The nuclear masses of \(^{7}\mathrm{Be},^{9} \mathrm{Be},\) and \(^{10} \mathrm{Be}\) are \(7.0147,9.0100\) and 10.0113 amu, respectively. Which of these nuclei has the largest binding energy per nucleon?

To calculate the binding energy, we will use the following formula: Binding Energy = (Total Mass of Individual Nucleons - Nuclear Mass) × 931.5 MeV We also know that the nuclear mass of \(^{7}\mathrm{Be},^{9} \mathrm{Be},\) and \(^{10}\mathrm{Be}\) are \(7.0147, 9.0100\) and 10.0113 amu, respectively. For each isotope, we need to first calculate the total mass of individual nucleons (protons and neutrons) and then calculate the binding energies.

For \(^{7}\mathrm{Be}\): Number of protons = 4 (it's beryllium) Number of neutrons = 7 - 4 = 3 Total mass of individual nucleons = (mass of proton × number of protons) + (mass of neutron × number of neutrons) = (4 × 1.0073) + (3 × 1.0087) = 7.0349 amu For \(^{9}\mathrm{Be}\): Number of protons = 4 (it's beryllium) Number of neutrons = 9 - 4 = 5 Total mass of individual nucleons = (mass of proton × number of protons) + (mass of neutron × number of neutrons) = (4 × 1.0073) + (5 × 1.0087) = 9.0373 amu For \(^{10}\mathrm{Be}\): Number of protons = 4 (it's beryllium) Number of neutrons = 10 - 4 = 6 Total mass of individual nucleons = (mass of proton × number of protons) + (mass of neutron × number of neutrons) = (4 × 1.0073) + (6 × 1.0087) = 10.0491 amu

For \(^{7}\mathrm{Be}\): Binding Energy = (7.0349 - 7.0147) × 931.5 = 18.83 MeV For \(^{9}\mathrm{Be}\): Binding Energy = (9.0373 - 9.0100) × 931.5 = 25.38 MeV For \(^{10}\mathrm{Be}\): Binding Energy = (10.0491 - 10.0113) × 931.5 = 35.20 MeV

For \(^{7}\mathrm{Be}\): Binding Energy per Nucleon = 18.83 MeV / 7 = 2.69 MeV/nucleon For \(^{9}\mathrm{Be}\): Binding Energy per Nucleon = 25.38 MeV / 9 = 2.82 MeV/nucleon For \(^{10}\mathrm{Be}\): Binding Energy per Nucleon = 35.20 MeV / 10 = 3.52 MeV/nucleon

Comparing the binding energy per nucleon values, we can see that: \(^{10}\mathrm{Be}\) has the largest binding energy per nucleon at 3.52 MeV/nucleon.