Binding Energy
The key concept behind the release of energy in fusion (and fission) reactions is binding energy.
Binding energy is the energy that is lost when a nucleus is created from protons and neutrons. If you
added up the total mass of the nucleons (protons and neutrons) that compose an atom, you would notice
that this sum is less than the actual mass of the atom. This missing mass, called the mass defect, is a
measure of the atom's binding energy. It is released during the formation of a nucleus from the
composing nucleons. This energy would have to be put back into the nucleus in order to decompose it
into its individual nucleons. The greater the binding energy per nucleon in the atom, the greater the
atom's stability. To calculate the binding energy of a nucleus, all you have to do is sum the mass of the
individual nucleons, and then subtract the mass of the atom itself. The mass leftover is then converted
into its energy equivalent. The relation between mass and energy is shown in Einstein's famous equation
E = mc2. However, we will just multiply the mass by a conversion factor to have the units of energy in
millions of electron volts (MeV), a standard unit of energy in nuclear physics. Therefore, the equation for
binding energy that you can use later is:
Eb = (Z × mH + N × mn - misotope) × 931.5 MeV/amu
Eb = binding energy, in MeV
Z = number of protons
mH = mass of a hydrogen atom (1.007825 atomic mass units, or amu)
N = number of neutrons
mn = mass of a neutron (1.008664904 amu)
misotope = actual mass of the isotope
931.5 Mev/amu = the conversion factor to convert mass into energy, in units of MeV
Remember how I said that the greater the binding energy per nucleon of an atom, the greater it's
stability? Well, above is a graph of the relative binding energy per nucleon vs. mass number (total number
of nucleons composing an atom). Notice that the nuclei of the light elements are generally less stable
than the heavier nuclei up to those with a mass number around 56. The nuclei of the heaviest elements
are less stable than the nuclei that have a mass number of around 56. From this, you can see that the
nuclei around iron are the most stable. This information implies two methods towards the converting of
mass into useful amounts of energy: fusion and fission.
Fusing two nuclei of very small mass, such as hydrogen, will create a more massive nucleus and release
a small amount of mass which appears as energy. Meanwhile, fissioning elements of great mass, like
uranium, will create two lower-mass and more stable nuclei while losing mass in the form of kinetic and/or
radiant energy. The calculation to find the energy released in these reactions is similar to calculating, and
related to, binding energy. If the reactants (the things that went into the reaction) are bound more weakly
than the products (the stuff that comes out of the reaction), then the reaction releases energy. Just sum
the masses of the reactants and subtract the sum of the masses of the products. As an example, lets take
a look at a step in the proton-proton reaction:
(2.) 2H + 1H
3H
+ gamma ray (y)
The fusion of the deuteron (2H) and another proton (a hydrogen nucleus) resulted in the formation of 3He
and a gamma ray. If you summed the masses 2H (2.0140 amu) and the proton (1.007825 amu), and
subtracted the isotopic mass of 3He (3.01603 amu), you would end up with 0.005795 amu of missing
mass. This is equivalent to 5.398MeV of released energy (not including any kinetic energy the reactants
had), in this case taking the form of a gamma ray and any additional kinetic energy of the products.
Nucleosynthesis
Introduction:
As the main sequence fusion cycles (proton-proton and CNO) transform more and more hydrogen to
helium, one of the most likely possibilities of fusion would involve two 4He nuclei fusing to create a
nucleus with an atomic mass of 8. However, there are no stable isotopes of any element with an atomic
mass of 8. 8Be in particular has a lifetime of only 10-17 seconds! At the temperatures in which the protonproton and CNO cycle occurs, 8Be will break apart before it is involved in any further fusion reactions.
This has become known as the beryllium bottleneck, because it is 8Be's instability that prevents the
heavier elements from being formed relatively immediately as helium is created.
Red Giant: Betelgeuse
The Triple-Alpha Process
When a large enough amount of hydrogen is converted to helium within the
core (in our sun, about 10% of its mass), the core may begin to collapse on
itself, increasing the density and temperature. When the temperature rises
above 100 million K, helium nuclei may be converted to carbon (12C) through
a very high, and extremely improbable, energy reaction called the triple-alpha
process (remember an alpha particle is really just a helium nucleus). This is
because the temperatures are high enough to fuse two 4He into the extremely
unstable 8Be at a large enough rate so that there is always a small amount of
Original image courtesy of
8Be. In the short amount of time that a 8Be nucleus exists, it may fuse w ith
NASA.
another 4He producing an "excited" carbon isotope with an atomic mass of
12. These carbon nuclei in their "excited" state are unstable, but they may
release a gamma ray before breaking apart, thus becoming the stable 12C nucleus. This usually begins
occuring during the red giant phase of a star (you will learn more about that later), at which point the
hydrogen fuel in the core has been used up, and the temperature rises enough to trigger the triple-alpha
process.
Supernova 1987A
Further Element Formation
After that, atoms of even higher mass may be created from the
fusion of carbon with other nucleons. For example:
13C
16O
17O
+ 4He
+ 4He
21Ne + 4He
20Ne
+ neutron (n)
+n
24Mg + n
This process of creating the heavier elements is called
nucleosynthesis. Elements up to iron may be created in this fashion
as well as through a variety of other fusion reactions. Elements
heavier than iron are formed through neutron capture, because the
fusion of iron with other elements must absorb energy, rather than
release it. This situation of neutron capture occurs during a
supernova (more on this later), creating up to the heaviest of natural
elements.
Original image courtesy of NASA.
Optional Unit VIII: Atomic Physics
A. Atomic Theory
Key Concepts
Rutherford's gold foil experiment, performed in conjunction with Geiger and Marsden, provided
evidence for the nucleus due to the scattering of alpha particles. The repulsion of some alpha
particles suggested that the nucleus is positively charged, containing protons
.
within the nucleus of the atom.
The atomic number describes the number of protons in the nucleus. For a neutral atom this is
also the number of electrons outside the nucleus.
Subtracting the atomic number from the atomic mass number gives the number of neutrons in the
nucleus.
Isotopes are atoms of the same element (i.e., they have the same number of protons, or the same
atomic number) which have a different number of neutrons in the nucleus. Isotopes of an element
have similar chemical properties.
Radioactive isotopes are called radioisotopes.
Most of the elements in the periodic table have several isotopes, found in varying proportions for
any given element.
The average atomic mass of an element takes into account the relative proportions of its isotopes
found in nature.
A nuclear binding force holds the nucleus of the atom together. The nuclear mass defect, a
slightly lower mass of the nucleus compared to the sum of the masses of its constituent matter, is
due to the nuclear binding energy holding the nucleus together.
The mass defect can be used to calculate the nuclear binding energy, with E = mc2.
The average binding energy per nucleon is a measure of nuclear stability. The higher the
average binding energy, the more stable the nucleus.
The Bohr model of the atom described the electrons as orbiting in discrete, precisely defined
circular orbits. Electrons can only occupy certain allowed orbitals. For an electron to occupy an
allowed orbit, a certain amount of energy must be available.
Each orbit is assigned a quantum number, with the lowest quantum numbers being assigned to
those orbitals closest to the nucleus. Only a specified maximum number of electrons can occupy
an orbital. Under normal circumstances, electrons occupy the lowest energy level orbitals closest
to the nucleus. By absorbing additional energy, electrons can be promoted to higher orbitals, and
release that energy when they return back to lower energy levels.
The Bohr model of the atom helped to offer one possible explanation for the emission spectrum
formed by hydrogen and other gases.
Photons are used to describe the wave-particle duality of light. The energy of a photon depends
upon its frequency. This helps to explain the photoelectric effect; only photons having a
sufficiently high energy are capable of dislodging an electron from the illuminated surface.
E = hv where E is the photon energy in J, v is the photon frequency in Hz, and h is Planck's
constant, 6.626 x 10-34 J/Hz.
Quantum theory offers a mathematical model to help explain the nature of the atom.
Quantum theory describes a region surrounding the nucleus which has the highest probability of
locating an electron. These orbital "clouds" have some unusual and interesting shapes.
Learning Outcomes
Students will increase their abilities to:
1. Define the following terms: atomic number, isotope, radioisotopes, nuclear binding force,
average binding energy, nuclear mass defect, nuclear binding energy, photon.
2. Use the atomic number of an element to determine the number of protons in a nucleus.
3. Infer the number of electrons in a neutral atom from the atomic number of an element.
4. Use the atomic mass number and the atomic number to determine the number of neutrons in
the nucleus of an atom.
5. Recognize that isotopes of an element have similar chemical properties, but different physical
properties.
6. Give an example of an element which contains isotopes and show how those isotopes differ
from each another.
7. Explain that the average atomic mass of an element takes into account the relative proportions
of its isotopes found in nature.
8. Explain some of the important characteristics of the Bohr model of the atom.
9. Identify, interpret, or explain the use of quantum numbers in orbital theory.
10. Show how the Bohr model of the atom offered explanations for some physical phenomena,
while failing to provide a suitable explanation for others
.
11. Explain how photons are used to describe the wave-particle duality of light.
12. Explain that quantum theory helps to explain the photoelectric effect, the Compton effect, and
other important physical principles which earlier theories did not account for adequately.
13. State that quantum theory describes a region surrounding the nucleus which has the highest
probability of locating an electron.
14. Describe some of the electron orbital descriptions provided by quantum theory.
Teaching Suggestions, Activities and Demonstrations
Caution: It is recommended that no experimentation be done with radioactive sources. Use
simulations, computer generated models, or audiovisual aids instead. Very low level sources
ofionizing radiation can be used if other simulations are not appropriate, but extreme care should
always be exercised. It is also important to label radioactive sources properly and to store them in a
safe, secure location.
1. Using a gas discharge tube or a cathode ray tube, connected to a high voltage power supply,
demonstrate what happens to the beam if it allowed to interact with: a) light striking the beam
at right angles, b) an object carrying a static charge, and c) a magnetic field.
Caution: Special care should be taken when using a high voltage power supply. Also, gas
discharge tubes may produce dangerous x-rays. Teachers should demonstrate this,
observing appropriate safety precautions. Roentgen's accidental discovery of x-rays can
be simulated by placing a fluorescent object near the gas discharge tube. An unexposed
sheet of 4 inch by 5 inch type 57 polaroid film (ISO 3 000) placed in the vicinity for a
prolonged period of time will also demonstrate this.
2. Prepare 30 small cubes painted green on all sides and another 30 cubes with five blue sides and
one red side. The 30 blue-red faces simulate radioactive nuclides after decay. Shake the red-blue
cubes and allow them to scatter on a flat surface. Any red sides facing up represent nuclides
that have undergone decay. Count the number of red sides facing up after each trial. Record the
results. Replace the cubes that have the red face up with a green cube (representing a stable
daughter nucleus) and continue shaking and scattering several more times. On each shake the
total number of cubes being shaken should be 30.
Record all observations. Generate a nuclear decay curve, plotting the number of blue
faces against the number of shakes. From the results, determine the half-life of the
sample.
Is there a statistical model to predict the half-life? What effect does the sample size have
on the decay curve and the half-life? (i.e., Start with 20 red-blue cubes instead of the 30
used in the first trial. Repeat the test.)
(To obtain statistically significant results, use more than 30 cubes.)
3. Research the biological effects of ionizing radiation. This activity can be done in conjunction with
Biology 30.
4. Develop a report on the solar wind, the results of solar flare activity, the aurora borealis, or
some other phenomenon related to cosmic rays.
5. The photoelectric effect and the Compton effect helped to give rise to quantum theory.
Research these two phenomena. Attempt to explain why theories which were prevalent at the
time failed to account for these phenomena.
6. If a group of students who have some experience in programming are interested, they can
develop a computer program to simulate radioactive decay.