Question

The atomic mass of $_{6}^{12}C$ is 12.000000 u and that of $_{6}^{13}C$ is 13.003354 u. The required energy to remove a neutron from $_{6}^{13}C$, if mass of neutron is 1.008665 u, will be :

• A. 62.5 MeV
• B. 6.25 MeV
• C. 4.95 MeV
• D. 49.5 MeV

Answer 1

The Correct Option is C

To determine the energy required to remove a neutron from $_{6}^{13}C$, we need to calculate the difference in binding energy between the isotopes $_{6}^{12}C$ and $_{6}^{13}C$. Here are the steps and calculations: 

1. The formula for binding energy (BE) of a nucleus is given by: \(\text{BE} = \left( Zm_p + Nm_n - M \right) \times 931.5 \, \text{MeV/u}\)where \( Z \) is the number of protons, \( N \) is the number of neutrons, \( m_p \) is the mass of a proton, \( m_n \) is the mass of a neutron, and \( M \) is the atomic mass of the nuclide.
2. For $_{6}^{12}C$, the atomic mass \( M = 12.000000 \, \text{u} \), \( Z = 6 \), \( N = 6 \).
3. For $_{6}^{13}C$, the atomic mass \( M = 13.003354 \, \text{u} \), \( Z = 6 \), \( N = 7 \).
4. We need to find the energy required to remove a neutron. Thus, calculate the binding energy difference: \(\Delta \text{BE} = \text{BE}_{_{6}^{13}\text{C}} - \text{BE}_{_{6}^{12}\text{C}}\)
5. Express the difference in masses: 
   \(\Delta M = M(_{6}^{13}\text{C}) - M(_{6}^{12}\text{C}) - m_n = 13.003354 \, \text{u} - 12.000000 \, \text{u} - 1.008665 \, \text{u}\)
6. Simplify the mass difference: 
   \(\Delta M = 13.003354 - 12.000000 - 1.008665 = -0.005311 \, \text{u}\)
7. Convert the mass difference to energy: 
   Using \(931.5 \, \text{MeV/u}\), we calculate the energy change: 
   \(\Delta \text{E} = |\Delta M| \times 931.5 \, \text{MeV/u} = 0.005311 \times 931.5 = 4.950 \, \text{MeV}\)

Therefore, the energy required to remove a neutron from $_{6}^{13}C$ is 4.95 MeV.

Answer 2

To remove a neutron from \( ^{13}_6C \), the nuclear reaction can be represented as:

\(^{13}_6C \rightarrow ^{12}_6C + \text{neutron}.\)

The mass defect \(\Delta m\) is given by:

\(\Delta m = \left(12.000000 + 1.008665\right) - 13.003354 = -0.00531 \, \text{u}.\)

The energy required for this process is calculated using:

\(E = \Delta m \times 931.5 \, \text{MeV/u}.\)

Substituting values:

\(E = 0.00531 \times 931.5 \approx 4.95 \, \text{MeV}.\)

The Correct answer is: 4.95 MeV