Understanding Binding Energy Per Nucleon

The question asks to identify which nucleus among the given options has the maximum binding energy per nucleon. Binding energy per nucleon is a crucial concept in nuclear physics that measures the stability of an atomic nucleus. A higher binding energy per nucleon indicates a more stable nucleus, meaning more energy is required to break the nucleus apart into its individual protons and neutrons (nucleons).

The Binding Energy Curve

The stability of nuclei can be visualized using the binding energy curve, which plots binding energy per nucleon against the mass number (A) of the nucleus.

• For very light nuclei (like Hydrogen isotopes), the binding energy per nucleon is relatively low.
• As the mass number increases, the binding energy per nucleon generally rises, reaching a peak in the region of intermediate mass numbers.
• Elements around the mass number A ≈ 50-60, such as Iron (Fe) and Nickel (Ni), exhibit the highest binding energy per nucleon. This peak signifies the region of greatest nuclear stability.
• For nuclei heavier than this peak region (like Lead), the binding energy per nucleon starts to decrease. This is due to the increasing disruptive effect of the electrostatic repulsion between the numerous protons within the nucleus, which outweighs the strong nuclear force's contribution per nucleon.

Analysis of Options

Let's examine the given options in the context of the binding energy curve:

• Option 1: $^4$He (Helium-4): This is a very light nucleus. Its binding energy per nucleon is around 7.07 MeV. It is stable but not the most stable.
• Option 2: $^{12}$C (Carbon-12): This is also a light nucleus. Its binding energy per nucleon is around 7.68 MeV. It is more stable than $^4$He but still significantly less than nuclei in the peak region.
• Option 3: $^{58}_{28}$Ni (Nickel-58): Nickel isotopes, particularly $^{58}$Ni, are located very close to the peak of the binding energy curve. $^{58}_{28}$Ni has a binding energy per nucleon of approximately 8.75 MeV, which is among the highest known.
• Option 4: $^{208}_{82}$Pb (Lead-208): This is a heavy nucleus. While $^{208}_{82}$Pb is a stable isotope, its binding energy per nucleon is around 7.87 MeV. Being in the heavier region, its binding energy per nucleon is lower than that of nuclei like Nickel.

Conclusion

Comparing the binding energies per nucleon, nuclei in the mass number range of 50-60, such as $^{58}_{28}$Ni, possess the maximum binding energy per nucleon. This makes them the most stable nuclei.