## Binding Energy Formula and Mass Defect

**Binding energy** can calculate the expected mass of any nuclei from the knowledge of the nuclear composition like masses of the electron, proton and neutron particles or nucleon and Einstein relativity equation or formula in physics or chemistry. Such binding energy calculation shows that the calculated mass of a nucleus is always greater than the experimentally determined mass. This difference comes from binding energy definition or calculation is called the mass defect, mass deficit, or mass decrement of the radioactive atom in nuclear chemistry.

For example, the helium nuclei (_{2}He^{4}) have the following data for binding energy and mass defect calculation, the mass of two protons and neutrons = 2.01456 and 2.01743 amu respectively. Therefore, the total mass = (2.01456 + 2.01743) = 4.03190 amu. But the actual mass of helium nuclei (_{2}He^{4}) = 4.0015 amu. Hence (4.03190 – 4.00150) = 0.03040 amu is the mass defect of helium. This mass defect converted or transformed into the energy by the Einstein relativity equation in physics or nuclear chemistry.

### Nuclear Binding Energy Curve of Chemical Elements

Binding energies vary greatly with the bound particles or the composition of the nucleus of periodic table elements. An idea of the relative stability of the stable nuclei of different chemical elements can be obtained by the plot of the binding energy per nucleon against the mass number of periodic table elements.

### Mass Defect Calculation

Let an atom has mas number = A and atomic number = Z. The atom contains Z number of proton and (A – Z) number of neutron particles in the nucleus. Therefore, the mass of the atom expected to, (A – Z)m_{n} + Zm_{p} + Zm_{e}. But the isotopic mass (M) is usually less than the theoretically calculated value. Therefore the difference is called mass defect equal to the value, D = (A -Z)m_{n} + Zm_{p} + Zm_{e} – M = (A – Z)m_{n} + Zm_{H} – M. Where m_{n} = mass of the neutron, mp = mass of the proton, m_{e} = mass of the electron, and mass of hydrogen atom = m_{H} = m_{p} + m_{e}.

According to the definition of binding energy, the energy required to hold together the constituent particles like proton, neutron, and electron in an atom calculated from the loss of mass of the particle or measure by the mass defect formula. The mass loss converted into energy according to the law of mass-energy conservation with the Einstein principle, E = mc^{2}. 1 amu mass produces 1.492 Ã— 10^{-3} erg or 931 MeV of energy.Â Therefore, the energy formula derived from the Einstein mass-energy equation which is responsible for holding the constituent particles together define as binding energy and calculate in MeV units = 931 Ã— D. For examples, the mass defect in the deuterium atom (mass = 2.01474 amu) = m_{p} + m_{n} + m_{e} – m = 0.00293. Therefore, the E_{B} = 931 Ã— 0.00293 = 2.72 MeV.

### Average Binding Energy Per Nucleon

The mass defect for helium atom = 0.03040 amu and E_{B} = 0.03040 Ã— 931 MeV = 28.3 MeV. But the helium nucleus contains a total of four nuclear particles or nucleons. The average binding energy per particles or nucleon = 1.36 MeV but the mean value = 28.3 MeV/4 = 7.07 MeV.

Atom | Binding Energy (MeV) | Mean E_{B} per Nucleon |

^{7}Li |
39.27 | 5.61 |

^{12}C |
92.30 | 7.69 |

^{20}Ne |
160.00 | 8.00 |

^{209}Bi |
1630.00 | 7.80 |

The mean binding energy formula per particles (electron, proton, and neutron) or nucleons different in different atom express above the table laying between the values 7.5 to 8.5 plotted against the atomic number of curve or graph. The plot shows that the binding energy per nucleon rises sharply from the isotopes of hydrogen to the next heavier chemical elements and produces a large amount of energy by bound or fusing together or nuclear fusion.