If `M_(o)` is the mass of an oxygen isotope `._(8)O^(17), M_(p)` and `M_(N)` are the masses of a proton and neutron respectively, the nuclear binding energy of the isotope is:

Options

A. `M_Oc^2`
B. `(M_O-17M_n)c^2`
C. `(M_O8M_n)c^2`
D. `(8M_p+9M_n-M_O)C^2`

Answer

To find the nuclear binding energy of the oxygen isotope \( _{8}^{17}O \), we will follow these steps:

### Step 1: Identify the number of protons and neutrons
For the oxygen isotope \( _{8}^{17}O \):
- The atomic number (number of protons, \( Z \)) is 8.
- The mass number (total number of nucleons, \( A \)) is 17.

To find the number of neutrons (\( N \)):
\[ N = A - Z = 17 - 8 = 9 \]

### Step 2: Calculate the mass of nucleons
The mass of nucleons can be calculated using the number of protons and neutrons:
\[ \text{Mass of nucleons} = (Z \cdot M_p) + (N \cdot M_n) \]
Substituting the values:
\[ \text{Mass of nucleons} = (8 \cdot M_p) + (9 \cdot M_n) \]

### Step 3: Calculate the mass defect
The mass defect (\( \Delta m \)) is the difference between the mass of the nucleons and the mass of the nucleus:
\[ \Delta m = \text{Mass of nucleons} - M_0 \]
Substituting the expression for mass of nucleons:
\[ \Delta m = (8 \cdot M_p + 9 \cdot M_n) - M_0 \]

### Step 4: Calculate the binding energy
The binding energy (\( BE \)) is given by the equation:
\[ BE = \Delta m \cdot c^2 \]
Substituting the expression for mass defect:
\[ BE = \left((8 \cdot M_p + 9 \cdot M_n) - M_0\right) \cdot c^2 \]

### Final Expression
Thus, the binding energy of the oxygen isotope \( _{8}^{17}O \) can be expressed as:
\[ BE = (8 \cdot M_p + 9 \cdot M_n - M_0) \cdot c^2 \]