Calculating Mass Defect of Oxygen-16

The mass defect (\( \Delta m \)) of a nucleus is the difference between the total mass of its constituent nucleons (protons and neutrons) and the actual measured mass of the nucleus. This missing mass is converted into energy that holds the nucleus together, known as the binding energy.

Understanding the Given Information

We are given the following information for the \( ^{16}_{8}O \) nucleus:

• Number of protons (Atomic number, \( Z \)): 8
• Number of nucleons (Mass number, \( A \)): 16
• Number of neutrons (\( N = A - Z \)): \( 16 - 8 = 8 \)
• Mass of electron, \( m_e = 0.00055u \) (Note: Electron mass is usually considered for atomic mass defect, but for nuclear mass defect, we typically use proton and neutron masses directly, assuming the given experimental mass is the nuclear mass).
• Mass of proton, \( m_p = 1.00727u \)
• Mass of neutron, \( m_n = 1.00866u \)
• Experimental mass of the \( ^{16}_{8}O \) nucleus = 15.99053u
• Conversion factor: \( 1u = 931.5 \) MeV/\( c^2 \)

Step 1: Calculate the Total Mass of Individual Nucleons

The total mass of the protons and neutrons if they were separate is calculated by summing the masses of 8 protons and 8 neutrons.

Total mass of nucleons = (Number of protons \( \times \) Mass of proton) + (Number of neutrons \( \times \) Mass of neutron)

Total mass of nucleons = \( (8 \times m_p) + (8 \times m_n) \)

Substituting the given values:

\( (8 \times 1.00727u) + (8 \times 1.00866u) \)

\( 8.05816u + 8.06928u \)

Total mass of nucleons = \( 16.12744u \)

Step 2: Calculate the Mass Defect in Atomic Mass Units (u)

The mass defect is the difference between the calculated total mass of individual nucleons and the experimental mass of the nucleus.

\( \Delta m = \) (Total mass of individual nucleons) - (Experimental mass of nucleus)

\( \Delta m = 16.12744u - 15.99053u \)

\( \Delta m = 0.13691u \)

Step 3: Convert the Mass Defect from u to MeV/c²

We use the given conversion factor \( 1u = 931.5 \) MeV/\( c^2 \) to convert the mass defect from atomic mass units to MeV/\( c^2 \).

\( \Delta m \) in MeV/\( c^2 \) = \( \Delta m \) in u \( \times \) Conversion factor

\( \Delta m \) in MeV/\( c^2 \) = \( 0.13691u \times 931.5 \) MeV/\( c^2 \)/u

\( \Delta m \) in MeV/\( c^2 \) \( \approx 127.539665 \) MeV/\( c^2 \)

Rounding this value gives approximately \( 127.5 \) MeV/\( c^2 \).

Thus, the mass defect of the \( ^{16}_{8}O \) nucleus is approximately \( 127.5 \) MeV/\( c^2 \).

Revision Table: Key Concepts

Additional Information: Mass Defect and Binding Energy

Mass defect is a crucial concept in nuclear physics. It arises from Einstein's mass-energy equivalence principle, \( E=mc^2 \). When protons and neutrons combine to form a nucleus, some mass is converted into energy. This energy, the binding energy, is released during the formation of the nucleus and is required to break the nucleus apart into its constituent nucleons.

• A larger binding energy per nucleon generally indicates a more stable nucleus.
• The mass defect calculation uses the masses of free protons and neutrons, not those bound within the nucleus.
• The electron mass is typically included when calculating the mass defect of an atom (difference between the mass of an atom and the sum of masses of its constituent protons, neutrons, and electrons). However, this question asks for the mass defect of the nucleus and provides the experimental nuclear mass, so electron mass is not needed for this specific calculation.

The process involves calculating the theoretical mass based on the number of protons and neutrons and their individual masses, and then comparing it to the experimentally measured mass of the nucleus. The difference is the mass defect, which is then converted to energy units (like MeV) using the conversion factor based on \( c^2 \).