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Step-by-Step Solution
1. Understanding Mass Defect
The mass of a nucleus, denoted by $M(A,Z)$, is found to be slightly less than the sum of the masses of its constituent protons and neutrons. The difference between these two masses is known as the mass defect.
2. Definition of Mass Defect
For a nucleus $_Z^A X$ containing $Z$ protons and $(A-Z)$ neutrons, the mass defect is given by:
$ \Delta m \;=\; Z\,M_{p} \;+\; (A - Z)\,M_{n} \;-\; M(A,Z) $
Here,
$M_{p}$ is the mass of a proton.
$M_{n}$ is the mass of a neutron.
$M(A,Z)$ is the actual mass of the nucleus.
3. Relation Between Mass Defect and Binding Energy
The binding energy (B.E.) of the nucleus is the energy equivalent of the mass defect. According to Einstein’s mass-energy equivalence, the binding energy is:
$ \text{B.E.} \;=\; \Delta m\,c^2 \quad \Longrightarrow \quad \frac{\text{B.E.}}{c^2} \;=\; \Delta m $
Substituting $ \Delta m $ from the definition of mass defect, we get:
$ \frac{\text{B.E.}}{c^2} = Z\,M_{p} \;+\; (A - Z)\,M_{n} \;-\; M(A,Z). $
4. Deriving the Expression for Nuclear Mass
Rearranging the above relation to solve for $M(A,Z)$:
$ M(A,Z) = Z\,M_{p} \;+\; (A - Z)\,M_{n} \;-\; \frac{\text{B.E.}}{c^2}. $
Thus, the nuclear mass is the sum of the proton masses and neutron masses minus the energy equivalent of the binding energy.
5. Conclusion
Therefore, the correct expression that relates the mass of the nucleus to the masses of its constituent nucleons and its binding energy is:
$ \boxed{M(A,Z) = Z\,M_{p} \;+\; (A - Z)\,M_{n} \;-\; \dfrac{\text{B.E.}}{c^2}} $
