A protonand a deuteron (, “heavy” hydrogen) 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-3nucleus and a gamma ray (a high-energy photon, which has kinetic energy but zero rest energy):
This is one of the thermonuclear or fusion reactions that takes place inside a star such as our Sun.
The mass of the proton is 1.0073 u(unified atomic mass unit,), the mass of the deuteron is 2.0136 u, the mass of the helium-3nucleus is 3.0155 u, and the gamma ray is massless. Although in most problems you solve in this course it is adequate to use values of constants rounded to two or three significant figures, in this problem you must keep at least six 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. (a) 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. What is the approximate initial total kinetic energy of the proton and deuteron required for the fusion reaction to proceed, in joules and electron volts ()? (b) Given the initial conditions found in part (a), what is the kinetic energy of theplus the energy of the gamma ray, in joules and in electron volts? (c) The net energy released is the kinetic energy of theplus the energy of the gamma ray found in part (b), minus the energy input that you calculated in part (a). What is the net energy release, in joules and in electron volts? Note that you do get back the energy investment made in part (a). (d) Kinetic energy can be used to drive motors and do other useful things. If a mole of hydrogen and a mole of deuterium underwent this fusion reaction, how much kinetic energy would be generated? (For comparison, aroundare obtained from burning a mole of gasoline.) (e) Which of the following potential energy curvesin Figure 6.87 is a reasonable representation of the interaction in this fusion reaction? Why?
As we will study later, the average kinetic energy of a gas molecule is, whereis the “Boltzmann constant,”, andis the absolute or Kelvin temperature, measured from absolute zero (so that the freezing point of water is). The approximate temperature required for the fusion reaction to proceed is very high. This high temperature, required because of the electric repulsion barrier to the reaction, is the main reason why it has been so difficult to make progress toward thermonuclear power generation. Sufficiently high temperatures are found in the interior of the Sun, where fusion reactions take place.
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