Calculate the nuclear binding energy (in \(\mathrm{J}\) ) and the binding energy per nucleon of the following isotopes: (a) \({ }_{3}^{7} \mathrm{Li}(7.01600 \mathrm{amu}),(\mathrm{b}){ }_{17}^{35} \mathrm{Cl}(34.95952 \mathrm{amu})\).

The concept of 'mass defect' refers to the difference between the predicted mass of a nucleus if it were assembled from free protons and neutrons and its actual mass. This discrepancy arises because energy is released when nucleons (protons and neutrons) bind together to form a nucleus, leading to a loss of mass. According to the mass-energy equivalence principle, this energy has mass, and so the combined mass of separate nucleons is actually greater than the mass of the nucleus they form.

The calculation of the mass defect involves summing the masses of the individual protons and neutrons and then subtracting the actual mass of the nucleus. The resulting value is very small and measured in atomic mass units (amu), a specialized unit used to measure atomic mass which reflects the scale and precision needed for these calculations.

E=mc² Explained

Mass-energy equivalence is a fundamental concept introduced by Albert Einstein, suggesting that mass and energy are two forms of the same thing and can be converted into each other. This principle is best known by Einstein's famous equation, E=mc², where E stands for energy, m for mass, and c for the speed of light in a vacuum. The equation shows that even a tiny amount of mass can be converted into a huge amount of energy, as the speed of light squared (c²) is an enormous number.

In the context of nuclear physics, mass-energy equivalence explains how the mass defect corresponds to the binding energy of a nucleus. The energy released when a nucleus forms is equal to its mass defect times the square of the speed of light. When calculating nuclear binding energy, we apply this concept to determine the energy stability of a nucleus.

The 'binding energy per nucleon' gives us a measure of the stability of a nucleus. It is calculated by dividing the total nuclear binding energy by the number of nucleons (both protons and neutrons) in the nucleus.

A higher binding energy per nucleon generally indicates a more stable nucleus, as it implies that more energy would be required to break the nucleus apart. This concept is important because it helps explain why certain isotopes are more stable than others and why some nuclei are candidates for nuclear power and weapons. The binding energy per nucleon peaks at elements around Iron on the periodic table, which is the reason why heavier elements can release energy through fission and lighter elements through fusion.

Core of Nuclear Physics

Einstein's equation, E=mc², is the cornerstone of nuclear physics, as it quantifies the relationship between mass and energy. In practice, nuclear binding energy calculations involve this equation to find the energy equivalent of the mass defect. It explains the vast energy contained in atomic nuclei and why nuclear reactions can release so much energy compared to chemical reactions.

By calculating the nuclear binding energy using Einstein's equation, scientists can make crucial predictions about nuclear reactions, such as the energy released in nuclear power plants or the explosive power of nuclear weapons.

The atomic mass unit (amu) is a standard unit of mass that quantifies the mass of atoms or molecules. It is defined as one twelfth of the mass of a carbon-12 atom, with one amu approximately equal to \(1.66053906660 \times 10^{-27}\) kilograms.

Atomic mass units are crucial in nuclear physics because they provide a convenient scale to measure and express very small masses, such as those of protons, neutrons, and entire nuclei. When calculating mass defects and nuclear binding energies, we convert from amu to kilograms to use international system (SI) units, which allows us to apply Einstein's equation accurately.

Isotopes are variants of a particular chemical element that have the same number of protons but different numbers of neutrons in their nuclei, resulting in different mass numbers. These differences in neutron number can significantly affect the stability of a nucleus.

For example, the isotopes of Lithium (\( _3^7Li \) ) and Chlorine (\( _{17}^{35}Cl \) ) mentioned in the exercise have a specific number of protons and neutrons, which dictates their unique mass and stability. The study of isotopes is essential in understanding the variety of nuclear behaviors in nature and in numerous applications ranging from medical imaging to archaeology.