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Question: The binding energy per nucleon for $C^{12}$ is 7.68 MeV and that for $C^{13}$ is 7.5 MeV. The energy...

The binding energy per nucleon for C12C^{12} is 7.68 MeV and that for C13C^{13} is 7.5 MeV. The energy required to remove a neutron from C13C^{13} is

A

5.34 MeV

B

5.5 MeV

C

9.5 MeV

D

9.34 MeV

Answer

5.34 MeV

Explanation

Solution

The energy required to remove a neutron from a nucleus is known as the neutron separation energy. This energy is equivalent to the difference in the total binding energies of the parent nucleus and the daughter nucleus (after removing the neutron).

The process can be represented as:

13C→12C+n+Eremove^{13}C \rightarrow ^{12}C + n + E_{remove}

Where EremoveE_{remove} is the energy required to remove a neutron. This energy is given by:

Eremove=Binding Energy of 13C−Binding Energy of 12CE_{remove} = \text{Binding Energy of } ^{13}C - \text{Binding Energy of } ^{12}C

The total binding energy of a nucleus is calculated by multiplying its binding energy per nucleon by the total number of nucleons (mass number, A).

For 12C^{12}C:

  • Number of nucleons (AC12A_{C12}) = 12
  • Binding energy per nucleon for 12C=7.68 MeV^{12}C = 7.68 \text{ MeV}
  • Total Binding Energy of 12C^{12}C (BEC12BE_{C12}) = 7.68 MeV/nucleon×12 nucleons7.68 \text{ MeV/nucleon} \times 12 \text{ nucleons}

BEC12=92.16 MeVBE_{C12} = 92.16 \text{ MeV}

For 13C^{13}C:

  • Number of nucleons (AC13A_{C13}) = 13
  • Binding energy per nucleon for 13C=7.5 MeV^{13}C = 7.5 \text{ MeV}
  • Total Binding Energy of 13C^{13}C (BEC13BE_{C13}) = 7.5 MeV/nucleon×13 nucleons7.5 \text{ MeV/nucleon} \times 13 \text{ nucleons}

BEC13=97.5 MeVBE_{C13} = 97.5 \text{ MeV}

Now, calculate the energy required to remove a neutron from 13C^{13}C:

Eremove=BEC13−BEC12E_{remove} = BE_{C13} - BE_{C12}

Eremove=97.5 MeV−92.16 MeVE_{remove} = 97.5 \text{ MeV} - 92.16 \text{ MeV}

Eremove=5.34 MeVE_{remove} = 5.34 \text{ MeV}