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. This is a classic nuclear physics problem that connects electrostatic energy to nuclear size. Let us first understand the question. 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 R. First, the given mass difference between the two nuclei is 0.002956 atomic mass units. Using the conversion factor of 931.5 MeV per atomic mass unit, we multiply these two values together and find that the energy difference is approximately 2.753 MeV. Oxygen-15 has eight protons while nitrogen-15 has only seven, so oxygen-15 experiences greater electrostatic repulsion among its protons. Using the standard electrostatic energy expression for a uniformly charged sphere, the energy difference between the two nuclei works out to 12.096 divided by R in units of MeV. Since the binding-energy difference equals the electrostatic-energy difference, we can now set up the equation 2.753 equals 12.096 divided by R. Rearranging, R equals 12.096 divided by 2.753, which gives us a nuclear radius of approximately 3.42 femtometres. Therefore, the nuclear radius of both nitrogen-15 and oxygen-15 is 3.42 femtometres, and the correct answer is option C. The key takeaway is that the binding-energy difference directly reveals the electrostatic energy difference, which in turn allows us to calculate the nuclear radius.