Question

If \(\mathrm{M}_{0}\) is the mass of an isotope, \({ }^{17}{ }_{8} \mathrm{O}, \mathrm{M}_{\mathrm{p}}\) and \(\mathrm{M}_{\mathrm{n}}\) are the masses of a Proton and neutron respectively, the binding energy of the isotope is (A) \(\left(\mathrm{M}_{0}-8 \mathrm{M}_{\mathrm{p}}\right) \mathrm{C}^{2}\) (B) \(\left(\mathrm{M}_{0}-8 \mathrm{M}_{p}-9 \mathrm{M}_{\mathrm{n}}\right) \mathrm{C}^{2}\) (C) \(\left(\mathrm{M}_{0}-17 \mathrm{M}_{\mathrm{n}}\right) \mathrm{C}^{2}\) (D) \(\mathrm{M}_{\mathrm{O}} \mathrm{C}^{2}\)

Short answer

The correct formula for the binding energy of the given isotope \({ }^{17}{ }_{8} \mathrm{O}\) is \( \left(\mathrm{M}_{0}-8 \mathrm{M}_{p}-9 \mathrm{M}_{\mathrm{n}}\right) \mathrm{C}^{2} \).

Step-by-step solution

Analyzing Option A

In this option, we have binding energy as \(\left(\mathrm{M}_{0}-8 \mathrm{M}_{\mathrm{p}}\right) \mathrm{C}^{2}\). Here, we're only considering the mass of protons. This doesn't make sense because an isotope has both protons and neutrons in its nucleus. Hence, this option is not correct.

Analyzing Option B

In this option, we have binding energy as \(\left(\mathrm{M}_{0}-8 \mathrm{M}_{p}-9 \mathrm{M}_{\mathrm{n}}\right) \mathrm{C}^{2}\). This option considers the mass of both protons and neutrons, which sounds reasonable. We also have 8 protons and 9 neutrons, which add up to the total of 17 nucleons in the isotope. This option seems promising.

Analyzing Option C

In this option, we have binding energy as \(\left(\mathrm{M}_{0}-17 \mathrm{M}_{\mathrm{n}}\right) \mathrm{C}^{2}\). Here, we're only considering the mass of neutrons and ignoring the mass of protons. This doesn't make sense because an isotope has both protons and neutrons in its nucleus. Hence, this option is not correct.

Analyzing Option D

In this option, we have binding energy as \(\mathrm{M}_{\mathrm{O}} \mathrm{C}^{2}\). Here, we're only considering the mass of Oxygen and not considering the masses of protons and neutrons. This doesn't make sense because an isotope's binding energy arises from the mass defect of nucleons. Hence, this option is not correct.

Conclusion

Based on the analysis of each option, Option B is the correct formula for the binding energy of the given isotope. The binding energy of \({ }^{17}{ }_{8} \mathrm{O}\) can be calculated using the formula: \( \left(\mathrm{M}_{0}-8 \mathrm{M}_{p}-9 \mathrm{M}_{\mathrm{n}}\right) \mathrm{C}^{2} \)

Key concepts

Isotopes

Isotopes are forms of the same element that contain the same number of protons but different numbers of neutrons in their nuclei. This means that while they will always have the same atomic number, isotopes can have different atomic masses. Isotopes are identified by their mass number, which is the sum of protons and neutrons in the nucleus.
For example, Oxygen has several isotopes. In the case of \({ }^{17}{ }_{8} \ ext{O}\), it contains 8 protons and 9 neutrons. Although the number of protons remains unchanged (which is characteristic for any element's isotopes), the difference in neutron count results in different mass numbers.

• Isotopes can have stable or unstable nuclei; unstable isotopes may undergo radioactive decay.
• The chemical properties of isotopes of the same element are typically similar since these properties are more dependent on electronic configuration than nuclear composition.

Understanding isotopes is crucial in nuclear physics, medicine (for radioactive tracers), and archaeology (carbon dating).

Mass Defect

Mass defect is a phenomenon observed in atoms where the mass of the nucleus is less than the sum of the individual masses of the protons and neutrons that comprise it. This difference in mass represents the binding energy holding the nucleus together, as explained by Einstein's famous equation, \(E=mc^2\).
This small variation in mass occurs because energy is released when a nucleus is formed from protons and neutrons due to the attractive forces among the nucleons. This lost mass is converted into energy, which acts as the binding energy.

• It is significant in nuclear reactions because it accounts for the large amounts of energy released in processes such as nuclear fusion and fission.
• Binding energy per nucleon is a measure of the stability of a nucleus, with higher values indicating greater stability.

In nuclear physics, understanding mass defect allows scientists to calculate the energy released in nuclear reactions and provides insights into the force holding nuclei together.

Nuclear Physics

Nuclear Physics is the branch of physics that studies atomic nuclei and their constituents. It explores the forces that hold the nucleus together, known as the nuclear force, which is stronger than any other forces observed in ordinary matter.
Nuclear physics is fundamental in the study and application of nuclear energy and nuclear reactions. It includes topics such as:

• Radioactivity – the process by which unstable nuclei decay, releasing radiation.
• Nuclear fusion and fission – reactions that involve the combination or splitting of atomic nuclei, respectively, which release vast amounts of energy.
• Particle accelerators – devices that accelerate charged particles to high speeds, often used to investigate nuclear reactions.

The concepts of binding energy and mass defect are central to nuclear physics, as they explain why energy is released during nuclear reactions. This field also has far-reaching implications for various technologies, including power generation, medical imaging, and treatments.