the number of nucleons in each atom (33 for 63Cu) and then calculate the energy required to separate them based on the binding energy per nucleon. This helps us estimate the energy involved in completely disassembling the atomic structure.

Examples & Analogies

Think of a tightly packed suitcase (the coin) full of clothes (nucleons). If you want to unpack that suitcase and separate everything individually, it takes a considerable amount of time and effort (energy). This exercise estimates how much energy would be needed to achieve that level of separation for an entire suitcase worth of atomic structure.

Exercise 4: Ratio of Nuclear Radii

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13.4 Obtain approximately the ratio of the nuclear radii of the gold isotope 197Au and the silver isotope 107Ag.

Detailed Explanation

To solve this exercise, we use the relationship for nuclear radius R as \( R = R_0 A^{1/3} \). From this equation, we can find the radii for both isotopes using their mass numbers. The ratio of the nuclear radii can then be easily calculated by dividing the radii of 197Au by those of 107Ag. This demonstrates how nuclear size changes with mass number and provides insight into the nuclear structure.

Examples & Analogies

Consider balloons (nuclei) that expand in size as you add air (nucleons). Just as balloons get bigger with more air, nuclei grow with more nucleons. This exercise helps visualize the connection between the number of nucleons and the corresponding change in nuclear radius.

Exercise 5: Q Value of a Nuclear Reaction

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13.5 The Q value of a nuclear reaction A + b fi C + d is defined by Q = [ m + m - m - m ]c2. Determine from the given data the Q-value of the following reactions and state whether the reactions are exothermic or endothermic.

Detailed Explanation

In this problem, we need to calculate the Q-value for provided nuclear reactions using the equation for Q, which considers the initial and final masses of nuclei. By substituting the mass values given for A, b, C, and d into the equation, we can determine whether energy is released (exothermic) or absorbed (endothermic) during the reaction. The sign of the Q value indicates the nature of the reaction.

Examples & Analogies

Think of a seesaw (the nuclear reaction). When there is more weight on one end (more reactants), it tips in that direction (either releasing or absorbing energy). This exercise illustrates how mass and energy balance influences the motion of the seesaw just like it influences nuclear reactions.

Exercise 6: Fission Energetics

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13.6 Suppose, we think of fission of a 56Fe nucleus into two equal fragments, 28Al. Is the fission energetically possible? Argue by working out Q of the process.

Detailed Explanation

In this exercise, we analyze whether the fission of 56Fe into two parts of 28Al is energetically favorable. By calculating the Q value using the actual masses of 56Fe and 28Al, we can determine if energy is released in the fission process which would indicate that the reaction is energetically possible. If Q is positive, the fission will release energy, thus supporting its feasibility.

Examples & Analogies

Consider a large boulder (56Fe) being split into two smaller rocks (28Al). If it takes more energy to break the boulder than the energy produced when split—think about it like breaking a branch for firewood (or energy)—then it wouldn't be worth it. This exercise evaluates the energy dynamics of splitting larger atoms.

Exercise 7: Energy from Fissioning 239Pu

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13.7 The fission properties of 239Pu are very similar to those of 235U. The average energy released per fission is 180 MeV. How much energy, in MeV, is released if all the atoms in 1 kg of pure 239Pu undergo fission?

Detailed Explanation

To calculate the total energy released in the fission of 1 kg of 239Pu nuclei, we first need to determine the number of 239Pu atoms in 1 kg. Knowing the atomic mass of 239Pu, we calculate how many nuclei present and multiply this number by the energy released per fission, which is 180 MeV. This way, we gain an understanding of the energy potential in a given mass of fissile material.

Examples & Analogies

Envision a factory assembly line where each product (atom) releases energy during its process. As you add more products (fissioning more atoms of 239Pu), the overall energy output accumulates rapidly, demonstrating just how potent this process is when scaled up.

Exercise 8: Energy from Fusion of Deuterium

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13.8 How long can an electric lamp of 100W be kept glowing by fusion of 2.0 kg of deuterium? Take the fusion reaction as 2H+2Hfi 3He+n+3.27 MeV.

Detailed Explanation

In this problem, we aim to find out how long a 100W lamp can run using the energy produced from the fusion of 2.0 kg of deuterium. Firstly, we calculate the number of fusion reactions that can take place in 2.0 kg of deuterium and determine the total energy output from those reactions. We then convert this total energy into time for the lamp based on its power consumption.

Examples & Analogies

Imagine fueling a car with gas (energy). The more gas you have, the longer you can drive before refueling. This exercise estimates how long the 'fuel' from deuterium fusion can power a device, similar to how long a gas tank can sustain your journey.

Exercise 9: Potential Barrier Height Calculation

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13.9 Calculate the height of the potential barrier for a head-on collision of two deuterons.

Detailed Explanation

In this exercise, we want to find the height of the potential barrier based on the Coulomb repulsion when two deuterons collide. We start by determining the charged interaction that occurs when deuterons come very close to each other. The height of this barrier will give insights into the conditions needed for fusion, such as how much energy is required to overcome this barrier.

Examples & Analogies

Think of two magnets (deuterons) that repel each other. The closer you try to bring them together without overcoming the repulsion requires energy, much like needing to push against a door that’s being held shut. This exercise shows how nuclear forces operate in colliding particles.

Exercise 10: Density Independence of Nuclear Matter

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13.10 From the relation R = R A1/3, where R is a constant and A is the mass number of a nucleus, show that the nuclear matter density is nearly constant (i.e. independent of A).

Detailed Explanation

Here, we analyze the relationship between the radius of a nucleus and its mass number. By using the formula R = R0 A^{1/3}, we see that the volume of the nucleus increases with mass number, but the density remains constant. This means as we add more nucleons, their structural arrangement keeps the nuclear density similar, illustrating that even larger nuclei do not become less dense as we increase their size.

Examples & Analogies

Imagine filling a balloon (the nucleus) with more air (nucleons). As you add more air, the structure (density) within the balloon remains consistent regardless of its size. This exercise uncovers the behavior of atoms on a larger scale similar to filling a balloon evenly.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Binding Energy: The energy needed to disassemble the nucleus into its individual components.

  • Nuclear Radius: The size of the nucleus, which increases with the mass number according to the formula R = R0 A^(1/3).

  • Q-value: Represents the energy released or absorbed in a nuclear reaction.

  • Mass Defect: The difference between the expected mass of a nucleus and its actual mass, related to binding energy.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • An example of binding energy calculation for 14N where the mass defect is derived from the constituents’ proton and neutron masses.

  • Calculating the energy release during the fission of 235U to demonstrate real-life applications of nuclear energy.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • Energy so grand, in binding we stand, separating nucleons, with forces so planned.

📖 Fascinating Stories

  • Imagine pushing apart friends who are closely bound; the energy to keep them together is their binding energy.

🧠 Other Memory Gems

  • Remember 'Nuclear Binds Secure' to recall that binding energy secures the nucleons together.

🎯 Super Acronyms

B.E.S.T. - Binding Energy is Stability's Test.

Flash Cards

Review key concepts with flashcards.

Glossary of Terms

Review the Definitions for terms.

  • Term: Binding Energy

    Definition:

    The energy required to separate nucleons within a nucleus.

  • Term: Nucleus

    Definition:

    The central part of an atom, containing protons and neutrons.

  • Term: Qvalue

    Definition:

    The difference in total energy released or absorbed in a nuclear reaction.

  • Term: Mass Defect

    Definition:

    The difference between the expected mass of a nucleus and its actual mass.

  • Term: Isotope

    Definition:

    Different forms of the same element with the same number of protons but different numbers of neutrons.

  • Term: Density

    Definition:

    Mass per volume of nuclear material, which remains constant across various nuclei.