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Chapter 26: Problem 70

The binding energy of deuteron is \(2.2 \mathrm{MeV}\) and that of \({ }_{2} \mathrm{He}^{4}\) is \(28 \mathrm{MeV}\). If two deuterons are fused to form one \({ }_{2} \mathrm{He}^{4}\), then the energy released is (a) \(30.2 \mathrm{MeV}\) (b) \(25.8 \mathrm{MeV}\) (c) \(23.6 \mathrm{MeV}\) (d) \(19.2 \mathrm{MeV}\)

Short Answer

Expert verified
The energy released is 23.6 MeV (option c).

Step by step solution

01

Understand the Problem

We need to calculate the energy released from the fusion of two deuterons into one helium-4 nucleus. We know the binding energy for deuteron is 2.2 MeV, and for helium-4, it's 28 MeV.
02

Calculate Total Binding Energy of Deuterons

Since a deuteron has a binding energy of 2.2 MeV, two deuterons will have a total binding energy of \(2 \times 2.2 \text{ MeV} = 4.4 \text{ MeV}\).
03

Determine Energy Required to Break Initial Bonds

To free both deuterons for fusion, 4.4 MeV of energy is initially "unlocked." This energy is part of the initial system energy.
04

Calculate Energy Released

When two deuterons fuse into one helium-4 nucleus, the helium-4 nucleus introduces a total binding energy of 28 MeV. Subtract the initial binding energy of the two deuterons: \(28 \text{ MeV} - 4.4 \text{ MeV} = 23.6 \text{ MeV}\). Thus, the energy released in the fusion reaction is 23.6 MeV.

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Key Concepts

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

Binding Energy
Binding energy is a fundamental concept in nuclear physics. It refers to the energy required to disassemble a nucleus into its separate components, which are protons and neutrons. In simpler terms, it's the glue that holds the nucleus together.
  • The higher the binding energy, the more stable the nucleus is.
  • Binding energy is typically expressed in mega electron volts (MeV), which is a unit of energy widely used in particle physics.
  • This concept is crucial for understanding nuclear reactions, as the difference in binding energies before and after the reaction dictates the energy absorbed or released during that reaction.
Understanding the binding energy allows us to calculate how much energy is released in nuclear fusion, such as when two lighter nuclei, like deuterons, combine to form a heavier nucleus, like helium-4. This energy release is a vital process in stars, powering the sun and providing energy for life on Earth.
Deuteron
A deuteron is the nucleus of deuterium, an isotope of hydrogen. It consists of one proton and one neutron. Although it is a stable isotope, it has a relatively low binding energy of 2.2 MeV compared to heavier nuclei.
  • Deuterons are important in nuclear fusion reactions because they can fuse to form heavier elements, such as helium-4.
  • This fusion of deuterons is one part of the process that powers stars, including our sun.
  • Despite their simple structure, deuterons offer an insight into the fundamental forces that operate within atomic nuclei.
When two deuterons come together to form a helium-4 nucleus, a significant amount of energy is released due to the difference in binding energy between the two initial deuterons and the resulting helium-4 nucleus.
Helium-4
Helium-4 is a very stable and common isotope of helium, consisting of two protons and two neutrons. It has a much higher binding energy of 28 MeV, which contributes to its stability.
  • This isotope plays a crucial role in nuclear fusion reactions, specifically those taking place in stars.
  • Its high binding energy means that forming helium-4 from lighter nuclei releases a sizeable amount of energy.
  • The synthesis of helium-4 through fusion reactions contributes to the production of the energy that fuels stars.
In the process of nuclear fusion, when deuterons fuse to create helium-4, the stable and high-energy characteristics of helium-4 make these reactions energetically favorable. This process is not only significant for energy creation in astrophysical settings but is also being investigated for potential applications in energy production on Earth.
Energy Released
The energy released during nuclear fusion reactions is a fascinating aspect of nuclear physics. When two atomic nuclei, like deuterons, fuse to form a heavier nucleus such as helium-4, a large amount of energy is freed. This occurs because the binding energy of the final nucleus is significantly higher than that of the initial reacting nuclei.
  • The difference in binding energy between the products and the reactants directly translates into the energy released.
  • In our example, two deuterons each with a binding energy of 2.2 MeV add up to 4.4 MeV for both. After fusion, the helium-4 nucleus formed has a total binding energy of 28 MeV.
  • Thus, the energy released is the difference: 28 MeV - 4.4 MeV = 23.6 MeV.
This release of energy is why nuclear fusion has such potential as a powerful energy source. It can liberate vast amounts of energy from relatively small amounts of fuel, explaining its role in the processes that power the sun and consequently, all life on Earth.

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Most popular questions from this chapter

In radioactive decay process, the negatively charged emitted \(\beta\) -particles are (a) the electrons present inside the nucleus (b) the electrons produced as a result of the decay of neutrons inside the nucleus (c) the electrons produced as a result of collissions between atoms (d) the electrons orbiting around the nucleus

Two radioactive samples have decay constant \(15 x\) and \(3 x\). If they have the same number of nuclei initially, the ratio of number of nuclei after a time \(\frac{1}{6 x}\) is (a) \(\frac{1}{e}\) (b) \(\frac{e}{2}\) (c) \(\frac{1}{e^{4}}\) (d) \(\frac{1}{e^{2}}\)

A sample of radioactive material has a mass \(m\), decay constant \(\lambda\) and molecular weight \(M .\) If \(N_{A}\) is Avogadro's number, the initial activity of the sample is (a) \(\lambda \mathrm{m}\) (b) \(\frac{\lambda m}{M}\) (c) \(\frac{\lambda m N_{A}}{M}\) (d) \(m M e^{\lambda}\)

In the following nuclear reaction, $$ { }_{6} \mathrm{C}^{11} \rightarrow{ }_{5} \mathrm{~B}^{11}+\beta^{+}+X $$ what does \(X\) stand for ? (a) A proton (b) A neutron (c) A neutrino (d) An electron

The binding energies of the atoms of elements \(A\) and \(B\) are \(E_{a}\) and \(E_{b}\) respectively. Three atoms of the element \(B\) fuse to give one atom of element \(A\). This fusion process is accompanied by release of energy \(e\). Then \(E_{a}, E_{b}\) and e are related to each other as (a) \(E_{a}+e=3 E_{b}\) (b) \(E_{a}=3 E_{b}\) (c) \(E_{a}-e=3 E_{b}\) (d) \(E_{a}+3 E_{b}+e=0\)
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