Mass Defect and Binding Energy per Nucleon
A bound nucleus has less rest mass than its separate protons and neutrons. This mass defect corresponds to its binding energy through E = mc squared. Binding energy per nucleon reaches its highest region near iron and nickel.
What a learner can do afterwards
- Calculates the mass defect and the binding energy of a stated nuclide
- Calculates binding energy per nucleon and identifies the high-binding region near iron and nickel
- Uses stated binding energies to determine whether a reaction with unchanged total protons and neutrons releases or requires energy
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You have used E = mc squared. A bound nucleus has less rest mass than its free protons and neutrons together. Mass defect = total free-nucleon mass minus nuclear mass. Binding energy is the energy needed for complete separation into free nucleons at rest. Nuclear masses exclude electrons. If using atomic masses, account for electrons consistently.
Suppose stated nuclear masses total 5000 MeV/c squared before binding and 4988 MeV/c squared after binding. The defect is 5000 - 4988 = 12 MeV/c squared. Binding energy = mass defect × c squared, so it is 12 MeV. Use the same mass units before subtracting.
For a mass defect in u, multiply by 931.5 to obtain binding energy in MeV. Divide total binding energy by mass number, the number of nucleons, to find binding energy per nucleon. For example, a stated total of 18 MeV across 3 nucleons gives 6 MeV per nucleon. This is an average for complete separation, not the exact energy needed to remove a particular nucleon.
The curve of binding energy per nucleon against mass number generally rises for light nuclei, peaks in the iron and nickel region, then falls for heavier nuclei. Iron-56 is near the maximum, not its unique highest point. There are local peaks. Nuclear attraction acts mainly between nearby nucleons. Electrical repulsion between protons acts over longer distances, helping explain the fall for heavy nuclei.
For a reaction with unchanged total protons and neutrons, add the binding energies of all products and compare with all starting nuclei. A larger final total releases energy; a smaller one requires energy. With nucleon count conserved, compare nucleon-weighted averages too. Many light-nucleus fusion and heavy-nucleus fission reactions release energy, but the stated products decide. An energy-releasing reaction may still face a barrier.
For breakup into stated fragments, subtract their total binding energy from the starting binding energy. A positive difference must be supplied. A nucleus bound by 20 MeV splitting into two fragments each bound by 3 MeV needs 20 - 2 × 3 = 14 MeV. Complete separation into free nucleons requires the whole binding energy.
Convert mass defect to binding energy, divide by nucleon count, and compare total binding energies for the stated reaction.
Lesson, questions and flashcards written for this dataset. Physics references were consulted for facts and calculation methods.
Readings written for this dataset. Flashcards draw on physics fact records. Video reference credited to Khan Academy: Mass defect and binding energy.
More to read
Choosing masses for a binding calculation
Check whether a mass table lists nuclei or neutral atoms before using it. A neutral atom includes electrons. Adding bare proton and neutron masses and subtracting a neutral atom's mass mixes two different sets of particles.
One consistent calculation subtracts a nuclear mass from the masses of its free protons and neutrons. Another uses hydrogen-atom masses in place of proton masses and subtracts the neutral atomic mass. The electron masses then cancel, apart from small electron-binding corrections.
The mass difference corresponds to the energy released when the bound system forms. Supplying that energy allows complete separation again. Dividing it by nucleon count gives an average, not a separate removal energy for every nucleon.
Check all the reaction products
Knowing that a nucleus is light or heavy does not fully describe a proposed reaction. Write down every starting nucleus and every product. Free protons and neutrons contribute zero binding energy.
When the total proton and neutron counts are unchanged, compare the two total binding energies. Greater binding in the products means lower total rest mass and an energy release. Lower binding means energy must be supplied. For several products, add their total binding energies rather than taking an unweighted average of their per-nucleon values.
The high-binding region near iron and nickel explains broad fusion and fission trends. It does not remove local differences between nuclei or the barriers that can prevent an energy-releasing reaction from happening readily.
Average binding and radioactive decay
A high binding energy per nucleon describes strong average binding. It does not prove that every possible decay is forbidden. The particular starting nucleus and products matter.
Protons repel electrically. The short-range nuclear attraction acts between nearby nucleons. Many light stable nuclei have similar proton and neutron counts; heavier stable nuclei generally have more neutrons than protons. Nuclei outside the stable combinations may change through radioactive decay, sometimes through several stages. Some radioactive nuclei survive for very long times.
Binding-energy subtraction works directly when total proton and neutron counts stay unchanged. For changes that alter those counts, compare the rest masses of all starting and final particles instead of applying that subtraction rule without adjustment.
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