# Atoms, Energy Levels Nuclear Energy

```Atoms, Energy Levels Nuclear Energy
Energy is the ability to do work. The SI unit of energy or
work is the Joule. Work is W = F d, so a F force of 1 Newton
pushing over a distance d of 1 meter does a work of 1 Joule.
In mechanics there are two kinds of energy. The kinetic energy of a mass m kilograms (kg) moving with a speed v meters
per second (v c) is KE = mv 2 /2. There is also potential
energy associated with the position of an object in a field of
force, a such as a gravitational or an electric field. The potential energy relative to ground level of a mass m at a height
h is P E = mgh where the acceleration of gravity is g = 9.8
m/s/s. In an electric field a charge q at some location has a
potential energy P E = qV where V is the voltage at that location. Heat is the transfer of internal energy, which is kinetic
and potential energy of the random motions of the atoms in
a material. Normally the unit heat is the kilocalorie, or food
calorie, which is the energy needed to raise the temperature of
1 kilogram of water by 1◦ Celsius (or 1.8◦ Fahrenheit). Joule
showed that 1 kilocalorie=4,186 Joules. An important unit of
energy in atomic and nuclear physics is the electron-Volt (eV).
1eV=1.6 × 10−19 Joules.
Eintein discovered that energy and mass are equivalent in the
formula E = mc2 . Since the speed of light c = 3 × 108 m/s,
m = 1 kilogram somehow is equivalent to the enormous amount
E = 9 × 1016 Joules=2.15 × 1013 food Calories≈ 2.5 × 107 tons
of TNT. So, if one could convert 1 kilogram of mass completely
into energy the explosion would release the energy equivalent
of 25 Megatons of TNT. Or, that energy could run a 1,000
Megawatt electric plant (large city plant) for almost 3 years.
Fermions and Bosons
Quantum physics shows that all particles are either fermions
or bosons. Fermions are particles of spin 1/2 or 3/2, or 5/2, ...
Electrons, protons and neutrons are fermions because they all
have spin 1/2. Fermions obey the Pauli Exclusion Principle,
which says that no two identical fermions can occupy the same
quantum state. One might say that they cannot be in the same
place at the same time. Electrons are elementary particles with
spin 1/2, negative charge −e = −1.6 × 10−19 Coulombs, and
mass me = 9.109 × 10−31 kg. Protons have a charge +e and a
mass mp = 1.673 × 10−27 kg, or about 1836 times more mssive
than the electron. Neutrons are electrically neutral and a mass
mn = 1.675 × 10−27 kg, or a little heavier than a proton.
Bosons are particles of spin 0, or 1, or 2, ... The most important
example are photons of light which have spin 1, a property
that leads to the polarization of light. Bosons do not obey the
Pauli Exclusion Principle. Many photons can occupy the same
quantum state, they can be in the same place at the same time,
as light beams normally do. This is what makes lasers possible.
Atoms
Atoms consist of a very small nucleus with a diameter of about
10−15 meters surrounded by a cloud of electrons. The diameter of the electron cloud is about 10−10 meters. The nucleus
contains a number Z (called the atomic number, or position in
the periodic table of the elements) of protons and some number
N of neutrons. In a normal, neutral, atom the electron cloud
around the nucleus contains the same number Z of electrons.
The chemical elements are Hydrogen (H) has 1 proton, Helium
(He) has 2 electrons, Lithium (Li) has 3 electrons, etc.
• Isotopes: But, There are isotopes of a chemical element that
differ by the number of neutron in the nucleus. So, a nuclear no+Z
tation is used to distinguish isotopes, N
X, so the important
Z
isotopes of hydrogen are: common hydrogen 11 H, deuterium
2
3
1 H, and tritium 1 H, beyond that they are completely unstable
and undergo radioactive decay. For helium there are common
helium 42 H, and a more rare 32 H. In this notation 10 n is a neutron, 11 p is a proton, and 0−1 e is an electron. Some isotopes are
unstable, or radioactive, and emit particles when they decay,
14
0
for example 14
6 C →7 N +−1 e, which is useful in radioactive
dating because it has a half life of 5,600 years. The electron
that is emitted with high energy is called a β particle. Notice
the conservation laws that the total of the upper numbers and
of the lower numbers are the same on both sides of the nuclear
reactions. In the fission reaction the final mass mf is less than
the initial mass mi , the difference appears as a large amount of
energy by E = (mf − m1 )c2 . The neutron produced can react
with other uranium 235 nuclei and generate a chain reaction.
This can be used for nuclear weapons or for power generation
in a nuclear reactor.
• Energy levels in atoms
Physical systems, specially atoms and nuclei exist with certain
special values of energy, called energy levels, that are characteristic of each system. For example the allowed energy levels
of the electron in the hydrogen atom are
−13.6
eV, n = 0, 1, 2, 3, . . .
n2
They are negative indicating that the electron is bound, or
trapped in the atom. If the electron escapes, and we have
En =
an ion, this corresponds to n = ∞, or E∞ = 0. These were
discovered by Niels Bohr, in what was the real beginning of
quantum physics. When the electron jumps from a high energy
level EH to a n empty lower energy level EL it emits a photon
(a particle of light) that carries away the energy of the jump.
The frequency f of the light is
hf = EH − EL , or
hc
= EH − EL , where λ is the wavelength.
λ
This may be the most important equation of quantum physics.
The energy levels of the H atom explain why the spectrum of
the light emitted by hydrogen has the discrete lines that are
observed. The jumps can be visualized as shown.
Jumps that end up in the n = 2 state produce the Balmer series
of visible spectral lines. The Lyman series is ultraviolet. An
energy level can consist of several quantum states depending
on the angular momentum (` = 0, 1 2, . .) and spin (up or
down). Due to the Pauli exclusion principle all the electrons in
an atom cannot go to the lowest energy level, but they must go
into higher levels as the lower ones get occupied. This is fundamental in the structure of atoms and the periodic table. An
electron can jump to an empty higher energy level by absorbing
a photon of the correct energy. Other photons are ignored. A
material can be transparent to visible light photons, like glass,
because they do not have energies that can be absorbed by the
glass.
• Energy bands in crystals
In solid crystals the energy levels become bands of energy levels.
In the figure below the lower valence bands have all their energy
levels occupied by electrons.
Therefore an electron cannot move because it would have to
move to an already occupied level, which is forbidden by the
Pauli exclusion principle. On the right the upper conduction
band is empty of electrons, and there is a large energy gap
between the valence and conduction bands. An electron could
move only if it acquires the energy to jump the gap, which
does not normally happen and it is an insulator. If the gap
is small it is a semiconductor, like silicon. If the two bands
overlap, the electrons in the valence band find empty levels
immediately above them, with no gap. Here we have a metal
which conducts electricity because the electrons can move into
the empty levels in the conduction band under the influence of
any applied electric field. Semiconductor electronics is based
on silicon crystals, which have four electrons in their outermost
orbits, doped with impurities. Phosphorus or arsenic atoms are
n type impurities because they have five outermost electrons
and are able to donate the fifth electron into the conduction
band, so there is a controlled amount of electrical conductivity.
Aluminum or gallium are p type impurities because they have
three outermost electrons and are able to accept an electron
from a nearby silicon atom, thus creating a vacancy (a ”hole”)
in the valence. This allows conductivity because when an electron jumps into the hole, it leaves behind another hole. It is as
if we had conduction by holes that act as positive charges. An
n − p junction can be used to make a diode. An n − p − n, or
p − n − p junction can make a transistor. Transistors can act
as electrically controlled switches. An open switch transistor
can represent a 0, and a closed one can represent a 1. This
is fundamental in computers because transistors can be made
very small, and a computer chip can contain millions of them.
Other systems have the same behavior but the energy levels
are more complicated. In atoms the energy jumps involve eVs.
Inside the nucleus the particles also have energy levels but the
jumps are much more powerful, measured in MeV( millions of
eVs).
• Uranium, Nuclear reactors
There are many types of nuclear reactions. An example is
238
4
234
92 U →2 He+92 U +γ, where γ is a gamma ray, and the nucleus
4
2 He is called an α particle. This radioactive decay occurs with
235
a slow half life of 4.5 × 109 years. 238
92 U and 92 U are two
important isotopes of uranium. In nature, say in a uranium
235
mine, 99.3% is 238
92 U and 0.7% is 92 U . One can enrich the
percentage of 235 using cascades of ultracentrifuges operating
in series, in what is a very slow and expensive process. The
uranium 235 isotope splits into pieces of smaller mass when hit
by a slow or fast neutron, a process called fission. For example
235
1
141
92
1
92 U +0 n →56 Ba +36 Kr + 30 n, where the Ba and Kr and
neutrons come out moving fast with a total kinetic energy of
about 200 MeV, in other words, it gets very hot. The neutrons
released could hit other 235
92 U and split them, leading to a chain
reaction. A sphere of pure 235
92 U with a critical mass of about
52 kilograms is needed to make an ”atomic” (really a nuclear)
bomb. A nuclear reactor using uranium as a fuel tries to achieve
a controlled fission reaction. One problem is that 238
92 U absorbs
fast neutrons that are lost to the chain reaction, although it
239
1
produces plutonium according to 238
92 U + fast 0 n →94 P u +
2 0−1 e. A sphere of pure 239
94 P u with acritical mass of about
10 kilograms is needed to make an plutonium bomb, but is
harder to do. The reactor is possible because 238
92 U ignores
slow neutrons, they just bounce off. So, in a nuclear reactor,
such as shown below, we need to slow does the neutrons. This
is done by surrounding the fuel pellets with a ”moderator”,
which is a material that slows down the neutrons so they will
not be absorbed by the 238
92 U . The reactor vessel shown above
contains the fuel rods that contain uranium, the moderator,
and control rods of neutron absorbing materials such as cadmium, boron, etc. that can be inserted to stop the reaction. A
more complete reactor is shown below.
The moderator gets very hot. In the first reactor figure, the
moderator is cooled by water in pipes that removes heat, becomes steam, and can drive a turbine to generate electricity.
This cooling water does not come into contact with the moderator of the fuel rods. In the second reactor, the moderator
(water or liquid sodium) flows through another vessel that contains water and the working steam is generated there. The
spent fuel rods are taken out and replaced. The spent fuel rods
are removed and stored and must be kept cool because there
are a lot of radioactive decays happening that release a lot of
heat.
If the moderator does not absorb neutrons one can use natural
uranium, without enriching it. Graphite and heavy water can
be used as moderators with natural uranium. Water is H2 O.
Heavy water has deuterium in place of the normal hydrogen.
The Chernobyl reactor, and the first reactor built in 1942 by
Enrico Fermi at the University if Chicago, used natural uranium with a graphite moderator. When the reactor vessel of
the Chernobyl reactor broke in 1986, the graphite caught fire,
and the smoke carried a lot of radioactive isotopes to the environment. Canadian reactors use heavy water as a moderator.
American, Japanese, and other reactors use normal ”light” water as a moderator. Then they must use enriched uranium, to
something like 3% to 5% instead of 0.7% 235
92 U .
The reactor vessel is surrounded by a very strong primary containment vessel. The Chernobyl reactor did not have a strong
primary containment vessel. The uranium fuel pellets are surrounded by a zirconium alloy to keep the radioactive decay
products encased. The fuel pellets must be cooled, even after
the fission is stopped by the control rods, because the readioactive by-products in the reactor’s fuel rods continued to generate
tremendous amounts of heat and have to be cooled by circulating water using water pumps driven by diesel generators. If
this fails a meltdown of the fuel pellets can happen and they
can melt through the reactor vessel, the containment vessel,
and the reactor building, and get into the environment, with
release of lots of dangerous radiation.
The zirconium resists oxidation (rusting) in the water moderator. However, if the moderator gets too hot, and turns into
steam, the zirconium at high temperature will oxidize, taking
oxygen from the water and releasing hydrogen. If the pressure
in the reactor and in the primary containment vessel gets too
high steam is released accompanied by hydrogen, which can
explode when it comes in contact with air. In March 2011 an
earth quake and tsunami in Japan caused a nuclear disaster
that is still unfolding at the Fukushima Dai-ichi nuclear plant
that consisted of 6 nuclear reactors. Within 5 seconds of the
eartquake the control rods flawlessly thrust upward into the 3
operational reactors and stopped the fission reactors. However,
the readioactive by-products in the reactor’s fuel rods continued to generate tremendous amounts of heat and had to be
cooled by circulating water using water pumps driven by diesel
generators. The problem was that the tsunami flooded the
generators, which stopped working. Without cooling water the
high pressure hydrogen generated in the reactor reactor vessel had to be vented (together with radioactive gases and dust)
into the containment vessel, from which it leaked to the reactor
building. Here the hydrogen came in contact with the oxygen
in the air and a spark caused it to explode. The roofs of reactors 1, 3, and 4 were blown away, and thousands of people had
to be evacuated. The future of nuclear energy is in question.
Nuclear fusion and stars
In nuclear fission large nuclei, like uranium, are split into smaller
nuclei, with a conversion of mass into energy. In nuclear fusion small nuclei, like hydrogen, can join to make larger nuclei, like helium. Nuclear fusion is the energy source of stars.
For example the following sequence of reactions is important
is stars. Firstly 11 H +11 H →21 H +01 e + ν, where 21 H is deuterium, 01 e is an antielectron, or positron, and ν is a neutrino,
which is a particle that only experiences weak interactions, and
therefore can easily penetrate through matter. The sun produces lots of neutrinos, but they can go trough the whole Earth
without interacting, which makes them very difficult to detect.
Secondly 11 H +21 H →32 He + γ, where γ is a very energetic
gamma ray photon. Thirdly 11 H +32 He →42 He +01 e + ν, or also
1
1
4
3
3
2 He +2 He →2 He +1 H +1 H. The net result is
1
1H
+11 H +11 H +11 H →42 He +01 e +01 e + ν + ν + γ.
In all these reactions we have positively charged nuclei that
repel each other. Therefore they must be moving at very high
speeds to overcome the repulsion and approach each other close
enough that the short range nuclear force can grab them together. This requires gases at huge temperatures, of the order
of 107 ◦ C, such as are found in the core of stars. So far, fusion reactions, also called thermonuclear reactions, have been
achieved on Earth in hydrogen bombs. However, it still has
not been possible to build a controlled thermonuclear reactor
because of difficulties in reaching the very high temperatures
required, and in building a container (a magnetic bottle) to
hold the hot gases. The most important candidate for a magnetic bottle to date is called a Tokamak.
• Coal burning power plants
A large city will be powered by one or more 1,000 megawatt
(MW)(109 Watt) electric power plants. Turkey point power
plant in Homestead consists of two 400 MW oil/natural gasfired generation units (Units 1 and 2) and two nuclear Westinghouse pressurized water reactors (Units 3 and 4), each supplying steam to one high pressure and two low-pressure turbines
with a power output rated at 693 MW for each unit. In 2007, it
added the 1,150 MW combined-cycle gas-fired Unit 5. It serves
the entire southern portion of Florida. With a combined capacity of 3,330 MW, the site is the largest generating station
in Florida and is the sixth largest power plant in the United
States.
A large coal train called a ”unit train” may be two kilometers
(over a mile) long, containing 100 cars with 100 short tons of
coal in each one, for a total load of 10,000 tons. Currently,
eighty unit trains leave Wyoming every day. A large electric
power plant under full load requires at least one coal delivery this size every day. Plants may get as many as three to
five trains a day. TVAs Kingston Fossil Plant near Knoxville,
Tennessee generates about 1,140 MW, or enough electricity to
supply 700,000 homes. To meet this demand, Kingston burns
about 14,000 tons of coal a day, an amount that would
fill 140 railroad cars. There are coal fired plants that burn
three times as much coal each day.
• Renewable Energy Sources
Wind power
Wind driven turbines are being used for power generation.
In the USA some 44,000 MW are being generated by wind
powered turbines, about 2.3% of US electric power generation.
Texas, with 10,000 MW of capacity, has the most installed wind
power capacity of any U.S. state, followed by Iowa with about
4,000 MW. China aims to have 100,000 MW of on-grid wind
power generating capacity by the end of 2015 and to generate
190 billion kilowatt hours (kWh) of wind power annually. Researchers from Harvard and Tsinghua University have found
that China could meet all of their electricity demands from
wind power through 2030.
Solar power
The solar constant, a measure of flux density, is the amount
of incoming solar electromagnetic radiation per unit area that
would be incident on a plane perpendicular to the rays, at
a distance of one astronomical unit (AU) (roughly the mean
distance from the Sun to the Earth). The solar constant includes all types of solar radiation, not just the visible light.
It is measured by satellite to be roughly 1.361 kilowatts per
square meter (kW/m2 ). So, the energy on 1 kilometer square
is about 1,361 MW, which is equivalent to a large generating
plant. However, it is not easy to harness that energy. U.S.
solar energy capacity increased by 17% in 2007, reaching the
total equivalent of 8,775 megawatts (MW). In 2007 the United
States installed 342 MW of solar photovoltaic (PV) electric
power, 139 thermal megawatts (MWth) of solar water heating,
762 MW of pool heating, and 21 MW of solar space heating
and cooling.
A number of different solar thermal technologies are in use in
the U.S. The figure above shows the Holaniku at Keahole Point
2 MW concentrating parabolic trough solar power installation
in Kona, Hawaii. The largest and oldest solar power plant in
the world is the 354 MW SEGS thermal power plant, in California. The 64 MW Nevada Solar One uses parabolic trough
technology in one of the largest solar plants in the world. The
Martin Next Generation Solar Energy Center is a hybrid 75megawatt (MW) parabolic trough solar energy plant which is
owned by Florida Power & Light Company (FPL). The solar
plant is a component of the 3,705 MW Martin County Power
Plant, which is currently the single largest fossil fuel burning power plant in the United States. Completed at the end
of 2010, it is located in western Martin County, Florida, just
north of Indiantown.
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