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Question: If M(A;Z), \[{{M}_{p}}\] and \[{{M}_{n}}\]denote the masses of the nucleus\[{{_{Z}}^{A}}X\] , proton...

If M(A;Z), Mp{{M}_{p}} and Mn{{M}_{n}}denote the masses of the nucleusZAX{{_{Z}}^{A}}X , proton and neutron respectively in units of u(1u=931.5MeV/c2{{c}^{2}})and BE represents its bonding energy in MeV, then:
A) M(A,Z)=ZMp+(AZ)MnBEM(A,Z)=Z{{M}_{p}}+(AZ){{M}_{n}}BE
B) M(A,Z)=ZMp+(AZ)Mn+BE/c2M(A,Z)=Z{{M}_{p}}+(AZ){{M}_{n}}+BE/{{c}^{2}}
C) M(A,Z)=ZMp+(A−Z)Mn−BE/c2M(A,Z)=Z{{M}_{p}}+(A-Z){{M}_{n}}-BE/{{c}^{2}}
D) M(A,Z)=ZMp+(A−Z)Mn+BE/c2M(A,Z)=Z{{M}_{p}}+(A-Z){{M}_{n}}+BE/{{c}^{2}}

Explanation

Solution

Proton is defined as the elementary particle which is identical to the nucleus of a hydrogen atom. It has a positive charge on it whose magnitude is equal to that of an electron. Whereas electrons are defined as the negatively charged particle which balances the positive charge of protons in the nucleus. Nucleus is defined as the collection of protons and neutrons. Protons and neutrons are in turn made up of particles called quarks.

Complete solution:
Bonding energy or bond energy is defined as the average of bond enthalpy which gives the strength of the chemical bond. It also provides stability to the compound. The energy is needed to break the bond and that energy is called bond energy. For example the bond energy present in hydrogen oxygen bonds in water molecules is the amount of energy which is required to split the bonds of hydrogen and oxygen in the molecule dividing by two.
So the formula for bond energy is the following$$$$
M(A,Z)=ZMp+(A−Z)Mn−BEc2M(A,Z)=Z{{M}_{p}}+(A-Z){{M}_{n}}-\dfrac{BE}{{{c}^{2}}}
BE=[ZMp+(A−Z)Mn−M(A,Z)]c2BE=[Z{{M}_{p}}+(A-Z){{M}_{n}}-M(A,Z)]{{c}^{2}}
Now takingc2{{c}^{2}} the towards left hand side we get,
BEc2=[ZMp+(A−Z)Mn−M(A,Z)]\dfrac{BE}{{{c}^{2}}}=[Z{{M}_{p}}+(A-Z){{M}_{n}}-M(A,Z)]
Now take M(A,Z) towards the left hand side and all the variables at the left hand side present will be taken towards the right hand side. So we get,
M(A,Z)=ZMp+(A−Z)Mn−BEc2M(A,Z)=Z{{M}_{p}}+(A-Z){{M}_{n}}-\dfrac{BE}{{{c}^{2}}}

Hence the correct option is (C).

Note: The bond energy always has a positive value because breaking any bond requires energy. Bond energy determines the strength of the bond so higher is the bond energy stronger will be the bond. If the bond energy is negative then the bond will not be able to exist as it will be unstable. So the bond energy will always have a positive value.