Exploring Fusion Reactions in Stars: Proton + Deuteron to 3He + Gamma Ray

In summary, the conversation discusses a fusion reaction that takes place inside a star, which produces helium-3 and a high-energy photon. The fusion reaction requires the proton and deuteron to come close enough together to touch, and the minimum kinetic energy needed for the reaction to take place is calculated. The energy of the gamma ray and the kinetic energy of the helium-3 nucleus are also calculated. The conversation highlights the challenges in building a practical fusion reactor and the potential energy gain from this reaction.
  • #1
Qnslaught
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Homework Statement



One of the thermonuclear or fusion reactions that takes place inside a star such as our Sun is the production of helium-3 (3He, with two protons and one neutron) and a gamma ray (high-energy photon, denoted by the lowercase Greek letter gamma, ) in a collision between a proton (1H) and a deuteron (2H, the nucleus of "heavy" hydrogen, consisting of a proton and a neutron):

1H + 2H 3He +
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 2.3 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 = 2Your answer is incorrect. eV
Kdeuteron = 3 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 = 4Your answer is incorrect. 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 = 5 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) = 6 eV

Homework Equations





The Attempt at a Solution



I tried using the rest energies to convert to kinetic energy, but it didn't work.
 
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  • #2
I also tried using the momentum principle and the energy principle, but I am still stuck. Can someone please help me?
 
  • #3
Can you give me a hint or something?

First, it is important to understand that the fusion reaction described in this problem is a two-body collision, where the total initial momentum is zero. This means that the final momentum must also be zero, as momentum is conserved in a closed system. From this, we can use the momentum principle to solve for the minimum kinetic energy of the proton and deuteron required for the reaction to take place.

We know that the center-to-center distance at collision is 2.3 x 10^-15 m, which is the minimum distance required for the strong interaction to occur. This means that the proton and deuteron must have enough kinetic energy to overcome the repulsive electrostatic force and reach this distance.

To solve for the minimum kinetic energy, we can use the equation for the Coulomb potential energy, U = kq1q2/r. In this case, q1 and q2 are the charges of the proton and deuteron, respectively, and k is the Coulomb constant. We can set this equal to the kinetic energy, K = 1/2mv^2, and solve for the minimum velocity, v, of the proton and deuteron at the moment of collision.

Once we have the velocities, we can use the equation for kinetic energy to solve for the minimum kinetic energy in terms of eV.

For part (c), we can use the energy principle, which states that the total energy before and after the collision must be equal. Since the initial total energy is just the sum of the kinetic energies of the proton and deuteron, and the final total energy is the sum of the kinetic energy of the helium-3 nucleus and the energy of the gamma ray, we can set these equal to each other and solve for the energy of the gamma ray.

Finally, for part (d), we can use the equation for kinetic energy to solve for the small kinetic energy of the helium-3 nucleus, using its mass and the velocity we found in part (b).

The gain in available energy in this reaction can be calculated by subtracting the initial total energy (Kproton + Kdeuteron) from the final total energy (KHe-3 nucleus + Egamma ray). This will give us the energy released in the reaction, which is the gain in available energy.

Overall, it is important to approach this problem using the principles of conservation of momentum and energy, as
 

FAQ: Exploring Fusion Reactions in Stars: Proton + Deuteron to 3He + Gamma Ray

1. What is fusion and how does it occur in stars?

Fusion is the process of combining two or more atomic nuclei to form a heavier nucleus. In stars, fusion occurs when extreme temperatures and pressures cause hydrogen atoms to collide and fuse together, creating helium atoms and releasing energy in the form of gamma rays.

2. Why is the proton + deuteron to 3He + gamma ray reaction important in studying fusion in stars?

This specific fusion reaction is important because it is one of the primary reactions responsible for the production of energy in stars. It is also the first step in the process of creating heavier elements, which are essential for the formation of planets and life.

3. How is the proton + deuteron to 3He + gamma ray reaction studied?

The reaction is studied through experiments in particle accelerators and through observations of stars using telescopes. Scientists also use theoretical models and computer simulations to understand the complex processes involved in fusion reactions.

4. What challenges do scientists face in studying fusion reactions in stars?

One of the main challenges is recreating the extreme temperatures and pressures found in stars in laboratory settings. Another challenge is understanding the complex interactions between particles during fusion reactions, which require advanced mathematical models and simulations.

5. What are the potential applications of studying fusion reactions in stars?

Studying fusion reactions in stars can provide valuable insights into the inner workings of the universe and help us better understand the processes that lead to the formation and evolution of stars and planets. It also has potential applications in developing fusion energy as a clean and sustainable source of power.

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