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

To find the binding energy (\( BE \)) of a nucleus, we use the equation: \[ BE = \Delta m \cdot c^2, \] where \( \Delta m \) is the mass defect and \( c \) is the speed of light. For the helium nucleus, the mass defect is: \[ \Delta m = (2m_p + 2m_n) - m_{\text{He}}, \] where: - \( m_p = 1.0073 \, \text{u} \) is the mass of a proton, - \( m_n = 1.0087 \, \text{u} \) is the mass of a neutron, - \( m_{\text{He}} = 4.0015 \, \text{u} \) is the mass of the helium nucleus. Substitute the values for the protons and neutrons: \[ \Delta m = (2 \cdot 1.0073 + 2 \cdot 1.0087) - 4.0015 = 4.0310 - 4.0015 = 0.0295 \, \text{u}. \] Now, to convert the mass defect into energy, we use the conversion factor \( 1 \, \text{u} = 931.5 \, \text{MeV/c}^2 \): \[ BE = 0.0295 \cdot 931.5 = 28.4 \, \text{MeV}. \] Final Answer: \[ \boxed{28.4 \, \text{MeV}}. \]