The atomic mass of `.^(16)O` is `15.995` amu while the individual masses of proton and neutron are `1.0073` amu and `1.0087` amu respectively. The mass of electron is `0.000548` amu. Calculate the binding energy of the oxygen nucleus.

Explanation

To calculate the binding energy of the oxygen nucleus, we will follow these steps:

### Step 1: Determine the number of protons, neutrons, and electrons in the oxygen-16 nucleus.
- Oxygen-16 has an atomic number of 8, which means it has 8 protons.
- The mass number (A) is 16, which means it has 16 - 8 = 8 neutrons.
- It also has 8 electrons.

### Step 2: Calculate the mass of the oxygen nucleus.
The mass of the oxygen nucleus can be calculated by subtracting the mass of the electrons from the atomic mass of oxygen.
- Mass of 8 electrons = 8 * 0.000548 amu = 0.004384 amu.
- Therefore, the mass of the nucleus = Atomic mass - Mass of electrons = 15.995 amu - 0.004384 amu = 15.990616 amu.

### Step 3: Calculate the theoretical mass of the nucleus.
The theoretical mass of the nucleus can be calculated using the masses of protons and neutrons.
- Mass of protons = 8 * 1.0073 amu = 8.0584 amu.
- Mass of neutrons = 8 * 1.0087 amu = 8.0696 amu.
- Total theoretical mass = Mass of protons + Mass of neutrons = 8.0584 amu + 8.0696 amu = 16.128 amu.

### Step 4: Calculate the mass defect (Δm).
The mass defect is the difference between the theoretical mass of the nucleus and the actual mass of the nucleus.
- Δm = Theoretical mass - Actual mass = 16.128 amu - 15.990616 amu = 0.137384 amu.

### Step 5: Convert the mass defect to energy (binding energy).
Using Einstein's equation, \(E = Δm \cdot c^2\), where \(c\) is the conversion factor from amu to MeV (931.5 MeV/amu).
- Binding energy (BE) = Δm * 931.5 MeV/amu = 0.137384 amu * 931.5 MeV/amu = 128.5 MeV.

### Step 6: Calculate the binding energy per nucleon.
To find the binding energy per nucleon, divide the total binding energy by the number of nucleons (which is 16).
- Binding energy per nucleon = Total binding energy / Number of nucleons = 128.5 MeV / 16 = 8.03125 MeV.

### Step 7: Convert the binding energy per nucleon to Joules.
To convert from MeV to Joules, use the conversion factor \(1 \text{ MeV} = 1.6 \times 10^{-13} \text{ Joules}\).
- Binding energy per nucleon in Joules = 8.03125 MeV * \(1.6 \times 10^{-13} \text{ J/MeV}\) = \(1.284 \times 10^{-12} \text{ J}\).

### Final Answer:
- The binding energy of the oxygen nucleus is approximately 128.5 MeV, and the binding energy per nucleon is approximately 8.03 MeV or \(1.284 \times 10^{-12} \text{ J}\).