The binding energy per nucleon for `C^(12)` is `7.68 MeV` and that for `C^(13)` is 7.47 MeV. What is the energy required to remove a neutron from `C^(13)` ?

Options

A. `0.21 MeV`
B. `2.52 MeV`
C. `4.95 MeV`
D. `2.75 MeV`

Answer

To find the energy required to remove a neutron from \( C^{13} \), we need to calculate the difference in binding energy between \( C^{13} \) and \( C^{12} \). Here’s how to do it step by step:

### Step 1: Understand the Binding Energy Concept
The binding energy per nucleon is the energy required to remove a nucleon from the nucleus. For a nucleus, the total binding energy can be calculated by multiplying the binding energy per nucleon by the number of nucleons.

### Step 2: Calculate the Total Binding Energy for \( C^{13} \)
The binding energy per nucleon for \( C^{13} \) is given as \( 7.47 \, \text{MeV} \). Since \( C^{13} \) has 13 nucleons (6 protons and 7 neutrons), the total binding energy \( E_{B}(C^{13}) \) can be calculated as follows:

\[ E_{B}(C^{13}) = \text{Number of nucleons} \times \text{Binding energy per nucleon} \]
\[ E_{B}(C^{13}) = 13 \times 7.47 \, \text{MeV} = 97.11 \, \text{MeV} \]

### Step 3: Calculate the Total Binding Energy for \( C^{12} \)
The binding energy per nucleon for \( C^{12} \) is given as \( 7.68 \, \text{MeV} \). Since \( C^{12} \) has 12 nucleons (6 protons and 6 neutrons), the total binding energy \( E_{B}(C^{12}) \) can be calculated as follows:

\[ E_{B}(C^{12}) = 12 \times 7.68 \, \text{MeV} = 92.16 \, \text{MeV} \]

### Step 4: Calculate the Energy Required to Remove a Neutron
The energy required to remove a neutron from \( C^{13} \) is the difference in binding energy between \( C^{13} \) and \( C^{12} \):

\[ \text{Energy required} = E_{B}(C^{13}) - E_{B}(C^{12}) \]
\[ \text{Energy required} = 97.11 \, \text{MeV} - 92.16 \, \text{MeV} = 4.95 \, \text{MeV} \]

### Final Answer
The energy required to remove a neutron from \( C^{13} \) is \( 4.95 \, \text{MeV} \).

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