The binding energy per nucleon for a deuteron and an $\alpha$-particle are $x_1$ and $x_2$ respectively. The energy $Q$ released in the following reaction is:
$_1H^2 + _1H^2 \rightarrow {_2}{He}^4 + Q$

$A$ nucleus of mass $M + \Delta m$ is at rest and decays into two daughter nuclei of equal mass. If $C$ is the speed of light,and $E_1$ is the binding energy per nucleon for the parent nucleus and $E_2$ is that for the daughter nuclei,then:

List-$I$ shows various functional dependencies of energy $(E)$ on the atomic number $(Z)$. Energies associated with certain phenomena are given in List-$II$. Choose the option that describes the correct match between the entries in List-$I$ to those in List-$II$.

List-$I$	List-$II$
$(P) \ E \propto Z^2$	$(1)$ energy of characteristic $x-$rays
$(Q) \ E \propto (Z-1)^2$	$(2)$ electrostatic part of the nuclear binding energy for stable nuclei with mass numbers in the range $30$ to $170$
$(R) \ E \propto Z(Z-1)$	$(3)$ energy of continuous $x-$rays
$(S) \ E$ is practically independent of $Z$	$(4)$ average nuclear binding energy per nucleon for stable nuclei with mass number in the range $30$ to $170$
	$(5)$ energy of radiation due to electronic transitions from hydrogen-like atoms

The masses of neutron, proton and deuteron in amu are $1.00893$, $1.00813$ and $2.01473$, respectively. The packing fraction of the deuteron in amu is

The binding energy per nucleon for a deuteron $(_{1}^{2}H)$ and an $\alpha -$ particle $(_{2}^{4}He)$ are $x_1$ and $x_2$ respectively. The energy $(Q)$ released in the reaction $_{1}^{2}H + {}_{1}^{2}H \to {}_{2}^{4}He + Q$ is

Derive the formula for the binding energy of a nucleus by explaining the mass defect of the nucleus and write the formula for the binding energy per nucleon.