The Physics
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Opus in profectus

# Fusion

## Problems

### practice

1. Hydrogen fusion in the sun is a multistep reaction, but the net result is that four hydrogen atoms fuse into one helium atom (plus a bunch of junk).

11H → 42He + 2(0+1e + 00γ + 00ν)

The mass of the sun is 1.99 × 1030 kg, 91% of which is hydrogen. Its power output is 3.85 × 1026 W. Determine…

1. the mass of four hydrogen atoms
2. the mass defect when four hydrogen atoms fuse into one helium atom (in atomic mass units and megaelectronvolts)
3. the rate at which the sun's mass is decreasing
4. the total mass destroyed if all the sun's hydrogen were converted into helium
5. the expected lifetime of the sun (assuming its power output will remain constant)
2. The fuel used in most high-yield thermonuclear weapons is solid lithium 6 deuteride (ρ = 820 kg/m3). These weapons, commonly known as "hydrogen bombs" or "H-bombs", use the energy released when a nuclueus of light lithium and heavy hydrogen, also known as deuterium, fuse to form two nuclei of ordinary helium (a two part process).

63Li + 21H → 2 42He

Given the reaction described above, determine…

1. the mass of one molecule of lithium 6 deuteride
2. the mass defect when one molecule of lithium 6 deuteride is transformed into two atoms of helium (in atomic mass units and megaelectronvolts)

A typical thermonuclear weapon has a yield on the order of several million tons of TNT or about as destructive one truck bomb for every person in Brooklyn. (One ton of TNT is equal to 4.184 GJ by definition.) Given a "small" hydrogen bomb with an explosive yield of one megaton, determine…

1. the mass destroyed in its detonation
2. the mass of the fuel required
3. the volume of the fuel required
3. Write something.
4. The text below describes some of the nuclear reactions that occurred during the detonation of the first hydrogen bomb code named "Mike" at Eniwetok Atoll in 1952. Identify the fusion reactions described in these two paragraphs and rewrite them in symbolic form; that is, as reaction equations.

All these processes, proceeding through microseconds, prepared Mike for thermonuclear burning. Now the escaping X-radiation of the fissioning sparkplug heated the compressed deuterium at its boundaries; the increasing thermal motion of the deuterium nuclei pushed them together until they passed the barrier of electrostatic repulsion between them and came within range of the nuclear strong force, at which point they began to fuse. Some fused to form a helium nucleus — an alpha particle — with the release of a neutron, the alpha and the neutron sharing an energy of 3.27 MeV(1). The neutron passed through the electrified mass of fusing deuterons and escaped, but the positively charged alpha dumped its energy into the heating deuterium mass and helped heat it further.

Other deuterium nuclei fused to form a tritium nucleus with the release of a proton, the triton and the proton sharing 4.03 MeV(2). The positively charged proton dumped more energy into the deuterium mass. The tritium nucleus fused in turn with another deuterium nucleus to form an alpha particle and a high-energy neutron that shared 17.59 MeV(3). The 14 MeV neutrons from this reaction began to escape the hot, compressed deuterium plasma and encountered the U238 nuclei of the vaporized uranium pusher. U238 fissions when it captures neutrons with energies above 1 MeV; so the U238 of the uranium pusher began to fission then under the intense neutron bombardment, flooding more X rays back into the deuterium mass from the outside just as the sparkplug fission reaction was radiating them from the inside, trapping the deuterium between two violent walls of heat and pressure. Deuterium-bred tritium fused with tritium as well, producing a helium nucleus and two neutrons that shared 11.27 MeV of energy(4). At lower orders of probability, deuterium captured a neutron and bred tritium(5); deuterium-bred helium fused with deuterium and made heavy [ordinary] helium plus a highly energetic proton(6), or captured a neutron and bred tritium plus a proton(7). All these reactions contributed to the force of the Mike explosion.

### numerical

1. This question is composed of three related parts.
1. What percentage of the original mass is converted to energy when two heavy hydrogen atoms fuse to form one helium atom?
2. Calculate the mass destroyed in the detonation of a typical hydrogen bomb if its explosive potential is equivalent to one million tons of TNT. (One ton of TNT possesses 4.184 GJ of chemical energy.)
3. Using your results from the previous two questions, determine the mass of nuclear fuel needed to make an "H-bomb" capable of leveling hundreds of square kilometers.

### statistical

1. The triple alpha process of fusing three helium nuclei to make one carbon nucleus takes place in stars whose core temperatures are about 108K (one hundred million kelvin). In a star whose mass is more than five times that of our sun, the fusion of helium nuclei (a.k.a. alpha particles) onto lighter nuclei continues element by element. Adding helium to carbon makes oxygen. Adding helium to oxygen makes neon. Add another helium and you get magnesium, then silicon, then sulfur, and so on. Each step raises the atomic number by 2 and the mass number by 4. The process gets a little confusing after calcium and terminates at iron. Beyond iron, the reactions consume energy — they become endothermic. The table below summarizes the simplest reactions in the alpha fusion chain. Compute the maximum possible energy of the emitted gamma ray in each step.
Alpha fusion chain
parents daughters Eγ(MeV)
a. 126C  +  42He 168O  +  00γ
b. 168O  +  42He 2010Ne  +  00γ
c. 2010Ne  +  42He 2412Mg  +  00γ
d. 2412Mg  +  42He 2814Si  +  00γ
e. 2814Si  +  42He 3216S  +  00γ
f. 3216S  +  42He 3618Ar  +  00γ
g. 3618Ar  +  42He 4020Ca  +  00γ
2. Carbon-carbon fusion occurs in stars whose cores are hotter than 6 × 108K (600 million kelvin). At these temperatures, carbon preferentially fuses with itself to produce heavier elements, mostly sodium (2311Na) and neon (2010Ne). The principal reactions are shown in the table below. Compute the mass-energy difference between the parent and daughter nuclei for each reaction and state whether the reaction is endothermic or exothermic.
Carbon-carbon fusion
parents daughters m
endo or exo?
a. 126C  +  126C 2412Mg  +  00γ
b. 126C  +  126C 2312Mg  +  10n
c. 126C  +  126C 2311Na  +  11p
d. 126C  +  126C 2010Ne  +  42He
e. 126C  +  126C 168O  +  242He
3. Oxygen-oxygen fusion occurs in stars whose cores are hotter than 109K (one billion kelvin). Silicon (2814Si) is the the major product from the nuclear fusion of oxygen. The principal reactions are shown in the table below. Compute the mass-energy difference between the parent and daughter nuclei for each reaction and state whether the reaction is endothermic or exothermic.
Oxygen-oxygen fusion
parents daughters m
endo or exo?
a. 168O  +  168O 3216S  +  00γ
b. 168O  +  168O 3116S  +  10n
c. 168O  +  168O 3115P  +  11p
d. 168O  +  168O 2814Si  +  42He
e. 168O  +  168O 2412Mg  +  242He