#12 "Nuclear Power"
http://www.phy6.org/stargaze/Snuclear.htm
Related site on nuclear weapons,
http://www.phy6.org/stargaze/Snucweap.htm
Also on the Sun's energy.
http://www.phy6.org/stargaze/Sun7enrg.htm
#13 Nuclear power in space,
https://en.wikipedia.org/wiki/Nuclear_power_in_space
http://www.eoearth.org/article/Nuclear power in space
#14 The natural reactor at Oklo,
http://en.wikipedia.org/wiki/Oklo_phenomenon.html
#15 "The Making of the Atomic Bomb " by Richard Rhodes, 886 pp.,Simon and Schuster 1988.
"Nuclear Renewal," is a short book about nuclear energy by the same author, reviewed at
http://www.phy6.org/outreach/books/NuclEnrg.htm
Answers to problems in "Nuclear Energy"
(S-8A-1) The Foundations: Atoms and Nuclei
(1) If chlorine consists of 25% Cl37 and 75% Cl35 , and A is Avogadro's number--what is the mass of A atoms of chlorine? (That would be the effective atomic weight in natural chlorine).
Solution
Out of 4 atoms, 3 will have atomic weight 35 and one will have 37. The average is the sum divided by 4 -- (105 + 37)/4 = 142/4 = 35.5
(2) Compile a glossary, defining briefly in alphabetical order in your own words:
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Alpha particle | Energetic helium nucleus, emitted by radioactive nuclei
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Atom | Elementary building block in the chemistry of matter
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Atomic weight | Mass of an atom, in units of a hydrogen atom mass.
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Avogadro's number | Number of atoms or molecules in a number of grams equal to the
atomic or molecular weight
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Beta particle | Fast electrons emitted by radioactive nuclei
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Electromagnetic radiation | A family of waves propagating in space, representing. oscillating electric and magnetic forces, e.g. light, radio.
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Electron | Light elementary particle, negatively charged, found in all atoms.
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Energy level | One of the energies at which, according to quantum laws, atoms
or nuclei may be found.
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Excited state of atom | A state of an atom with more energy than the lowest
"ground state"
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Excited state of atomic nucleus | A state of the atomic nucleus with more energy than
the stable (or most stable) "ground state"
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Frequency of EM wave | Number of oppositely directed excursions of the electric or magnetic force
aAt a point in space where the wave passes
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Gamma rays | Electromagnetic radiation of very short waves, emitted by nuclei
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Ground state | The lowest energy state of an atom or nucleus
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Half life | For a radioactive element, the time needed for half of it to decay
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Ion | Atom or molecule which has lost one or more electrons, or attached extra ones.
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Isotope | Variety of a chemical element with the same no. of protons & neutrons
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Molecule | A chemical combination of two or more atoms.
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Molecular weight | The sum of atomic weights of a molecule
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Neutrino | Uncharged and nearly massless elementary particle; may carry energy
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Neutron | Uncharged nucleon, similar to proton.
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Nuclear radiation | Waves or particles emitted by unstable atomic nuclei.
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Nucleus (of atom) | Core of an atom, electrically positive and with most of the mass.
Photon Quantity of energy associated with the emission or absorption of
an electromagnetic wave.
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Planck's constant, | A physical constant appearing in equations of quantum physics.
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Proton | A positive particle; neutrons and protons form the atom's nucleus
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Quantum mechanics | Rules of mechanics on the atomic and nuclear scale
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Radiation, | General name for either electromagnetic or nuclear radiation.
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(3) Very high energy ions from space ("cosmic radiation") arrive at the top of the Earth's magnetosphere, collide with atoms and splash out fragments, some of which are neutrons. Neutrons do not feel magnetic forces, but electrons and protons can get trapped, though those splashed from the atmosphere always return and hit the atmosphere again.
Is this a credible explanation to the "radiation belt" trapped in the magnetic field of the Earth?
Yes. Particles from the atmosphere always return and are absorbed by the atmosphere, but neutrons may decay in flight and yield energetic protons (also electrons) which could appear on a magnetically trapped orbit. The original Van Allen belt is believed to originate that way.
(4) A certain radioactive isotope has a half-life of 2 days. How long approximately does it take until only 1/1000 of it remains in a given sample?
About 20 days, or 10 half-lives, because (1/2)10 = 1/1024
(5) Hydrogen (forming H2 molecules) weighs about 90 gram per cubic meter. How many molecules of hydrogen are in one cubic micron (a micron is the millionth part of the meter)?
If A is Avogadro's number 6.022 1023 then 2 gram hydrogen contains A molecules, and 90 gram
contain 45A. A cubic micron is 10-12 cubic meters, so the number is
N = 45 (6.022 1023) 10-12 = 271 1053 = 2.71 107
or about 27 million molecules
(S-8A-2) Nuclear Binding Energy
(1) Why can't one find in our environment elements whose atoms weigh 300 times as much as the proton, or more?
Such nuclei contain too many protons repelling each other, and in spite of the strong nuclear attraction between their particles, are unstable.
(2) Compile a glossary, defining briefly in alphabetical order in your own words:
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Alpha radioactivity | Nuclear instability leading to the emission of alpha particles
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Beta Radioactivity | Nuclear instability leading to the emission of electrons, from conversion of neutrons to proton-electron pairs (plus neutrino)
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Binding energy | The energy holding a nucleus together--the amount needed
to completely break it apart.
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Controlled nuclear fusion | Combination of light nuclei to heavier ones, in the lab
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Core of the Sun | The central region of the Sun where energy is generated | |
Curve of binding energy | The graph of nuclear binding energy against mass. | |
Daughter isotope | An isotope resulting from radioactive decay.
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Deuterium | The heavy isotope of hydrogen, contains proton + neutron
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Mass spectrometer | Instrument to measure the mass of nuclei, by deflecting a beam of ions magnetically or timing their flight | |
Nuclear fusion | Nuclear reaction joining light nuclei to form heavier ones. | |
Positron | The electron's positive counterpart (can be created in the lab)
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Controlled nuclear fusion | Combination of light nuclei to heavier ones, in the lab
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Short range force | A force which decreases with distance r faster than 1/r2
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Strong (nuclear) force | A short range attraction in the nucleus, holding protons and
neutrons
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Weak (nuclear) force | A weaker short-range nuclear force, tries to balance number of
neutrons and protons.
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(3) What is the source of the Sun's energy?
Nuclear fusion of hydrogen in the Sun's core, producing helium
(4) Why is the binding energy of the nucleus given a negative sign?
The energy of a nucleus is what is extra energy available; zero energy means all particles are independently spread out. A bound nucleus needs energy input to reach "zero energy" state, so its energy is negative.
(5)
(a) The atomic weight of deuterium (2H) is 2.0140, of Helium 4He 4.0026 (in units of the proton mass), and the "rest energy" E=mc2 of the proton is 938.3 Mev (million ev, with 1 ev = one electron-volt; see #9). How many ev are released when two atoms of deuterium combine to one of 4He, by nuclear fusion?
2 (2,0140) – 4.0026 = 0.0254 atomic mass units
Mass converted to energy
E = mc2 = 0.0254 (938.3)Mev = 23.8 Mev = 2.38 107 ev
(b) If 1 ev = 1.60 10-19joule and Avogadro's number is A = 6.022 1023, how many joules are released by the fusion of 4 grams of deuterium?
4 gram helium contain A atoms, so the energy released is
E = (6.022 1023)(2.38 107)(1.60 10-19) joule
23 + 7 – 19 = 11
(6.022)(2.38)(1.60) = 22.93
So
E = 22.93 1011 joule =2.293 1012 joule
(c) One gram of TNT can release 3.8 kilocalories of energy, each of which is equivalent to 4184 joules. How many tons of TNT are required to release the energy calculated above?
1 gram TNT = (3.8) (4184) = 1.59 104 joule
(2.293 1012)/(1.59 104 ) = 1.442 108 gram = 144.2 ton TNT
(6) Here is another application of Einstein's equation E=mc2. You better be familiar with scientific notation for very small and very large numbers before trying to solve this, and be sure to check all steps of the calculation.
The Sun loses mass all the time, by at least two mechanisms.
First, it radiates sunlight energy E, and by the equivalence of energy and mass, the process must also reduce its mass. The energy radiated at the Earth's orbit--150 million kilometers from the Sun--is about 1300 watt ("the solar constant") per square metre of area perpendicular to the Sun's rays, and the velocity of light is about c = 300,000 km/sec.
Second, it also emits the solar wind. For reasons which after 70 years are still unclear, the uppermost atmosphere of the Sun ("solar corona") is very hot, about a million degrees centigrade, explaining why atoms in that layer tend to be stripped of most or all of their electrons--e.g. iron atoms missing a dozen electrons, which requires a tremendous amount of buffeting.
The Sun's gravity cannot hold down a gas so hot. Instead, the topmost solar atmosphere is constantly blown away as the solar wind--a rarefied stream of free ions and electrons, moving outwards at about 400 km/second The density of that wind at the Earth's orbit is about 10 protons per cubic centimeter (taking into account the presence of helium ions), and the mass of a proton is about 1.673 10-27 kilograms.
Which of the two processes causes the Sun a greater mass loss?
============
Solution let us compare the mass loss due to either process through an area of 1 square metre at the Earth's orbit, perpendicular to the flow of sunlight, during one second. Working in metres, seconds and kilograms, c = 3 108 metre/sec, and the energy flow is 1300 joule/sec. If m is the mass lost during that time through he chosen area (by conversion to radiant solar energy)
m = E/c2 = 1300 / 9.1016 = 1.444 10–14 kilograms
The solar wind passing through the same area includes all the matter contained in a column of cross section 1 metre2 and of length c = 400 kilometers or 4 105 metres. One cubic metre contains 106 cubic centimeters and the mass of 107 protons. The flow through the area is therefore 4 1012 protons, with a mass 6.69 10–15 kilograms.
The loss due to sunlight is therefore greater by about a factor of two. Still, it is remarkable how close these two numbers are to each other--one dictated by processes in the innermost core of the Sun, the other by processes in its outermost layer.
Coincidence, you say?
(a similar calculation can be found in http://www.phy6.org/stargaze/Lsun7erg.htm#massloss )
(7) An object (e.g. a spaceship) ejected from the surface of Earth needs v 1 = 11.3 km/s to escape Earth's gravity ("escape velocity"),
A neutron has rest energy E1 = mc2 = 939.535 MeV (million electron volts). If the velocity of light is 300,000 km/sec (close enough) and a neutron is ejected from the Earth's surface with just enough velocity to escape gravity, what is its energy in MeV (or in electron volts, eV)? Use the non-relativistic expression when deriving the kinetic energy E1 of the escaping neutron (it is accurate enough).
Solution: If m is the mass of the neutron,
E0 = mc2 = 9.39535 108 ev
E1 = m v12 / 2
Dividing the 2nd equation by first, with all velocities in meters/second:
E1/ E0 = E1/ 9.39535 108 = (0.5) (v1/c) 2
= (0.5) (1.13 104 / 3 108)2
= 0.5 (0.376666 10–4) 2
= 0.5 (0.1418777 10–8)
= 0.070939 10–8
E1 = (9.39535 108)( 0.070939 10–8) = 0.6665 eV
This is less than 1 eV! Radiation belt particles have energies of the order of MeV, and even electrons of the polar aurora have of the order of 10,000 eV (thermal energy of air molecules in your room is about 0.03 eV). Gravitational energy is therefore completely negligible by comparison--or in other words, the electromagnetic forces on particles in space tend to be much, much bigger than their gravitational forces.
(S-8A-3) Fission of Heavy Nuclei
(1) (For this problem, solve first problem (5) in the preceding section)
Assuming a U235 nucleus releases 200 Mev in a fission event (counting some secondary processes, see #10; the total averages 215 Mev), how many tons of TNT are needed to obtain the energy yielded by complete fission of 1 gram U235 ?
If A = 6,022 1023 is Avogadro's number, 1 gram of U2355 contains A/235 atoms. By (b) of preceding problem (5), each atom yields (2 108 ev)(1.6 10–19) joule. The total energy released is
(6.022 1023)(2 108)(1.6 10–19)joule /235 =
=(6.022 . 2 . 1.6 / 235) 1012
= 6.2 1010 joule
By (c) of the preceding problem (5), 1 gram of TNT holds 3.8 kilo-calories or
1.59 . 104 joule
So the energy released is the same as
(8.2/1.59) 10 (10–4) gram = 5.16 106 gram = 5.16 ton TNT
(2) Compile a glossary, defining briefly in alphabetical order in your own words:
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Barn (unit) | Area of 10–24 square cm., unit of nuclear cross section.
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Cascade for isotope enrichment | Teaming of many isotope separators for enrichment
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Chain reaction (nuclear) | A fission reaction in which each fission produces at least one additional fission
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Critical mass | A mass of nuclear fuel sufficient for a chain reaction. | |
Cross section | (for nuclear interaction) Equivalent target area in a nucleus for an
incoming particle to produce a reaction. | |
Curve of binding energy | The graph of nuclear binding energy against mass. | |
Delayed neutrons | Neutrons emitted from fission with 1-2 second delay
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Enrichment (of uranium) | Technology raising the fraction of the U235 isotope.
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Fission (nuclear) | The splitting of an atomic nucleus into two large fragments.
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Fission fragments | Nuclei of lighter elements, produced by nuclear fission.
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Fuel rods | Rods containing fuel, inserted into a nuclear reactor.
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Graphite | A form of carbon, used as moderator in nuclear fission.
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Heavy water | Water in which deuterium replaces hydrogen
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Isotope separation by centrifuges | Isotope separation by a gas centrifuge. | | |
Isotope separation by porous partitions | Separation of isotopes by gas flow through porous partitions.
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Photon | A packet of energy formed when an electromagnetic wave is absorbed.
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Plutonium | An artificial element of atomic weight 94, common nuclear fuel.
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"Poisoning" of a nuclear reactor | The accumulation of neutron-absorbing fission fragments, reducing or stopping fission in a reactor.
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Prompt neutrons | Neutrons emitted promptly from nuclear fission, about 98%
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Reprocessing of nuclear fuel | Chemical separation of fission product from unburned nuclear fuel and artificial fuel isotopes.
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Thermal neutron | A neutron slowed down by a moderator to thermal energies, much below 1 eV.
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(S-8A-4) Controlling the Nuclear Reaction
(1) Why is the nuclear power industry interested in elements such as deuterium (2H), carbon (12C), cadmium, Thorium (232T), Uranium (238U), (235U) and (233U), Plutonium (239Pu),
Deuterium and carbon are preferred moderators in nuclear reactors
Deuterium and the related nucleus tritium (3H) are also candidates for controlled fusion
Thorium 232T can absorb a neutron from uranium fission and turn into 233U, a usable nuclear fuel.
235U is a nuclear fuel found in nature as 0.7% of uranium. Natural or enriched, it can fuel nuclear reactors. Uranium enriched in 235U is also used in nuclear bombs.
238U is the most common isotope of uranium in nature
239Pu is an artificial isotope of element 94, produced (in steps) by neutron absorption in 238U.
(2) Compile a glossary, defining briefly in alphabetical order in your own words:
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Breeder reactor | A nuclear reactor producing new fuel by neutron capture.
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Cadmium | A metal used in reactor control, since it avidly consumes neutrons.
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Chernobyl accident | The destruction in 1986 of a nuclear power reactor in Chernobyl, Ukraine
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Containment building | building A building with thick walls enclosing a nuclear reactor, confining any waste released in an accidental meltdown. | |
Control rods | Rods loaded with cadmium thrust into a nuclear reactor, to control the rate of fission
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Fast neutrons | The Unmoderated neutrons from nuclear fission, useful in converting 238U into 239Pu , and also in nuclear bombs. | |
Meltdown | Destruction of the core of a reactor by uncontrolled heat release. | |
Oklo phenomenon | Natural fission in uranium deposits, which occurred in Oklo,
Gabon, about 1.5 billion yeas ago.
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Prompt critical nuclear reactor | A nuclear reactor losing control, by maintaining a chain reaction with prompt neutrons alone
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Thorium cycle | Nuclear power cycle using 233U produced from Thorium
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Three Mile Island accident | A partial meltdown in 1979 of a nuclear power reactor at Three Mile Island, near Harrisburg, Pennsylvania. | |
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