Question

Atomic masses of two oxygen isotopes \(^{16}_{8}\text{O}\) and \(^{18}_{8}\text{O}\) are \(15.99491\ \text{u}\) and \(17.99916\ \text{u}\), respectively, where \(\text{u}\) is the atomic mass unit. Masses of proton and neutron are given by \(1.00727\ \text{u}\) and \(1.00866\ \text{u}\), respectively. The speed of light is \(c\). What is the difference between the binding energies of \(^{18}_{8}\text{O}\) and \(^{16}_{8}\text{O}\) nuclei in units of \(\text{u } c^2\)?

• A. \(0.01307\)
• B. \(2.00425\)
• C. \(0.99559\)
• D. \(3.01291\)

Hint:

Notice that the proton terms cancel out completely when subtracting the binding energies.
The difference is simply the mass of the two extra neutrons minus the actual difference in the isotopic masses.
This significantly reduces the arithmetic load.

Answer 1

The Correct Option is A

To find the difference in binding energies of the two oxygen isotopes \(^{16}_{8}\text{O}\) and \(^{18}_{8}\text{O}\), we will first calculate the binding energy for each isotope and then find the difference.

Step 1: Calculate the mass defect for each isotope.

The mass defect (\(\Delta m\)) is the difference between the sum of the individual masses of the protons, neutrons, and electrons and the actual isotopic mass.

For \(^{16}_{8}\text{O}\):

• Number of protons = 8
• Number of neutrons = 8
• Number of electrons = 8
• Mass of 8 protons = \(8 \times 1.00727\ \text{u}\)
• Mass of 8 neutrons = \(8 \times 1.00866\ \text{u}\)
• Mass of 8 electrons (approximately negligible compared to nucleus) can be ignored here for simplicity in nuclear calculations.
• Total nucleon mass = \(8 \times 1.00727 + 8 \times 1.00866 = 8.05816 + 8.06928 = 16.12744\ \text{u}\)
• Mass defect for \(^{16}_{8}\text{O}\), \(\Delta m_{16} = 16.12744 - 15.99491 = 0.13253\ \text{u}\)

For \(^{18}_{8}\text{O}\):

• Number of protons = 8
• Number of neutrons = 10
• Number of electrons = 8
• Mass of 8 protons = \(8 \times 1.00727\ \text{u}\)
• Mass of 10 neutrons = \(10 \times 1.00866\ \text{u}\)
• Mass of 8 electrons (again negligible here)
• Total nucleon mass = \(8 \times 1.00727 + 10 \times 1.00866 = 8.05816 + 10.0866 = 18.14476\ \text{u}\)
• Mass defect for \(^{18}_{8}\text{O}\), \(\Delta m_{18} = 18.14476 - 17.99916 = 0.14560\ \text{u}\)

Step 2: Calculate the binding energy for each isotope.

The binding energy (\(E\)) is given by Einstein's equation \(E = \Delta m \times c^2\).

• Binding energy of \(^{16}_{8}\text{O}\), \(E_{16} = \Delta m_{16} \times c^2 = 0.13253 \times c^2 \text{u c}^2\)
• Binding energy of \(^{18}_{8}\text{O}\), \(E_{18} = \Delta m_{18} \times c^2 = 0.14560 \times c^2 \text{u c}^2\)

Step 3: Calculate the difference in binding energies.

• Difference in binding energies = \(E_{18} - E_{16} = (0.14560 - 0.13253)\ \text{u c}^2 = 0.01307\ \text{u c}^2\)

Conclusion: The difference between the binding energies of \(^{18}_{8}\text{O}\) and \(^{16}_{8}\text{O}\) nuclei is \(0.01307\ \text{u c}^2\).