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By the end of this section, you will be able to:
  • Calculate the mass defect and binding energy for a wide range of nuclei
  • Use a graph of binding energy per nucleon (BEN) versus mass number ( A ) graph to assess the relative stability of a nucleus
  • Compare the binding energy of a nucleon in a nucleus to the ionization energy of an electron in an atom

The forces that bind nucleons together in an atomic nucleus are much greater than those that bind an electron to an atom through electrostatic attraction. This is evident by the relative sizes of the atomic nucleus and the atom ( 10 −15 and 10 −10 m , respectively). The energy required to pry a nucleon from the nucleus is therefore much larger than that required to remove (or ionize) an electron in an atom. In general, all nuclear changes involve large amounts of energy per particle undergoing the reaction. This has numerous practical applications.

Mass defect

According to nuclear particle experiments, the total mass of a nucleus ( m nuc ) is less than the sum of the masses of its constituent nucleons (protons and neutrons). The mass difference, or mass defect    , is given by

Δ m = Z m p + ( A Z ) m n m nuc

where Z m p is the total mass of the protons, ( A Z ) m n is the total mass of the neutrons, and m nuc is the mass of the nucleus. According to Einstein’s special theory of relativity, mass is a measure of the total energy of a system ( E = m c 2 ). Thus, the total energy of a nucleus is less than the sum of the energies of its constituent nucleons. The formation of a nucleus from a system of isolated protons and neutrons is therefore an exothermic reaction—meaning that it releases energy. The energy emitted, or radiated, in this process is ( Δ m ) c 2 .

Now imagine this process occurs in reverse. Instead of forming a nucleus, energy is put into the system to break apart the nucleus ( [link] ). The amount of energy required is called the total binding energy (BE)    , E b .

Binding energy

The binding energy is equal to the amount of energy released in forming the nucleus, and is therefore given by

E b = ( Δ m ) c 2 .

Experimental results indicate that the binding energy for a nucleus with mass number A > 8 is roughly proportional to the total number of nucleons in the nucleus, A . The binding energy of a magnesium nucleus ( 24 Mg ), for example, is approximately two times greater than for the carbon nucleus ( 12 C ).

The figure shows a reaction. The LHS shows a nucleus plus binding energy. This nucleus is a cluster of closely packed protons and neutrons and is labeled, smaller mass. On the RHS is a nucleus with loosely packed protons and neutrons, labeled, separated nucleons, greater mass.
The binding energy is the energy required to break a nucleus into its constituent protons and neutrons. A system of separated nucleons has a greater mass than a system of bound nucleons.

Mass defect and binding energy of the deuteron

Calculate the mass defect and the binding energy of the deuteron. The mass of the deuteron is m D = 3.34359 × 10 −27 kg or 1875.61 MeV/ c 2 .

Solution

From [link] , the mass defect for the deuteron is

Δ m = m p + m n m D = 938.28 MeV / c 2 + 939.57 MeV / c 2 1875.61 MeV / c 2 = 2.24 MeV / c 2 .

The binding energy of the deuteron is then

E b = ( Δ m ) c 2 = ( 2.24 MeV / c 2 ) ( c 2 ) = 2.24 MeV .

Over two million electron volts are needed to break apart a deuteron into a proton and a neutron. This very large value indicates the great strength of the nuclear force. By comparison, the greatest amount of energy required to liberate an electron bound to a hydrogen atom by an attractive Coulomb force (an electromagnetic force) is about 10 eV.

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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