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: \( \left(\mathrm{M}_{0}-8 \mathrm{M}_{p}-9 \mathrm{M}_{\mathrm{n}}\right) \mathrm{C}^{2} \)