The atomic mass of `._(7)N^(15)` is `15.000108 "amu"` and that of `._(8)O^(16)` is `15.994915 "amu"`. The minimum energy required to remove the least tightly bound proton is ( mass of proton is `1.007825 "amu"`)

Options

A. 0.013018 MeV
B. 12.13 MeV
C. 13.018 MeV
D. 12.13 eV

Answer

To find the minimum energy required to remove the least tightly bound proton from the nucleus of oxygen-16, we can follow these steps:

### Step 1: Identify the Reaction
We are considering the reaction where one proton is removed from the oxygen-16 nucleus to form nitrogen-15. The reaction can be represented as:
\[ _{8}^{16}O \rightarrow _{7}^{15}N + p \]
where \( p \) is the proton being removed.

### Step 2: Calculate the Mass Defect (Δm)
The mass defect is calculated using the formula:
\[ \Delta m = m_{\text{reactants}} - m_{\text{products}} \]
In our case:
- The mass of the reactant (oxygen-16) is \( 15.994915 \, \text{amu} \).
- The mass of the products (nitrogen-15 and a proton) is:
\[ m_{\text{products}} = m_{N} + m_{p} = 15.000108 \, \text{amu} + 1.007825 \, \text{amu} \]

Calculating the mass of the products:
\[ m_{\text{products}} = 15.000108 + 1.007825 = 16.007933 \, \text{amu} \]

Now, substituting into the mass defect formula:
\[ \Delta m = 15.994915 - 16.007933 = -0.013018 \, \text{amu} \]

### Step 3: Calculate the Q-value
The Q-value of the reaction, which represents the energy released (or absorbed), is given by:
\[ Q = \Delta m \cdot c^2 \]
where \( c^2 \) is approximately \( 931.5 \, \text{MeV/amu} \).

Substituting the values:
\[ Q = -0.013018 \, \text{amu} \cdot 931.5 \, \text{MeV/amu} \]

Calculating this:
\[ Q = -12.126267 \, \text{MeV} \]

### Step 4: Interpret the Result
The negative sign indicates that energy is absorbed in the process of removing the proton. Therefore, the minimum energy required to remove the least tightly bound proton is:
\[ \text{Energy required} = 12.126267 \, \text{MeV} \approx 12.13 \, \text{MeV} \]

### Final Answer
The minimum energy required to remove the least tightly bound proton is approximately **12.13 MeV**.

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