Question
$${m_p}$$ denotes the mass of a proton and $${m_n}$$ that of a neutron. A given nucleus of binding energy $$BE,$$ contains $$Z$$ protons and $$N$$ neutrons. The mass $$m\left( {N,Z} \right)$$ of the nucleus is given by
A.
$$m\left( {N,Z} \right) = N{m_n} + Z{M_p} - BE{c^2}$$
B.
$$m\left( {N,Z} \right) = N{m_n} + Z{M_p} + BE{c^2}$$
C.
$$m\left( {N,Z} \right) = N{m_n} + Z{M_p} - \frac{{BE}}{{{c^2}}}$$
D.
$$m\left( {N,Z} \right) = N{m_n} + Z{M_p} + \frac{{BE}}{{{c^2}}}$$
Answer :
$$m\left( {N,Z} \right) = N{m_n} + Z{M_p} - \frac{{BE}}{{{c^2}}}$$
Solution :
Binding energy of a nucleus containing $$N$$ neutrons and $$Z$$ protons is
$$\eqalign{
& BE = \left[ {N{m_n} + Z{m_p} - m\left( {N,Z} \right)} \right]{c^2} \cr
& \Rightarrow \frac{{BE}}{{{c^2}}} = N{m_n} + Z{m_p} - m\left( {N,Z} \right) \cr
& \Rightarrow m\left( {N,Z} \right) = N{m_n} + Z{m_p} - \frac{{BE}}{{{c^2}}} \cr} $$