Binding energy per nucleon for `C^(12)` is `7.68 MeV` and for `C^(13)` is `7.74 MeV`. The energy required to remove a neutron from `C^(13)` is .

Options

A. `5.34 MeV`
B. `5.5 MeV`
C. `9.5 MeV`
D. `9.34 MeV`

Answer

To find the energy required to remove a neutron from \( C^{13} \), we can follow these steps:

### Step 1: Understand the Binding Energy Concept
Binding energy is the energy required to disassemble a nucleus into its constituent protons and neutrons. The binding energy per nucleon gives us an idea of how tightly the nucleons are bound together.

### Step 2: Identify the Given Values
We have the following binding energies:
- Binding energy per nucleon for \( C^{12} \) = \( 7.68 \, \text{MeV} \)
- Binding energy per nucleon for \( C^{13} \) = \( 7.74 \, \text{MeV} \)

### Step 3: Calculate the Total Binding Energies
To find the total binding energies for \( C^{12} \) and \( C^{13} \):
- Total binding energy for \( C^{12} \) (which has 12 nucleons) = \( 12 \times 7.68 \, \text{MeV} \)
- Total binding energy for \( C^{13} \) (which has 13 nucleons) = \( 13 \times 7.74 \, \text{MeV} \)

### Step 4: Perform the Calculations
1. Calculate the total binding energy for \( C^{12} \):
\[ \text{Total Binding Energy for } C^{12} = 12 \times 7.68 = 92.16 \, \text{MeV} \]

2. Calculate the total binding energy for \( C^{13} \):
\[ \text{Total Binding Energy for } C^{13} = 13 \times 7.74 = 100.62 \, \text{MeV} \]

### Step 5: Find the Energy Required to Remove a Neutron
The energy required to remove a neutron from \( C^{13} \) is equal to the difference in total binding energies between \( C^{13} \) and \( C^{12} \):
\[ E = \text{Total Binding Energy for } C^{13} - \text{Total Binding Energy for } C^{12} \]
\[ E = 100.62 \, \text{MeV} - 92.16 \, \text{MeV} = 8.46 \, \text{MeV} \]

### Final Answer
The energy required to remove a neutron from \( C^{13} \) is \( 8.46 \, \text{MeV} \).

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