The mass of `""_(7)N^(15)` is 15.00011 amu, mass of `""_(8)O^(16)` is 15.99492 amu and `m_P = 1.00783` amu. Determine binding energy of last proton of

Options

A. `2.13 MeV`
B. `0.13 MeV`
C. `10 MeV`
D. `12.13 MeV`

Answer

To determine the binding energy of the last proton in the formation of \( _{8}^{16}O \) from \( _{7}^{15}N \) and a proton, we can follow these steps:

### Step 1: Write the reaction
We know that the formation of oxygen from nitrogen and a proton can be represented as:
\[ _{7}^{15}N + _{1}^{1}p \rightarrow _{8}^{16}O \]

### Step 2: Identify the masses
From the problem, we have the following masses:
- Mass of \( _{7}^{15}N = 15.00011 \) amu
- Mass of \( _{8}^{16}O = 15.99492 \) amu
- Mass of proton \( m_p = 1.00783 \) amu

### Step 3: Calculate the mass defect (\( \Delta m \))
The mass defect is calculated using the formula:
\[ \Delta m = \text{mass of reactants} - \text{mass of products} \]
Here, the mass of reactants is the sum of the mass of nitrogen and the mass of the proton, and the mass of products is the mass of oxygen:
\[ \Delta m = (m_N + m_p) - m_O \]
Substituting the values:
\[ \Delta m = (15.00011 + 1.00783) - 15.99492 \]
Calculating this gives:
\[ \Delta m = 16.00794 - 15.99492 = 0.01302 \text{ amu} \]

### Step 4: Convert mass defect to energy
To convert the mass defect into energy, we use the relation:
\[ E = \Delta m \cdot c^2 \]
Where \( c^2 \) can be converted using the conversion factor \( 1 \text{ amu} = 931.5 \text{ MeV} \):
\[ E = 0.01302 \text{ amu} \times 931.5 \text{ MeV/amu} \]
Calculating this gives:
\[ E \approx 12.13 \text{ MeV} \]

### Step 5: Conclusion
The binding energy of the last proton in the formation of \( _{8}^{16}O \) from \( _{7}^{15}N \) and a proton is approximately \( 12.13 \text{ MeV} \).