Question

The mass of proton, neutron, and helium nucleus are respectively \( 1.0073 \, \text{u}, 1.0087 \, \text{u}, \, 4.0015 \, \text{u} \). The binding energy of the helium nucleus is:

• A. \( 14.2 \, \text{MeV} \)
• B. \( 56.8 \, \text{MeV} \)
• C. \( 28.4 \, \text{MeV} \)
• D. \( 7.1 \, \text{MeV} \)

Hint:

The binding energy of a nucleus can be computed by first calculating the mass defect (the difference between the total mass of nucleons and the mass of the nucleus) and then converting it to energy using the relation \( 1 \, \text{u} = 931.5 \, \text{MeV/c}^2 \).

Answer 1

The Correct Option is C

The binding energy (\( BE \)) of a nucleus is calculated using the equation \( BE = \Delta m \cdot c^2 \), where \( \Delta m \) is the mass defect and \( c \) is the speed of light. For a helium nucleus, the mass defect (\( \Delta m \)) is determined by subtracting the nucleus's mass from the combined mass of its constituent protons and neutrons: \( \Delta m = (2m_p + 2m_n) - m_{\text{He}} \). The given values are: \( m_p = 1.0073 \, \text{u} \) (proton mass), \( m_n = 1.0087 \, \text{u} \) (neutron mass), and \( m_{\text{He}} = 4.0015 \, \text{u} \) (helium nucleus mass). Substituting these values yields \( \Delta m = (2 \cdot 1.0073 + 2 \cdot 1.0087) - 4.0015 = 4.0310 - 4.0015 = 0.0295 \, \text{u} \). To convert this mass defect to energy, we use the conversion factor \( 1 \, \text{u} = 931.5 \, \text{MeV/c}^2 \), resulting in a binding energy of \( BE = 0.0295 \cdot 931.5 = 28.4 \, \text{MeV} \). Final Answer: \[ \boxed{28.4 \, \text{MeV}} \]