In a fusion reaction, the nuclei of two atoms join to form a single atom of a di

Question

In a fusion reaction, the nuclei of two atoms join to form a single atom of a different element. In such a reaction, a fraction of the rest energy of the original atoms is converted to kinetic energy of the reaction products. A fusion reaction that occurs in the Sun converts hydrogen to helium. Since electrons are not involved in the reaction, we focus on the nuclei. Hydrogen and deutrim (heavy hydrogen) can react to form helium plus a high-energy photon called a gamma ray: Objects involved in the reaction: Particle H (proton) H (deuterium) 1 3He (helium) 2 # of protons # of neutrons Charge Rest Mass (atomic mass units) 1.0073 2.0136 3.0155 +2e gamma ray Although in most problems you solve in this course you should use values of constants rounded to 2 or 3 significant figures, in this problem you must keep at least 5 significant figures throughout your calculation. Problems involving mass changes require many significant figures because the changes in mass are small compared to the total mass. In this problem you must use the following values of constants, accurate to 5 significant figures: Constant c (speed of light) e (charge of a proton) atomic mass unit Value to 5 significant figures 2.9979e8 m/s 1.6022e-19 coulomb 1.6605e-27 kg 8.9875e9 N-m2/c2 4nco The diagrams below represent different states in the fusion process: He 612 2H A proton (H nucleus) and a deuteron (2H nucleus) start out far apart. An experimental apparatus shoots them toward each other (with equal and opposite momenta). If they get close enough to make actual contact with each other, they can react to form a helium-3 nucleus and a gamma ray (a high energy photon, which has kinetic energy but zero rest energy). Consider the system containing all particles. Work out the answers to the following questions on paper, using symbols (algebra), before plugging numbers into your calculator.

Explanation / Answer

(1) We will apply the conservation of energy
Kinetic energy of proton + kinetic energy of deuterium = Potential energy + Final kinetic energy
Final kinetic energy = Kinetic energy of proton + kinetic energy of deuterium - Potential energy
Potential energy = KQ2 /2r = (9*109)*(1.6*10-19)2 /(2*0.9*10-15) = 1.28*10-13 J
Final kinetic energy = 2.19*10-13 + 1.09*10-13 - 1.28*10-13 = 2*10-13 J
(2) Now for kinetic energy of the reactants
Kf = final kinetic energy + (mP + mde- mhe)C2
=  2*10-13 + (1.007+ 2.0136 - 3.0155)*(1.6605*10-27)*(3*108)2
= 9.622*10-13 J

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