If \( m \), \( m_n \) and \( m_p \) are the masses of a \( {}^{A}_{Z}\mathrm{X} \) nucleus, a neutron and a proton, respectively, then
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( m\lt (A-Z)m_n+Zm_p \)
The nucleus \( {}^{A}_{Z}\mathrm{X} \) is made of \( Z \) protons and \( A-Z \) neutrons, so the total mass of its separate constituents is \( (A-Z)m_n+Zm_p \).
Option (1) \( m\lt (A-Z)m_n+Zm_p \) — Correct. The mass of a nucleus is always less than the total mass of its constituents. The difference is the mass defect \( \Delta M \), and its energy equivalent \( \Delta M\,c^{2} \) is the binding energy released when the nucleus forms.
Option (2) \( m=(A-Z)m_n+Zm_p \) — Incorrect. Equality would mean zero mass defect and zero binding energy. But the nuclear mass is always found to be smaller, for example by 0.13691 u for oxygen-16.
Option (3) \( m=(A-Z)m_p+Zm_n \) — Incorrect. The masses are swapped. The mass \( m_p \) belongs with the \( Z \) protons and \( m_n \) with the \( A-Z \) neutrons. It also states an equality, while the nuclear mass is less than the total mass of its constituents.
Option (4) \( m\gt (A-Z)m_n+Zm_p \) — Incorrect. This would make the mass defect negative. The nucleus would then hold more energy than its separate nucleons and would not stay bound.
NCERT (2026-27):
Nuclear Binding Energy – “In Section 13.2 we have seen that the nucleus is made up of neutrons and protons. Therefore it may be expected that the mass of the nucleus is equal to the total mass of its individual protons and neutrons. However, the nuclear mass M is found to be always less than this.”
Nuclear Binding Energy – “… The difference in mass of a nucleus and its constituents, ΔM, is called the mass defect, and is given by
\( \Delta M=[Zm_p+(A-Z)m_n]-M \qquad (13.7) \)”