Question

Given mass of neutron \( 1.0087 \, u \), mass of proton \( 1.0073 \, u \), mass of \( ^4He \) \( = 4.0018 \, u \), find the binding energy of He.

• A. \( 27.8 \, \text{MeV} \)
• B. \( 28.1 \, \text{MeV} \)
• C. \( 29.5 \, \text{MeV} \)
• D. \( 30.2 \, \text{MeV} \)

Hint:

The binding energy can be calculated using the mass defect, which is the difference between the total mass of individual nucleons and the actual mass of the nucleus. Convert the mass defect to energy using \( E = \Delta m c^2 \).

Answer 1

The Correct Option is B

Step 1: Mass defect.
The binding energy of a nucleus can be found using the mass defect. The mass defect is the difference between the sum of the masses of individual nucleons (protons and neutrons) and the actual mass of the nucleus. For the \( ^4He \) nucleus, it consists of 2 protons and 2 neutrons. The total mass of individual nucleons is: \[ \text{Total mass of nucleons} = (2 \times \text{mass of proton}) + (2 \times \text{mass of neutron}) \] Substituting the given values: \[ \text{Total mass of nucleons} = (2 \times 1.0073 \, u) + (2 \times 1.0087 \, u) = 2.0146 \, u + 2.0174 \, u = 4.0320 \, u \] Now, the actual mass of the \( ^4He \) nucleus is given as \( 4.0018 \, u \). The mass defect is: \[ \text{Mass defect} = \text{Total mass of nucleons} - \text{Mass of } ^4He \] \[ \text{Mass defect} = 4.0320 \, u - 4.0018 \, u = 0.0302 \, u \]
Step 2: Convert mass defect to energy.
To convert the mass defect into binding energy, use Einstein’s equation \( E = \Delta m c^2 \). The energy corresponding to 1 atomic mass unit (1 \( u \)) is approximately 931.5 MeV. Thus, the binding energy is: \[ \text{Binding energy} = 0.0302 \, u \times 931.5 \, \text{MeV/u} = 28.1 \, \text{MeV} \]
Final Answer: \( 28.1 \, \text{MeV} \).