Fusion Reaction: Kinetic Energy Gain & Energy Source

[SpeedyKeeper]

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 deuterium (heavy hydrogen) can react to form helium plus a high-energy photon called a gamma ray:




Objects involved in the reaction: Particle # of protons # of neutrons Charge Rest Mass (atomic mass units)
1H (proton) 1 0 +e 1.0073
2H (deuterium) 1 1 +e 2.0136
3He (helium) 2 1 +2e 3.0155
gamma ray 0 0 0 0

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 Value to 5 significant figures
c (speed of light) 2.9979e8 m/s
e (charge of a proton) 1.6022e-19 coulomb
atomic mass unit 1.6605e-27 kg
8.9875e9 N·m2 /C2


A proton (1H 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.



Compare the initial state and final states of the system. Which quantities have changed?

rest energy
potential energy
kinetic energy


The deuterium nucleus starts out with a kinetic energy of 1.38e-13 joules, and the proton starts out with a kinetic energy of 2.77e-13 joules. The radius of a proton is 0.9e-15 m; assume that if the particles touch, the distance between their centers will be twice that. What will be the total kinetic energy of both particles an instant before they touch?
4 joules

B: Reaction to make helium
Now that the proton and the deuterium nucleus are touching, the reaction can occur.
Take the final state from the previous process to be the initial state of the system for this new process.



Compare the initial state and final states of the system. Which quantities have changed?

rest energy
kinetic energy
potential energy




What is the kinetic energy of the reaction products (helium nucleus plus photon)?
joules

C: Gain of kinetic energy:
What was the gain of kinetic energy in this reaction? (The products have more kinetic energy than the original particles did when they were far apart. How much more?)
joules

D: Fusion as energy source
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?
joules
(For comparison, around 1e6 joules are obtained from burning a mole of gasoline.)

Homework Equations

The Attempt at a Solution

 

[DaveC426913]

Sooooooo...we'll just ... fill in the blanks for you then?

Flash forward 10 years and you'll be operating the fusion reactor at the end of my street?

 

[kerimek]

A: The initial state of the system contains two particles with rest energy and kinetic energy, while the final state contains one particle with rest energy and kinetic energy and a photon with kinetic energy but zero rest energy. Therefore, the quantities that have changed are rest energy, potential energy, and kinetic energy.

B: The total kinetic energy of both particles an instant before they touch can be calculated by adding their individual kinetic energies, which are 1.38e-13 joules and 2.77e-13 joules respectively. This gives a total kinetic energy of 4.15e-13 joules.

C: The kinetic energy of the reaction products can be calculated by subtracting the initial kinetic energy (4.15e-13 joules) from the final kinetic energy of the helium nucleus and the photon. This gives a kinetic energy of 1.38e-13 joules.

D: To find the gain of kinetic energy in this reaction, we can subtract the initial kinetic energy (4.15e-13 joules) from the final kinetic energy of the reaction products (1.38e-13 joules). This gives a gain of kinetic energy of 2.77e-13 joules.

If a mole of hydrogen and a mole of deuterium underwent this fusion reaction, the total kinetic energy generated would be 2.77e-13 joules x 6.02e23 moles = 1.67e11 joules. This is significantly more energy than is obtained from burning a mole of gasoline, making fusion a potentially powerful energy source.