Question

Calculate the binding energy per nucleon of a) \({ }_{2}^{4} \mathrm{He}(4.002603 \mathrm{u})\) b) \({ }_{2}^{3} \mathrm{He}(3.016030 \mathrm{u})\) c) \({ }_{1}^{3} \mathrm{H}(3.016050 \mathrm{u})\) d) \({ }_{1}^{2} \mathrm{H}(2.014102 \mathrm{u})\)

Short answer

Question: Determine the binding energy per nucleon for the following atoms: \({ }_{2}^{4} \mathrm{He}\), \({ }_{2}^{3} \mathrm{He}\), \({ }_{1}^{3} \mathrm{H}\), and \({ }_{1}^{2} \mathrm{H}\). Answer: Follow the step-by-step solution provided above to calculate the binding energy per nucleon for each atom.

Step-by-step solution

Find the binding energy of \({ }_{2}^{4} \mathrm{He}\)

1. Calculate the total mass of protons and neutrons before binding: 2 protons (\(2 \times 1.007825 \mathrm{u}\)) and 2 neutrons (\(2 \times 1.008665 \mathrm{u}\)). 2. Calculate the mass defect: \((2 \times 1.007825 + 2 \times 1.008665) - 4.002603 \mathrm{u}\). 3. Convert mass defect to binding energy: \([mass\ defect]\times 931.5\ MeV/u\). 4. Calculate the binding energy per nucleon: \([binding\ energy]/[number\ of\ nucleons]\).

Find the binding energy of \({ }_{2}^{3} \mathrm{He}\)

1. Calculate the total mass of protons and neutrons before binding: 2 protons (\(2 \times 1.007825 \mathrm{u}\)) and 1 neutron (\(1.008665 \mathrm{u}\)). 2. Calculate the mass defect: \((2 \times 1.007825 + 1.008665) - 3.016030 \mathrm{u}\). 3. Convert mass defect to binding energy: \([mass\ defect]\times 931.5\ MeV/u\). 4. Calculate the binding energy per nucleon: \([binding\ energy]/[number\ of\ nucleons]\).

Find the binding energy of \({ }_{1}^{3} \mathrm{H}\)

1. Calculate the total mass of protons and neutrons before binding: 1 proton (\(1.007825 \mathrm{u}\)) and 2 neutrons (\(2 \times 1.008665 \mathrm{u}\)). 2. Calculate the mass defect: \((1.007825 + 2 \times 1.008665) - 3.016050 \mathrm{u}\). 3. Convert mass defect to binding energy: \([mass\ defect]\times 931.5\ MeV/u\). 4. Calculate the binding energy per nucleon: \([binding\ energy]/[number\ of\ nucleons]\).

Find the binding energy of \({ }_{1}^{2} \mathrm{H}\)

1. Calculate the total mass of protons and neutrons before binding: 1 proton (\(1.007825 \mathrm{u}\)) and 1 neutron (\(1.008665 \mathrm{u}\)). 2. Calculate the mass defect: \((1.007825 + 1.008665) - 2.014102 \mathrm{u}\). 3. Convert mass defect to binding energy: \([mass\ defect]\times 931.5\ MeV/u\). 4. Calculate the binding energy per nucleon: \([binding\ energy]/[number\ of\ nucleons]\). After calculating the binding energy per nucleon for each atom, you will have the answer to the exercise.

Key concepts

Nuclear Physics

Nuclear physics is the field of physics that studies the constituents and interactions of atomic nuclei. At the core of nuclear physics is the understanding of how protons and neutrons, collectively known as nucleons, are bound together in the nucleus. This domain is fundamental to our understanding of the universe as it sheds light on the force that allows the nucleus of an atom to remain intact, despite the repulsive electromagnetic force between the positively charged protons.

The energy that binds these nucleons together is known as binding energy, and it is a key topic within nuclear physics. The stronger the binding energy, the more stable the nucleus. Different isotopes, or forms of elements with varying numbers of neutrons, have different binding energies which influence their properties and behaviors. Nuclear physics also explores nuclear reactions such as fission, where a heavy nucleus splits, and fusion, where light nuclei combine, both of which are crucial for understanding both natural phenomena such as stars and potential energy sources here on Earth.

Mass Defect

Mass defect is a concept in nuclear physics that explains the difference in mass between the combined mass of an atom's nucleons (its protons and neutrons) when separate and the actual mass of its nucleus. Interestingly, the mass of a nucleus is always slightly less than the sum of the individual masses of the nucleons.

This 'missing' mass is called the mass defect, and it is what gets converted into binding energy – the energy needed to hold the nucleus together – according to Einstein's famous equation, E=mc². This equation indicates that mass and energy are interchangeable and that a small amount of mass can be transformed into a large amount of energy. In nuclear reactions, this concept is vital, as the mass defect correlates with the energy released or absorbed during these reactions.

Nucleon

A nucleon is one of the two types of subatomic particles found in the nucleus of an atom: the proton and the neutron. Protons carry a positive electric charge, whereas neutrons are neutral, hence their names. Each nucleon is made up of smaller particles known as quarks, which are held together by the strong nuclear force mediated by particles called gluons.

Nucleons are essential components of atomic nuclei, and the number of protons (atomic number) determines an element's chemical properties, while the total number of nucleons (mass number) gives the atomic mass. When it comes to nuclear stability, the ratio of protons to neutrons is of considerable importance. For students, understanding nucleons is fundamental to grasping concepts such as isotopes, radioactivity, and nuclear reactions.

Atomic Mass Unit

The atomic mass unit (u) is a standard unit of mass that quantifies mass on an atomic or molecular scale. It is defined as one twelfth the mass of a carbon-12 atom at rest and in its ground state. One atomic mass unit is approximately equal to 1.66053906660 x 10^-27 kilograms.

In nuclear physics and chemistry, using the atomic mass unit makes it possible to express masses of atoms and subatomic particles with numbers that are more manageable. For instance, the mass of a proton is roughly 1.007825 u, and a neutron is 1.008665 u, reflecting the fact that they are very close in mass but slightly heavier than one u due to the binding energy involved in holding the nucleons in the nucleus. This system lends itself to precise calculation and comparison of the mass defect and binding energy in atoms.