<< Chapter < Page Chapter >> Page >
  • Define and discuss binding energy.
  • Calculate the binding energy per nucleon of a particle.

The more tightly bound a system is, the stronger the forces that hold it together and the greater the energy required to pull it apart. We can therefore learn about nuclear forces by examining how tightly bound the nuclei are. We define the binding energy    (BE) of a nucleus to be the energy required to completely disassemble it into separate protons and neutrons . We can determine the BE of a nucleus from its rest mass. The two are connected through Einstein’s famous relationship E = ( Δ m ) c 2 size 12{E= \( Δm \) c rSup { size 8{2} } } {} . A bound system has a smaller mass than its separate constituents; the more tightly the nucleons are bound together, the smaller the mass of the nucleus.

Imagine pulling a nuclide apart as illustrated in [link] . Work done to overcome the nuclear forces holding the nucleus together puts energy into the system. By definition, the energy input equals the binding energy BE. The pieces are at rest when separated, and so the energy put into them increases their total rest mass compared with what it was when they were glued together as a nucleus. That mass increase is thus Δ m = BE / c 2 size 12{Δm="BE"/c rSup { size 8{2} } } {} . This difference in mass is known as mass defect . It implies that the mass of the nucleus is less than the sum of the masses of its constituent protons and neutrons. A nuclide A X size 12{"" lSup { size 8{A} } X} {} has Z size 12{Z} {} protons and N size 12{N} {} neutrons, so that the difference in mass is

Δ m = ( Zm p + Nm n ) m tot . size 12{Δm= \( ital "Zm" rSub { size 8{p} } + ital "Nm" rSub { size 8{n} } \) - m rSub { size 8{"tot"} } } {}

Thus,

BE = ( Δ m ) c 2 = [ ( Zm p + Nm n ) m tot ] c 2 , size 12{"BE"= \( Δm \) c rSup { size 8{2} } = \[ \( ital "Zm" rSub { size 8{p} } + ital "Nm" rSub { size 8{n} } \) - m rSub { size 8{"tot"} } \]c rSup { size 8{2} } } {}

where m tot size 12{m rSub { size 8{"tot"} } } {} is the mass of the nuclide A X , m p is the mass of a proton, and m n is the mass of a neutron. Traditionally, we deal with the masses of neutral atoms. To get atomic masses into the last equation, we first add Z size 12{Z} {} electrons to m tot size 12{m rSub { size 8{"tot"} } } {} , which gives m A X size 12{m left ("" lSup { size 8{A} } X right )} {} , the atomic mass of the nuclide. We then add Z size 12{Z} {} electrons to the Z size 12{Z} {} protons, which gives Zm 1 H size 12{ ital "Zm" left ("" lSup { size 8{1} } H right )} {} , or Z size 12{Z} {} times the mass of a hydrogen atom. Thus the binding energy of a nuclide A X size 12{"" lSup { size 8{A} } X} {} is

BE = { [ Zm ( 1 H ) + Nm n ] m ( A X ) } c 2 .

The atomic masses can be found in Appendix A , most conveniently expressed in unified atomic mass units u ( 1 u = 931 . 5 MeV / c 2 size 12{1" u"="931" "." 5 "MeV"/c rSup { size 8{2} } } {} ). BE is thus calculated from known atomic masses.

The image shows some spherical protons and neutrons pulled out from a nucleus. The work done to pull them apart is binding energy.
Work done to pull a nucleus apart into its constituent protons and neutrons increases the mass of the system. The work to disassemble the nucleus equals its binding energy BE. A bound system has less mass than the sum of its parts, especially noticeable in the nuclei, where forces and energies are very large.

Things great and small

Nuclear Decay Helps Explain Earth’s Hot Interior

A puzzle created by radioactive dating of rocks is resolved by radioactive heating of Earth’s interior. This intriguing story is another example of how small-scale physics can explain large-scale phenomena.

Radioactive dating plays a role in determining the approximate age of the Earth. The oldest rocks on Earth solidified about 3 . 5 × 10 9 size 12{3 "." 5 times "10" rSup { size 8{9} } } {} years ago—a number determined by uranium-238 dating. These rocks could only have solidified once the surface of the Earth had cooled sufficiently. The temperature of the Earth at formation can be estimated based on gravitational potential energy of the assemblage of pieces being converted to thermal energy. Using heat transfer concepts discussed in Thermodynamics it is then possible to calculate how long it would take for the surface to cool to rock-formation temperatures. The result is about 10 9 size 12{"10" rSup { size 8{9} } } {} years. The first rocks formed have been solid for 3 . 5 × 10 9 size 12{3 "." 5 times "10" rSup { size 8{9} } } {} years, so that the age of the Earth is approximately 4 . 5 × 10 9 size 12{4 "." 5 times "10" rSup { size 8{9} } } {} years. There is a large body of other types of evidence (both Earth-bound and solar system characteristics are used) that supports this age. The puzzle is that, given its age and initial temperature, the center of the Earth should be much cooler than it is today (see [link] ).

The figure shows that the center of the Earth cools by three heat transfer methods. Convection heat transfer in the center region, then conduction heat transfer moves thermal energy to the surface, and finally radiation heat transfer from the surface to space.
The center of the Earth cools by well-known heat transfer methods. Convection in the liquid regions and conduction move thermal energy to the surface, where it radiates into cold, dark space. Given the age of the Earth and its initial temperature, it should have cooled to a lower temperature by now. The blowup shows that nuclear decay releases energy in the Earth’s interior. This energy has slowed the cooling process and is responsible for the interior still being molten.

We know from seismic waves produced by earthquakes that parts of the interior of the Earth are liquid. Shear or transverse waves cannot travel through a liquid and are not transmitted through the Earth’s core. Yet compression or longitudinal waves can pass through a liquid and do go through the core. From this information, the temperature of the interior can be estimated. As noticed, the interior should have cooled more from its initial temperature in the 4 . 5 × 10 9 size 12{4 "." 5 times "10" rSup { size 8{9} } } {} years since its formation. In fact, it should have taken no more than about 10 9 size 12{4 "." 5 times "10" rSup { size 8{9} } } {} years to cool to its present temperature. What is keeping it hot? The answer seems to be radioactive decay of primordial elements that were part of the material that formed the Earth (see the blowup in [link] ).

Nuclides such as 238 U and 40 K have half-lives similar to or longer than the age of the Earth, and their decay still contributes energy to the interior. Some of the primordial radioactive nuclides have unstable decay products that also release energy— 238 U size 12{"" lSup { size 8{"238"} } U} {} has a long decay chain of these. Further, there were more of these primordial radioactive nuclides early in the life of the Earth, and thus the activity and energy contributed were greater then (perhaps by an order of magnitude). The amount of power created by these decays per cubic meter is very small. However, since a huge volume of material lies deep below the surface, this relatively small amount of energy cannot escape quickly. The power produced near the surface has much less distance to go to escape and has a negligible effect on surface temperatures.

A final effect of this trapped radiation merits mention. Alpha decay produces helium nuclei, which form helium atoms when they are stopped and capture electrons. Most of the helium on Earth is obtained from wells and is produced in this manner. Any helium in the atmosphere will escape in geologically short times because of its high thermal velocity.

Questions & Answers

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Physics 101. OpenStax CNX. Jan 07, 2013 Download for free at http://legacy.cnx.org/content/col11479/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Physics 101' conversation and receive update notifications?

Ask