Semi difficult fusion problem, thanks

In summary, the given conversation discusses a fusion reaction in which a proton and deuteron collide to form a helium-3 nucleus and a gamma ray. The strong interaction plays a crucial role in this reaction, as the particles must come into contact for it to occur. The minimum kinetic energy required for the proton and deuteron to react is 355414 eV and 177920 eV, respectively. The energy of the resulting gamma ray can be found using the equation E = pc, and the kinetic energy of the helium-3 nucleus is very small compared to the gamma ray's energy. The gain in available energy in this reaction is determined by subtracting the total kinetic energy of the final products from the initial total kinetic energy.
  • #1
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Homework Statement


a) 1H + 2H 3He + gamma
The rest mass of the proton is 1.0073 u (unified atomic mass unit, 1.66 10-27 kg), the rest mass of the deuteron is 2.0136 u, the rest mass of the helium-3 nucleus is 3.0155 u, and the gamma ray is a high-energy photon, whose mass is zero. The strong interaction has a very short range and is essentially a contact interaction. For this fusion reaction to take place, the proton and deuteron have to come close enough together to touch. The approximate radius of a proton or neutron is about 110-15 m.

(b) In this situation where the initial total momentum is zero, what minimum kinetic energy must the proton have, and what minimum kinetic energy must the deuteron have, in order for the reaction to take place? Express your results in eV. Assume that the center to center distance at collision is 2.7 10-15 m. You will find that the proton and deuteron have speeds much smaller than the speed of light (which you can verify if you like after calculating their kinetic energies). Keep in mind what you see in the diagram you drew in part (a). You may find it useful to remember that kinetic energy can be expressed either in terms of speed or in terms of the magnitude of momentum. It is very important to do the analysis symbolically; don't plug in numbers until the very end. If you try to do the problem numerically, and/or ignore part (a), you will probably not be able to complete the analysis.
Kproton = ??eV
Kdeuteron = ??eV
(c) Becaus the helium-3 nucleus is massive, its kinetic energy is very small compared to the energy of the massless photon. Therefore, what will be the energy of the gamma ray in eV? The relationship E2 - (pc)2 = (mc2)2 is valid for any particle, including a massless photon, so the momentum of a photon is p = E/c, where E is the photon energy. You may need to consider the momentum principle as well as the energy principle in your analysis.
Egamma ray = ??eV

(d) Now that you know the energy of the gamma ray, calculate the (small) kinetic energy of the helium-3 nucleus. Hint: You will find that the speed of the helium-3 nucleus is very small compared to the speed of light.
KHe-3 nucleus =?? eV

(e) You see that there is a lot of energy in the final products of the fusion reaction, which is why scientists and engineers are working hard to try to build a practical fusion reactor. The problem is the difficulty and energy cost in getting the electrically charged reactants close enough to fuse (the proton and deuteron in this reaction). If these problems can be overcome, what is the gain in available energy in this reaction?
(KHe-3 nucleus+Egamma ray) - (Kproton+Kdeuteron) = ??eV

Homework Equations


Ef = Ei
Uf = Ki = 1/2 m v1^2 + 1/2 m v2^2
Ptot = 0
P^2*(1/2m1 + 1/2m2)


The Attempt at a Solution



I think initial total kinetic is 4.2666E-14 and beyond that I'm obviously doing something wrong...
 
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  • #2
ok
Kproton = 355414 eV
Kdeuteron = 177920eV

still trying to find the rest though
i think c involves a quadratic
 
  • #3


I would like to offer some guidance and clarification on the problem at hand.

Firstly, the given reaction is a fusion reaction, where two particles (proton and deuteron) combine to form a heavier particle (helium-3) and release energy in the form of a gamma ray. This reaction is a type of nuclear reaction that occurs in the core of stars, including our sun. It is also a potential source of energy for future power generation.

Now, let's break down the problem and approach it step by step. In part (a), we are given the reaction equation and the rest masses of the particles involved. The key concept here is conservation of energy, where the total energy of the reactants (proton and deuteron) must equal the total energy of the products (helium-3 and gamma ray). This means that the sum of the kinetic energies and the rest energies of the reactants must equal the sum of the kinetic energies and the rest energies of the products.

Moving on to part (b), we are asked to find the minimum kinetic energies of the proton and deuteron for the reaction to take place. To do this, we need to consider the momentum principle, which states that the initial total momentum of the system is equal to the final total momentum of the system. In this case, the initial total momentum is zero, since the particles are at rest, and the final total momentum is also zero, since the products are also at rest. This means that the momentum of the proton and deuteron must be equal in magnitude but opposite in direction, so that they cancel each other out. We can use this information to solve for the minimum kinetic energies of the particles.

In part (c), we are asked to find the energy of the gamma ray. This can be done using the energy principle, which states that the initial total energy of the system is equal to the final total energy of the system. We know the initial total energy, which is the sum of the kinetic energies and rest energies of the reactants, and we know the final total energy, which is the energy of the gamma ray. Using the given equation and the fact that the photon has zero rest mass, we can solve for the energy of the gamma ray.

In part (d), we are asked to find the kinetic energy of the helium-3 nucleus. This can be done using the same approach as in part (c), where we use
 

1. What is a semi-difficult fusion problem?

A semi-difficult fusion problem refers to the challenge of merging two or more atomic nuclei to form a heavier nucleus, commonly known as nuclear fusion. This process requires a significant amount of energy and precise conditions, making it a difficult task to achieve.

2. Why is fusion considered a difficult problem?

Fusion is considered a difficult problem because it requires extreme temperatures and pressures to overcome the repulsive forces between atomic nuclei. Additionally, controlling the fusion reaction and sustaining it for a significant amount of time is also a major challenge.

3. What are the potential benefits of solving the semi-difficult fusion problem?

If we can successfully solve the semi-difficult fusion problem, it can provide a virtually limitless source of clean energy. Fusion reactions produce little to no radioactive waste and do not emit greenhouse gases, making it a potential solution to the world's energy crisis.

4. What are some current approaches to solving the semi-difficult fusion problem?

There are several approaches to solving the semi-difficult fusion problem, including magnetic confinement fusion, inertial confinement fusion, and laser-driven fusion. Each approach has its own set of challenges and advantages, and scientists continue to research and develop new methods for achieving fusion.

5. Is it possible to solve the semi-difficult fusion problem in the near future?

While significant progress has been made in fusion research, it is difficult to predict when the semi-difficult fusion problem will be solved. Many factors, such as funding, technological advancements, and international collaborations, will play a role in the timeline for achieving fusion energy. However, with continued research and development, it is possible that fusion could become a viable source of energy in the future.

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