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Numerade Educator

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Problem 27 Hard Difficulty

Consider the deuterium-tritium fusion reaction with the tritium nucleus at rest:
$$\frac{2}{1} \mathrm{H}+\frac{3}{1} \mathrm{H} \longrightarrow_{2}^{4} \mathrm{He}+\frac{1}{0} \mathrm{n}$$
(a) From Equation $13.1,$ estimate the required distance of closest approach. (b) What is the Coulomb potential energy (in electron volts) at this distance? (c) If the deuteron has just enough energy to reach the distance of closest approach, what is the final velocity of the combined deuterium and tritium nuclei in terms of the initial deuteron velocity, $v_{0} ?$ (d) Use energy methods to estimate the minimum initial deuteron energy required to achieve fusion. (e) Why does the fusion reaction occur at much lower deuteron energies than that calculated in (d)?

Answer

(a) $2.70\times10^{-15}m$ (b) 720 KeV (c) $v_F=\frac{v_0 m_D}{m_D+m_T}$ (d) 1.2 MeV (e) Quantum tunneling can occur

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Video Transcript

in this question. We have a duty you excite that is hitting towards a stationary treating you and this I actually produces, right, uh, video four or no for particle plus a do trump using the equation 13.1. We want to estimate what is the required distance off? Closest approach, right? So far, Fusion to occur. We expect that, uh, atoms, So really you quite Bedi touching. So we expect them to be almost touching. And the distance between your two centres of Russia we'll speak be the radius off this to nucleus edit together. So if our is to us uh, treat and disability distance off closest approach and this radius can be estimated using the equation to remove one which is already goes to are not you say constant went right by eight about one where eesti mess number. So in this case we have are not and the mess number 42 years to mess number for treating this tree rich and Arnaud is a constant off 1.2. For me, it's a 1.2 times £10 minus 15. Meet us. This should give us about 2.7 have said Paul minus 15 meters. So this is so descends off. Closest approach, not a calculated blue glowing potential energy. At this distance, potential energy is given this key Que one que two over are eight, so que is just a constant 8.99 said about night. Q one and Q two are the respective charges. 42 new clients. So if both off them I have only one proton, right, so goof off them have a single elementary charge each to just 1.6 times sent almost 19 clubs. No, just square that if anybody are, which is the distance of closest approach is 2.77 15. Meet us. Should get about 1.15 Understand four minutes, 13 juice what we want in terms off electron foot. So we have to divide this by 0.6 Singapore mice. 19. To convert it into electoral votes, every should get about 3.2 times said about five electron faults. This is can also be return. A similar and 20 kilo electron Foot's cave. No. 46 parts. Assuming debt, the Duccio has just enough energy to reach the distance off closest approach, but it means that after the fusion, the new place off the helium and the neutron would not, uh, have we're not separate now. The reason why is because at the medium of energy it means that a d center of mass free both off this new place are moving together with the same amount of momentum. And he after the collision, they should stick together and not move right. This is in the center of mass room for minimum energy, which means that this to clients must be stuck together and moving with the same velocity. So save dusty. BF Sialkot Is he let free If and the final state, it would be these two new clients moving together with some velocity v f maybe want to find what is Thesis PF So, using the conservation of momentum, we know that the initial momentum of the system, it's just do you do TV? So we have the initial due to last year's fee, not times myself Deuterium. This is the initial momentum. The final momentum is, uh, alpha particle. No, just massive teach deuterium plus best softy to times. The final will all see VF. So this is the final mentum and this must be equal to each other rights, and therefore, this must be equal. Be not times miss off duty review water by missile due to your classmates off. Treat you now the final part of this question, I'm gonna use energy conservation to estimate what is the minimum due to energy required to achieve fusion, there is it must be able to reach this state off. Closest approach. So we're gonna use energy conservation by energy conservation, initial energy. She's just the kinetic energy off of deuterium. So we have meself deuterium and the last TV Not This must be equals to the final energy off the system just before refuse. Right? We just went through duty. Elementary, Tamar movie together, Eddie. Distance off. Closest approach. We have the kinetic energy off this combined, miss, and we also have the potential cologne potential energy. So this let me just call this you so no to find What is the initial energy? Well, you want to do is run a substitute in the f so from a previous expression for V f well, that this is not over Mt Plus t square, so we can actually cancel one off the m d blast empty terms and, uh, what we want. Sorry. This is Huff and Fi Square. Should be a square. Were here. So what we really want is to get it off the audit fee, no terms and bring it to the left inside. All right. So bringing this this time to the left in sight, we were half one minus over and d plus t aftereffect arising out. Half MDV, not square. Yes, it goes to you and therefore off MD fee not square. Did you show? Amount off kinetic energy? She goes to you, you gotta buy one plus and the over mt plus mt in. Ah, empty. It's approximately about two times de mess off hydrogen, whereas empty. It's about three times teammates off hydrogen, so we can actually simplify this fraction. Empty over. Empty glass. Empty to be about 2/5. It's about 25 and so we'll be half. Sorry. This is one minus right here. It's a minus. Sign one minus 2/5. Which is what? Uh, Children. So you have treat helps you. And remember, that's how you the potential energy skip. It s 100 and 20 k V So this should give us about 1.2 and even I would give it at some. Some of this fusion reactions can still occur. Even toe. EDS. The 22 on energies that this right. And the reason behind that is that they are quantum particles must remember that they are very, very small. And the, uh, they for obey the rules off quantum cacs more rather than the classical ideas. So even though they might not have enough energy as you know, quantum particles stay exists. It's probably t waves, so they have a probability of existing, even though Dayton, Ohio, enough energy, this to a non zero probability that they can be close enough to each other at a given point type. And that is actually what we call quantum tunneling. And this tunneling allows the particles to actually be close enough to each other, even though they might not have enough energy to overcome the Coolum potential energy and therefore allowing the fusion to actually occur even at the temperatures