Question

If \[M(A,\,Z),\,{M_p}\,and\,{M_n}\] denote the masses of the nucleus, proton and neutron respectively in units of \[u = (1u = 931.5MeV/{c^2})\] and \[BE\] represents its binding energy in \[MeV,\] then.
A) \[M(A,\,Z) = Z{M_p} + (A - Z){M_n} - BE\]
B) \[M(A,\,Z) = Z{M_p} + (A - Z){M_n} + BE/{c^2}\]
C) \[M(A,\,Z) = Z{M_p} + (A - Z){M_n} - BE/{c^2}\]
D) \[M(A,\,Z) = Z{M_p} + (A - Z){M_n} + BE\]

Answer 1

Hint: In this question first of all we will give the definitions of the mass defect and binding energy using these definitions we will get their expression and use them to get the desired result.
Mass defect: Mass of the actual atom is always less than the sum of masses of its constituent’s particles so the difference between the mass of its constituent’s particles and the actual mass of the atom is called the mass defect.
Binding energy: It is defined as the energy required to separate the nucleons of the nucleus of an atom apart from each other.

Formula used:
(i) Mass defect: \[\Delta m = Z{M_P} + \left( {A - Z} \right){M_n} - M\left( {A,Z} \right)\]
Where \[Z,{M_p},{M_n},\,and\,A\] are the atomic number, mass of a proton, mass of neutron and mass number of atoms.
(ii) Binding energy: \[BE = \Delta m{c^2}\] where \[BE\] is binding energy and \[\Delta m\] is mass defect and \[c\] is the speed of light.

Complete step by step answer:
By using formula of mass defect
\[\Delta m = Z{M_P} + \left( {A - Z} \right){M_n} - M\left( {A,Z} \right)...................\left( 1 \right)\]
With the help of Einstein’s formula \[E = m{c^2}\] we have the mass defect \[\Delta m\] that gives the energy as follows:
From the formula \[BE = \Delta m{c^2}\] we have on rearranging it
\[\Rightarrow \Delta m = \dfrac{{BE}}{{{c^2}}}\]
Substituting this value of \[\Delta m\] in eq. \[\left( 1 \right)\], we will have
$\Rightarrow \dfrac{{BE}}{{{c^2}}} = Z{M_P} + \left( {A - Z} \right){M_n} - M\left( {A,Z} \right) $
$\Rightarrow M\left( {A,Z} \right) = Z{M_P} + \left( {A - Z} \right){M_n} - \dfrac{{BE}}{{{c^2}}}$

Therefore, option (C) is correct.

Note:
Before solving this question we should have knowledge about the atomic mass, atomic number, and the relation between them. Here the rearrangement of the equation should be done very carefully so that there is no error in the solution to the question.