Nuclear Radius from Binding Energy

Full transcript

In this problem, we will find the radius of a nucleus using the difference in binding energy between two closely related nuclei, nitrogen-15 and oxygen-15. We are given the electrostatic energy of protons in a spherical nucleus, along with the measured masses of neutron, hydrogen-1, nitrogen-15, and oxygen-15. The difference in binding energies is purely due to electrostatic energy, and we need to find the nuclear radius. First, the given mass difference is 0.002956 atomic mass units. Multiplying by 931.5 MeV per atomic mass unit, the energy difference works out to approximately 2.753 MeV. Oxygen-15 has eight protons while nitrogen-15 has only seven, so oxygen experiences greater electrostatic repulsion. Using the standard electrostatic energy expression, the energy difference between the two nuclei equals 12.096 divided by R. Since the binding energy difference equals the electrostatic energy difference, we set 2.753 equal to 12.096 divided by R. Solving for R gives us approximately 3.42 femtometres. Therefore, the nuclear radius is 3.42 femtometres, which corresponds to option C. The key takeaway is that the binding energy difference directly gives us the electrostatic energy difference, and from that we can calculate the nuclear radius.