Nuclear binding energies

Einstein s equation for the relation between mass and energy also allows us to 

This quantity of energy is called the nuclear binding energy for carbon-12. In general, the nuclear binding energy is the quantity of energy required to break up ] mol of nuclei into their individual nucleons  

Binding energies are commonly expressed in millions of electron volts, that is, in mega-electron volts (MeV)  

A particularly useful factor converts a given mass defect in atomic mass units to its energy equivalent in electron volts  

Earlier we found the mass defect of the C nucleus to be 0.098940 amu. Therefore, the binding energy per C nucleus, expressed in MeV, is 

There are at present 116 known chemical elements. However, there are well over 2000 known nuclear species as a result of several isotopes being known for each element. About three-fourths of the nuclear species are unstable and undergo radioactive decay. Protons and neutrons are the particles which are found in the nucleus. For many purposes, it is desirable to describe the total number of nuclear particles without regard to whether they are protons or neutrons. The term nucleon is used to denote both of these types of nuclear particles. In general, the radii of nuclides increase as the mass number increases with the usual relationship being expressed as 

Any nuclear species is referred to as a nuclide. Thus, H, 23uNa, 12SC, 23892U are different recognizable species or nuclides. A nuclide is denoted by the symbol for the atom with the mass number written to the upper left, the atomic number written to the lower left, and any charge on the species, q to the upper right. For example, 

As was described earlier in this chapter, the model of the atom consists of shells of electrons surrounding the nucleus, which contains protons and, except for the isotope 1H, a certain number of neutrons. 

Each type of atom is designated by the atomic number, Z, and a symbol derived from the name of the element. The mass number, A, is the whole number nearest to the mass of that species. For example, the mass number of H is 1, although the actual mass of this isotope is 1.00794 atomic mass units (amu). Because protons and neutrons have masses that are essentially the same (both are approximately 1 atomic mass unit, amu), the mass number of the species minus the atomic number gives the number of neutrons, which is denoted as N. Thus, for 157N, the nucleus contains seven protons and eight neutrons. 

When atoms are considered to be composed of their constituent particles, it is found that the atoms have lower masses than the sum of the masses of the particles. For example, 42He contains two electrons, two protons, and two neutrons. These particles have masses of 0.0005486, 1.00728, and 1.00866 amu, respectively, which gives a total mass of 4.03298amu for the particles. However, the actual mass of 42He is 4.00260 amu, so there is a mass defect of 0.030377 amu. That disappearance of mass occurs because the particles are held together with an energy that can be expressed in terms of the Einstein equation, 

We can compare the stability of nuclides of different elements by determining the 

A quantitative measure of nuclear stability is the nuclear binding energy, which is the energy required to break up a nucleus into its component protons and neutrons. This quantity represents the conversion of mass to energy that occurs during an exothermic nuclear reaction. 

The concept of nuclear binding energy evolved from studies of nuclear properties showing that the masses of nuclei are always less than the sum of the masses of the nucleons, which is a general term for the protons and neutrons in a nucleus. For example, the 9F isotope has an atomic mass of 18.9984 amu. The nucleus has 9 protons and 10 neutrons and therefore a total of 19 nucleons. Using the known masses of the ]H atom (1.007825 amu) and the neutron (1.008665 amu), we can carry out the following analysis. The mass of 9 H atoms (i.e., the mass of 9 protons and 9 electrons) is 

Therefore, the atomic mass of an F atom calculated from the known numbers of electrons, protons, and neutrons is 

This value is larger than 18.9984 amu (the measured mass of F) by 0.1587 amu. 

A = energy of product — energy of reactants Am = mass of product - mass of reactants Thus, the change in mass is 

This value is larger than 18.99840 amn (the measnred mass of 9F) by 0.15868 amn. 

Student Annotation Remember that joule is a derived unit  

The mass of a nucleus is given by Eq. 3.18 in terms of the masses of its constituents. That same equation also defines the binding energy of the nucleus  

The factor c, which multiplies the mass to transform it into energy, will be omitted from now on. It wUl always be implied that multiplication or division by is necessary to obtain energy from mass or vice versa. Thus, Eq. 3.18 is rewritten as 

The binding energy B A,Z) is equal to the ener necessary to break the nucleus apart into its constituents, Z free protons and N free neutrons. 

As mentioned in Section 3.4, atomic masses rather than nuclear masses are measured in most cases. For this reason, Eq. 3.19 will be expressed in terms of atomic masses by adding the appropriate masses of atomic electrons. If one adds and subtracts Zm in Eq. 3.19, 

Bg = binding energy of the electron in the hydrogen atom B A,Z) = binding energy of all the electrons of the atom whose nucleus has mass Mf (A,Z) 

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In the previous section we saw that the stability of a nucleus is affected by its neutron/proton ratio. Even among those nuclei that we consider stable, however, there is a variation in the forces which hold the nucleus together. In order to study this variation in nuclear binding energy, let us consider the process of building a nucleus from protons and neutrons. For an example, let us look at the hypothetical reaction... 

Similar calculations can be made for other nuclei. A significant comparison between nuclear binding energies can be made if we divide the total binding energy of each nucleus by the... 

Calculate the nuclear binding energy in electronvolts for a helium-4 nucleus, given the following masses 4He, 4.0026/ u H, 1.0078imu n, 1.0087mu. 

STRATEGY The nuclear binding energy is the energy released in the formation of the nucleus from its nucleons. Use H atoms instead of protons to account for the masses of the electrons in the He atom produced. Write the nuclear equation for the formation of the nuclide from hydrogen atoms and neutrons, and calculate the difference in masses between the products and the reactants convert the result from a multiple... 

Nuclear binding energies are determined by applying Einstein s formula to the mass difference between the nucleus and its components. Iron and nickel have the highest binding energy per nucleon. 

FIGURE 17.20 The variation of the nuclear binding energy per nucleon. The maximum binding energy per nucleon occurs near iron and nickel. Their nuclei have the lowest energies of all because their nucleons are most tightly bound. (The vertical axis is binr/A)... 

As described in Chapter 2, nuclei with more than one nucleon are held together by the strong nuclear force. Energy must be provided to overcome this force and remove a nucleon from a nucleus. This energy is called the nuclear binding energy. 

The isotopic molar masses of all stable and many unstable isotopes have been determined using mass spectrometry, as described in Section 2-. These masses can be found in standard data tables. We provide values as needed for calculations. Example illustrates the calculation of nuclear binding energies from isotopic molar... 

The most abundant isotope of helium has two neutrons and an isotopic molar mass of 4.00260 g/mol. Compute the nuclear binding energy of this nuclide. 

Georgia State University. Nuclear Binding Energy. Available online. URL http //hyperphysics.phy-astr.gsu.edu/hbase/ nucene/nucbin.html. 

Nuclear binding energy is the energy equivalent (in E = mc2) of the difference between the mass of the nucleus of an atom and the sum of the masses of its uncombined protons and neutrons. For example, the mass of a He nucleus is 4.0026 amu. The mass of a free proton is 1.00728 amu, and that of a free neutron is 1.00866 amu. The free particles exceed the nucleus in mass by... 

The mass defect for helium-4 is 0.0304 g/mol. Determine the nuclear binding energy in joules per mole for 1 mol of helium-4. 

The difference in mass is significant. It would show up on any reasonably precise balance. Thus, the mass of the nucleus of carbon-12 is significantly less than the mass of its component nucleons. The difference in mass between a nucleus and its nucleons is known as the mass defect. What causes this mass defect It is caused by the nuclear binding energy the energy associated with the strong force that holds a nucleus together. 

You can use Einstein s equation to calculate the nuclear binding energy for carbon-12. 

The difference between the nuclear binding energy of the reactant nuclei and the product nuclei represents the energy change of the nuclear reaction. 

There are four naturally occurring isotopes of iron ( Fe 5.82%, Fe 91.66%, Fe 2.19%, Fe 0.33%), and nine others are known. The most abundant isotope ( Fe) is the most stable nuclear configuration of all the elements in terms of nuclear binding energy per nucleon. This stability, in terms of nuclear equilibrium established in the last moments of supernova events, explains the widespread occurrence of iron in the cosmos. The isotope Fe has practical applications, most notably in Mossbauer spectroscopy, which has been widely exploited to characterize iron coordination complexes. 

Fig. 4.2. Valley of nuclear stability and nuclear binding energy. Top Beyond Z = 20, the distribution of stable isotopes curves downwards in the (N, Z) plane, showing that stable nuclei grow richer in neutrons as their atomic numberincreases. Bottom The binding energy per nucleon, A / A, is a measure of how robust a nuclear species is in the face of attempts to break it up. This curve reaches a peak around iron.

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